Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00003301/00001
## Material Information- Title:
- Analysis of a proposed six inch diameter Split Hopkinson Pressure Bar
- Creator:
- Jerome, Elisabetta Lidia
- Publication Date:
- 1991
- Language:
- English
- Physical Description:
- ix, 121 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Aluminum ( jstor )
Amplitude ( jstor ) Diameters ( jstor ) Inertia ( jstor ) Specimens ( jstor ) Strain gauges ( jstor ) Strain rate ( jstor ) Stress waves ( jstor ) Wave propagation ( jstor ) Wavelengths ( jstor ) Aerospace Engineering, Mechanics and Engineering Science thesis Ph. D Concrete -- testing ( lcsh ) Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF Materials -- Dynamic testing ( lcsh ) Materials -- Fatigue testing ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1991.
- Bibliography:
- Includes bibliographical references (leaves 82-85)
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Elisabetta Lidia Jerome.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 001693379 ( ALEPH )
25223136 ( OCLC ) AJA5458 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

ANALYSIS OF A PROPOSED SIX INCH DIAMETER SPLIT HOPKINSON PRESSURE BAR By ELISABETTA LIDIA JEROME A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991 ACKNOWLEDGMENTS I wish to thank my committee for their support and advice during this educational process. In particular I thank Dr. C. Allen Ross for his advice, support, and encouragement. Without him this effort would not have been possible. I would also like to thank my husband David for his patience and understanding during the many nights and weekends I spent at my desk or in front of the computer. I would like to acknowledge the financial support of the Air Force Engineering and Services Center at Tyndall Air Force Base. Finally, I wish to acknowledge the help and support of Sverdrup Technology Inc. during my educational endeavors. TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................... ii LIST O F FIG U R ES................................................................................................ v A B ST R A C T .............................................................................................................. viii CHAPTERS 1 INTRODUCTION.................................................................................... 1 O bjective.............................................................................................. 1 Background .......................................... ................ .............................. 1 A approach .............................................................................................. 3 2 MATHEMATICAL BACKGROUND ...................................... .......... 5 SHPB Assumptions ........................................................................... 5 Equations of Motion ............................................. ........................ 8 Solutions of the Frequency Equation............... .......... ........... .. 11 Transient Behavior............................................................................ 19 3 MATHEMATICAL AND NUMERICAL ANALYSIS............................ 21 Scaling Law s.......................................................... ............................. 21 Wave Propagation in an SHPB .......................................... ........... .. 25 Modified Pochhammer-Chree Method....................................... 26 Finite Difference Numerical Simulation..................................... 35 Experimental Method ............................................................. 37 Com prison of R results ........................................................................... 38 Inertia and Friction Effects ............................................................. 70 4 SIX-INCH SHPB FEASIBILITY .............................................................. 78 Sum m ary ............................................................................................... 78 Conclusions and Recommendations ................................... ........... 80 page -.............................................. 82 REFERENCES APPENDICES A Modified Pochhammer-Chree Program and Sample Outputs......... 86 B Ratio of Fourier Coefficients for Given a/A of 3 Bar Diameters...... 105 C Miscellaneous HULL Plots ................................................................... 107 D HULL Sample Input Files ..................................................................... 114 E Fortran Listing for Inertia Correction............................................... 117 BIOGRAPHICAL SKETCH ....................................................................................... 121 LIST OF FIGURES page 1. Compressive SHPB Schematic ................................................... ............... 1 2. Cylindrical Coordinates ..................................................... ......................... 9 3. Pochhammer-Chree Solutions for First Three Modes................................... 15 4. Tyndall 2-inch SHPB. Experimental Trace. Aluminum Specimen Length and Diameter 1.254 inch ...................................................... ............. 30 5. Modified Pochhammer-Chree Results. Incident Stress Normalized to 1.0.................... ........................................................................................ .. 3 1 6. Normalized Longitudinal Stress Given as a Function of Radial Location for Several Fourier T erm s .......................................................................................... 34 7. Experimental Results. 2-inch SHPB. Stress Pulses for a 6061-T6511 Aluminum Specimen, Diameter = 3.18 cm, Length = 3.18 cm.................. 39 8. Experimental Results. 2-inch SHPB. Stress Pulses for a 6061-T6511 Aluminum Specimen, Diameter = 2.54 cm, Length = 2.54 cm.................. 40 9. Experimental Results. 2-inch SHPB. Stress Pulses for a 6061-T6511 Aluminum Specimen, Diameter = 0.56 cm, Length = 0.56 cm.................. 41 10. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses for an Aluminum Specimen, Diameter = 3.18 cm, Length = 3.18 cm....... 42 11. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses for an Aluminum Specimen, Diameter = 2.54 cm, Length = 2.54 cm........ 43 12. HULL Output. 2-inch SHPB. Stress (TYY) and (EYY) Strain Pulses for an Aluminum Specimen, Diameter = 0.56 cm, Length = 0.56 cm........ 44 13. Experimental Traces at Three Incident Stress Levels (60 MPa-Top, 100 MPa-Middle, 167 MPa-Bottom). Specimen Diameter 1.253 in., Length = .625 in., 2-inch SHPB.................................. ............ 46 14. HULL Output. 2-inch SHPB. 2-inch Diameter and Length, Aluminum Specimen, 257 usec Pulse ......................................................... 48 a 1 15. Modified Pochhammer-Chree Results. A 125 6-inch SHPB. A 125 6-inch SHPB. 750 sec Pulse.................................................................................................. 49 a 3 16. Modified Pochhammer-Chree Results. A 250 6-inch SHPB. 500 pusec Pu lse ..................................................................................................... 50 pWag a 3 17. HULL Output. a = 250 500/sec Pulse, Station 1 (Top) Bar Center, Station 2 (Bottom) Bar Surface ....................................... 51 a 3 18. Modified Pochhammer-Chree Results. A-125 6-inch SHPB. 250 usec Pulse.. .............................................................................................. 52 a 3 19. HULL Output at Incident Bar Strain Gage. a = 125' 6-inch SH PB 250 ysec Pulse.......................................................................................... 53 20. Experimental Traces at Three Incident Bar Locations for Two Striker Bar Lengths. 16-inch (Top) and 25-inch (Bottom), 2-inch SHPB .................... 55 21. HULL Output. 2-inch Bar, 250 /sec Pulse, All Stress Components on the Surface at Incident Bar Strain Gage.................................. .......... .. 56 22. HULL Output. 6-inch Bar, 500 usec Pulse, All Stress Components on the Surface at Incident Bar Strain Gage.................................. .......... .. 57 23. Modified Pochhammer-Chree Results at Incident Bar Strain G age. 2-inch Bar ............................................................................................ 58 24. HULL Output. 6-inch Bar, 250 usec Pulse. Surface (Top) and Center (Bottom) Transmitted Stresses ........................... 59 25. HULL Output. 6-inch Bar, 500 ,usec Pulse. Surface (Top) and Center (Bottom) Transmitted Stresses ........................... 60 26. HULL Output. 2-inch Bar, Longitudinal Stress Near the Impacted End (Top) and at the Strain Gage Location (Bottom) ................. 62 27. HULL Output. 6-inch Bar, All Stresses Near Impacted End (Top) and at the Strain Gage Location (Bottom) .................................. ............. 63 28. HULL Output. 2-inch Bar, All Stresses Near the Impacted End (Top) and at the Strain Gage Location (Bottom) .................................. ............. 64 29. HULL Output. 2-inch Bar, All Stresses at Three Locations at the Bar-Specimen Interface ........................................................................ 66 30. HULL Output. 2-inch Bar, Stresses in Aluminum Specimen at Two Locations Near the Surface (Top), Near the Center (Bottom). Compare with Figure 29..................................................... .................................... 67 31. HULL Output. 2-inch Bar, Transmitted Stresses at Three Radial Locations. Surface (Top), Middle (Middle), Center (Bottom). Compare with Figure 29 ..................................... .......................... .......... 68 32. HULL Output. 2-inch Bar, Transmitted Stress with Zero Friction Between Specimens and SHPB ............................................. .............. 74 33. HULL Output. 2-inch Bar, Transmitted Stress with Infinite Friction Between Specim en and SHPB......................................................................... 75 34. HULL Output. 6-inch Bar, Transmitted Stress with Zero Friction............ 76 page 35. HULL Output. 6-inch Bar, Transmitted Stress with Infinite Friction....... 77 36. Modified Pochhammer-Chree Results. Radial Variations of Reflected Stresses. 2-inch SHPB (Top), 6-inch SHPB (Bottom) ................................ 88 37. Modified Pochhammer-Chree Results. Radial Variations of Transmitted Stresses. 2-inch SHPB (Top), 6-inch SHPB (Bottom).................................. 89 38. HULL Output. 2-inch SHPB. Radial and Longitudinal Velocities at Incident Bar Specimen Interface. (Bar Surface: Top, Middle of the Bar: Bottom, Bar Center: Right)............................................ 108 39. HULL Output. 6-inch SHPB. All Transmitted Stresses at Bar Surface.............................................................................................................. 109 40. HULL Output. 6-inch SHPB. All Transmitted Stresses at Bar Center. Compare with Figure 37 ...................................................................... 110 41. HULL Output. 2-inch SHPB. 1-inch Aluminum Specimen. Incident and Reflected Stresses............ ................................................. 111 42. HULL Output. 2-inch SHPB. 1-inch Aluminum Specimen. All Stresses W within the Specimen ..................................................................... 112 43. HULL Output. 2-inch SHPB. 1-inch Aluminum Specimen. All stresses in the Transmitter Bar................................................................. 113 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ANALYSIS OF A PROPOSED SIX INCH DIAMETER SPLIT HOPKINSON PRESSURE BAR By Elisabetta Lidia Jerome August 1991 Chairman: C. Allen Ross Major Department: Aerospace Engineering, Mechanics and Engineering Science The objective of this investigation was to determine whether a 6-inch diameter Split Hopkinson Pressure Bar (SHPB) is feasible; whether the assumptions made in a typical 2- to 3-inch SHPB still apply; and what chan- ges, problems, and issues would be associated with an SHPB that size. The effort included a mathematical and a numerical analysis. Experimental data from a 2-inch SHPB were used to compare and validate analytical models. A methodology based on the Pochhammer-Chree frequency equation was developed by the author to study the wave propagation and wave inter- action in any size SHPB. Additionally, a second order accurate finite dif- ference code was used to mathematically understand the wave motion in a typical bar. The effects of friction and inertia were analyzed numerically, and general guidelines and corrections were developed. This investigation concluded that a 6-inch SHPB is theoretically feasible and, in fact, identical to any existing system if the input pressure pulse duration is scaled appropriately. The modified Pochhammer-Chree method can be used to assess the validity of any size SHPB and to study the wave interactions. If the specimen length and diameter are kept in the same order of magnitude, one need not worry about friction in the proposed 6-inch SHPB. A correction will be needed to account for inertia, especially at strain-rates of 50 to 100 sec' and above. The Split Hopkinson Pressure Bar has been and remains a reliable, accurate, and useful tool to study material response to loading, even for a new proposed system six inches in diameter. CHAPTER 1 INTRODUCTION Objective The objective of this investigation is to determine whether a 6-inch diameter Split Hopkinson Pressure Bar (SHPB) is feasible; whether the assumptions made in a typical 2- to 3-inch SHPB still apply; and what major changes, problems and issues would be associated with an SHPB that size. STRAIN GAGE INCIDENT BAR STRAIN GAGE TRANSMITTER BAR Figure 1. Compressive SHPB Schematic Background The Split Hopkinson Pressure Bar may be used to study materials at high strain-rates in tension, shear and compression. A compressive SHPB is shown schematically in Figure 1. An SHPB system consists of a specimen sandwiched between and in contact with two elastic bars--incident and trans- mitter bar. The incident bar is impacted by a striker bar at a known velocity V STRIKER BAR causing a pressure wave to travel down the bar. This wave is partially trans- mitted and partially reflected at the incident bar/specimen interface, and partially transmitted and partially reflected at the specimen/transmitter bar interface. Strain gages attached to the incident and transmitter bars record the strains associated with the reflected, transmitted, and incident pulses. From these strain data, stresses, strains, and strain-rates in the specimen can be computed as a function of time. Although different sizes of SHPBs have been built, the largest one in existence in the USA is 3 inches in diameter. The main reason behind this size limitation is that it is generally assumed that the wave propagation in the bar is one-dimensional, implying a high length to diameter ratio. A simpler form of the SHPB has been around since the late 1940s, when techniques and instrumentation were not as sophisticated as they are today. Historically, there may not have been a need to test a material specimen larger than 3 inches in diameter. However, in recent times the need for more economical structures to resist impulsive loadings associated with earthquakes, accidental explosions, and effects of conventional explosives has prompted more research into strain rate effects on the properties of concrete. Considerable data have been generated in 2- to 3-inch diameter SHPB systems on concrete specimens containing small or scaled down aggregate. The question which arises from such small tests is, does the strain rate sensitivity, found for the small laboratory specimens, apply to the standard 6-inch diameter large aggregate test specimens which are cored or cast in the field? There is a need to use a 6-inch diameter SHPB type apparatus to test concrete specimens, which will match the concrete specimens which are cored directly in the field. The SHPB has proven to be a very effective and useful tool to conduct high strain-rate material response studies since Kolsky [1] described its use in 1949. Detailed discussions of how a typical system works can be found in numerous papers including Hauser, Simmons and Dorn [2], Lindholm [3], Malvern and Ross [4], Robertson, Chou, and Rainey [5], and many others. Because of the inherent difficulties associated with a system of this kind, an idealized one-dimensional wave theory is commonly assumed where stress is uniform over each cross section, plane cross sections remain plane during motion, and stress and strain are uniform throughout the length of the specimen. The above assumptions are adequate if the length of the bar greatly exceeds the diameter, in order to ensure no end effects and the loading pulse is much larger than the transit time of the specimen. However, factors like dispersion, radial and axial inertia effects, and friction between the specimen and the incident and transmitter bars can cause erroneous results if care is not used in either conducting the experiment or analyzing the data. Approximate corrections for the calculated one-dimensional stress have been formulated to account for inertial and frictional effects. Kolsky [6] introduced a correction for radial inertia. Davies and Hunter [7] dis- cussed both friction and inertia effects. Other papers of interest on this subject are those by Rand [8], Dharan and Hauser [9], Samanta [10], Jahsman [11], Chiu and Neubert [12], and Young and Powell [13]. The original SHPB apparatus was devised to study materials in com- pression. More recently it was modified to test specimens in both tension and torsion. Examples of these techniques are discussed by Nicholas [14], Lawson [15], Baker and Yew [16], Jones [17], Okawa [18], Ross, Nash, and Friesenhahn [19], Rajendran and Bless [20], and Ross [21]. Follansbee and Frantz [22] used the Pochhammer-Chree frequency equation to correct the recorded pulses for dispersion as they travel down the bar assuming only the first longitudinal vibrational mode is excited. Approach Before undertaking any research it is imperative to know and under- stand what has already been done on the subject. This ensures no duplica- tion of work and gives a good working foundation from which to start the research effort. The wave propagation in an SHPB is governed by the equations of motion. Generally, it is assumed the motion is one-dimensional, and there- fore is uniform in the radial and the circumferential direction. Although the one-dimensional wave theory has been shown to be accurate for the existing systems, a mathematical analysis is needed for the proposed 6-inch SHPB to assess whether this simple theory still applies. The two-dimensional axisym- metric cylindrical equations of motion need to be analyzed and solved (approximately), by various methods to understand the wave motion in a bar. There are no known exact solutions to the two-dimensional wave propagation problem in a finite cylindrical bar, and therefore one must turn to a numerical scheme, namely a finite difference approximation. Modern computers make it possible to solve large problems in a reasonable amount of time. The plan was to first numerically simulate an existing SHPB in order to compare the results with experimental data and have confidence in the reliability of the numerical approximation. The second step would then be to simulate the proposed 6-inch system and gather information on the wave motion. The final step will be a comprehensive look at the different methodologies, approximations, and assumptions to better understand the implications of increasing the size of the SHPB system. From this one can assess whether the data reduction of the current SHPBs still applies to the large bars, and what are the problems associated with a system two or three times the size of the existing ones. CHAPTER 2 MATHEMATICAL BACKGROUND SHPB Assumptions As discussed earlier, a one-dimensional wave theory is generally assumed in the analysis of an SHPB. The one-dimensional theory assumes longitudinal spatial variations only. Applying Newton's law for a displace- ment u in the x direction of the diagram below, gives the equations of motion for an elastic system a --- 8a ---- +- dx ax I-- -- dx aa (a2u -aA + (a + dx)A = pAdx ax atz or upon simplification au a2u ax =p at2 and Hooke's law du = Ee =-E, ax where a is the stress, E is Young's modulus, E is strain, t is time, A is the cross-sectional area, and p is the density of the material. Equation (la) can be rewritten as follows a2u / 2U Wt (E/p ) ax2 (lb) where E/p is the familiar elastic wave speed squared, or E/p =Co The general solution to (Ib) is of the form u = f(x + Cot) + g(x Cot) corresponding to forward and backward propagating waves. Assuming u = g(x Cot) only, differentiate u and obtain Ou u , Sg' and -cog ax at where g' denotes differentiation of the function w.r.t. the argument (x-Cot). Eliminating g' between these two expressions yields Bu 9u S= --Co = -CoE at ax Strains are usually measured in an SHPB experiment at some distance from the specimen. To obtain the interface displacements, these recorded strains must be time-shifted to give the strains at the interfaces, and then integrated over time to give S= f Co (ei r) dt and u2 = Co tdt where the subscripts i, r, and t indicate incident, reflected and transmitted. Additionally, ul and u2 are the specimen's displacements at the incident and output bars interfaces respectively. The negative signs give positive dis- placements for compressive (negative) strains. Let L be the length of the specimen; then the average strain in the specimen is Es = (U2 Ui)/L. This is sometimes approximated for short specimens and slowly changing stress by assuming equal stresses at the two interfaces, which implies that Et = Ei + Er. The specimen strain then becomes s= (-2co/L) f trdt and the specimen strain rate is s = (-2co/L) Er. By using Hooke's law, the stresses on either face of the specimen are os2 = E A and Usl = E (Ei + Er) A where is the ratio of the bar and specimen cross section areas. If the As interface stresses are not assumed to be equal, then Et # Ei + Er and the specimen strain and strain rate are given by S=( Co/L) f ( i +Er) dt and s = (-co/L) (E i + r) There are some problems associated with such a simplified one- dimensional theory, even for long slender rods, and these problems are accentuated for larger diameter and shorter length SHPB systems. First, longitudinal waves are dispersive, (wave speed is a function of frequency), which means the waveform that is recorded at a distance from the specimen is not the same waveform that actually arrives at the specimen. Wave disper- sion can be corrected for, but is time consuming and the correction methods are not exact. The one-dimensional theory also neglects stress and displace- ment variations across the cross section. Since strains are measured only at the surface, this can also lead to errors. Equations of Motion To assess the validity of this simple theory, one can study the equa- tions of motion that govern the wave motion in a circular cross section SHPB. The equations of motion are given in tensor form as d2u V-T+pb= dt2 (2) where T is the stress tensor, b are the body forces (which are usually safely d2 neglected), and y is the acceleration term. In cylindrical coordinates (Figure 2), equation (2) becomes 02Ur dUrr rr UOO 1 UrO 8Urz p + + at2 8r r r 80 Oz p + r+ +2 (3) at2 8r r aO 8z r Suz a"rz + 1 BOoz + a0z +g at2 Or r -0 aZ r Figure 2. Cylindrical Coordinates Equations (3) represent the three-dimensional motion in a cylindrical bar. The modes of vibration associated with an isotropic circular cylinder are in general very difficult to calculate and they depend on the specific boundary conditions as well as on the form of the input loading. If the axis of the cylinder coincides with the z-axis, two general types of vibrational modes exist. One set is symmetric and one is asymmetric. In this analysis, the assumption is made that since the striker bar impacts the incident bar squarely, then the motion should be axisymmetric. This means the solution is independent of theta; then u., a, and oz vanish, and the second of equa- tions (3) is satisfied identically. It also means that the only possible modes of vibration will be symmetric. Using Redwood's [23] notation, these modes will be called Mm,n where m is either 1 (symmetric) or 2 (asymmetric) and n ranges from 1 (Young's modulus mode) to infinity. Therefore, in this study of an SHPB, only M1,n modes are considered. The constitutive rela- tions for an elastic material can be written as aur (Ur az St +2 U)r (fr aUz ) Or =('t+ ")fi 7+-f 9Ur Uz) arz a= ut + u) az ar (4) ouz (r OUr uz = (1 + 2u)z + +r aOrr aUrz aeeoo = r- + + rr Or 9z where A and P are Lame's constants. Making use of (4) and neglecting theta dependence, equations (3) can thus be reduced to only two equations 2Ur [2ur 1 Our Ur 2Uz a2ur p = ( + 2) + (2 2+u) --+ at2 ar r Or r rz az2 (5) 2z 2Uz 1 Uz O2Uz 2Ur 1 OUr p- =2 2 r + + 2,u) + ( + u) ++r r - t r r Or 0z2 L-z r 9z (5 The boundary conditions require no stress at the bar surface or arr =0 and Or = 0 at r = a where a is the radius of the bar. Also at a free end, Urz = 0 and Ozz = 0. For a colinear impact of one bar against another, the initial conditions can be in the form of a step change in pressure az = -PH(t), arz = 0 where P is a uniform pressure and H(t) is the Heaviside step function, or as a harmonic displacement of uz = sinw t and arz = 0. The condition orz = 0 is correct only if the two bars are of identical diameter and material, so that the interface is a plane of symmetry of the stress state, until a reflected wave reaches the interface. Equations (5) have been solved for an infinitely long bar by Pochhammer and Chree for the propagation of sinusoidal waves and studied by numerous investigators over the years [6, 24, 25, 26, 27, 28, 29]. The solution is called the frequency equation and has the form 2y Jo(ha) (2y2 P) Ji(Ka) a = 0 (6) 2u 2 Jo(ha) Jo(ha) uy-a JI(Ka) aa A + 2/ aa where h2 = (2r/A)2 [pc /(A + 2u) 1],K2 = (27r/A)2 [(pc/p) 1], A = wave- length, y = 2 Jo and J1 are the zero and first order Bessel functions A Cp respectively, w is the angular frequency, and cp is the phase velocity. Solutions of the Frequency Equation There are an infinite number of solutions to Equation (6), each cor- responding to a unique mode of vibration. The solutions are exact only for an infinitely long cylinder although, for a cylinder whose length greatly exceeds its diameter, the errors are small. Furthermore, the solutions are not explicit but rather a function relating the phase velocity cp, the wavelength A and Poisson's ratio v. Now (6) can be written in implicit form as 2Uy -J1 (Ka) 2y Jo (ha) 2y2 J (a). (7) 2u 2Jo(ha) 2 Jo (ha) =0 [ a2 A + 2/s By making use of the following properties of Bessel functions, a (Jo(ha))= -Jl(ha) A (Ji (ha)) = hJo (ha) (J (ha)) /a, equation (7) can be rewritten as 4py (KJo(Ka) Ji(Ka)/a) (-KJi(ha)) ((2y2 ) Ji (Ka) L2p. (-hJo (ha) + J1 (ha)/a) +2Jo (ha) = 0, A + 2,u which finally becomes y2 JK o(a) 1 1 2+) + (K- 2) Jo(ha)= (8) J1 (ca) 2 a 2 hJ1 (ha) Equation (8) has been solved numerically by several investigators [25, 27, 30]. It involves assuming a value of Poisson's ratio and then for varying values of frequencies, ranging from zero to some large number, equation (8) is numerically evaluated to obtain the values of the phase velocity. This turns out to be a very laborious task even with the aid of high speed computers. Results are usually shown in graphical form, namely in curves relating the phase velocity and the frequency, both in nondimensional form for general applicability. Figure 3 [22] shows the solution to equation (8) for a Poisson's ratio of .29. Only the first three vibrational modes are plotted namely M1,1, M ,2 and M ,. Recall that equation (8) only applies to symmetric vibrational modes. Mode M is of most interest since it appears to be the only mode excited in a typical SHPB. As the frequency of this mode approaches zero, or as the wavelength approaches infinity, the phase velocity approaches (v- ), which is the elastic, one-dimensional bar-wave speed, co. As the frequency becomes very large, the phase velocity ap- proaches cr, the velocity of Rayleigh surface waves. Although it may not be clear from Figure 3, Cp, the phase velocity, approaches cr from the down side, implying that the phase velocity reaches a minimum which is less than the Rayleigh wave speed at some intermediate frequency. Recall that 2 A+2 2 2 E Cd- Ct- Co - P P c P h2 = (27t/A)2 (pc/(A + 2,u) 1) and K2 = (23r/A)2 (pc/ 1) and it is easily shown that co < ca and for M,1 cp < co. Then for the M1,1 mode the parameter h is always imaginary since, for all frequencies, cp < ed where Cd is the dilatational wave speed in an infinite medium. a For very low frequencies K is real, but past roughly = 0.65, when c becomes less than the shear wave speed in an infinite medium ct, K is imagi- nary. This means that at low frequencies ( < 0.65), the displacement is mainly due to plane transverse waves, since the set of dilatational waves exists only as a surface disturbance. At high frequencies the total displace- ment becomes increasingly like a pure surface disturbance [33]. Mode M1,2 and higher may be of interest in a "larger" SHPB and at high frequencies. These modes have what are called cut-off frequencies, i.e., frequencies at which the phase velocity becomes infinite and no real disturbance propagates in that mode. From equation (8) this implies either that y = 0.0 or that J1(Ka) = 0.0. If y = 0.0, equation (8) becomes 1 o 1 1 2 Jo (ha) 2 ct a 2 c2 hJ1 (ha) since ic-=--- 1 = 1~ 1~ , 2 2 2 K2 ) C2 and K 2 = -O for Cp = oo, where wco is the cut-off frequency. ct 2 Following the same reasoning h = -, for cp = oo and thus one can write Cd" tJO (ti 0a (9) \Jo -a c2 Wco J1 ad c Cd Cd If Ji (Ka) = 0.0, it can be said that J, (Ct a = 0.0 (10) [Ct ) Equations (9) and (10) can now be used to determine the cut-off frequencies for any mode M,n . From the solutions of the frequency equation, equation (8), it is possible to calculate the radial and longitudinal displacements for each mode at every frequency, and Poisson's ratio within the cylinder. Bancroft [25] calculated and plotted the amplitude of the longitudinal displacement a and showed that they are (as expected) uniform for =0.0 and the motion a is confined to the outside surface when approaches infinity. This agrees with the observations made earlier about mode M,.1 response in general. Numerous other approximate methods of varying degrees of com- plexity have been developed to mathematically treat an SHPB system. They fall somewhere in between the simple one-dimensional theory and the two- dimensional, coupled differential equations just presented. Unfortunately, those that show good agreement with the Pochhammer-Chree solution have very complicated descriptive equations, while those simple equations agree with the exact theory only over a very limited frequency range. 1.4 1.2 Cd/Co o ^ MODE M1 0 1,3 C 0 1.0 P= = SM Cr = RAYLEIGH VELOCITY 0.8 C2= MODE M P C/Co 1,1 0.6 0.4 0.0 0.5 1.0 1.5 2.0 a/A Figure 3. Pochhammer-Chree Solutions for First Three Modes. When computers were not readily available and usable, a large num- ber of approximate theories were developed. Most important to this study are the different methods and approaches used to derive these theories. As already discussed, the simplest analysis is to assume purely axial stress, uniform over each cross section. This leads to the familiar one-dimensional wave equation in terms of the axial displacement u,, 2Uz a2 (11) at2 pz which predicts that waves of all frequencies travel at the same constant velocity co = v'O. A better approximation to this theory introduces a cor- rection for the radial motion by considering the inertia of the cross section [23]. The approach involves the use of Hamilton's principle which states that the first variation of the integral of the Lagrangian, (T V), with respect to time is zero. Specifically 6 f (T- V) dt =0 (12) and (T V) can be expressed as f0 pA -+(vR) 2 a EA a dz ] (13) L0 Iii pA ( at ( az atj -^j2 az kinetic energy potential (strain) energy where L is the length of the cylinder, R is the radius of gyration about the z-axis, v is Poisson's ratio, and the velocities in the z and r directions are respectively _uz 8ur au_ and aur at at az But ur can be assumed in this approximation to be of the form -v r-z so that 8ur a-Uz S- -vr t at 0z 0t " The result of substituting equation (13) into (12) and solving via integration by parts is as follows a2Uz 2 a4uz a 2u P a-- (vR) a2a2t- =E (14) P at2 t Z2 ( 14 This equation gives a better approximation of the exact theory than equation (11) at low frequencies. However, for short wavelengths the errors become considerable. A third approximate theory developed by Love [29] is based on the exact characteristic equation (8), where the Bessel functions are expanded in a power series. If a, the radius of the cylinder, is small enough so that ha and Ka are small compared to unity, then powers of ha and Ka higher than the second can be neglected. That is Jo(Ka) = 1 4 (Ka)2 Ji(Ka) = V2 (a) and the phase velocity becomes Cp = 1 v2a2) (15) and may be rewritten as v where c a (16) S=1 v2,2- where Co= V p Co X) a This theory agrees fairly well with the exact theory up to values of = 1 but then rapidly diverges. The assumptions that Kca and ha are small compared to unity imply that the wavelengths of the vibrations are large compared to the radius of the cylinder. Another common theory was developed by Mindlin and Herrmann [26]. It considers shear stresses and strains by assuming first that the radial displacement is of the form Ur = (r/a) u(z,t) with us = 0 and uz = w(z,t). Forces and moments are then calculated from standard engineering mechanics formulae and corrected for shear and iner- tia by introducing factors K1 and K2. This leads to equations of the form 2 2 2u 2 22 Au 2 a2U a Kp --- 8K2( + ) u 4aK2 pa 2 az2 z 3t (17) au 2 a2a) 2 232) 2aa + a2 (A + 2u) =p a az az" at By substituting u = A exp (-iyz) exp (iwt) and w = B exp (-iyz) exp (iwt) into equation (17), two equations in A and B are obtained, from which A and B can be eliminated to again obtain a characteristic equation which relates phase velocity and frequency for the first two modes. By adjusting K2 and K1, a very good approximation can be obtained for mode M1,i Mode M1,2, on the other hand, shows considerable deviation from the exact theory. Many other theories have been developed and, in general, their com- plexity is directly proportional to their accuracy. These analyses are impor- tant because of the insight they give into the response of a cylinder subjected to impulsive loading. Knowing how stresses and displacements are dis- tributed is crucial to the study of wave propagation and SHPB systems. Transient Behavior Thus far, the discussion has been limited to continuous waves or at least to pulses more than just a few cycles in length. However, in an SHPB experiment, a very short pulse is propagated through the cylinder, giving rise to a transient behavior which cannot be completely determined from the continuous theories; therefore, a different approach must be taken. One way to tackle the transient problem is to use Fourier analysis. By decompos- ing the pulse into its continuous component, continuous wave theory can be used on each component at any point down the bar and the new pulse can be obtained by adding the pieces together again. This method is not exact, but it does give a close approximation, especially at some distance from the source. Choosing a mathematical function that describes the input pulse and that can be represented by a reasonably simple Fourier series is very dif- ficult. Davies [27] used a trapezoidal pulse while Kolsky [28] assumed an error function. In both cases the Pochhammer-Chree curves are used to obtain the phase velocity for each pulse, and in calculating the new shape of the wave-forms, only the fundamental mode M1,1 was assumed to be ex- cited. Another approach taken by some investigators in studying transient behavior is based on the concept of "dominant groups" and the method of stationary phase. The idea is that if at the beginning everything is in phase, any time after that all waves are out of phase, and therefore interfere destructively. When the combined effects of all these waves are studied, the 20 main contribution comes from a small "dominant" group whose phase velocities, periods, and wavelengths are almost the same. By focusing on this special group, one can obtain approximate propagation of longitudinal, flexural, and torsional waves in a cylinder. In this study a modified Pochhammer-Chree method is developed using the Fourier analysis approach. It is straightforward and a much simpler numerical simulation than the existing complicated computer codes. This new approach is presented in the next section. CHAPTER 3 MATHEMATICAL AND NUMERICAL ANALYSIS Scaling Laws In order to analyze the propagation of waves in a cylindrical bar, one should ideally solve the three-dimensional equations of motion. Unfor- tunately, there is no known closed form solution to the problem. Even if axisymmetry is assumed, the two-dimensional equations of motion cannot be easily solved. The problem is further compounded when a different material is introduced, such as when a specimen is introduced in an SHPB. The addition of two interfaces between dissimilar materials creates reflections and transmissions which cause waves to superpose, cancel each other, and in general interact with each other in a very complicated way. Pochhammer and Chree independently solved the equations for the propagation of a sinusoidal wave in an infinite bar. The solution, discussed in chapter 2, is the so called frequency equation; it has an infinite number of solutions, one for each mode, and results in a function relating phase velocity, wave length and Poisson's ratio. These solutions are valid and exact for infinitely long bars. If the bar is long enough to eliminate end effects (approximately 10 diameter lengths) and if the interest is in the transient pulse in its first passage, then the infinite assumption is reasonable. The Pochhammer-Chree solutions can be plotted in nondimensional form as shown in Figure 3. It is interesting to note that since A = CpT, then a a ac a S and thus c where T is the pulse duration and is defined as T /c, where L is the striker bar length. Now if a is multiplied by a as T = 2 Ls/co, where L is the striker bar length. Now if a is multiplied by a constant factor and T is also multiplied by that same factor, then a plot of a vs. -- will not change. What this means is that mathematically, under 0 the assumptions made, if the diameter is multiplied by a constant and the duration of the input pulse is simultaneously multiplied by the same con- a a stant, then the solution for the original a is the same for both values. This applies to a sinusodial, continuous input where the wavelength is con- stant, and travels at only one wavespeed relative to the bar radius: a long wave in a large bar will have the same wave speed as a short wave in a small a bar, as long as -A stays constant. What happens if the input is a transient pulse? A simple mathemati- cal function can be chosen to describe this type of input which can be represented by a Fourier series. A transient pulse is composed of a spectrum of frequencies; the higher frequency components travel more slow- ly than the lower frequency components, and thus lag behind and cause the initial sharp pulse to spread. This spreading is called dispersion. The Poch- hammer-Chree solutions give the velocity of each wave depending on its frequency. Thus, one may correct a given pulse for dispersion by repre- senting the pulse by a series of frequency components, calculating how far each frequency component has traveled in a certain time, and then reas- semble the pulse. This task basically amounts to correcting for phase chan- ges within each term of the Fourier series. It is important to be able to correct for dispersion since the SHPB pulses are recorded on strain gages at distances typically 30 to 60 inches from the specimen. Interest is in the response at the specimen itself, but in many cases a strain gage cannot be physically placed there. The dispersion correction technique allows one to predict the shape of the pulse as it travels from the specimen to the strain gage. Further, following the same kind of reasoning as for the continuous wave case, a C a t can be written where n is the number of points A, c nAtco taken to represent the transient pulse and At is the time interval between them. In essence, nAt is the period of the transient pulse, and if a n At stays constant, the case is analogous to the continuous case. This means as long as the bar radius and the input pulse length are increased or decreased by the same amount, the response characteristics of a SHPB system will not change. This fact is important to this study because the main concern is the feasibility of a 6-inch SHPB system. What the Pochhammer-Chree equa- tions show is that if the incident pulse is of large enough duration relative to the bar radius, then, as far as the longitudinal waves of the M,, mode are concerned, there is mathematically no difference between a 2-, 3-, 6- or 60-inch diameter SHPB. One must then determine what pulse duration would be needed to make a new 6-inch system equivalent to the existing 2- and 3-inch bars and whether the stress, strain and displacement distribution of that pulse are satisfactory for a usable SHPB. As mentioned earlier, for a typical SHPB system, a uniform stress and strain distribution is assumed, and only strains on the surface are measured. It is then very important to identify under which conditions the one-dimen- sional assumption is acceptable. Davies [27] computed the ratio of the longitudinal displacement at the surface to that of the longitudinal displace- a ment along the bar axis and showed that for less than 0.1 their dif- ference is less than 5 percent. Thus for a continuous pulse, with one value of a A, one-dimensionality is assured if A < 0.1. On the other hand, a tran- sient wave contains many different frequencies, and in fact the majority of the a s will be greater then 0.1. Thus, if the transient wave f(t)is represented with a Fourier series expansion, we would generally express it in a form such as A o0 Ao + Dncos(nwot-qo'.) f(t)- 2 Dn cos (n ao t - 2 1 where Ao and Dn are Fourier coefficients and p'n is the phase angle which is a function of the wave speed and the wavelength. Specifically, from [31] =c(ncox Co Pn = (P+ (nwx) [( 1i o c [-n- J where nwo is the angular frequency and c, is the phase velocity of the nth component. Equation (8) and Figure 3 give the relationship between c, and the frequency. The amplitudes in a Fourier series expansion are in decreasing order, meaning that Ao is the largest and Dn+1 is less then Dn. Furthermore, the frequencies of each component are also in order, they start at zero and go to infinity. In summing the Fourier terms, each term will contribute less and less to the overall pulse. This is why truncating the series at some large number, say 55, results in a very small error. Also, at some a point the nth component will have an A 0.1 To ensure one-dimen- a sionality in a SHPB the terms preceding the one with A = 0.1 must sum up to give almost the complete pulse. In other words, the amplitude D, of the a nth component having =0.1 must be a small fraction of the original amplitude Ao. Notice that it does not matter at which component a X reaches 0.1, but rather the value of the amplitude at that point must be small. If it is at the 10th or the 100th term in the Fourier series it makes no difference. The only question then is how small should this Dn amplitude be with respect to Ao, or what percentage of the whole pulse must be accounted for when a reaches 0.1. One way to answer this question is to look at the values for the existing SHPB systems at the University of Florida and at a Tyndall AFB. A short computer code was written to calculate and Fourier amplitude values for an assumed trapezoidal input based on the Pochhammer-Chree solutions [32]. It was found (see Appendix B) that for each system the amplitudes of the Fourier components had fallen to 5-9 a percent of the original value by the time A reached 0.1. Therefore, a good minimum amplitude ratio to use for the proposed 6-inch system is about 10 percent. This would assure the one-dimensionality of the response. In conclusion, because of the form of the equations involved, a pres- sure bar system can be mathematically scaled up or down to any degree, without affecting the characteristics of the wave motion. Analyses of the bars and input pulses used both at Tyndall AFB and at the University of Florida, showed that a 6-inch system would be equivalent to the existing 2- and 3-inch systems, if the input pulse duration was roughly 500 microseconds. The length of the striker bar is related to the pulse duration as 2L T = where co is the elastic wave speed and Ls is the striker bar length. The elastic wave speed co for steel is approximately 5080 m/sec. This means that for a 500 microsecond incident pulse in a steel SHPB, a 1.27 m (50- inch) long striker bar would be required. Wave Propagation in an SHPB Having an understanding of this very important fact, one can now turn to a more in-depth look at wave propagation in a cylindrical bar, to broaden the understanding of the overall problem. Three different methodologies were followed to study this problem. The goal was to compare and contrast results, understanding the drawbacks, assumptions and reliability of each one. These three approaches were : A new numerical method developed by the author using the Pochhammer-Chree frequency solution and referred to as a modified Pochhammer-Chree method. A finite difference numerical simulation of the governing equa- tions contained in an existing computer code. Experimental results from the SHPB at Tyndall AFB. The following sections are a general description of these three approaches. Modified Pochhammer-Chree Method As previously discussed, Pochhammer and Chree independently solved the two-dimensional equations of motion for an infinitely long bar assuming a sinusoidal waveform. Their solution is what is referred to as the frequency equation for longitudinal motion of an elastic rod. It turns out to be a function relating Poisson's ratio, wavelength, and wave speed, and can be solved for each mode of a cylindrical bar for a particular material. This solution is neither simple nor explicit. Beginning with the Pochhammer- Chree frequency equation and assuming only the first mode (MI,1) is ex- cited, the wave speed of any wave of a given frequency in the bar of interest, since it is dependent only on its wavelength, can be calculated. Using this wavespeed and corresponding wavelength along with given material proper- ties, the longitudinal and radial distribution of displacements and stresses may be determined from the assumed displacement functions for the "lon- gitudinal" vibrations. One can now turn to the transient pulse being propagated in the bar and describe it in terms of Fourier series; that means representing the waveform by the sum of an infinite number of pulses, each one having a distinct amplitude, a distinct wavelength and traveling at a unique speed. This wave speed can be calculated from the frequency equa- tion. Therefore, each piece of the input pulse can be followed as it travels down the bar, and at any time, the complete waveform can be reconstructed numerically by adding up all the pieces. This methodology is currently being used to correct for dispersion in the SHPB at the University of Florida and at Tyndall AFB. To carry the logic a bit further, it appears one can still treat the pulses individually as they encounter the boundary between the specimen and the incident bar, partially reflect and partially transmit through, and eventually encounter the second interface where more reflec- tions and transmissions take place. The basic assumption is that as the single Fourier component encounters an interface, its amplitude splits ac- cording to the characteristic impedance values of the two media, (which will be derived later), but it will maintain its original frequency and velocity. In other words, each wave component divides each time it encounters an inter- face, new components are formed each with a different amplitude, but all with the same frequency. Note also that the sum of these Fourier com- ponents produce the total incident, reflected, and transmitted pulses whose wavelengths will also not change. The implications of this assumption are crucial to the conclusion that one can theoretically follow each wave component through multiple reflec- tions, transmissions, and any length of travel, and at any time know its amplitude and location. With that information one can then reconstruct the pulse at any time or location and obtain the shape of the response. A computer program has been written for an SHPB following the logic just described and is included in Appendix A. A trapezoidal input pulse was assumed to represent the longitudinal strain at r=a and 2 meters from the specimen. Fifty-five terms were carried through in the Fourier analysis. At the onset all the terms in the series are assumed to be in phase and their relative phases are calculated as they travel down the bar. After multiple reflections, the pulses are reconstructed at the two strain gage locations on the incident and transmitter bars (both 1 meter from the specimen). At any interface between dissimilar materials, velocities and forces must be con- tinuous. Assuming normal incidence waves only, in a perfectly elastic medium, then the relationship between the incident, transmitted, and reflected stresses can be shown to be at = [2 Alp2 c2/(A2p2 c2 + Alpi ci )] oi r = [(A2p2 c2 Al p cl)/(A2p2 c2 + Al pi c) ] i where A is the bar cross-sectional area, p is the density, and c is the wavespeed. Subscripts 1 and 2 pertain to the two materials, and the wave is moving from material 1 to material 2. Each term in the Fourier series has a different wavespeed, and since only Mode MI,i is considered in the analysis, these wavespeeds are always less than or equal to the infinite elastic speed VEIp. For the sake of consistency in comparing the various approaches, this effort was based entirely upon a compressive SHPB whose incident and transmitter bars are made of steel, and a work hardened aluminum specimen of varying dimensions was assumed. Thus, the characteristic impedance pc is approximately three times greater in the bars than in the specimen. Conse- quently, an incident compressive wave will be reflected as a tension wave at the incident bar-specimen interface and the transmitted part will still be in compression. As this compressive pulse reaches the end of the specimen at the transmitter bar, it will be both reflected and transmitted in compression in accordance with the mismatch of characteristic impedances of the bar and the specimen. This means that a compressive wave is now "trapped" in the specimen. It loses amplitude at each reflection, but it remains compressive. It also means that every time the wave reaches the specimen-incident bar interface it will also transmit into the incident bar as a compressive pulse. The compressive pulse traveling out of the specimen into the incident bar superposes itself onto the reflecting tensile part and thus reduces its mag- nitude. As the compression pulse makes multiple reflections and transmissions within the specimen, the magnitude of the original tensile reflection of the incident pulse is reduced by the magnitude of the trans- mitted compressive pulse from the specimen into the incident bar. The length of the compressive incident pulse is of finite time, which means the tensile reflection and the compressive transmission pulses are also of finite length, and in fact are of identical duration as the incident wave. This implies the wavelengths of these transient pulses do not change as they cross interfaces, only their amplitudes are changed. These phenomena are clearly seen in experimental traces (see for example Figure 4), and in the Modified Pochhammer-Chree results (see for example Figure 5) obtained from the program discussed in the previous page; the reflected tensile wave starts out at a certain value, and then in the time required to transit twice the specimen length, it jumps to a lower level due to the arrival of the first portion of the compressive pulse from the specimen. The amplitudes of the reflected and the transmitted pulses add up to the amplitude of the incident pulse. In the experiment of Figure 4 the specimen was plastically deformed, while for Figure 5 the specimen remains elastic. It should be mentioned at this point that in order to be able to directly compare the results, the dimensions of the bars and the locations of the strain gages were the same in all methodologies. Specifically, since the Tyndall SHPB is 2 inches in diameter and the gages are 40 inches on either side of the specimen, both the modified Pochhammer-Chree method and the finite difference method were run with those values. For the 6-inch SHPB, the numerical methods assumed strain gages 80 inches on either side of the specimen. C-, So F: C-, 1: I- edW 'SS381S o o o o 0 0 0 0 ln in o I -4 Z Tz I0 II OE -I o z rM o to ro NT O d dI I I qtowoooWm6l N iii iii TY- S3SS3u.S 03ZnnryVION As mentioned earlier, the results of the Pochhammer-Chree frequency equation also lead to the variation of the stresses and displacements in the radial direction. These variations also depend on the wave speed and the wavelength of the pulse, and there are an infinite number of solutions each associated with a particular mode of vibration. After some tedious deriva- tions [33] the amplitudes W(r) and U(r) of the assumed longitudinal and radial displacements uz = W(r) f(t,z) and ur = U(r) f(t,z) are ^ Ji (ha) haJo(hr) (1 -f x) /caJo(Kr) W(r)= iyc +I ha J J(ha) (x 1) J (Kc a) J1 (ha) x( ha J (hr) (1 x) 1 /aJ Ji ( r) U 0 ha (x- )a J, (ha) (x -1) (2x- 1) J1 (K a) where c = a constant determined by the amplitude of vibration, (Co) 1-2v h=y(fpx- 1)V2, K/=y(2x- 1)'/ With this information the radial, shear, and longitudinal stresses can be calculated as 2 [^J(ha) Jo (hr) a Ji(hr) =2 ha (1 x) ha1(ha) r J (ha)+ fix 1 Jo (Kr) xfx- la J1(K r) 1 x J1 (K a) 1 x r Ji(/ca) ur = -2 ip y ch J1 (ha) (J1(ha) J1 ( a) f (t,z) -2 J(ha) Jo (ha) J1 (K a) z= -2uyc (ha) [(l+x-fix) haJl(ha) + (1-x)a J( ]f(t,z) Again assuming only Mode M1,1 is excited, the values of the wave speed for each component of the Fourier series under consideration can be calculated. Thus, if the bar is divided radially into a number of increments, it is possible to calculate the stress and displacement distributions in the bar at any point at any time. These calculations can be combined with those of the preceding discussion of the Modified Pochhammer-Chree equation and the result is a complete description of the elastic wave propagation in an SHPB. Care must be taken when solving the equations involving Bessel func- tions numerically, since as the argument becomes negative, modified Bessel functions must be used. Specifically Jo (ix) = Io (x) and Ji (ix) = i II (x), or in general , In (x) = (i)-n Jn (ix) a Davies [27] showed that if the ratio is kept below 0.1 then the radial variation is less than 5%. In an SHPB there are an infinite number of pulses a whose W values span from zero all the way to infinity. As discussed earlier, it is sufficient to make sure that the amplitude of the Fourier components have fallen to a small fraction of the initial value, when the value of - reaches and goes beyond 0.1. That is, most of the contributing components a of the Fourier series must fall below = 0.1. A subroutine was added to the Modified Pochhammer-Chree program to calculate the radial variations for the longitudinal and the radial stresses. The results of the longitudinal stress for a 2-inch and a 6-inch bar are plotted in Figure 6. As expected, the lower ( lower values of n), Fourier series terms S1- C' ii /H I C-0 t o C I / / / - I I J I- I of cu Ci Ocn 0e I UU LL IP i0 0o SLI I I I I Hn- 1) 011 I II I I I I I I II I I I II I -o 000000000 000000000 SS3I.S IVNIanlIIONO1i 3:ZInVVON (the ones whose values are low), show a nearly uniform distribution of a stress, while as the wavelengths become smaller ( larger ), the radial variations become progressively greater to a point where they change sign. If the amplitudes of these higher terms have fallen to a very small value, then their contribution to the overall pulse will be negligible. Plots of the radial variations of reflected and transmitted stresses are shown in Appendix A for a 2 and a 6-inch SHPB (Figures 36 and 37). It is also apparent that as the diameter of the bar is increased and the input wavelength is kept the same, the stress will show uniformity for only a few terms. This fact is additional proof that the one-dimensionality of an SHPB is dependent on the ratio of the radius to the wavelength, and if kept the same as for the existing SHPBs, then the wave propagations should be mathematically identical. Although the higher Fourier terms are highly nonuniform, their con- tributions are very small. Furthermore, if one normalizes the amplitude values of the radial and the shear stresses to those of the longitudinal stres- ses at the center, the results show the radial and shear stresses to be of many orders of magnitude smaller than the longitudinal values. This can also be observed in the finite difference calculations, and it is further proof of the one-dimensional nature of the wave propagation in the bar. Also, as expected, when the diameter of the bar is increased to 6 inches, the radial and shear stresses also increase, but remain in the same order of magnitude, and can thus still be safely neglected. Finite Difference Numerical Simulation The two-dimensional equations of motion in cylindrical coordinates can be solved using an existing hydrocode such as HULL [34]. HULL has a two-dimensional Lagrangian module that solves conservation of mass, momentum, and energy simultaneously with the definitions of the deforma- tion tensor, constitutive relations, and the stress deviators. A centered finite difference approximation is employed. The topic of constitutive rela- tions is the most crucial as it includes the relationship between the elastic and plastic deviatoric strains and stresses, and the relationship between pressure, density, and internal energy. The accuracy of any hydrocode is highly dependent on the implementation of the constitutive relations. In this effort only metals were considered, namely steel and aluminum. Both are modeled in HULL as elastic/plastic materials, although the stress levels were kept low enough to keep the steel bars in the elastic region. Both materials are also considered isotropic. Stresses are decom- posed into a spherical and a deviatoric component, and the generalized Hooke's law is used. Yielding is treated following the von Mises criterion, which assumes that yielding starts when the combined-stress distortion ener- gy equals the yield value of distorsion energy in a tension or compression test. Specifically, it relates the second deviatoric stress invariant to a specified yield function. Once yield is reached, the deviatoric stresses are corrected and reset to be normal to the yield surface in stress space. The total stresses can then be calculated. Using HULL, a trapezoidal pressure pulse was input on the incident bar and the specimen was specified as an elastic/strain-hardening material. The HULL output stations included the locations of the strain gages on the bars, plus one within the specimen itself. Sliding interfaces with no friction were used between the two materials for most calculations. To study the effects of friction, locked interfaces that allow no sliding to occur were used. The bars are made of steel and remain elastic at all times. The first runs were made to model as closely as possible the experimental data available for aluminum from the Tyndall AFB SHPB. Appendix D contains a HULL sample input file. HULL was also run to investigate the complicated wave interaction phenomena at the specimen-bar interfaces, especially in cases when the specimen is smaller in diameter than the bars. The next step was to study the effects of a 6-inch diameter bar, with two input pulses. One input pulse with a wavelength long enough to give a one-dimensional response was used. Additionally, one with a much shorter wavelength was used to show how the stresses and displacements vary in the radial direction, and disperse after a short travel, making the conventional SHPB analysis invalid. Experimental Method How does one know how good or even reliable the HULL output is? In order to answer this question, a direct comparison was sought between existing experimental data obtained at Tyndall AFB, and the calculated results from the HULL code. Specifically, numerous compression tests were run on aluminum 6061-T6511 specimens between steel incident and trans- mitter bars. The input pressure was varied to assess strain rate sensitivity, and stresses and strains were recorded at strain gages located in the incident and transmitter bars. It was found that (as already known) this particular aluminum is strain rate independent; that is, the stress strain curve is not affected by the rate at which the load is applied. This is very important because the HULL code does not take strain rate into account. Therefore, a comparison will be meaningful between the experimental data and the com- puter results for an aluminum specimen. The available data was obtained from a 2-inch diameter SHPB and varying specimen sizes. HULL was then run with a new material added to the material library to match as closely as possible the 6061-T6511 aluminum properties. In particular, aluminum is modeled as an elastic/plastic material, and one would expect to observe that stress strain relationship in the output. One should be able to recreate the relationship between stress and strain since that is what the SHPB is designed and used for. The rationale was to prove the validity of the numerical tool by comparing existing data with the results of the code, and then move on to the proposed 6-inch bar and have the confidence to trust and draw conclusions from those results. Comparison of Results A number of compressive tests were conducted at Tyndall AFB using a 2- inch SHPB on aluminum specimens. The diameter and length of the specimen, as well as the input pressure were varied to assess their influence. The results for three cases are shown in Figures 7, 8, and 9. To compare experimental and numerical results, HULL was run matching as closely as possible the conditions in the Tyndall tests. One needs to realize however, that no matter how good the numerical model is, there are some parameters one cannot predict or reproduce. For example, the input pulse is not in reality a perfect square pulse, nor is it perfectly uniform along the cross section as numerically modeled. Furthermore, the constitutive equations for steel and especially for aluminum (which goes beyond yield in the course of the test), are theoretical models and by no means exact. Finally, the effects of friction are not included in the numerical scheme. Having said that, one is then looking for a qualitative match in the data, more so than exact values of stresses or strains. What this means is that certain trends that have been observed experimentally should show up numerically. For example, as the specimen gets smaller compared to the incident bar, experimental results show that the transmitted stress also be- comes smaller. Consequently, the reflected stress increases and its shape becomes more uniform as the initial high peak starts to blend in with the rest of the reflected pulse. The HULL calculations also show these phenomena. Although the actual amplitude values do not match, the relative change between tests does. In general, the experimental data gives higher trans- mitted stresses, but it does so consistently. Experimental results are shown in Figures 7, 8, and 9 and can be compared to the numerical results from HULL shown in Figures 10, 11, and 12. The results of the HULL calculations show very good agreement with the experiments, which point out the validity of the HULL code. Ei 00 I- *G '0 rij 4O -! ce C 0 4"S *4- I-3 o o o 0 o 4 1 edW 'SS3UIS In j o o i- o o o a o o o0 0 0 0 0 0 I I edW 'SS3EIS S E I.- C/3 ct2 6 r- ~II U-I 13 -j eE ..c~ P, o0 Uo *a <* -t a 03 o ia a Uj - I- 1 I- S: o . < In SI -- a E z a - cu a iu Cu RdW 'SS3u.LS 2.01E+00 1.6gE+00 1.21E+00 8.01E-01 4.01E-01 0.0 -4.01E-01 -8.01E-01 -1.21E+00 -1.61E+00 -2.01E+00 0.00 1.00E-03 8.00E-04 6.00E-04 4.00E-04 2.00E-04 00 -2.00E-04 -4.00E-04 -6.0OE-04 -8.00E-04 -1.00E-03 - 0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.1 T (MSEC) SPUT HOPKINSON BAR 0.4 CM ZONING Problem 4.0000 SSTA= 1 XO= 2.33 YO= 99.80 MAX= 5.372E-04 MIN=-8.220E-04 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0,88 0.99 1.1 T (MSEC) Figure 10. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses for an Aluminum Specimen, Diameter = 3.18 cm, Length = 3.18 cm STA= 1 XO= 2.33 YO= 99.80 MAX= 1.287E+00 MIN=-1.969E+00 2.01E+00 1.61E+00 1.21E+00 8.01E-01 4.01E-01 0.0 -4.01E-01 -8.01E-01 -1.21E+00 -1.61E+00 -2.01E+00 0.00 1.00E-03 8.00E-04 6.00E-04 4 00E-04 2.00E-04 0.0 -2.00E-04 -4.00E-04 -6.00E-04 -8.00E-04 -1.00E-03 0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.1 T (MSEC) SPUT HOPKINSON BAR 0.4 CM ZONING Problem 3.0000 STA= 1 XO= 2.33 YO= 99.80 I MAX= 6.267E-04 MIN--8.215E-04 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 T (MSEC) Figure 11. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses for an Aluminum Specimen, Diameter = 2.54 cm, Length = 2.54 cm STA= 1 XO= 2.33 YO= 99.80 MAX= 1.500E+00 MIN=-1.968E+00 2.01E+00 1.61E+00 1.21E+00 8.01E-01 4.01E-01 0.0 -4.01E-01 -8.01E-01 - 1.21E+00 -1.61E+00 -2.01E+00 0.00 1 00E-03 8 OOE-04 * 8.00E-04 4 00E-04 4.00E-04 2.00E-04 0.0 -2.00E-04 -4 00E-04 -6.00E-04 -8.00E-04 -1.00E-03 - 0.00 STA= 1 X0= 2.33 YO= 99.80 MAX= 1.666E+00 MIN=-1.973E+00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.' T (MSEC) SPLIT HOPKINSON BAR 0.4 CM ZONING Problem 6.0000 STA= 1 XO= 2.33 YO= 99.80 MAX= 6.938E-04 MIN--8.238E-04 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 T (MSEC) 0.99 1.1 Figure 12. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses for an Aluminum Specimen, Diameter = 0.56 cm, Length = 0.56 cm It makes sense physically that as the specimen gets smaller, more and more of the incident pulse is reflected back, to the limit case when there is no specimen and the reflected trace is identical and reverse of the incident. It also seems reasonable that when the specimen diameter is less then half of the SHPB diameter, little of the incident pulse gets transmitted into the specimen, which means that even less of this transmitted pulse will get back to the reflected pulse and therefore its basic shape will not be changed. Hence the appearance of a distinct peak in the reflected trace is reduced when the specimen is small. It has already been shown in Figure 5 that the Modified Pochhammer- Chree method also shows the characteristic "peak" in the reflected trace even though the solution is elastic. The only difference between an elastic and an elastic-plastic specimen is in the magnitude of the stress that may be applied. Therefore, the only difference in the shape of the traces in the modified Pochhammer-Chree method will be a higher compressive reflec- tion out of the specimen, resulting in a lower reflective plateau after the initial "peak", than for the corresponding non-elastic case. To show this, three experimental tests were run with the same specimen configuration at different levels of stress; one level was kept in the aluminum elastic region, the other two exceeded the yield value for aluminum. Figure 13 shows how after the initial peak in the elastic case (top figure), the reflected pulse almost goes down to zero, while as the input stress is increased (middle and bottom figures), the plateau after the peak increases also. Another issue to be explored experimentally and numerically is the effect of decreasing the pulse duration with respect to the diameter of the bar. As discussed, the equations governing an SHPB show that increasing the ratio will affect two important things. The first thing is that the pulse will disperse more; that is, the contribution from the slower traveling com- ponents will be greater, and the overall waveform will distort and spread out more as it travels. The second thing is that the wave propagation will be less one dimensional in nature, which means that the longitudinal stress will have <-REFLECTED 0.0 o1 <-INCIDENT -TRANSMITTED 50 -25.0 i-- In -50.0 - 0.0 250.Ous 500.Ous 750.Ous TIME <-REFLECTED 0.0 i- -50.0 - /<-TRANSMITTED <-INCIDENT -100.0 -- I I I I I I I I I I I I 0.09 250.Ous 500.0us 750.0us TIME I T r n I m I I I I I I I I I _ 100.0 - -.-REFE CTED U) TIME Figure 13. Experimental Traces at Three Incident Stress Levels (60 MPa-Top, S a-iddle, 167 MPa-Bottom). Specimen Diameter 1.253 Length = .625 in, 2-inch S B 0.05 250.0us S00,us 750.0us Figure 13. Experimental Traces at Three Incident Stress Levels (60 MPa-Top, 100 MPa-Middle, 167 MPa-Bottom). Specimen Diameter 1.253 in., Length = .625 in., 2-inch SHPB larger variations along the radius and the radial and shear stresses will be a less negligible as increases. Numerically one can observe the effects of decreasing the input wavelength by using HULL and the Modified Pochham- mer-Chree method. The following runs shown in Table 1, were made to test the methodologies against what it is analytically believed to be true. Table 1. HULL and Modified Pochhammer-Chree Runs DIAMETER WAVELENGTH a FIGURES (inches) microsecondss) A 2 250 1/125 3, 14 6 750 1/125 15 6 500 3/250 16, 17 6 250 3/125 18,19 Figures 5, 15, 16, and 18 show results from the Modified Pochhammer- Chree program, while Figures 14, 17, and 19 are HULL output traces. Notice that HULL appears to have two different coordinate systems from one figure to the next. The reason for this is that the runs were made on different computer systems (VAX and CRAY) and each system has a slightly different version of HULL. In one z is the longitudinal coordinate, and in the other, y is the longitudinal coordinate. This discrepancy should not a cause confusion. It can be seen that the pulses disperse more when s is greater. a The results also confirm that if X is the same, regardless of the bar 2.41E+00 STA= 1 XO= 2.33 YO= 99.80 1.81E+00 MAX= 1.743E+00 MIN=-3.323E+00 1.21E+00 6.01E-01 S0.0 -6.01E-01 S-1.21E+00- -1.81E+00 -2.41E+00 -3.01E+00 -3.61E+00 0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.1 T (MSEC) SPUT HOPKINSON BAR 0.4 CM ZONING Problem 5.0000 1.50E-03 STA= 1 XO= 2.33 YO= 99.80 1 20E-03 MAX= 7.268E-04 MIN--1.386E-03 9.00E-04- 3.00E-04 0.0 -3.00E-04 -6.00E-04 -9.00E-04 -1 20E-03 -1 50E-03 0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.8B 0.99 1.1 T (MSEC) Figure 14. HULL Output. 2-inch SHPB. 2-inch Diameter and Length, Aluminum Specimen, 257psec Pulse o d000o dd000o d-0 - I I I I I I l I I I I S3SS3 IS C3ZrrnVW ON I C | - C3 3f CO 0 0 YC '- r. I-S LL) U- a w Oi I > Q 0I 0 I I I I I I I I 1 I I 0 0 0 0 0^ 8 --i s r "^ ^ ^ ' n^nr-o Sc~c o SSMuLS iaMGIONI O3ZrWNON T (MSEC) Figure 17. HULL Output. = 500 psec Pulse, Station 1 (Top) Bar Center, Station 2 (Bottom) Bar Surface I O2 c 0 0 E I 0. Z< CD W 00 z z N E I I I I I I I I I I I I I I I I 0 LI u4) N7 -P 4n I l I I I TI I S3SS3ylS a3ZnVMviON I ~ : II 1+ 0 o Bo T ai 00 x =0 N LV N 00 t 0 I Cb 0 W 4 X 0 0 0 0 0 0 e'4 0 0 oi N N ID I I (B)) ssais diameter, the waveforms are mathematically the same; this is shown in Figures 5 and 15. Figures 18 and 19 indicate that an input pulse of 250 microseconds for a 6-inch SHPB is not recommended. It appears dispersion becomes a problem, even after a very short time, and also the one-dimen- sionality of the motion is questionable. Experimentally, one can achieve different wavelengths by changing the length of the striker bar. The result- ing waveforms for a 16-inch and a 25-inch striker bar are shown in Figure 20 ( all three bar locations are on the incident bar ). Notice the more pronounced dispersion for the shorter wavelength. As discussed, along with more wave dispersion, the one dimen- sionality of the motion in a bar is also affected by the change of A values. a Again, one can turn to the two numerical schemes and observe how, as - increases, the radial and shear stresses increase, and the longitudinal stress shows more variations along the radial direction. Figures 21, and 22 show all stress components on the surface of the incident bar at the strain gage location for a 2-inch bar and a 250 microsecond pulse, and a 6-inch bar with a 500 microsecond pulse. Incident and reflected stresses are shown. Al- though the radial and shear components of the stresses are higher for higher values of -, they nevertheless remain negligible when compared to the longitudinal stress. Figures 23, 24, and 25 show the pulses at the center and at the surface of the transmitter bar at the strain gage location. Figure 23 is the result of the modified Pochhammer-Chree method for a 2-inch bar and 250 microseconds. Figures 24 and 25 are HULL output for 250 and 500 microsecond input pulses respectively. Figure 23 shows a difference of about 1.5% between the two curves. The variation in stress along the cross section is roughly 5% for Figure 24 and 2.5% for Figure 25. Therefore, the a uniformity of the longitudinal stress is degraded as A is increased, although for the proposed 6-inch bar and a 500 microseconds pulse, the wave motion should be very close to one-dimensional. distances from striker bar end 0.0001 0.0003 0.0005 0.0007 Time (sec) 0.0001 0.0003 0.0005 0.0007 Time (sec) Figure 20. Experimental Traces at Three Incident Bar Locations for Two Striker Bar Lengths. 16-inch (Top) and 25-inch (Bottom), 2-inch SHPB -1 --I -0.0001 -1 -I- -0.0001 0.0009 0.0009 2.41E+00 1.81E+00 1.21E+00 6.01E-01 0.0 -6.01E-01 -1.21E+00 -1.81E+00 II -2.41E+00 -3.01E+00 -3.61E+00 - 0.00 REFLECTED STA- 1 XO= 2.33 YO= 99.80 7- 7 YY DENT ---* 0.11 0.22 0.33 0.44 0.55 T (MSEC) 0.66 0.77 0.8B 0.99 1.1 Figure 21. HULL Output 2-inch Bar, 250 usec Pulse, All Stress Components on the Surface at Incident Bar Strain Gage 0.75 T (MSEC) Figure 22. HULL Output. 6-inch Bar, 500 ,sec Pulse, All Stress Components on the Surface at Incident Bar Strain Gage Wm I uJm -LU N o S I I I I I I I I I I I I I 0 LLJ I- o-' d^ 1 c o-- Io 66I I I I I - I ill iiI -I r 2 -- \ i ^----- * ^~~~~~ ~~ rM LC MK ^ O ior ac -i iK o d d d d l ^ -^ S3SS3I1S 1N3aIONI a3ZInVI-NON 3.00x 10-' STA= 4 XO = 7.410E+00 MAX = 2173E-01 YO 4.998F-'.; -,: ",:"'-, o.oo " 03.0010 -6.00x10" -9.DO00xD' -1.20 -1.50 -1.80 Z -2.10 -2.40 -2.70 0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 '.20 1.35 15 TIME (MSEC) 3.00x 0-' O.03.00 x 10-' YO-----------4----------------------------------------------- STA= 5 XO = 1.900E-01 MAX 2 69 l1 YO 4.99BE+02 MI --2. 1 0 0.00 Z -6.00x I ' -9.00x1 0' -9.00 ID" -1.20 Z LA 2 -1.50 -2.10 -1.80 L HP = "hoop" stress I -2.40 2 -------- i -------- i -------- i -------- -------- -2,70 0.00 0.15 0.30 0.45 0.60 0.75 0..90 1.05 1.20 1.35 1. TIME (MSEC) Figure 24. HULL Output 6-inch Bar, 250 ,sec Pulse. Surface (Top) and Center (Bottom) Transmitted Stresses 1.21E+00 8.01E-01 4.01E-01 0.0 -4.01E-01 -8.01E-01 -1.21E+00 -1.61E+00 -2.01E+00 -2.41E+00 -2.81E+00 0.00 1.21E+00 8.01E-01 4.01E-01 0.0 -4.01E-01 -8.01E-01 -1.21E+00 -1.61E+00 -2.01E+00 -2.41E+00 -2.81E+00 0.00 0.15 0.30 0.45 0.60 0.75 T (MSEC) 0.15 0.30 0.45 .60 0.75 0.90 1.05 1.20 1.35 1.5 T (MSEC) Figure 25. HULL Output 6-inch Bar, 500 ,sec Pulse. Surface (Top) and Center (Bottom) Transmitted Stresses STA= 3 XO= 7.41 YO= 399.80 MAX= 6.370E-01 MIN=-2.442E+00 STA- 4 X0= MAX- 5.477E-01 0.90 1.05 1.20 1.35 1. A tool like HULL offers the advantage of being able to "look" at the wave response, stresses, accelerations, etc., anywhere in the bar. In particular, two questions regarding the operation of an SHPB have been of interest for some time. The first one has to do with the initial oscillations that are observed experimentally and whether they are due to something inherent to the equations of motion, or whether they are the result of some experimental characteristic like friction, or striker misalignment. The second question has to do with what really goes on at the interfaces between the specimen and the bars, especially when the specimen is smaller in diameter. To answer the first question one can look at the HULL stress output at points very close to the impacted ends. Figures 26 through 28 show curves for various diameter bars and various input pulses at two locations on the inci- dent bar, one near the impacted end, and another one meter from it. The a following observations can be made; in the cases where A is larger, the oscillations in the input pulse are more pronounced and they take longer to damp out. Further, if one compares these results with the experimental traces (Figure 20), the similarity in the response is striking. Since the finite difference calculation does not take into account friction, misalignments, etc., it can be deduced that the so called "end effects" are due to inherent characteristics of the wave motion not to the way the experiment is carried out. That means higher frequency modes are excited at the impacted end of a the bar. If one refers back to Figure 3, the cases with larger show more oscillations because the response is further to the right on the x-axis, and thus more modes of vibrations are possible. Nevertheless, as predicted by Davies [27] and others, these high frequency modes damp out quickly, so that after about ten diameter lengths the only mode of vibration present is mode M1,1. The second question presents more of a problem; the wave interac- tion at the bar-specimen interfaces is extremely complicated, especially if the specimen yields at some point during the process. Looking at the stress a co a, I- c=._e C.4 o o i- s i3J oo o o o o o o o o a x X X X X xxx xx x X X X 0 0 0 0 0 x O O O x 0 0 0 0 0 0o 0o 0 0 0 0 6 o o 0 0 0 0 'o (33/3NNO) ZZi (33/3cNa) ZZL 1 .0 STA= 1 XO = 7 410E+00 MAX = 9.756E-0 YO 5.000E+00 MIN --3 328E+0( 1.00 5.00' 10-' 0.00 -5 .00 0 ' -100 2 z z -1.50 2.50 z -2.00 z z HP "hoop" stress a 0.00 0.i5 .30 045 0.60 0.75 0D 1.05 1 20 135 TIME (MSEC) 1.50 STA= 2 XO = 1.900E-01 MAX 9. YO 3.002E+02 MIN 3. 1.00 5.00x 10- - 0.00 -5.00x 10' - 1 .00 -1.50 Z -2.00 -2.50 -3.50 0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.3 TIME (MSEC) Figure 27. HULL Output. 6-inch Bar, All Stresses Near Impacted End (Top) and at the Strain Gage Location (Bottom) 64 1 50 Sl X0 2 32tO MAx 1 217E00 YO 5000E+00 MIN ---3 25E 0N 5.00 10 p 1Z S00 5 00O '1 - 150 -2. 00 r z 2 50 -J.oo ;z HP = "hoop" stress -= o -3 s0 000 0a15 030 0 45 060 0.75 090 1,0 1 20 135 I 5 TIME (MSEC) SPLIT HOPKINSON BAR LOCKED NODES Problem 5 1000 3 50 3.50 I 2 xO 2 .32E+00 MAX 2 474E+00 O 1 002E402 MIN --3 323E 00 2.80 -1 4( z 7 0C. 10-' 0 00 0 15 0 0 0 45 060 0.75 090 1.05 1 20 1 5 I TIME (MSEC) Figure 28. HULL Output. 2-inch Bar, All Stresses Near the Impacted End (Top) and at the Strain Gage Location (Bottom) components along one such interface from a HULL calculation, it is very difficult to make any definite statement. To help in the understanding of the complicated stress wave interaction, Figure 29 shows the important features and magnitudes of all stress components along the incident bar-specimen interface. The longitudinal stress (uniform and in compression, of mag- nitude 167 MPa sometime prior to this interface), at the interface varies from almost zero at the bar surface to 100 to 80 MPa at the center. Shear and radial stresses become of the same order of magnitude as the lon- gitudinal stress. What is worth noticing is that at the center aeo or the "hoop" stress is very large, in fact larger than the input stress. Also at this point the displacements in the longitudinal and radial direction are small. It appears then, that shear and circumferential stresses play an important role at the edge where the specimen ends against the bar face. In a sense, the incoming stress has to "squeeze" itself through a smaller cylinder therefore it is not surprising to find other than longitudinal stresses at the interface. What is remarkable is how quickly it recovers to an almost uniform state of stress within the specimen only one centimeter from the interface (Figure 30), and in the transmitter bar at the strain gage location (Figure 31), where the longitudinal stress is the only non zero component. Figures 30 and 31 cor- respond to the same SHPB as in Figure 29. Notice that a00 in Figure 30 is only about 50 MPa while it peaked at 220 MPa at the interface; that amounts to a reduction of 77% in one centimeter. Appendix C contains more HULL output traces which can further explain the above discussion. Finally, Bertholf and Karnes [23] pointed out that the first interface between the bar and the specimen effectively filters out most of the oscilla- tions in the incident pulse, and thus in the cases when these oscillations are large for one reason or another, it is wise to use only the transmitted stress to calculate the response in the specimen. The analytical tools also show this filtering effect of the specimen, and it appears the use of the transmitted pulse only, may be a worthwhile alternative for the proposed 6-inch SHPB. 0 6 soo -2l01E-0. 001E- S-tcE-or -S 0E-Of -0I-0'o O0 - OI4-01 -40 E-on wow- sor-ol 2 076-of I 00 -0 t o6 01 O 0:2 O. 0 0o** 0 s o00 o 0.8 000 T (inco Figure 29. HULL Output 2-inch Bar, All Stresses at Three Locations at the Bar-Specimen Interface STATION 1 STATION LOCATIONS 3 SPECIMEN t STATION 3 I H.0 \ I ot 0oi oU 0* o aO or o 0 0 i T acm) STATION 2 ooo~~~~~~~ ~ 05621aJ11 ss ooa o HULL Output 2-inch Bar, Stresses in Aluminum Specimen at Two Locations Near the Surface (Top), Near the Center (Bottom). Compare with Figure 29 9 00x10 ' 2.00 1.50 5.00x10- -1.00 -1.50 -2.00 -250 -3.00 0. Figure 30. 3,0010* 5A 3 X 2.328a*00 WX 2 774E+08 Y 3.000E+02 MIN ---4 50E+SO -.80. 10 -s50.to' -540o000 0o1 030 0o45 0.60 0.75 0.90 .0 1.20 1,35 TnE (.EC) 3 0010, 2 70., ' -2 TO"ID* -3600-' 4 lD' E r - 00oo 0' I 01.10' -2 70l0' -360 l00 " -454010 -' Figure 31. HULL Output. 2-inch Bar, Transmitted Stresses at Three Radial Locations. Surface (Top), Middle (Middle), Center (Bottom). Compare with Figure 29 The two analytical methodologies used have advantages and disadvantages, and the use of one over another depends on the particular need and applica- tion. The finite difference solution is the most accurate if the constitutive relationships of the materials involved are adequate. It is also the most time consuming and expensive. The Modified Pochhammer-Chree method developed in this study gives a good qualitative look at the wave propagation in a typical SHPB in a very short time. It can be reliably used to assess the one-dimensionality of the elastic wave motion, as well as dispersion effects and cross sectional variations and values of all the stress components. It is a tool that can also be used to investigate if a particular experimental setup is affected by friction or inertia by predicting what would happen in the ab- sence of friction and inertia. Furthermore, a typical HULL calculation took anywhere between three and six central processing unit (CPU) hours on a CRAY Y-MP computer at an average cost of $800.00 per job! The modified Pochhammer-Chree method takes seconds to run on most machines at an average cost of less than $2.00. Although the Modified Pochhammer-Chree method assumes a purely elastic response, it does not affect its usefulness since one would not use any of these analytical tools to investigate true material response, but rather to study the reliability of a certain SHPB configuration. These numerical codes are not intended to replace the ex- perimental method, but rather to study the validity of the data reduction, the implications of a geometric change, the effects of a different input pulse, and so on. After all, if the true constitutive relations of the specimen material were known one would not need to study its response with an SHPB. The Modified Pochhammer-Chree method can give the results it was designed to give with any elastic specimen, as long as the characteristic impedance of the material and the bars are known. Inertia and Friction Effects The analytical methods discussed are useful tools to study the wave propagation of any SHPB, and within their limitations and assumptions can assess the one dimensionality and therefore the usefulness of different bar sizes and configurations. However, the reliability of an SHPB can be influenced by radial and axial inertia and by friction between the specimen and the elastic bars. Neither the modified Pochhammer-Chree method nor the finite difference code HULL take those into account. Approximate cor- rections have been derived to account for the effects of radial and axial inertia [7] and friction [8]. The objective of this investigation is to under- stand when and how inertia and friction become important factors and whether the SHPB diameter is a key parameter on inertia and/or friction contributions. In other words, one would like to know whether increasing the diameter of a SHPB will increase the effects of inertia or the effects of friction. Inertia effects become more pronounced as either the initial stress in the input bar becomes high, or the strain rate is increased [24]. The former causes very large stress and strain gradients which from the equations of motion, make the inertia term become important [35]. High strain-rates do not appear to cause inertia effects per se, rather the way the higher strain- rates are achieved can cause inertia effects to become important. Specifical- ly, if one increases the input stress to obtain a higher strain-rate, and this input stress is in the form of a step function, then apparently the combina- tion of high input stress, and the oscillations caused by radial waves resulting from the short rise time of the input, causes inertia effects to become quite important. Bertholf and Karnes [24] eliminated one of these two causes by, numerically, using a ramp input function with a much longer rise time. Their results show that doing this effectively eliminates the inertia effects, even though the same levels of strain rates were achieved. This means that the magnitude of the input stress is not a major cause of inertia effects, rather the short rise time associated with the high input seems to significantly contribute to inertia effects. Therefore, since in an SHPB one is limited to a step input pressure, there appears to be a maximum achievable strain rate beyond which the validity of the experimental data becomes questionable. Bertholf and Karnes calculated a maximum allowable strain rate for their SHPB. They claim that since their numerical calculations are based on constitutive equations which are rate independent, their results can be ap- plied directly to different diameter systems. For a 1-inch diameter bar, a length L over diameter d (L/d) = 0.3 in the specimen, and a step input stress, Bertholf and Karnes calculated a maximum strain rate of 400 1/sec. For a ramp input, this value is much higher for the reasons discussed earlier. If the diameter is increased by a factor n, then the new maximum strain rate is 400/n 1/sec. This scaling cannot be applied directly to the proposed 6-inch bar because the L/d of the specimen will not necessarily be 0.3. In fact it will be about one for concrete specimens to ensure the same aggregate density in all directions. Therefore one needs to go back to the equations that Davies and Hunter [7] derived to calculate the contribution of inertia based on the kinetic energy due to both axial and radial motion. Their result is (L2 2 d2 d'e (19) Us = Ub+Ps(Vs (1 where Ub is the measured stress, subscript s pertains to the specimen and L2 d2 Sand v--are the axial and radial inertia contributions respectively. 6 8 Equation (19) can be rewritten in nondimensional form as S- ab d2 E/d t2 ps d2 L L (20) Ub Ub 6 Ld J The left hand side is the error in the measured stress. If one assumes a nominal Poisson's ratio of 0.3, the contribution of the term L/d can be calculated relative to Bertholf and Karnes L/d of 0.3. This leads to the conclusion that for a nominal L/d = 1.0, the error in the measured stress will increase by a factor of 40. This means that the maximum strain-rate they calculated should be reduced even further. This would lead to unacceptably low strain-rates. Therefore, the thing to do is accept the errors due to inertia effects, and correct them using equation (19) in the data reduction algorithm. Furthermore, if one looks at typical strain rate plots from actual SHPB experiments, it appears that in most cases the strain rate very quickly levels off to some constant value. What that means is that the derivative of the strain rate will eventually approach zero, and it may be that the inertia effects will not be as important after the initial 100 or so microseconds. Friction presents more of a problem in the sense that its effects are not easily quantifiable or easy to predict. It is shown from numerical cal- culations that the measured stress will be larger than the true stress, result- ing in an artificially high strength in the material being analyzed. But how much? Early work and studies done by Davies and Hunter [7] and by Saman- ta [10] show that in general, if the diameter and the length of the specimen are of the same order of magnitude, then friction contributes to only a few percent error. It appears that the important parameter is the ratio of the area constrained by friction over the volume of material in the specimen. In other words, a thin specimen will be more affected by friction than a thick one. Since the area constrained by friction is d2/2 (two faces) and the volume is n d2 L/4 then the key parameter in determining friction effects is 2/L. The larger this value the greater the error due to friction. This means that increasing the diameter of the SHPB bars and specimen will not worsen friction effects. Although an exact value for friction effects is not obtainable, one can assess the order of magnitude of the error by running HULL with "locked" interfaces between the bars and the specimen, and then compare the results with the other runs where the interfaces were friction- less. (There is no option in HULL for modeling friction). The results then bound the friction problem as the "locked" case represents a worst case scenario. Figures 32 and 33 show the transmitted stress for a 2-inch bar with zero and infinite friction, respectively. Figures 34 and 35 show the same thing for a 6-inch bar. As expected, the cases with infinite friction show a higher transmitted stress, implying an artificially higher strength in the specimen. It is interesting to notice that the error due to friction in the 2-inch bar is about 16.4% and in the 6-inch bar is only 5.6%. This agrees with the criterion derived above which states that the error is proportional to 2/L. It appears the larger SHPB set up will be less susceptible to friction in light of the fact that the specimen length will be increased along with the diameter. However, one should not imply from this that friction introduces errors in the existing systems of the order of 10-20%. That figure is an upper bound for a case where the specimen is attached to the bars. In reality, the friction coefficient will be somewhere less then 1.0, especially if care is taken in lubricating the ends. Therefore, the errors due to friction are negligible in typical SHPBs and will be so also for a proposed 6-inch system. If, however, one was to test thin specimens, then the data reduction should take friction effects into account. 2.51E+00 2.01E+00 1.51E+00 1.01E+00 5.01E-01 0.0 -5.01E-01 -1.01E+00 -1.51E+00 -2.01E+00 -2.51E+00 0.00 0.11 0.22 0.33 0.44 0.55 T (MSEC) 0.66 0.77 0.88 0.99 1.1 Figure 32. HULL Output. 2-inch Bar, Transmitted Stress with Zero Friction Between Specimens and SHPB STA= 3 XO= 2.33 YO= 300.08 N SI o _ >jw o a in F i 0 w N o0 0 0 ox 0 0 0 0 *N 0a ( ) ss (GN) SS3H.S 0.75 T (MSEC) Figure 34. HULL Output. 6-inch Bar, Transmitted Stress with Zero Friction T b b 0 - X 0 0 0 o 0 0 o 0 6 0 I I I 77 0 "- --- -- ^ o IN N N 0 o 00 I *I (e) 0 -- i -- -- ] --6-- -- .0 CHAPTER 4 SIX-INCH SHPB FEASIBILITY Summary The objective of this effort was to assess the feasibility of a 6-inch SHPB. There is an abundance of experimental data for existing SHPB sys- tems, and it was used in this study to test the accuracy and reliability of the numerical finite difference code HULL. The first part of this work researched the multitude of efforts con- ducted on the subject in the past years. In particular, it was established that it was important to look at the two-dimensional equations of motion as the simplified one-dimensional case may not be adequate for a larger system. The Pochhammer-Chree solution for an infinitely long bar was used as the baseline for the mathematical analysis. Early papers on inertia and friction effects were studied, and the methodology extended to the 6-inch bar sys- tem, primarily to have a feel for the magnitude of the errors introduced by those factors. Taking a closer look at the governing equations and at the Pochham- mer-Chree frequency equation, it became apparent that the problem of wave propagation in a cylindrical bar is independent of scale if the ratio of the diameter over the wavelength of the traveling pulse is kept constant. This is not a new idea, but the implications in this study are crucial. It means that in theory at least, an SHPB can be scaled up or down as long as the input pulse is also scaled up or down. It does not mean the SHPB response is one dimensional, rather it implies that since the simple one-dimensional analysis has been proven reliable for the existing systems, it should apply just as well to the proposed 6-inch SHPB. The next part of this effort looked at three methodologies in order to compare and contrast the results. The goal was to have a firm understanding of the implications of increasing the size of an SHPB. The first two included a finite difference numerical technique, and experimental traces obtained at Tyndall AFB. The third method, called the Modified Pochhammer-Chree method for elastic systems, consisted of computationally following each Fourier component of the input pulse as it reflects and transmits at the bar-specimen interfaces. Each pulse has a unique velocity and wavelength, which enables one to calculate its position in time. The complete pulse can then be reconstructed at any time. Furthermore, the distribution of dis- placements and stresses along the radial direction can also be calculated if the wavespeeds of the Fourier components are known. It was shown how this methodology gives surprisingly good results, considering the many or- ders of magnitude difference in computing time compared to the finite difference technique. All three methodologies reaffirmed the initial statement of scale in- variance of any SHPB. It was shown how if the input pressure pulse has a wavelength of 500 microseconds, a 6-inch SHPB will behave as a one-dimen- sional wave propagation system, and thus all the current data reduction schemes and the current assumptions will be valid. Finally, the effects of friction and inertia were analyzed. It turns out friction causes only a few percent error, even for a large 6-inch bar. Inertia however, will affect the results of an SHPB, and must be accounted for in the data analysis of a new 6-inch system. Inertia effects increase as strain-rate increases. Conclusions and Recommendations The following conclusions can be drawn from this analytical study. 1. A 6-inch SHPB is theoretically feasible, and in fact identical to any existing system if the input pressure pulse duration is scaled appropriately. Specifically, a 500 microsecond wavelength is recommended. 2. The modified Pochhammer-Chree method can be used to assess the validity and study the wave propagation in any size SHPB. The results give the incident, transmitted and reflected pulses for different diameters, wavelengths and materials. This method can be applied to any input pres- sure if it can be described by a Fourier series. 3. If the specimen length and diameter are kept in the same order of magnitude, one need not worry about friction in the proposed 6-inch SHPB. It will cause only a small error in the analysis. 4. The new proposed 6-inch SHPB will need a correction in the data analysis to account for inertia. This is true especially if the input pulse is a step function and if strain rates higher then 10-20 1/sec are desired. Davies and Hunter [7] derived an equation to correct for inertia which has been shown to be fairly accurate. It is recommended this equation be incor- porated in the existing data reduction algorithm. A Fortran subroutine has been written by the author, is included in Appendix E, and can be directly integrated in the existing data reduction programs in use at the University of Florida and at Tyndall AFB. By following the guidelines in steps 1 through 4, the test engineer has in effect removed or corrected for any rate, friction, or two-dimensional effects due to the testing apparatus itself, and thus the results will permit the 81 observation of the true material behavior. The intrinsic characteristics of the material will be separated from any other effects inherent to the way the specimen is being tested. Therefore, the Split Hopkinson Pressure Bar has been and remains a reliable, accurate and useful tool to study material response to loading, even for a new proposed system 6 inches in diameter. REFERENCES 1. Kolsky, H., "An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading," Proceedings of the Physics Society, Vol. 62, pp. 676-700, 1949. 2. Hauser, F.E., Simmons, J.A., and Dorn, J.E., "Response of Metals to High Velocity Deformation," Proceedings of the Metallurgical Society Conference, Vol. 9, pp. 93-101, 1960. 3. Lindholm, U.S., "Some Experiments with the Split Hopkinson Pressure Bar," Journal of the Mechanics and Physics of Solids, Vol. 12, pp. 317-335, 1964. 4. Malvern, L.E., and Ross C.A., "Dynamic Response of Concrete and Concrete Structures," Final Technical Report, AFOSR Contract F49620-83-K007, University of Florida, Gainesville, FL, May 1986. 5. Robertson, K.D., Chou, S. and Rainey, J.H., "Design and Operating Characteristics of a Split Hopkinson Pressure Bar Apparatus," Techni- cal Report AMMRC TR 71-49, Army Materials and Mechanics Re- search Center, Watertown, MA, November 1971. 6. Kolsky, H., Stress Wave in Solids, Dover Publications, New York, 1963. 7. Davies, E.D.H., and Hunter, S.C., "The Dynamic Compression Testing of Solids by the Method of the Split Hopkinson Pressure Bar," Journal of the Mechanics and Physics of Solids, Vol. 11, pp. 155-179, 1963. 8. Rand, J.L., "An Analysis of the Split-Hopkinson Pressure Bar," U.S. Naval Ordnance Laboratory, White Oak, MD, NOLTR67-156, 1967. 9. Dharan, C.K.H. and Hauser, F.E., "Determination of Stress-Strain Char- acteristics at Very High Strain Rates," Experimental Mechanics, Vol. 10, pp. 370-382, 1970. 10. Samanta, S.K., "Dynamic Deformation of Aluminum and Copper at Elevated Temperatures," Journal of the Mechanics and Physics of Solids. Vol. 19, pp. 117-124, 1971. 11. Jahsman, W.E., "Re-examination of the Kolsky Technique for Measuring Dynamic Material Behavior," Journal of Applied Mechanics, Vol. 38, pp. 75-82, 1971. 12. Chiu, S.S. and Neubert, V.H., "Difference Method for Wave Analysis of the Split Hopkinson Pressure Bar with a Viscoelastic Specimen", Jour- nal of the Mechanics and Physics of Solids. Vol. 15, pp. 177-193, 1967. 13. Young, C. and Powell, C.N., "Lateral Inertia Effects on Rock Failure in Split Hopkinson-Bar Experiments," 20th U.S. Symposium on Rock Mechanics, Austin, TX, pp. 299-303, 1979. 14. Nicholas, T., "An Analysis of the Split Hopkinson Bar Technique for Strain-Rate-Dependent Material Behavior", Journal of Applied Mechanics. Vol. 1, pp. 277-282, March 1973. 15. Lawson, J.E., "An Investigation of the Mechanical Behavior of Metals at High Strain Rates in Torsion," Ph.D. Dissertation, Air Force Institute of Technology, Wright-Patterson AFB, OH, June 1971. 16. Baker, W.E. and Yew, C.H., "Strain-Rate Effects in the Propagation of Torsional Plastic Waves", Journal of Applied Mechanics, Vol. 33, pp. 917-923, 1966. 17. Jones, R.P.N., "The Generation of Torsional Stress Waves in a Circular Cylinder", Quarterly Journal of Mechanics and Applied Mathematics, Vol. 12, pp. 208-211, 1959. 18. Okawa, K., "Mechanical Behavior of Metals Under Tension-Compres- sion Loading at High Strain Rate," International Journal of Plasticity, Vol. 1, pp. 347-358, 1985. 19. Ross, C.A., Nash, P.T., Friesenhahn, G.H., "Pressure Waves in Soils Using a Split-Hopkinson Pressure Bar," Final Technical Report, ESL- TR-81-29, AFESC, Tyndall AFB, FL, July 1986. 20. Rajendran, A.M., and Bless, S.J., "High Strain Rate Material Behavior," Technical Report, AFWAL-TR-85-4009, AFWAL, Kirtland AFB, NM, December 1985. 21. Ross, C.A., "Split Hopkinson Pressure Bar Tests," Final Technical Report, ESL-TR-88-82, AFESC, Tyndall AFB, FL, March 1989. 22. Follansbee, P.S. and Frantz, C., "Wave Propagation in the Split Hopkin- son Pressure Bar", ASME Journal of Engineering Materials and Tech- nology, Vol. 105, pp. 61-66, 1983. 23. Redwood, M., Mechanical Waveguides, Pergamon Press, New York, 1960. 24. Bertholf, L.D. and Karnes, C.H., "Two-dimensional Analysis of the Split Hopkinson Pressure Bar System," Journal of the Mechanics and Physics of Solids, Vol. 23,pp. 1-19, 1975. 25. Bancroft D., "The Velocity of Longitudinal Waves in Cylindrical Bars," Physical Review, Vol. 59, pp. 588-593, April 1941. 26. Mindlin, R.D. and Herrmann G., A One-Dimensional Theory of Com- pressional Waves in an Elastic Rod, Pro. 1st U.S. Nat. Cong. App. Mech., Chicago, pp. 187-191, 1951. 27. Davies, R.M., "A Critical Study of the Hopkinson Pressure Bar," Philosophical Transactions Royal Society, Vol. A240, pp. 375-457, 1948. 28. Kolsky, H., "The Propagation of Stress Pulses in Viscoelastic Solids," Phil. Mag. Vol. 1, pp. 693-710, 1956. 29. Love, A. E. H., 1927, The Mathematical Theory of Elasticity, Dover Publications, New York. 30. Karal, F.C., "Propagation of Elastic Waves in a Semi-infinite Cylindrical Rod Using Finite Difference Methods," Journal of Sound Vibration, Vol. 13, pp. 115-145, 1970. 31. Gong, J.C., Malvern, L.E., and Jenkins, D.A., "Dispersion Investigation in the Split Hopkinson Pressure Bar," Journal of Engineering Materials and Technol ogy, Vol. 112, pp. 309-314, July 1990. 32. Jerome, E.L., "Feasibility of a 6-inch Split Hopkinson Pressure Bar (SHPB)," Final Technical Report, ESL-TR-90-44, AFESC, Tyndall AFB, FL, August 1990. 33. Hopkinson, B., "A Method of Measuring the Pressure Produced in the Detonation of High Explosives or by the Impact of Bullets," Philosophi- cal Transactions of the Royal Society of London, Vol. 213, pp. 437-456, 1914. 34. Osborne, J.J., and Matuska, D.A., "HULL, Finite Difference Code," Orlando Technology, Inc., Shalimar, Florida, 1987. 35. Rinehart, J.S., Stress Transients in Solids, HyperDynamics, Santa Fe, New Mexico, May, 1975. 36. Chree, C., "The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Coordinates, Their Solutions and Applications," Cambridge Philosophical Society, Transactions, Vol. 14, 1889, pp.250-369. 37. Pochhammer, L., "On the Propagation Velocities of Small Oscillations in an Unlimited Isotropic Circular Cylinder," Journal fur die Reine und Angewandte Mathematik, Vol. 81, 1876, pp. 324-326. APPENDIX A MODIFIED POCHHAMMER-CHREE PROGRAM AND SAMPLE OUTPUTS This program uses the Pochhammer-Chree frequency equation solu- tion to the two-dimensional equations of motion in cylindrical coordinates to analyze the wave propagation of an arbitrary trapezoidal input pulse in a bar of varying dimensions and material properties. The wave is carried through a typical compressive SHPB. Multiple reflections and transmissions at the interfaces between the specimen and bars are calculated and the waves reconstructed in the incident and in the transmitter bar. The program logic is organized as follows: The user is asked to enter the areas, densities, and wave speeds for the bars and the specimen. This information is used to calculate the charac- teristic impedances of the materials. The user is asked to enter the pulse duration, number of time steps (each 0.5e-6 seconds long), the diameter of the bar, and the dimensions of the specimen. The input pulse is represented via a Fourier series. 55 terms are retained. Amplitude of the incident pulse is normalized to 1.0. Each com- ponent has a unique phase angle, frequency, and amplitude and they are all in phase at the onset. The Pochhammer-Chree solution for mode M1,i is used to calculate the wave speed of each Fourier component in the bar material as well as in the specimen material. Each pulse is then started at a distance of 2 meters from the specimen and is followed to the incident bar-specimen interface through the multiple reflections and transmissions that occur at that point and is fol- lowed into the specimen to the specimen-transmitter bar interface where more reflections and transmissions take place. Since the speed of each 86 component of the pulse is known, and since the characteristic impedances are known, the program calculates the amount that gets reflected and trans- mitted each time, as well as how long it takes for each component to travel any distance. All 55 terms are treated in this way, and then the waveform is reconstructed both in the incident and in the transmitter bar at the strain gage locations both 1 meter from the specimen. The output is in numerical form. The incident, transmitted, and reflected waves are represented by values one-half microsecond apart. These data can be easily plotted with any available software. LOTUS 123 was used by the author. Additionally, the program calculates the cross sectional variations of the stresses derived from the Pochhammer-Chree frequency solution. Be- ssel functions are evaluated numerically. For each radial value the 55 terms making up the traveling wave are added for each time increment. The por- tion of each stress dependent on r, discussed in chapter 3, are essentially multiplied to each Fourier component at each time step. All the terms, at their particular frequency are added up in time to form the complete waveform at each radial location. Longitudinal, radial, and shear stresses are calculated, and their amplitudes are all normalized to the longitudinal stress at the center. Figures 36 and 37 are examples of cross sectional variations of the longitudinal stress for a 2-inch and a 6-inch SHPB. Reflected and trans- mitted pulses are shown. 88 It) 0 0 v) oo N 0 0 0 N tla 0 0 u < 8 o 7)0 a a 7 o N 5 - S 0c / C) Mo. r V) m -JJ 1 I T- o Lu I (n ^ I P S S c 0 l C3 0 d a 0 0 0 0i0 Sc3WUS O3iKSNVil Si0 ? ? UU) CC F F1 S 5 S O21Sn5 T 5 L LL) -J ss 03 ,. a 0al CG'lu 0- -S 0-- "OI 3- ^^-- s 1 Z i \ =s 1 . .^ i ^ \ 1 i na\ C THIS PROGRAM USES POCHHAMMER AND CHREE C FREQUENCY SOLUTION OF THE TWO DIMENSIONAL C EQUATIONS OF MOTION IN CYLINDRICAL COORDINATES C TO ANALYZE THE WAVE PROPAGATION OF AN ARBITRARY C TRAPEZOIDAL INPUT PULSE IN A BAR OF VARYING C DIMENSIONS AND MATERIAL PROPERTIES. C C FURTHERMORE, IT CALCULATES THE RESPONSE DUE C TO THE MULTIPLE REFLECTIONS AND TRANSMISSIONS C AS THE WAVE ENCOUNTERS A SPECIMEN OF DIFFERENT C MATERIAL, SAME DIAMETER, AND SPECIFIED LENGTH. C C THE OUTPUT CONSISTS OF THE INCIDENT, REFLECTED C AND TRANSMITTED PULSES AT STATIONS ONE METER C FROM THE SPECIMEN BOTH ON THE INCIDENT AND ON THE C TRANSMITTER BAR. SOLUTIONS ARE GIVEN AT THE CENTER C OF THE BAR (FILE FOR001.DAT), AND AT THE SURFACE C (FILE FOR010.DAT) TO CHECK FOR RADIAL UNIFORMITY C OF THE MOTION. ALL AMPLITUDES ARE NORMALIZED. C dimension d(3000,40),f(3000),phl(200),ph2(200) dimension time(3000,40),u(3000),fr(3000),c(200),dl(200) dimension frr(3000),ut(3000),frrr(3000),utt(3000) dimension sig(100,200) s-0.014 pi-3.14159 C WAVE SPEEDS, AREAS, AND DENSITIES ARE ENTERED TO C CALCULATE CHARACTERISTIC IMPEDANCES write(*,*)' input speeds, densities, and areas' write(*,*)' col,co2,(m/sec),rol,ro2,A1,A2' write(*,*)' bar is 1, specimen is 2' Read(*,* )col,co2,rol, ro2,al,a2 a-2.0/(pi**2*s) tr-2*al*ro2*co2/(ro2*co2*a2+rol*col*al) ref-(a2*ro2*co2-al*rol*col)/( ro2*co2*a2+rol*col*al) tr2=2*a2*rol*col/( ro2*co2*a2+rol*col*al) ref2-ref write(*,*)tr,ref,tr2,ref2 n=-1 write(*,*)' input nn,wo (1/sec),diameter (mm),spec. length(m)' read(*,*)nn,wo,diam,zl2 dia-diam/1000.0 C STRAIN GAGE LOCATION IS ASSUMED AT ONE METER C FROM IMPACTED END, AND ONE METER FROM EITHER C SIDES OF THE SPECIMEN zl-1.0 213-1.0 z24-1.0 C 55 TERMS ARE USED IN THE FOURIER ANALYSIS imm-55 do 5 j-l,mm n=n+2 ne-(n-1)/2 C CALCULATE THE AMPLITUDES OF EACH PULSE C AS THEY CROSS THE INTERFACES BACK AND FORTH C C AT STATION ON THE INCIDENT BAR: d(j,1)=(-a*(-1)**ne/n**2)*sin(n*pi*s) d(j,2)=ref*d(j,1) c THIS NEXT PIECE IS THE FIRST REFLECTION BACK C WHICH WILL ADD TO THE REFLECTED STRESS d(j,3)=tr*ref2*tr2*d(j,l) d(j,6)=d(j,3)*ref2**2 C AT STATION ON THE TRANSMITTER BAR: d(j,4)=tr*tr2*d(j,1) c PLUS WHAT BOUNCES BACK AND FORTH IN THE SPECIMEN C AND ADDS TO THE TRANSMITTED STRESS d(j,5)=tr*ref2*ref2*tr2*d(j,1) d(j,7)=d(j,5)*ref2**2 5 continue C SUBROUTINE ANGLE CALCULATES PHASE CHANGES, VELOCITIES C AND WAVELENGTHS FOR EACH TERM IN THE FOURIER SERIES call angle(dia,col,co2,wo,nim,phl,ph2,c,dl) C SUBROUTINE CROSS CALCULATES THE CROSS SECTIONAL VARIATION C OF THE DISPLACEMENTS AND STRESSES. ONLY LONGITUDINAL STRESSES C ARE BROUGHT BACK INTO THE MAIN PROGRAM C SIGMA RR AND SIGMA ZZ ARE OUTPUTED TO FILE FOR002.DAT call cross(dia,c,dl,mm,sig) c c CALCULATE THE TIMES EACH PULSE TAKES TO C REACH THE STRAIN GAGES n-i do 10 j-l,mm n-n+2 time(n,l)=zl*((-1.0/col)+phl(n)) time(n,2)=time(n,l)+2.0*z13*((-1.0/col)+phl(n)) time(n,3)=time(n,2)+z12*( (-1.0/co2)+ph2(n))*2.0 time(n,4)=time(n,l)+z24*2.0*( (-.0/col)+phl(n)) 1 +zl2*((-1.0/co2)+ph2(n)) time(n,5)=time(n,4)+2*z12*((-1.0/co2)+ph2(n)) time(n,6)=2*z12*ph2(n)/(0.5*1.e-6) 10 continue |

Full Text |

PAGE 2 Q $1$/<6,6 2) $ 352326(' 6,; ,1&+ ',$0(7(5 63/,7 +23.,1621 35(6685( %$5 %\ (/,6$%(77$ /,',$ -(520( ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(17 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$ PAGE 3 $&.12:/('*0(176 ZLVK WR WKDQN P\ FRPPLWWHH IRU WKHLU VXSSRUW DQG DGYLFH GXULQJ WKLV HGXFDWLRQDO SURFHVV ,Q SDUWLFXODU WKDQN 'U & $OOHQ 5RVV IRU KLV DGYLFH VXSSRUW DQG HQFRXUDJHPHQW :LWKRXW KLP WKLV HIIRUW ZRXOG QRW KDYH EHHQ SRVVLEOH ZRXOG DOVR OLNH WR WKDQN P\ KXVEDQG 'DYLG IRU KLV SDWLHQFH DQG XQGHUVWDQGLQJ GXULQJ WKH PDQ\ QLJKWV DQG ZHHNHQGV VSHQW DW P\ GHVN RU LQ IURQW RI WKH FRPSXWHU ZRXOG OLNH WR DFNQRZOHGJH WKH ILQDQFLDO VXSSRUW RI WKH $LU )RUFH (QJLQHHULQJ DQG 6HUYLFHV &HQWHU DW 7\QGDOO $LU )RUFH %DVH )LQDOO\ ZLVK WR DFNQRZOHGJH WKH KHOS DQG VXSSRUW RI 6YHUGUXS 7HFKQRORJ\ ,QF GXULQJ P\ HGXFDWLRQDO HQGHDYRUV PAGE 4 7$%/( 2) &217(176 SDJH $&.12:/('*0(176 LL /,67 2) ),*85(6 Y $%675$&7 YLLL &+$37(56 ,1752'8&7,21 2EMHFWLYH %DFNJURXQG $SSURDFK 0$7+(0$7,&$/ %$&.*5281' 6+3% $VVXPSWLRQV (TXDWLRQV RI 0RWLRQ 6ROXWLRQV RI WKH )UHTXHQF\ (TXDWLRQ 7UDQVLHQW %HKDYLRU 0$7+(0$7,&$/ $1' 180(5,&$/ $1$/<6,6 6FDOLQJ /DZV :DYH 3URSDJDWLRQ LQ DQ 6+3% 0RGLILHG 3RFKKDPPHU&KUHH 0HWKRG )LQLWH 'LIIHUHQFH 1XPHULFDO 6LPXODWLRQ ([SHULPHQWDO 0HWKRG &RPSDULVRQ RI 5HVXOWV ,QHUWLD DQG )ULFWLRQ (IIHFWV 6,;,1&+ 6+3% )($6,%,/,7< 6XPPDU\ &RQFOXVLRQV DQG 5HFRPPHQGDWLRQV LLL PAGE 5 SDJH 5()(5(1&(6 $33(1',&(6 $ 0RGLILHG 3RFKKDPPHU&KUHH 3URJUDP DQG 6DPSOH 2XWSXWV % 5DWLR RI )RXULHU &RHIILFLHQWV IRU *LYHQ D$ RI %DU 'LDPHWHUV & 0LVFHOODQHRXV +8// 3ORWV +8// 6DPSOH ,QSXW )LOHV ( )RUWUDQ /LVWLQJ IRU ,QHUWLD &RUUHFWLRQ %,2*5$3+,&$/ 6.(7&+ ,9 PAGE 6 /,67 2) ),*85(6 SDJH &RPSUHVVLYH 6+3% 6FKHPDWLF &\OLQGULFDO &RRUGLQDWHV 3RFKKDPPHU&KUHH 6ROXWLRQV IRU )LUVW 7KUHH 0RGHV 7\QGDOO LQFK 6+3% ([SHULPHQWDO 7UDFH $OXPLQXP 6SHFLPHQ /HQJWK DQG 'LDPHWHU LQFK 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV ,QFLGHQW 6WUHVV 1RUPDOL]HG WR 1RUPDOL]HG /RQJLWXGLQDO 6WUHVV *LYHQ DV D )XQFWLRQ RI 5DGLDO /RFDWLRQ IRU 6HYHUDO )RXULHU 7HUPV ([SHULPHQWDO 5HVXOWV LQFK 6+3% 6WUHVV 3XOVHV IRU D 7 $OXPLQXP 6SHFLPHQ 'LDPHWHU FP /HQJWK FP ([SHULPHQWDO 5HVXOWV LQFK 6+3% 6WUHVV 3XOVHV IRU D 7 $OXPLQXP 6SHFLPHQ 'LDPHWHU FP /HQJWK FP ([SHULPHQWDO 5HVXOWV LQFK 6+3% 6WUHVV 3XOVHV IRU D 7 $OXPLQXP 6SHFLPHQ 'LDPHWHU FP /HQJWK FP +8// 2XWSXW LQFK 6+3% 6WUHVV 7< PAGE 7 SDJH +8// 2XWSXW A VHF 3XOVH 6WDWLRQ 7RSf %DU &HQWHU 6WDWLRQ %RWWRPf %DU 6XUIDFH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV f LQFK 6+3% VHF 3XOVH D A +8// 2XWSXW DW ,QFLGHQW %DU 6WUDLQ *DJH A \\M LQFK 6+3% VHF 3XOVH ([SHULPHQWDO 7UDFHV DW 7KUHH ,QFLGHQW %DU /RFDWLRQV IRU 7ZR 6WULNHU %DU /HQJWKV LQFK 7RSf DQG LQFK %RWWRPf LQFK 6+3% +8// 2XWSXW LQFK %DU VHF 3XOVH $OO 6WUHVV &RPSRQHQWV RQ WKH 6XUIDFH DW ,QFLGHQW %DU 6WUDLQ *DJH +8// 2XWSXW LQFK %DU VHF 3XOVH $OO 6WUHVV &RPSRQHQWV RQ WKH 6XUIDFH DW ,QFLGHQW %DU 6WUDLQ *DJH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV DW ,QFLGHQW %DU 6WUDLQ *DJH LQFK %DU +8// 2XWSXW LQFK %DU VHF 3XOVH 6XUIDFH 7RSf DQG &HQWHU %RWWRPf 7UDQVPLWWHG 6WUHVVHV +8// 2XWSXW LQFK %DU VHF 3XOVH 6XUIDFH 7RSf DQG &HQWHU %RWWRPf 7UDQVPLWWHG 6WUHVVHV +8// 2XWSXW LQFK %DU /RQJLWXGLQDO 6WUHVV 1HDU WKH ,PSDFWHG (QG 7RSf DQG DW WKH 6WUDLQ *DJH /RFDWLRQ %RWWRPf +8// 2XWSXW LQFK %DU $OO 6WUHVVHV 1HDU ,PSDFWHG (QG 7RSf DQG DW WKH 6WUDLQ *DJH /RFDWLRQ %RWWRPf +8// 2XWSXW LQFK %DU $OO 6WUHVVHV 1HDU WKH ,PSDFWHG (QG 7RSf DQG DW WKH 6WUDLQ *DJH /RFDWLRQ %RWWRPf +8// 2XWSXW LQFK %DU $OO 6WUHVVHV DW 7KUHH /RFDWLRQV DW WKH %DU6SHFLPHQ ,QWHUIDFH +8// 2XWSXW LQFK %DU 6WUHVVHV LQ $OXPLQXP 6SHFLPHQ DW 7ZR /RFDWLRQV 1HDU WKH 6XUIDFH 7RSf 1HDU WKH &HQWHU %RWWRPf &RPSDUH ZLWK )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVVHV DW 7KUHH 5DGLDO /RFDWLRQV 6XUIDFH 7RSf 0LGGOH 0LGGOHf &HQWHU %RWWRPf &RPSDUH ZLWK )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK =HUR )ULFWLRQ %HWZHHQ 6SHFLPHQV DQG 6+3% +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK ,QILQLWH )ULFWLRQ %HWZHHQ 6SHFLPHQ DQG 6+3% +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK =HUR )ULFWLRQ 9, PAGE 8 SDJH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK ,QILQLWH )ULFWLRQ 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV 5DGLDO 9DULDWLRQV RI 5HIOHFWHG 6WUHVVHV LQFK 6+3% 7RSf LQFK 6+3% %RWWRPf 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV 5DGLDO 9DULDWLRQV RI 7UDQVPLWWHG 6WUHVVHV LQFK 6+3% 7RSf LQFK 6+3% %RWWRPf +8// 2XWSXW LQFK 6+3% 5DGLDO DQG /RQJLWXGLQDO 9HORFLWLHV DW ,QFLGHQW %DU 6SHFLPHQ ,QWHUIDFH %DU 6XUIDFH 7RS 0LGGOH RI WKH %DU %RWWRP %DU &HQWHU 5LJKWf +8// 2XWSXW LQFK 6+3% $OO 7UDQVPLWWHG 6WUHVVHV DW %DU 6XUIDFH +8// 2XWSXW LQFK 6+3% $OO 7UDQVPLWWHG 6WUHVVHV DW %DU &HQWHU &RPSDUH ZLWK )LJXUH +8// 2XWSXW LQFK 6+3% LQFK $OXPLQXP 6SHFLPHQ ,QFLGHQW DQG 5HIOHFWHG 6WUHVVHV ,OO +8// 2XWSXW LQFK 6+3% LQFK $OXPLQXP 6SHFLPHQ $OO 6WUHVVHV :LWKLQ WKH 6SHFLPHQ +8// 2XWSXW LQFK 6+3% LQFK $OXPLQXP 6SHFLPHQ $OO VWUHVVHV LQ WKH 7UDQVPLWWHU %DU YLL PAGE 9 $EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH 6FKRRO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ $1$/<6,6 2) $ 352326(' 6,; ,1&+ ',$0(7(5 63/,7 +23.,1621 35(6685( %$5 %\ (OLVDEHWWD /LGLD -HURPH $XJXVW &KDLUPDQ & $OOHQ 5RVV 0DMRU 'HSDUWPHQW $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH 7KH REMHFWLYH RI WKLV LQYHVWLJDWLRQ ZDV WR GHWHUPLQH ZKHWKHU D LQFK GLDPHWHU 6SOLW +RSNLQVRQ 3UHVVXUH %DU 6+3%f LV IHDVLEOH ZKHWKHU WKH DVVXPSWLRQV PDGH LQ D W\SLFDO WR LQFK 6+3% VWLOO DSSO\ DQG ZKDW FKDQn JHV SUREOHPV DQG LVVXHV ZRXOG EH DVVRFLDWHG ZLWK DQ 6+3% WKDW VL]H 7KH HIIRUW LQFOXGHG D PDWKHPDWLFDO DQG D QXPHULFDO DQDO\VLV ([SHULPHQWDO GDWD IURP D LQFK 6+3% ZHUH XVHG WR FRPSDUH DQG YDOLGDWH DQDO\WLFDO PRGHOV $ PHWKRGRORJ\ EDVHG RQ WKH 3RFKKDPPHU&KUHH IUHTXHQF\ HTXDWLRQ ZDV GHYHORSHG E\ WKH DXWKRU WR VWXG\ WKH ZDYH SURSDJDWLRQ DQG ZDYH LQWHUn DFWLRQ LQ DQ\ VL]H 6+3% $GGLWLRQDOO\ D VHFRQG RUGHU DFFXUDWH ILQLWH GLIn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f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n PLWWHU EDU 7KH LQFLGHQW EDU LV LPSDFWHG E\ D VWULNHU EDU DW D NQRZQ YHORFLW\ O PAGE 12 FDXVLQJ D SUHVVXUH ZDYH WR WUDYHO GRZQ WKH EDU 7KLV ZDYH LV SDUWLDOO\ WUDQVn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n FXVVHG ERWK IULFWLRQ DQG LQHUWLD HIIHFWV 2WKHU SDSHUV RI LQWHUHVW RQ WKLV VXEMHFW DUH WKRVH E\ 5DQG >@ 'KDUDQ DQG +DXVHU >@ 6DPDQWD >@ -DKVPDQ >@ &KLX DQG 1HXEHUW >@ DQG PAGE 14 7KH ZDYH SURSDJDWLRQ LQ DQ 6+3% LV JRYHUQHG E\ WKH HTXDWLRQV RI PRWLRQ *HQHUDOO\ LW LV DVVXPHG WKH PRWLRQ LV RQHGLPHQVLRQDO DQG WKHUHn IRUH LV XQLIRUP LQ WKH UDGLDO DQG WKH FLUFXPIHUHQWLDO GLUHFWLRQ $OWKRXJK WKH RQHGLPHQVLRQDO ZDYH WKHRU\ KDV EHHQ VKRZQ WR EH DFFXUDWH IRU WKH H[LVWLQJ V\VWHPV D PDWKHPDWLFDO DQDO\VLV LV QHHGHG IRU WKH SURSRVHG LQFK 6+3% WR DVVHVV ZKHWKHU WKLV VLPSOH WKHRU\ VWLOO DSSOLHV 7KH WZRGLPHQVLRQDO D[LV\P PHWULF F\OLQGULFDO HTXDWLRQV RI PRWLRQ QHHG WR EH DQDO\]HG DQG VROYHG DSSUR[LPDWHO\f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fV ODZ IRU D GLVSODFHn PHQW X LQ WKH [ GLUHFWLRQ RI WKH GLDJUDP EHORZ JLYHV WKH HTXDWLRQV RI PRWLRQ IRU DQ HODVWLF V\VWHP D GD D fÂ§ G[ G[ fG[ fÂ§R$ D GD G[ G[f$ S$G[ Z ?GW RU XSRQ VLPSOLILFDWLRQ GD G[ ODf DQG +RRNHfV ODZ D ( e PAGE 16 ZKHUH R LV WKH VWUHVV ( LV PAGE 17 ZKHUH WKH VXEVFULSWV L U DQG W LQGLFDWH LQFLGHQW UHIOHFWHG DQG WUDQVPLWWHG $GGLWLRQDOO\ XL DQG 8 DUH WKH VSHFLPHQfV GLVSODFHPHQWV DW WKH LQFLGHQW DQG RXWSXW EDUV LQWHUIDFHV UHVSHFWLYHO\ 7KH QHJDWLYH VLJQV JLYH SRVLWLYH GLVn SODFHPHQWV IRU FRPSUHVVLYH QHJDWLYHf VWUDLQV /HW / EH WKH OHQJWK RI WKH VSHFLPHQ WKHQ WKH DYHUDJH VWUDLQ LQ WKH VSHFLPHQ LV eV 8 8Lf/ 7KLV LV VRPHWLPHV DSSUR[LPDWHG IRU VKRUW VSHFLPHQV DQG VORZO\ FKDQJLQJ VWUHVV E\ DVVXPLQJ HTXDO VWUHVVHV DW WKH WZR LQWHUIDFHV ZKLFK LPSOLHV WKDW H H eU 7KH VSHFLPHQ VWUDLQ WKHQ EHFRPHV HV FR/f feUGW DQG WKH VSHFLPHQ VWUDLQ UDWH LV ÂV fÂ§ &R/ f eU %\ XVLQJ +RRNHfV ODZ WKH VWUHVVHV RQ HLWKHU IDFH RI WKH VSHFLPHQ DUH 2V ( $ ? fÂ§ HW DQG FUVL ( $F 9$6 Â‘eL eYf ZKHUH fÂ§ LV WKH UDWLR RI WKH EDU DQG VSHFLPHQ FURVV VHFWLRQ DUHDV ,I WKH $V LQWHUIDFH VWUHVVHV DUH QRW DVVXPHG WR EH HTXDO WKHQ eW r e eU DQG WKH VSHFLPHQ VWUDLQ DQG VWUDLQ UDWH DUH JLYHQ E\ eV a &T/f If e^ eW eUf GW DQG HV F/f eW Hc eA PAGE 18 7KHUH DUH VRPH SUREOHPV DVVRFLDWHG ZLWK VXFK D VLPSOLILHG RQHn GLPHQVLRQDO WKHRU\ HYHQ IRU ORQJ VOHQGHU URGV DQG WKHVH SUREOHPV DUH DFFHQWXDWHG IRU ODUJHU GLDPHWHU DQG VKRUWHU OHQJWK 6+3% V\VWHPV )LUVW ORQJLWXGLQDO ZDYHV DUH GLVSHUVLYH ZDYH VSHHG LV D IXQFWLRQ RI IUHTXHQF\f ZKLFK PHDQV WKH ZDYHIRUP WKDW LV UHFRUGHG DW D GLVWDQFH IURP WKH VSHFLPHQ LV QRW WKH VDPH ZDYHIRUP WKDW DFWXDOO\ DUULYHV DW WKH VSHFLPHQ :DYH GLVSHUn VLRQ FDQ EH FRUUHFWHG IRU EXW LV WLPH FRQVXPLQJ DQG WKH FRUUHFWLRQ PHWKRGV DUH QRW H[DFW 7KH RQHGLPHQVLRQDO WKHRU\ DOVR QHJOHFWV VWUHVV DQG GLVSODFHn PHQW YDULDWLRQV DFURVV WKH FURVV VHFWLRQ 6LQFH VWUDLQV DUH PHDVXUHG RQO\ DW WKH VXUIDFH WKLV FDQ DOVR OHDG WR HUURUV (TXDWLRQV RI 0RWLRQ 7R DVVHVV WKH YDOLGLW\ RI WKLV VLPSOH WKHRU\ RQH FDQ VWXG\ WKH HTXDn WLRQV RI PRWLRQ WKDW JRYHUQ WKH ZDYH PRWLRQ LQ D FLUFXODU FURVV VHFWLRQ 6+3% 7KH HTXDWLRQV RI PRWLRQ DUH JLYHQ LQ WHQVRU IRUP DV f a a GX 9 f 7 SE S fÂ§ + GW f ZKHUH 7 LV WKH VWUHVV WHQVRU E DUH WKH ERG\ IRUFHV ZKLFK DUH XVXDOO\ VDIHO\ Ga QHJOHFWHGf DQG fÂ§U LV WKH DFFHOHUDWLRQ WHUP ,Q F\OLQGULFDO FRRUGLQDWHV )LJXUH f HTXDWLRQ f EHFRPHV GAXU G2Q &KU 8GG A -B G2A GU] GU U U G G] GnXH G" -B GRHH GRWf] 4MT GU U 62 G] U 3 G2U] -B G2JÂ G2H] 2Q GU U G G] U f PAGE 19 GL U )LJXUH &\OLQGULFDO &RRUGLQDWHV (TXDWLRQV f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n WLRQV f LV VDWLVILHG LGHQWLFDOO\ ,W DOVR PHDQV WKDW WKH RQO\ SRVVLEOH PRGHV RI YLEUDWLRQ ZLOO EH V\PPHWULF 8VLQJ 5HGZRRGfV >@ QRWDWLRQ WKHVH PRGHV ZLOO EH FDOOHG 0P Q ZKHUH P LV HLWKHU V\PPHWULFf RU DV\PPHWULFf DQG Q UDQJHV IURP PAGE 20 &U] AÂ8U Â8]A Â] GU D] fÂ§ $ ÂLf $ 8U W Â8U 1 U ÂU f GD UU GDU] RHH U fÂ§ fÂ§ Df GU G] ZKHUH $ DQG S DUH /DPHfV FRQVWDQWV 0DNLQJ XVH RI f DQG QHJOHFWLQJ WKHWD GHSHQGHQFH HTXDWLRQV f FDQ WKXV EH UHGXFHG WR RQO\ WZR HTXDWLRQV ÂXU Q R [ SfÂ§U $ Lf DW Ân8U Â8U 8U f a GU U GU U 8] Ff8U $ Lf L ÂUÂ] G] G8] GX] ÂX] G8U Â8U S DW 7 GU U GU $ Lf $ Lf G] ÂUÂ] U G] f 7KH ERXQGDU\ FRQGLWLRQV UHTXLUH QR VWUHVV DW WKH EDU VXUIDFH RU 2UU DQG 2Q DW U D ZKHUH D LV WKH UDGLXV RI WKH EDU $OVR DW D IUHH HQG DU] DQG D]] )RU D FROLQHDU LPSDFW RI RQH EDU DJDLQVW DQRWKHU WKH LQLWLDO FRQGLWLRQV FDQ EH LQ WKH IRUP RI D VWHS FKDQJH LQ SUHVVXUH A fÂ§3+Wf RU] ZKHUH 3 LV D XQLIRUP SUHVVXUH DQG +Wf LV WKH +HDYLVLGH VWHS IXQFWLRQ RU DV D KDUPRQLF GLVSODFHPHQW RI X] VLQR} W DQG DU] 7KH FRQGLWLRQ DU] LV FRUUHFW RQO\ LI WKH WZR EDUV DUH RI LGHQWLFDO GLDPHWHU DQG PDWHULDO VR WKDW WKH LQWHUIDFH LV D SODQH RI V\PPHWU\ RI WKH VWUHVV VWDWH XQWLO D UHIOHFWHG ZDYH PAGE 21 UHDFKHV WKH LQWHUIDFH (TXDWLRQV f KDYH EHHQ VROYHG IRU DQ LQILQLWHO\ ORQJ EDU E\ 3RFKKDPPHU DQG &KUHH IRU WKH SURSDJDWLRQ RI VLQXVRLGDO ZDYHV DQG VWXGLHG E\ QXPHURXV LQYHVWLJDWRUV RYHU WKH \HDUV > @ 7KH VROXWLRQ LV FDOOHG WKH IUHTXHQF\ HTXDWLRQ DQG KDV WKH IRUP f ZKHUH K MU$f ?SF@; MXf ?@N U$f >SF@Qf fÂ§ @ $ ZDYHn OHQJWK \ \U fÂ§ -R DQG -L DUH WKH ]HUR DQG ILUVW RUGHU %HVVHO IXQFWLRQV &Q 3 UHVSHFWLYHO\ D! LV WKH DQJXODU IUHTXHQF\ DQG FS LV WKH SKDVH YHORFLW\ 6ROXWLRQV RI WKH )UHTXHQF\ (TXDWLRQ 7KHUH DUH DQ LQILQLWH QXPEHU RI VROXWLRQV WR (TXDWLRQ f HDFK FRUn UHVSRQGLQJ WR D XQLTXH PRGH RI YLEUDWLRQ 7KH VROXWLRQV DUH H[DFW RQO\ IRU DQ LQILQLWHO\ ORQJ F\OLQGHU DOWKRXJK IRU D F\OLQGHU ZKRVH OHQJWK JUHDWO\ H[FHHGV LWV GLDPHWHU WKH HUURUV DUH VPDOO )XUWKHUPRUH WKH VROXWLRQV DUH QRW H[SOLFLW EXW UDWKHU D IXQFWLRQ UHODWLQJ WKH SKDVH YHORFLW\ F WKH ZDYHOHQJWK $ DQG 3RLVVRQfV UDWLR Y 1RZ f FDQ EH ZULWWHQ LQ LPSOLFLW IRUP DV f PAGE 22 %\ PDNLQJ XVH RI WKH IROORZLQJ SURSHUWLHV RI %HVVHO IXQFWLRQV -R KDff -LKDf -L KDf f KKDf MKDffD HTXDWLRQ f FDQ EH UHZULWWHQ DV : F-RFDf -LFDfDf F-LKDff ? D! S Ir M -L NDf L K-R KDf -L KDfDf $SDU $ IL -RKDf ZKLFK ILQDOO\ EHFRPHV \ N -R FDf B -L NDf D U \f -R KDf K-LKDf f (TXDWLRQ f KDV EHHQ VROYHG QXPHULFDOO\ E\ VHYHUDO LQYHVWLJDWRUV > @ ,W LQYROYHV DVVXPLQJ D YDOXH RI 3RLVVRQfV UDWLR DQG WKHQ IRU YDU\LQJ YDOXHV RI IUHTXHQFLHV UDQJLQJ IURP ]HUR WR VRPH ODUJH QXPEHU HTXDWLRQ f LV QXPHULFDOO\ HYDOXDWHG WR REWDLQ WKH YDOXHV RI WKH SKDVH YHORFLW\ 7KLV WXUQV RXW WR EH D YHU\ ODERULRXV WDVN HYHQ ZLWK WKH DLG RI KLJK VSHHG FRPSXWHUV 5HVXOWV DUH XVXDOO\ VKRZQ LQ JUDSKLFDO IRUP QDPHO\ LQ FXUYHV UHODWLQJ WKH SKDVH YHORFLW\ DQG WKH IUHTXHQF\ ERWK LQ QRQGLPHQVLRQDO IRUP IRU JHQHUDO DSSOLFDELOLW\ )LJXUH >@ VKRZV WKH VROXWLRQ WR HTXDWLRQ f IRU D 3RLVVRQfV UDWLR RI 2QO\ WKH ILUVW WKUHH YLEUDWLRQDO PRGHV DUH SORWWHG QDPHO\ 0 Y 0M DQG 0M 5HFDOO WKDW HTXDWLRQ f RQO\ DSSOLHV WR V\PPHWULF YLEUDWLRQDO PRGHV 0RGH 0 LV RI PRVW LQWHUHVW VLQFH LW DSSHDUV WR EH WKH RQO\ PRGH H[FLWHG LQ D W\SLFDO 6+3% $V WKH IUHTXHQF\ RI WKLV PRGH DSSURDFKHV ]HUR RU DV WKH ZDYHOHQJWK DSSURDFKHV LQILQLW\ WKH SKDVH YHORFLW\ DSSURDFKHV 9HS f ZKLFK LV WKH HODVWLF RQHGLPHQVLRQDO EDUZDYH PAGE 23 VSHHG F $V WKH IUHTXHQF\ EHFRPHV YHU\ ODUJH WKH SKDVH YHORFLW\ DSn SURDFKHV FU WKH YHORFLW\ RI 5D\OHLJK VXUIDFH ZDYHV $OWKRXJK LW PD\ QRW EH FOHDU IURP )LJXUH FS WKH SKDVH YHORFLW\ DSSURDFKHV FU IURP WKH GRZQ VLGH LPSO\LQJ WKDW WKH SKDVH YHORFLW\ UHDFKHV D PLQLPXP ZKLFK LV OHVV WKDQ WKH 5D\OHLJK ZDYH VSHHG DW VRPH LQWHUPHGLDWH IUHTXHQF\ 5HFDOO WKDW B $ X B M( &G S &W a S Fra S K MW$f SFA$ MXf f DQG N MW$f S&SnS f DQG LW LV HDVLO\ VKRZQ WKDW F &G DQG IRU 0LL FS A F 7KHQ IRU WKH 0LL PRGH WKH SDUDPHWHU K LV DOZD\V LPDJLQDU\ VLQFH IRU DOO IUHTXHQFLHV FS &G ZKHUH &D LV WKH GLODWDWLRQDO ZDYH VSHHG LQ DQ LQILQLWH PHGLXP )RU YHU\ ORZ IUHTXHQFLHV N LV UHDO EXW SDVW URXJKO\ ZKHQ FS EHFRPHV OHVV WKDQ WKH VKHDU ZDYH VSHHG LQ DQ LQILQLWH PHGLXP F F LV LPDJLn QDU\ 7KLV PHDQV WKDW DW ORZ IUHTXHQFLHV Af WKH GLVSODFHPHQW LV PDLQO\ GXH WR SODQH WUDQVYHUVH ZDYHV VLQFH WKH VHW RI GLODWDWLRQDO ZDYHV H[LVWV RQO\ DV D VXUIDFH GLVWXUEDQFH $W KLJK IUHTXHQFLHV WKH WRWDO GLVSODFHn PHQW EHFRPHV LQFUHDVLQJO\ OLNH D SXUH VXUIDFH GLVWXUEDQFH >@ 0RGH 0M DQG KLJKHU PD\ EH RI LQWHUHVW LQ D ODUJHU 6+3% DQG DW KLJK IUHTXHQFLHV 7KHVH PRGHV KDYH ZKDW DUH FDOOHG FXWRII IUHTXHQFLHV LH IUHTXHQFLHV DW ZKLFK WKH SKDVH YHORFLW\ EHFRPHV LQILQLWH DQG QR UHDO GLVWXUEDQFH SURSDJDWHV LQ WKDW PRGH )URP HTXDWLRQ f WKLV LPSOLHV HLWKHU WKDW \ RU WKDW -AFDf ,I \ HTXDWLRQ f EHFRPHV Wf ? 2n&2 f IO ? ,f FR OFfD &W-RKDf K-LKDf PAGE 24 VLQFH .a n ? &W ? fn &? DQG N IRU FS } ZKHUH FMRFR LV WKH FXWRII IUHTXHQF\ &L )ROORZLQJ WKH VDPH UHDVRQLQJ K r IRU FS rr DQG WKXV RQH FDQ ZULWH 47 &2 &2 O&G ? D W -R A &2 FR A &G -O &2 FR ? &G ,I -L FDf LW FDQ EH VDLG WKDW f -[ &2 FR fÂ§ D f (TXDWLRQV f DQG f FDQ QRZ EH XVHG WR GHWHUPLQH WKH FXWRII IUHTXHQFLHV IRU DQ\ PRGH 0LQ )URP WKH VROXWLRQV RI WKH IUHTXHQF\ HTXDWLRQ HTXDWLRQ f LW LV SRVVLEOH WR FDOFXODWH WKH UDGLDO DQG ORQJLWXGLQDO GLVSODFHPHQWV IRU HDFK PRGH DW HYHU\ IUHTXHQF\ DQG 3RLVVRQfV UDWLR ZLWKLQ WKH F\OLQGHU %DQFURIW >@ FDOFXODWHG DQG SORWWHG WKH DPSOLWXGH RI WKH ORQJLWXGLQDO GLVSODFHPHQW DQG VKRZHG WKDW WKH\ DUH DV H[SHFWHGf XQLIRUP IRU A DQG WKH PRWLRQ LV FRQILQHG WR WKH RXWVLGH VXUIDFH ZKHQ DSSURDFKHV LQILQLW\ 7KLV DJUHHV ZLWK WKH REVHUYDWLRQV PDGH HDUOLHU DERXW PRGH 0 ` UHVSRQVH LQ JHQHUDO 1XPHURXV RWKHU DSSUR[LPDWH PHWKRGV RI YDU\LQJ GHJUHHV RI FRPn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n EHU RI DSSUR[LPDWH WKHRULHV ZHUH GHYHORSHG 0RVW LPSRUWDQW WR WKLV VWXG\ DUH WKH GLIIHUHQW PHWKRGV DQG DSSURDFKHV XVHG WR GHULYH WKHVH WKHRULHV $V DOUHDG\ GLVFXVVHG WKH VLPSOHVW DQDO\VLV LV WR DVVXPH SXUHO\ D[LDO VWUHVV XQLIRUP RYHU HDFK FURVV VHFWLRQ 7KLV OHDGV WR WKH IDPLOLDU RQHGLPHQVLRQDO ZDYH HTXDWLRQ LQ WHUPV RI WKH D[LDO GLVSODFHPHQW X] GX] DW (S DX] D] LLf PAGE 26 ZKLFK SUHGLFWV WKDW ZDYHV RI DOO IUHTXHQFLHV WUDYHO DW WKH VDPH FRQVWDQW YHORFLW\ &R $ EHWWHU DSSUR[LPDWLRQ WR WKLV WKHRU\ LQWURGXFHV D FRUn UHFWLRQ IRU WKH UDGLDO PRWLRQ E\ FRQVLGHULQJ WKH LQHUWLD RI WKH FURVV VHFWLRQ >@ 7KH DSSURDFK LQYROYHV WKH XVH RI +DPLOWRQfV SULQFLSOH ZKLFK VWDWHV WKDW WKH ILUVW YDULDWLRQ RI WKH LQWHJUDO RI WKH /DJUDQJLDQ 7 9f ZLWK UHVSHFW WR WLPH LV ]HUR 6SHFLILFDOO\ 7 9f GW DQG 7 9f FDQ EH H[SUHVVHG DV f / fnR ?S$ GX A X5f Gf8U ?G] GOM ($ GX] 9 G] f f G] NLQHWLF HQHUJ\ SRWHQWLDO VWUDLQf HQHUJ\ ZKHUH / LV WKH OHQJWK RI WKH F\OLQGHU 5 LV WKH UDGLXV RI J\UDWLRQ DERXW WKH ]D[LV Y LV 3RLVVRQfV UDWLR DQG WKH YHORFLWLHV LQ WKH ] DQG U GLUHFWLRQV DUH UHVSHFWLYHO\ GX] DW DQG GX [ GW %XW XU FDQ EH DVVXPHG LQ WKLV DSSUR[LPDWLRQ WR EH RI WKH IRUP WKDW fÂ§Y U GX] G] VR G8] GXU aGW fÂ§YU G] DW f PAGE 27 7KH UHVXOW RI VXEVWLWXWLQJ HTXDWLRQ f LQWR f DQG VROYLQJ YLD LQWHJUDWLRQ E\ SDUWV LV DV IROORZV 3 X5f G 8] G] GK f 7KLV HTXDWLRQ JLYHV D EHWWHU DSSUR[LPDWLRQ RI WKH H[DFW WKHRU\ WKDQ HTXDWLRQ f DW ORZ IUHTXHQFLHV +RZHYHU IRU VKRUW ZDYHOHQJWKV WKH HUURUV EHFRPH FRQVLGHUDEOH $ WKLUG DSSUR[LPDWH WKHRU\ GHYHORSHG E\ /RYH >@ LV EDVHG RQ WKH H[DFW FKDUDFWHULVWLF HTXDWLRQ f ZKHUH WKH %HVVHO IXQFWLRQV DUH H[SDQGHG LQ D SRZHU VHULHV ,I D WKH UDGLXV RI WKH F\OLQGHU LV VPDOO HQRXJK VR WKDW KD DQG .D DUH VPDOO FRPSDUHG WR XQLW\ WKHQ SRZHUV RI KD DQG .D KLJKHU WKDQ WKH VHFRQG FDQ EH QHJOHFWHG 7KDW LV -R[Df FDf -LFDf FDf DQG WKH SKDVH YHORFLW\ EHFRPHV } L Â‘! ? S 87A\D f DQG PD\ EH UHZULWWHQ DV &R 9a-Oa nD` 9$ ZKHUH F f PAGE 28 7KLV WKHRU\ DJUHHV IDLUO\ ZHOO ZLWK WKH H[DFW WKHRU\ XS WR YDOXHV RI A EXW WKHQ UDSLGO\ GLYHUJHV 7KH DVVXPSWLRQV WKDW .D DQG KD DUH VPDOO FRPSDUHG WR XQLW\ LPSO\ WKDW WKH ZDYHOHQJWKV RI WKH YLEUDWLRQV DUH ODUJH FRPSDUHG WR WKH UDGLXV RI WKH F\OLQGHU $QRWKHU FRPPRQ WKHRU\ ZDV GHYHORSHG E\ 0LQGOLQ DQG +HUUPDQQ >@ ,W FRQVLGHUV VKHDU VWUHVVHV DQG VWUDLQV E\ DVVXPLQJ ILUVW WKDW WKH UDGLDO GLVSODFHPHQW LV RI WKH IRUP XU UDf X]Wf ZLWK XH DQG X] Z]Wf )RUFHV DQG PRPHQWV DUH WKHQ FDOFXODWHG IURP VWDQGDUG HQJLQHHULQJ PHFKDQLFV IRUPXODH DQG FRUUHFWHG IRU VKHDU DQG LQHUn WLD E\ LQWURGXFLQJ IDFWRUV .L DQG 7KLV OHDGV WR HTXDWLRQV RI WKH IRUP f G 8 ff A G X D.L Q fÂ§ $ Xf X fÂ§ D. $ fÂ§ SD fÂ§ G] G] VW X [ G- G-D$ fÂ§ D $ f fÂ§ S D fÂ§ G] G]f W f %\ VXEVWLWXWLQJ X $ H[S aL\]f H[S NXWf DQG Z % H[S L\]f H[S LFXWf LQWR HTXDWLRQ f WZR HTXDWLRQV LQ $ DQG % DUH REWDLQHG IURP ZKLFK $ DQG % FDQ EH HOLPLQDWHG WR DJDLQ REWDLQ D FKDUDFWHULVWLF HTXDWLRQ ZKLFK UHODWHV SKDVH YHORFLW\ DQG IUHTXHQF\ IRU WKH ILUVW WZR PRGHV %\ DGMXVWLQJ DQG .L D YHU\ JRRG DSSUR[LPDWLRQ FDQ EH REWDLQHG IRU PRGH 0LL 0RGH 0L RQ WKH RWKHU KDQG VKRZV FRQVLGHUDEOH GHYLDWLRQ IURP WKH H[DFW WKHRU\ PAGE 29 0DQ\ RWKHU WKHRULHV KDYH EHHQ GHYHORSHG DQG LQ JHQHUDO WKHLU FRPn SOH[LW\ LV GLUHFWO\ SURSRUWLRQDO WR WKHLU DFFXUDF\ 7KHVH DQDO\VHV DUH LPSRUn WDQW EHFDXVH RI WKH LQVLJKW WKH\ JLYH LQWR WKH UHVSRQVH RI D F\OLQGHU VXEMHFWHG WR LPSXOVLYH ORDGLQJ .QRZLQJ KRZ VWUHVVHV DQG GLVSODFHPHQWV DUH GLVn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n LQJ WKH SXOVH LQWR LWV FRQWLQXRXV FRPSRQHQW FRQWLQXRXV ZDYH WKHRU\ FDQ EH XVHG RQ HDFK FRPSRQHQW DW DQ\ SRLQW GRZQ WKH EDU DQG WKH QHZ SXOVH FDQ EH REWDLQHG E\ DGGLQJ WKH SLHFHV WRJHWKHU DJDLQ 7KLV PHWKRG LV QRW H[DFW EXW LW GRHV JLYH D FORVH DSSUR[LPDWLRQ HVSHFLDOO\ DW VRPH GLVWDQFH IURP WKH VRXUFH &KRRVLQJ D PDWKHPDWLFDO IXQFWLRQ WKDW GHVFULEHV WKH LQSXW SXOVH DQG WKDW FDQ EH UHSUHVHQWHG E\ D UHDVRQDEO\ VLPSOH )RXULHU VHULHV LV YHU\ GLIn ILFXOW 'DYLHV >@ XVHG D WUDSH]RLGDO SXOVH ZKLOH .ROVN\ >@ DVVXPHG DQ HUURU IXQFWLRQ ,Q ERWK FDVHV WKH 3RFKKDPPHU&KUHH FXUYHV DUH XVHG WR REWDLQ WKH SKDVH YHORFLW\ IRU HDFK SXOVH DQG LQ FDOFXODWLQJ WKH QHZ VKDSH RI WKH ZDYHIRUPV RQO\ WKH IXQGDPHQWDO PRGH 0M ZDV DVVXPHG WR EH H[n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n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fV UDWLR 7KHVH VROXWLRQV DUH YDOLG DQG H[DFW IRU LQILQLWHO\ ORQJ EDUV ,I WKH EDU LV ORQJ HQRXJK WR HOLPLQDWH HQG HIIHFWV DSSUR[LPDWHO\ GLDPHWHU OHQJWKVf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n VWDQW DQG WUDYHOV DW RQO\ RQH ZDYHVSHHG UHODWLYH WR WKH EDU UDGLXV D ORQJ ZDYH LQ D ODUJH EDU ZLOO KDYH WKH VDPH ZDYH VSHHG DV D VKRUW ZDYH LQ D VPDOO EDU DV ORQJ DV VWD\V FRQVWDQW :KDW KDSSHQV LI WKH LQSXW LV D WUDQVLHQW SXOVH" $ VLPSOH PDWKHPDWLn FDO IXQFWLRQ FDQ EH FKRVHQ WR GHVFULEH WKLV W\SH RI LQSXW ZKLFK FDQ EH UHSUHVHQWHG E\ D )RXULHU VHULHV $ WUDQVLHQW SXOVH LV FRPSRVHG RI D VSHFWUXP RI IUHTXHQFLHV WKH KLJKHU IUHTXHQF\ FRPSRQHQWV WUDYHO PRUH VORZn O\ WKDQ WKH ORZHU IUHTXHQF\ FRPSRQHQWV DQG WKXV ODJ EHKLQG DQG FDXVH WKH LQLWLDO VKDUS SXOVH WR VSUHDG 7KLV VSUHDGLQJ LV FDOOHG GLVSHUVLRQ 7KH 3RFK KDPPHU&KUHH VROXWLRQV JLYH WKH YHORFLW\ RI HDFK ZDYH GHSHQGLQJ RQ LWV IUHTXHQF\ 7KXV RQH PD\ FRUUHFW D JLYHQ SXOVH IRU GLVSHUVLRQ E\ UHSUHn VHQWLQJ WKH SXOVH E\ D VHULHV RI IUHTXHQF\ FRPSRQHQWV FDOFXODWLQJ KRZ IDU HDFK IUHTXHQF\ FRPSRQHQW KDV WUDYHOHG LQ D FHUWDLQ WLPH DQG WKHQ UHDVn VHPEOH WKH SXOVH 7KLV WDVN EDVLFDOO\ DPRXQWV WR FRUUHFWLQJ IRU SKDVH FKDQn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f D Waa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n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n VLRQDO DVVXPSWLRQ LV DFFHSWDEOH 'DYLHV >@ FRPSXWHG WKH UDWLR RI WKH ORQJLWXGLQDO GLVSODFHPHQW DW WKH VXUIDFH WR WKDW RI WKH ORQJLWXGLQDO GLVSODFHn PHQW DORQJ WKH EDU D[LV DQG VKRZHG WKDW IRU BDA $ OHVV WKDQ WKHLU GLIn IHUHQFH LV OHVV WKDQ SHUFHQW 7KXV IRU D FRQWLQXRXV SXOVH ZLWK RQH YDOXH RI D $ RQHGLPHQVLRQDOLW\ LV DVVXUHG LI A 2Q WKH RWKHU KDQG D WUDQn VLHQW ZDYH FRQWDLQV PDQ\ GLIIHUHQW IUHTXHQFLHV DQG LQ IDFW WKH PDMRULW\ RI WKH V ZLOO EH JUHDWHU WKHQ 7KXV LI WKH WUDQVLHQW ZDYH IWf L V PAGE 34 UHSUHVHQWHG ZLWK D )RXULHU VHULHV H[SDQVLRQ ZH ZRXOG JHQHUDOO\ H[SUHVV LW LQ D IRUP VXFK DV $ IWf a=a n= 'Q &26 QFWf W SnQ f = ZKHUH $ DQG 'Q DUH )RXULHU FRHIILFLHQWV DQG Snf LV WKH SKDVH DQJOH ZKLFK LV D IXQFWLRQ RI WKH ZDYH VSHHG DQG WKH ZDYHOHQJWK 6SHFLILFDOO\ IURP >@ 3 Q n Q R [1 9 Fr ZKHUH QDf LV WKH DQJXODU IUHTXHQF\ DQG FQ LV WKH SKDVH YHORFLW\ RI WKH QWK FRPSRQHQW (TXDWLRQ f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f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f WKDW IRU HDFK V\VWHP WKH DPSOLWXGHV RI WKH )RXULHU FRPSRQHQWV KDG IDOOHQ WR SHUFHQW RI WKH RULJLQDO YDOXH E\ WKH WLPH A UHDFKHG 7KHUHIRUH D JRRG PLQLPXP DPSOLWXGH UDWLR WR XVH IRU WKH SURSRVHG LQFK V\VWHP LV DERXW SHUFHQW 7KLV ZRXOG DVVXUH WKH RQHGLPHQVLRQDOLW\ RI WKH UHVSRQVH ,Q FRQFOXVLRQ EHFDXVH RI WKH IRUP RI WKH HTXDWLRQV LQYROYHG D SUHVn VXUH EDU V\VWHP FDQ EH PDWKHPDWLFDOO\ VFDOHG XS RU GRZQ WR DQ\ GHJUHH ZLWKRXW DIIHFWLQJ WKH FKDUDFWHULVWLFV RI WKH ZDYH PRWLRQ $QDO\VHV RI WKH EDUV DQG LQSXW SXOVHV XVHG ERWK DW 7\QGDOO $)% DQG DW WKH 8QLYHUVLW\ RI )ORULGD VKRZHG WKDW D LQFK V\VWHP ZRXOG EH HTXLYDOHQW WR WKH H[LVWLQJ DQG LQFK V\VWHPV LI WKH LQSXW SXOVH GXUDWLRQ ZDV URXJKO\ PLFURVHFRQGV 7KH OHQJWK RI WKH VWULNHU EDU LV UHODWHG WR WKH SXOVH GXUDWLRQ DV / 7 fÂ§ ZKHUH FR LV WKH HODVWLF ZDYH VSHHG DQG /V LV WKH VWULNHU EDU OHQJWK &R 7KH HODVWLF ZDYH VSHHG F* IRU VWHHO LV DSSUR[LPDWHO\ PVHF 7KLV PHDQV WKDW IRU D PLFURVHFRQG LQFLGHQW SXOVH LQ D VWHHO 6+3% D P LQFKf ORQJ VWULNHU EDU ZRXOG EH UHTXLUHG :DYH 3URSDJDWLRQ LQ DQ 6+3% +DYLQJ DQ XQGHUVWDQGLQJ RI WKLV YHU\ LPSRUWDQW IDFW RQH FDQ QRZ WXUQ WR D PRUH LQGHSWK ORRN DW ZDYH SURSDJDWLRQ LQ D F\OLQGULFDO EDU WR EURDGHQ WKH XQGHUVWDQGLQJ RI WKH RYHUDOO SUREOHP 7KUHH GLIIHUHQW PHWKRGRORJLHV ZHUH IROORZHG WR VWXG\ WKLV SUREOHP 7KH JRDO ZDV WR FRPSDUH DQG FRQWUDVW PAGE 36 UHVXOWV XQGHUVWDQGLQJ WKH GUDZEDFNV DVVXPSWLRQV DQG UHOLDELOLW\ RI HDFK RQH 7KHVH WKUHH DSSURDFKHV ZHUH f $ QHZ QXPHULFDO PHWKRG GHYHORSHG E\ WKH DXWKRU XVLQJ WKH 3RFKKDPPHU&KUHH IUHTXHQF\ VROXWLRQ DQG UHIHUUHG WR DV D PRGLILHG 3RFKKDPPHU&KUHH PHWKRG f $ ILQLWH GLIIHUHQFH QXPHULFDO VLPXODWLRQ RI WKH JRYHUQLQJ HTXDn WLRQV FRQWDLQHG LQ DQ H[LVWLQJ FRPSXWHU FRGH f ([SHULPHQWDO UHVXOWV IURP WKH 6+3% DW 7\QGDOO $)% 7KH IROORZLQJ VHFWLRQV DUH D JHQHUDO GHVFULSWLRQ RI WKHVH WKUHH DSSURDFKHV 0RGLILHG 3RFKKDPPHU&KUHH 0HWKRG $V SUHYLRXVO\ GLVFXVVHG 3RFKKDPPHU DQG &KUHH LQGHSHQGHQWO\ VROYHG WKH WZRGLPHQVLRQDO HTXDWLRQV RI PRWLRQ IRU DQ LQILQLWHO\ ORQJ EDU DVVXPLQJ D VLQXVRLGDO ZDYHIRUP 7KHLU VROXWLRQ LV ZKDW LV UHIHUUHG WR DV WKH IUHTXHQF\ HTXDWLRQ IRU ORQJLWXGLQDO PRWLRQ RI DQ HODVWLF URG ,W WXUQV RXW WR EH D IXQFWLRQ UHODWLQJ 3RLVVRQfV UDWLR ZDYHOHQJWK DQG ZDYH VSHHG DQG FDQ EH VROYHG IRU HDFK PRGH RI D F\OLQGULFDO EDU IRU D SDUWLFXODU PDWHULDO 7KLV VROXWLRQ LV QHLWKHU VLPSOH QRU H[SOLFLW %HJLQQLQJ ZLWK WKH 3RFKKDPPHU &KUHH IUHTXHQF\ HTXDWLRQ DQG DVVXPLQJ RQO\ WKH ILUVW PRGH 0M Mf LV H[n FLWHG WKH ZDYH VSHHG RI DQ\ ZDYH RI D JLYHQ IUHTXHQF\ LQ WKH EDU RI LQWHUHVW VLQFH LW LV GHSHQGHQW RQO\ RQ LWV ZDYHOHQJWK FDQ EH FDOFXODWHG 8VLQJ WKLV ZDYHVSHHG DQG FRUUHVSRQGLQJ ZDYHOHQJWK DORQJ ZLWK JLYHQ PDWHULDO SURSHUn WLHV WKH ORQJLWXGLQDO DQG UDGLDO GLVWULEXWLRQ RI GLVSODFHPHQWV DQG VWUHVVHV PD\ EH GHWHUPLQHG IURP WKH DVVXPHG GLVSODFHPHQW IXQFWLRQV IRU WKH ORQn JLWXGLQDO YLEUDWLRQV 2QH FDQ QRZ WXUQ WR WKH WUDQVLHQW SXOVH EHLQJ SURSDJDWHG LQ WKH EDU DQG GHVFULEH LW LQ WHUPV RI )RXULHU VHULHV WKDW PHDQV UHSUHVHQWLQJ WKH ZDYHIRUP E\ WKH VXP RI DQ LQILQLWH QXPEHU RI SXOVHV HDFK RQH KDYLQJ D GLVWLQFW DPSOLWXGH D GLVWLQFW ZDYHOHQJWK DQG WUDYHOLQJ DW D XQLTXH VSHHG 7KLV ZDYH VSHHG FDQ EH FDOFXODWHG IURP WKH IUHTXHQF\ HTXDn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n WLRQV DQG WUDQVPLVVLRQV WDNH SODFH 7KH EDVLF DVVXPSWLRQ LV WKDW DV WKH VLQJOH )RXULHU FRPSRQHQW HQFRXQWHUV DQ LQWHUIDFH LWV DPSOLWXGH VSOLWV DFn FRUGLQJ WR WKH FKDUDFWHULVWLF LPSHGDQFH YDOXHV RI WKH WZR PHGLD ZKLFK ZLOO EH GHULYHG ODWHUf EXW LW ZLOO PDLQWDLQ LWV RULJLQDO IUHTXHQF\ DQG YHORFLW\ ,Q RWKHU ZRUGV HDFK ZDYH FRPSRQHQW GLYLGHV HDFK WLPH LW HQFRXQWHUV DQ LQWHUn IDFH QHZ FRPSRQHQWV DUH IRUPHG HDFK ZLWK D GLIIHUHQW DPSOLWXGH EXW DOO ZLWK WKH VDPH IUHTXHQF\ 1RWH DOVR WKDW WKH VXP RI WKHVH )RXULHU FRPn SRQHQWV SURGXFH WKH WRWDO LQFLGHQW UHIOHFWHG DQG WUDQVPLWWHG SXOVHV ZKRVH ZDYHOHQJWKV ZLOO DOVR QRW FKDQJH 7KH LPSOLFDWLRQV RI WKLV DVVXPSWLRQ DUH FUXFLDO WR WKH FRQFOXVLRQ WKDW RQH FDQ WKHRUHWLFDOO\ IROORZ HDFK ZDYH FRPSRQHQW WKURXJK PXOWLSOH UHIOHFn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f $W DQ\ LQWHUIDFH EHWZHHQ GLVVLPLODU PDWHULDOV YHORFLWLHV DQG IRUFHV PXVW EH FRQn WLQXRXV $VVXPLQJ QRUPDO LQFLGHQFH ZDYHV RQO\ LQ D SHUIHFWO\ HODVWLF PAGE 38 PHGLXP WKHQ WKH UHODWLRQVKLS EHWZHHQ WKH LQFLGHQW WUDQVPLWWHG DQG UHIOHFWHG VWUHVVHV FDQ EH VKRZQ WR EH D W $O S &$S & $O S? &O f 2 2U & fÂ§ $LSL &Of$S & $O S? &Of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n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n QLWXGH $V WKH FRPSUHVVLRQ SXOVH PDNHV PXOWLSOH UHIOHFWLRQV DQG PAGE 39 WUDQVPLVVLRQV ZLWKLQ WKH VSHFLPHQ WKH PDJQLWXGH RI WKH RULJLQDO WHQVLOH UHIOHFWLRQ RI WKH LQFLGHQW SXOVH LV UHGXFHG E\ WKH PDJQLWXGH RI WKH WUDQVn PLWWHG FRPSUHVVLYH SXOVH IURP WKH VSHFLPHQ LQWR WKH LQFLGHQW EDU 7KH OHQJWK RI WKH FRPSUHVVLYH LQFLGHQW SXOVH LV RI ILQLWH WLPH ZKLFK PHDQV WKH WHQVLOH UHIOHFWLRQ DQG WKH FRPSUHVVLYH WUDQVPLVVLRQ SXOVHV DUH DOVR RI ILQLWH OHQJWK DQG LQ IDFW DUH RI LGHQWLFDO GXUDWLRQ DV WKH LQFLGHQW ZDYH 7KLV LPSOLHV WKH ZDYHOHQJWKV RI WKHVH WUDQVLHQW SXOVHV GR QRW FKDQJH DV WKH\ FURVV LQWHUIDFHV RQO\ WKHLU DPSOLWXGHV DUH FKDQJHG 7KHVH SKHQRPHQD DUH FOHDUO\ VHHQ LQ H[SHULPHQWDO WUDFHV VHH IRU H[DPSOH )LJXUH f DQG LQ WKH 0RGLILHG 3RFKKDPPHU&KUHH UHVXOWV VHH IRU H[DPSOH )LJXUH f REWDLQHG IURP WKH SURJUDP GLVFXVVHG LQ WKH SUHYLRXV SDJH WKH UHIOHFWHG WHQVLOH ZDYH VWDUWV RXW DW D FHUWDLQ YDOXH DQG WKHQ LQ WKH WLPH UHTXLUHG WR WUDQVLW WZLFH WKH VSHFLPHQ OHQJWK LW MXPSV WR D ORZHU OHYHO GXH WR WKH DUULYDO RI WKH ILUVW SRUWLRQ RI WKH FRPSUHVVLYH SXOVH IURP WKH VSHFLPHQ 7KH DPSOLWXGHV RI WKH UHIOHFWHG DQG WKH WUDQVPLWWHG SXOVHV DGG XS WR WKH DPSOLWXGH RI WKH LQFLGHQW SXOVH ,Q WKH H[SHULPHQW RI )LJXUH WKH VSHFLPHQ ZDV SODVWLFDOO\ GHIRUPHG ZKLOH IRU )LJXUH WKH VSHFLPHQ UHPDLQV HODVWLF ,W VKRXOG EH PHQWLRQHG DW WKLV SRLQW WKDW LQ RUGHU WR EH DEOH WR GLUHFWO\ FRPSDUH WKH UHVXOWV WKH GLPHQVLRQV RI WKH EDUV DQG WKH ORFDWLRQV RI WKH VWUDLQ JDJHV ZHUH WKH VDPH LQ DOO PHWKRGRORJLHV 6SHFLILFDOO\ VLQFH WKH 7\QGDOO 6+3% LV LQFKHV LQ GLDPHWHU DQG WKH JDJHV DUH LQFKHV RQ HLWKHU VLGH RI WKH VSHFLPHQ ERWK WKH PRGLILHG 3RFKKDPPHU&KUHH PHWKRG DQG WKH ILQLWH GLIIHUHQFH PHWKRG ZHUH UXQ ZLWK WKRVH YDOXHV )RU WKH LQFK 6+3% WKH QXPHULFDO PHWKRGV DVVXPHG VWUDLQ JDJHV LQFKHV RQ HLWKHU VLGH RI WKH VSHFLPHQ PAGE 40 675(66 03D )LJXUH 7\QGDOO LQFK 6,,3% ([SHULPHQWDO 7UDFH $OXPLQXP 6SHFLPHQ /HQJWK DQG 'LDPHWHU LQFK PAGE 41 1250$/,=(' 675(66(6 ,1&+ 6+3% 2 7,0( PLFURVHFRQGV )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV ,QFLGHQW 6WUHVV 1RUPDOL]HG WR PAGE 42 $V PHQWLRQHG HDUOLHU WKH UHVXOWV RI WKH 3RFKKDPPHU&KUHH IUHTXHQF\ HTXDWLRQ DOVR OHDG WR WKH YDULDWLRQ RI WKH VWUHVVHV DQG GLVSODFHPHQWV LQ WKH UDGLDO GLUHFWLRQ 7KHVH YDULDWLRQV DOVR GHSHQG RQ WKH ZDYH VSHHG DQG WKH ZDYHOHQJWK RI WKH SXOVH DQG WKHUH DUH DQ LQILQLWH QXPEHU RI VROXWLRQV HDFK DVVRFLDWHG ZLWK D SDUWLFXODU PRGH RI YLEUDWLRQ $IWHU VRPH WHGLRXV GHULYDn WLRQV >@ WKH DPSOLWXGHV :Uf DQG 8Uf RI WKH DVVXPHG ORQJLWXGLQDO DQG UDGLDO GLVSODFHPHQWV X] :Uf r IW]f DQG XU 8Uf r IW]f DUH :Uf L \F -LKDf KD KD -R KUf [f N D -S N Uf -L KDf [ f -LN Df 8Uf fÂ§ \ F -L KDf KD [ Ofr KD -L KUf B IW [f N D -L F Uf -L KDf [f [ OfA -LFDf ZKHUH F D FRQVWDQW GHWHUPLQHG E\ WKH DPSOLWXGH RI YLEUDWLRQ [ W F` ?&R9f 3 Y 9 f K \ S[ Of9 \[Of 9 :LWK WKLV LQIRUPDWLRQ WKH UDGLDO VKHDU DQG ORQJLWXGLQDO VWUHVVHV FDQ EH FDOFXODWHG DV f a-L KDf 2Q IL\ F A [f KD -R KUf -LKDf IL[ f D -LKUf U -MKDf 3 [ a -R Uf B 3 [ a D -L r Uf OIW [ [ -L MF Df [ U -LFDf P RU] ?IL \ F K -L KDf I -L KUf -L KDf -L N Uf -L N Df I I W]f nV-LKDf -RKUf -R N Uf Df UF KDf O[n[fKD-KDf L[ffnDMODf IW]f PAGE 43 $JDLQ DVVXPLQJ RQO\ 0RGH 0c c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n WLRQV QXPHULFDOO\ VLQFH DV WKH DUJXPHQW EHFRPHV QHJDWLYH PRGLILHG %HVVHO IXQFWLRQV PXVW EH XVHG 6SHFLILFDOO\ -L[f ,[f DQG -L L[f L % [f RU LQ JHQHUDO ,Q ;f LffQ -Q L[f 'DYLHV >@ VKRZHG WKDW LI WKH UDWLR LV NHSW EHORZ WKHQ WKH UDGLDO YDULDWLRQ LV OHVV WKDQ b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f )RXULHU VHULHV WHUPV PAGE 44 1250$/,=(' /21*,78',1$/ 675(66 675(66(6 )25 ',))(5(17 )285,(5 7(506 $ fÂ§LQHfÂ§ VHF% fÂ§LQHfÂ§ VHF Q O$%f 5$',$/ /2&$7,21 D5 )LJXUH 1RUPDOL]HG /RQJLWXGLQDO 6WUHVV *LYHQ DV D )XQFWLRQ RI 5DGLDO /RFDWLRQ IRU 6HYHUDO )RXULHU 7HUPV PAGE 45 WKH RQHV ZKRVH A YDOXHV DUH ORZf VKRZ D QHDUO\ XQLIRUP GLVWULEXWLRQ RI VWUHVV ZKLOH DV WKH ZDYHOHQJWKV EHFRPH VPDOOHU ODUJHU f WKH UDGLDO YDULDWLRQV EHFRPH SURJUHVVLYHO\ JUHDWHU WR D SRLQW ZKHUH WKH\ FKDQJH VLJQ ,I WKH DPSOLWXGHV RI WKHVH KLJKHU WHUPV KDYH IDOOHQ WR D YHU\ VPDOO YDOXH WKHQ WKHLU FRQWULEXWLRQ WR WKH RYHUDOO SXOVH ZLOO EH QHJOLJLEOH 3ORWV RI WKH UDGLDO YDULDWLRQV RI UHIOHFWHG DQG WUDQVPLWWHG VWUHVVHV DUH VKRZQ LQ $SSHQGL[ $ IRU D DQG D LQFK 6+3% )LJXUHV DQG f ,W LV DOVR DSSDUHQW WKDW DV WKH GLDPHWHU RI WKH EDU LV LQFUHDVHG DQG WKH LQSXW ZDYHOHQJWK LV NHSW WKH VDPH WKH VWUHVV ZLOO VKRZ XQLIRUPLW\ IRU RQO\ D IHZ WHUPV 7KLV IDFW LV DGGLWLRQDO SURRI WKDW WKH RQHGLPHQVLRQDOLW\ RI DQ 6+3% LV GHSHQGHQW RQ WKH UDWLR RI WKH UDGLXV WR WKH ZDYHOHQJWK DQG LI NHSW WKH VDPH DV IRU WKH H[LVWLQJ 6+3%V WKHQ WKH ZDYH SURSDJDWLRQV VKRXOG EH PDWKHPDWLFDOO\ LGHQWLFDO $OWKRXJK WKH KLJKHU )RXULHU WHUPV DUH KLJKO\ QRQXQLIRUP WKHLU FRQn WULEXWLRQV DUH YHU\ VPDOO )XUWKHUPRUH LI RQH QRUPDOL]HV WKH DPSOLWXGH YDOXHV RI WKH UDGLDO DQG WKH VKHDU VWUHVVHV WR WKRVH RI WKH ORQJLWXGLQDO VWUHVn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n WLRQ WHQVRU FRQVWLWXWLYH UHODWLRQV DQG WKH VWUHVV GHYLDWRUV $ FHQWHUHG PAGE 46 ILQLWH GLIIHUHQFH DSSUR[LPDWLRQ LV HPSOR\HG 7KH WRSLF RI FRQVWLWXWLYH UHODn WLRQV LV WKH PRVW FUXFLDO DV LW LQFOXGHV WKH UHODWLRQVKLS EHWZHHQ WKH HODVWLF DQG SODVWLF GHYLDWRULF VWUDLQV DQG VWUHVVHV DQG WKH UHODWLRQVKLS EHWZHHQ SUHVVXUH GHQVLW\ DQG LQWHUQDO HQHUJ\ 7KH DFFXUDF\ RI DQ\ K\GURFRGH LV KLJKO\ GHSHQGHQW RQ WKH LPSOHPHQWDWLRQ RI WKH FRQVWLWXWLYH UHODWLRQV ,Q WKLV HIIRUW RQO\ PHWDOV ZHUH FRQVLGHUHG QDPHO\ VWHHO DQG DOXPLQXP %RWK DUH PRGHOHG LQ +8// DV HODVWLFSODVWLF PDWHULDOV DOWKRXJK WKH VWUHVV OHYHOV ZHUH NHSW ORZ HQRXJK WR NHHS WKH VWHHO EDUV LQ WKH HODVWLF UHJLRQ %RWK PDWHULDOV DUH DOVR FRQVLGHUHG LVRWURSLF 6WUHVVHV DUH GHFRPn SRVHG LQWR D VSKHULFDO DQG D GHYLDWRULF FRPSRQHQW DQG WKH JHQHUDOL]HG +RRNHf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n PLWWHU EDUV 7KH LQSXW SUHVVXUH ZDV YDULHG WR DVVHVV VWUDLQ UDWH VHQVLWLYLW\ DQG VWUHVVHV DQG VWUDLQV ZHUH UHFRUGHG DW VWUDLQ JDJHV ORFDWHG LQ WKH LQFLGHQW DQG WUDQVPLWWHU EDUV ,W ZDV IRXQG WKDW DV DOUHDG\ NQRZQf WKLV SDUWLFXODU DOXPLQXP LV VWUDLQ UDWH LQGHSHQGHQW WKDW LV WKH VWUHVV VWUDLQ FXUYH LV QRW DIIHFWHG E\ WKH UDWH DW ZKLFK WKH ORDG LV DSSOLHG 7KLV LV YHU\ LPSRUWDQW EHFDXVH WKH +8// FRGH GRHV QRW WDNH VWUDLQ UDWH LQWR DFFRXQW 7KHUHIRUH D FRPSDULVRQ ZLOO EH PHDQLQJIXO EHWZHHQ WKH H[SHULPHQWDO GDWD DQG WKH FRPn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f DUH WKHRUHWLFDO PRGHOV DQG E\ QR PHDQV H[DFW )LQDOO\ WKH HIIHFWV RI IULFWLRQ DUH QRW LQFOXGHG LQ WKH QXPHULFDO VFKHPH +DYLQJ VDLG WKDW RQH LV WKHQ ORRNLQJ IRU D TXDOLWDWLYH PDWFK LQ WKH GDWD PRUH VR WKDQ H[DFW YDOXHV RI VWUHVVHV RU VWUDLQV :KDW WKLV PHDQV LV WKDW FHUWDLQ WUHQGV WKDW KDYH EHHQ REVHUYHG H[SHULPHQWDOO\ VKRXOG VKRZ XS QXPHULFDOO\ )RU H[DPSOH DV WKH VSHFLPHQ JHWV VPDOOHU FRPSDUHG WR WKH LQFLGHQW EDU H[SHULPHQWDO UHVXOWV VKRZ WKDW WKH WUDQVPLWWHG VWUHVV DOVR EHn FRPHV VPDOOHU &RQVHTXHQWO\ WKH UHIOHFWHG VWUHVV LQFUHDVHV DQG LWV VKDSH EHFRPHV PRUH XQLIRUP DV WKH LQLWLDO KLJK SHDN VWDUWV WR EOHQG LQ ZLWK WKH UHVW RI WKH UHIOHFWHG SXOVH 7KH +8// FDOFXODWLRQV DOVR VKRZ WKHVH SKHQRPHQD $OWKRXJK WKH DFWXDO DPSOLWXGH YDOXHV GR QRW PDWFK WKH UHODWLYH FKDQJH EHWZHHQ WHVWV GRHV ,Q JHQHUDO WKH H[SHULPHQWDO GDWD JLYHV KLJKHU WUDQVn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f 22( ( 22( Â‘ 22( Â‘ 22( 22( 67$ ; < 7 06(&f )LJXUH +8// 2XWSXW LQFK 6+3% 6WUHVV 7< PAGE 53 .%f 63/,7 +23.,1621 %$5 &0 =21,1* 3UREOHP ( ( ( ( ( 22( Â‘ ( 22( 22( 67$ ; < 0$; ( 0,1 ( 22( 7 06(&f )LJXUHOO +8// 2XWSXW LQFK 6+3% 6WUHVV 7< PAGE 54 .%f 63/,7 +23.,1621 %$5 &0 =21,1* 3UREOHP 2 n n Â‘ 7 Â‘ Â‘ 7 2%% 7 06(&f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n WLRQ RXW RI WKH VSHFLPHQ UHVXOWLQJ LQ D ORZHU UHIOHFWLYH SODWHDX DIWHU WKH LQLWLDO SHDN WKDQ IRU WKH FRUUHVSRQGLQJ QRQHODVWLF FDVH 7R VKRZ WKLV WKUHH H[SHULPHQWDO WHVWV ZHUH UXQ ZLWK WKH VDPH VSHFLPHQ FRQILJXUDWLRQ DW GLIIHUHQW OHYHOV RI VWUHVV RQH OHYHO ZDV NHSW LQ WKH DOXPLQXP HODVWLF UHJLRQ WKH RWKHU WZR H[FHHGHG WKH \LHOG YDOXH IRU DOXPLQXP )LJXUH VKRZV KRZ DIWHU WKH LQLWLDO SHDN LQ WKH HODVWLF FDVH WRS ILJXUHf WKH UHIOHFWHG SXOVH DOPRVW JRHV GRZQ WR ]HUR ZKLOH DV WKH LQSXW VWUHVV LV LQFUHDVHG PLGGOH DQG ERWWRP ILJXUHVf WKH SODWHDX DIWHU WKH SHDN LQFUHDVHV DOVR $QRWKHU LVVXH WR EH H[SORUHG H[SHULPHQWDOO\ DQG QXPHULFDOO\ LV WKH HIIHFW RI GHFUHDVLQJ WKH SXOVH GXUDWLRQ ZLWK UHVSHFW WR WKH GLDPHWHU RI WKH EDU $V GLVFXVVHG WKH HTXDWLRQV JRYHUQLQJ DQ 6+3% VKRZ WKDW LQFUHDVLQJ WKH UDWLR ZLOO DIIHFW WZR LPSRUWDQW WKLQJV 7KH ILUVW WKLQJ LV WKDW WKH SXOVH ZLOO GLVSHUVH PRUH WKDW LV WKH FRQWULEXWLRQ IURP WKH VORZHU WUDYHOLQJ FRPn SRQHQWV ZLOO EH JUHDWHU DQG WKH RYHUDOO ZDYHIRUP ZLOO GLVWRUW DQG VSUHDG RXW PRUH DV LW WUDYHOV 7KH VHFRQG WKLQJ LV WKDW WKH ZDYH SURSDJDWLRQ ZLOO EH OHVV RQH GLPHQVLRQDO LQ QDWXUH ZKLFK PHDQV WKDW WKH ORQJLWXGLQDO VWUHVV ZLOO KDYH PAGE 56 675(66 03D 675(66 03D D &/ FQ FQ 8 WU Xf V 2XV 2XV 2XV 7,0( 2V 2XV 2XV 2XV V 2XV 2XV 2XV 7,0( )LJXUH ([SHULPHQWDO 7UDFHV DW 7KUHH ,QFLGHQW 6WUHVV /HYHOV 03D7RS 03D0LGGOH 03D%RWWRPf 6SHFLPHQ 'LDPHWHU LQ /HQJWK LQ LQFK 6+3% PAGE 57 ODUJHU YDULDWLRQV DORQJ WKH UDGLXV DQG WKH UDGLDO DQG VKHDU VWUHVVHV ZLOO EH OHVV QHJOLJLEOH DV A LQFUHDVHV 1XPHULFDOO\ RQH FDQ REVHUYH WKH HIIHFWV RI GHFUHDVLQJ WKH LQSXW ZDYHOHQJWK E\ XVLQJ +8// DQG WKH 0RGLILHG 3RFKKDP PHU&KUHH PHWKRG 7KH IROORZLQJ UXQV VKRZQ LQ 7DEOH ZHUH PDGH WR WHVW WKH PHWKRGRORJLHV DJDLQVW ZKDW LW LV DQDO\WLFDOO\ EHOLHYHG WR EH WUXH 7DEOH +8// DQG 0RGLILHG 3RFKKDPPHU&KUHH 5XQV ',$0(7(5 LQFKHVf :$9(/(1*7+ PLFURVHFRQGVf D $ ),*85(6 )LJXUHV DQG VKRZ UHVXOWV IURP WKH 0RGLILHG 3RFKKDPPHU &KUHH SURJUDP ZKLOH )LJXUHV DQG DUH +8// RXWSXW WUDFHV 1RWLFH WKDW +8// DSSHDUV WR KDYH WZR GLIIHUHQW FRRUGLQDWH V\VWHPV IURP RQH ILJXUH WR WKH QH[W 7KH UHDVRQ IRU WKLV LV WKDW WKH UXQV ZHUH PDGH RQ GLIIHUHQW FRPSXWHU V\VWHPV 9$; DQG &5$ PAGE 58 .%f 63/,7 +23.,1621 %$5 &0 =21,1* 3UREOHP 7 06(&f )LJXUH +8// 2XWSXW LQFK 6+3% LQFK 'LDPHWHU DQG /HQJWK $OXPLQXP 6SHFLPHQ VHF 3XOVH PAGE 59 1250$/,=(' 675(66(6 ,1&+ 6+3% ,1&+ $/80,180 63(&,0(1 7,0( 7,0( )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV LQFK 6+3% UVHF 3XOVH PAGE 60 ,=(' ,1&,'(17 675(66 ,1&+ 6+3% ,1&+ $/80,180 63(&,0(1 )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV A LQFK 6+3% ÂVHF 3XOVH PAGE 61 )LJXUH +8// 2XWSXW A VHF 3XOVH 6WDWLRQ 7RSf %DU &HQWHU 6WDWLRQ %RWWRPf %DU 6XUIDFH PAGE 62 1250$8=(' 675(66(6 ,1&+ 6+3% ,1&+ $/80,180 63(&,0(1 7,0( PLFURVHFRQGV )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV A LQFK 6+3% ÂVHF 3XOVH PAGE 63 675(66 .%f 62 [ 2f [ 2 7,0( 06(&f /IL 8f )LJXUH +8// 2XWSXW DW ,QFLGHQW %DU 6WUDLQ *DJH A SSU LQFK 6+3% VHF 3XOVH PAGE 64 GLDPHWHU WKH ZDYHIRUPV DUH PDWKHPDWLFDOO\ WKH VDPH WKLV LV VKRZQ LQ )LJXUHV DQG )LJXUHV DQG LQGLFDWH WKDW DQ LQSXW SXOVH RI PLFURVHFRQGV IRU D LQFK 6+3% LV QRW UHFRPPHQGHG ,W DSSHDUV GLVSHUVLRQ EHFRPHV D SUREOHP HYHQ DIWHU D YHU\ VKRUW WLPH DQG DOVR WKH RQHGLPHQn VLRQDOLW\ RI WKH PRWLRQ LV TXHVWLRQDEOH ([SHULPHQWDOO\ RQH FDQ DFKLHYH GLIIHUHQW ZDYHOHQJWKV E\ FKDQJLQJ WKH OHQJWK RI WKH VWULNHU EDU 7KH UHVXOWn LQJ ZDYHIRUPV IRU D LQFK DQG D LQFK VWULNHU EDU DUH VKRZQ LQ )LJXUH DOO WKUHH EDU ORFDWLRQV DUH RQ WKH LQFLGHQW EDU f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n WKRXJK WKH UDGLDO DQG VKHDU FRPSRQHQWV RI WKH VWUHVVHV DUH KLJKHU IRU KLJKHU YDOXHV RI WKH\ QHYHUWKHOHVV UHPDLQ QHJOLJLEOH ZKHQ FRPSDUHG WR WKH ORQJLWXGLQDO VWUHVV )LJXUHV DQG VKRZ WKH SXOVHV DW WKH FHQWHU DQG DW WKH VXUIDFH RI WKH WUDQVPLWWHU EDU DW WKH VWUDLQ JDJH ORFDWLRQ )LJXUH LV WKH UHVXOW RI WKH PRGLILHG 3RFKKDPPHU&KUHH PHWKRG IRU D LQFK EDU DQG PLFURVHFRQGV )LJXUHV DQG DUH +8// RXWSXW IRU DQG PLFURVHFRQG LQSXW SXOVHV UHVSHFWLYHO\ )LJXUH VKRZV D GLIIHUHQFH RI DERXW b EHWZHHQ WKH WZR FXUYHV 7KH YDULDWLRQ LQ VWUHVV DORQJ WKH FURVV VHFWLRQ LV URXJKO\ b IRU )LJXUH DQG b IRU )LJXUH 7KHUHIRUH WKH XQLIRUPLW\ RI WKH ORQJLWXGLQDO VWUHVV LV GHJUDGHG DV A LV LQFUHDVHG DOWKRXJK IRU WKH SURSRVHG LQFK EDU DQG D PLFURVHFRQGV SXOVH WKH ZDYH PRWLRQ VKRXOG EH YHU\ FORVH WR RQHGLPHQVLRQDO PAGE 65 9ROWV 9ROWV GLVWDQFHV IURP VWULNHU EDU HQG 7LPH VHFf )LJXUH ([SHULPHQWDO 7UDFHV DW 7KUHH ,QFLGHQW %DU /RFDWLRQV IRU 7ZR 6WULNHU %DU /HQJWKV LQFK 7RSf DQG LQFK %RWWRPf LQFK 6+3% PAGE 66 675(66 .%f 7 06(&f % )LJXUH +8// 2XWSXW LQFK %DU VHF 3XOVH $OO 6WUHVV &RPSRQHQWV RQ WKH 6XUIDFH DW ,QFLGHQW %DU 6WUDLQ *DJH PAGE 67 675(66 .f )LJXUH +8// 2XWSXW LQFK %DU VHF 3XOVH $OO 6WUHVV &RPSRQHQWV RQ WKH 6XUIDFH DW ,QFLGHQW %DU 6WUDLQ *DJH PAGE 68 1250$/,=(' ,1&,'(17 675(66(6 ,1&,'(17 675(66(6 &(17(5 685)$&( 2) 7+( %$5 7,0( PLFURVHFRQGV )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV DW ,QFLGHQW %DU 6WUDLQ *DJH LQFK %DU PAGE 69 7,0( 06(&f )LJXUH +8// 2XWSXW LQFK %DU QVHF 3XOVH 6XUIDFH 7RSf DQG &HQWHU %RWWRPf 7UDQVPLWWHG 6WUHVVHV PAGE 70 )LJXUH +8// 2XWSXW LQFK %DU VHF 3XOVH 6XUIDFH 7RSf DQG &HQWHU %RWWRPf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n GHQW EDU RQH QHDU WKH LPSDFWHG HQG DQG DQRWKHU RQH PHWHU IURP LW 7KH IROORZLQJ REVHUYDWLRQV FDQ EH PDGH LQ WKH FDVHV ZKHUH A LV ODUJHU WKH RVFLOODWLRQV LQ WKH LQSXW SXOVH DUH PRUH SURQRXQFHG DQG WKH\ WDNH ORQJHU WR GDPS RXW )XUWKHU LI RQH FRPSDUHV WKHVH UHVXOWV ZLWK WKH H[SHULPHQWDO WUDFHV )LJXUH f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n WLRQ DW WKH EDUVSHFLPHQ LQWHUIDFHV LV H[WUHPHO\ FRPSOLFDWHG HVSHFLDOO\ LI WKH VSHFLPHQ \LHOGV DW VRPH SRLQW GXULQJ WKH SURFHVV /RRNLQJ DW WKH VWUHVV PAGE 72 7= '<1(&&f 7== '<1(&&f 7,0( 06(&f 7,0( 06(&f )LJXUH +8// 2XWSXW LQFK %DU /RQJLWXGLQDO 6WUHVV 1HDU WKH ,PSDFWHG (QG 7RSf DQG DW WKH 6WUDLQ *DJH /RFDWLRQ %RWWRPf PAGE 73 675(66 .%f )LJXUH +8// 2XWSXW LQFK %DU $OO 6WUHVVHV 1HDU ,PSDFWHG (QG 7RSf DQG DW WKH 6WUDLQ *DJH /RFDWLRQ %RWWRPf PAGE 74 675(66 .f 675(66 .%f 7,0( 06(&f )LJXUH +8// 2XWSXW LQFK %DU $OO 6WUHVVHV 1HDU WKH ,PSDFWHG (QG 7RSf DQG DW WKH 6WUDLQ *DJH /RFDWLRQ %RWWRPf PAGE 75 FRPSRQHQWV DORQJ RQH VXFK LQWHUIDFH IURP D +8// FDOFXODWLRQ LW LV YHU\ GLIILFXOW WR PDNH DQ\ GHILQLWH VWDWHPHQW 7R KHOS LQ WKH XQGHUVWDQGLQJ RI WKH FRPSOLFDWHG VWUHVV ZDYH LQWHUDFWLRQ )LJXUH VKRZV WKH LPSRUWDQW IHDWXUHV DQG PDJQLWXGHV RI DOO VWUHVV FRPSRQHQWV DORQJ WKH LQFLGHQW EDUVSHFLPHQ LQWHUIDFH 7KH ORQJLWXGLQDO VWUHVV XQLIRUP DQG LQ FRPSUHVVLRQ RI PDJn QLWXGH 03D VRPHWLPH SULRU WR WKLV LQWHUIDFHf DW WKH LQWHUIDFH YDULHV IURP DOPRVW ]HUR DW WKH EDU VXUIDFH WR WR 03D DW WKH FHQWHU 6KHDU DQG UDGLDO VWUHVVHV EHFRPH RI WKH VDPH RUGHU RI PDJQLWXGH DV WKH ORQn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f DQG LQ WKH WUDQVPLWWHU EDU DW WKH VWUDLQ JDJH ORFDWLRQ )LJXUH f ZKHUH WKH ORQJLWXGLQDO VWUHVV LV WKH RQO\ QRQ ]HUR FRPSRQHQW )LJXUHV DQG FRUn UHVSRQG WR WKH VDPH 6+3% DV LQ )LJXUH 1RWLFH WKDW RRH LQ )LJXUH LV RQO\ DERXW 03D ZKLOH LW SHDNHG DW 03D DW WKH LQWHUIDFH WKDW DPRXQWV WR D UHGXFWLRQ RI b LQ RQH FHQWLPHWHU $SSHQGL[ & FRQWDLQV PRUH +8// RXWSXW WUDFHV ZKLFK FDQ IXUWKHU H[SODLQ WKH DERYH GLVFXVVLRQ )LQDOO\ %HUWKROI DQG .DUQHV >@ SRLQWHG RXW WKDW WKH ILUVW LQWHUIDFH EHWZHHQ WKH EDU DQG WKH VSHFLPHQ HIIHFWLYHO\ ILOWHUV RXW PRVW RI WKH RVFLOODn WLRQV LQ WKH LQFLGHQW SXOVH DQG WKXV LQ WKH FDVHV ZKHQ WKHVH RVFLOODWLRQV DUH ODUJH IRU RQH UHDVRQ RU DQRWKHU LW LV ZLVH WR XVH RQO\ WKH WUDQVPLWWHG VWUHVV WR FDOFXODWH WKH UHVSRQVH LQ WKH VSHFLPHQ 7KH DQDO\WLFDO WRROV DOVR VKRZ WKLV ILOWHULQJ HIIHFW RI WKH VSHFLPHQ DQG LW DSSHDUV WKH XVH RI WKH WUDQVPLWWHG SXOVH RQO\ PD\ EH D ZRUWKZKLOH DOWHUQDWLYH IRU WKH SURSRVHG LQFK 6+3% PAGE 76 63(&,0(1 )LJXUH +8// 2XWSXW LQFK %DU $OO 6WUHVVHV DW 7KUHH /RFDWLRQV DW WKH %DU6SHFLPHQ ,QWHUIDFH PAGE 77 675(66 .%f 6n7&(66 .%f )LJXUH +8// 2XWSXW LQFK %DU 6WUHVVHV LQ $OXPLQXP 6SHFLPHQ DW 7ZR /RFDWLRQV 1HDU WKH 6XUIDFH 7RSf 1HDU WKH &HQWHU %RWWRPf &RPSDUH ZLWK )LJXUH PAGE 78 675(66 '<1(&&f 675(66 '<1(&&f )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVVHV DW 7KUHH 5DGLDO /RFDWLRQV 6XUIDFH 7RSf 0LGGOH 0LGGOHf &HQWHU %RWWRPf &RPSDUH ZLWK )LJXUH PAGE 79 7KH WZR DQDO\WLFDO PHWKRGRORJLHV XVHG KDYH DGYDQWDJHV DQG GLVDGYDQWDJHV DQG WKH XVH RI RQH RYHU DQRWKHU GHSHQGV RQ WKH SDUWLFXODU QHHG DQG DSSOLFDn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n VHQFH RI IULFWLRQ DQG LQHUWLD )XUWKHUPRUH D W\SLFDO +8// FDOFXODWLRQ WRRN DQ\ZKHUH EHWZHHQ WKUHH DQG VL[ FHQWUDO SURFHVVLQJ XQLW &38f KRXUV RQ D &5$< <03 FRPSXWHU DW DQ DYHUDJH FRVW RI SHU MRE 7KH PRGLILHG 3RFKKDPPHU&KUHH PHWKRG WDNHV VHFRQGV WR UXQ RQ PRVW PDFKLQHV DW DQ DYHUDJH FRVW RI OHVV WKDQ $OWKRXJK WKH 0RGLILHG 3RFKKDPPHU&KUHH PHWKRG DVVXPHV D SXUHO\ HODVWLF UHVSRQVH LW GRHV QRW DIIHFW LWV XVHIXOQHVV VLQFH RQH ZRXOG QRW XVH DQ\ RI WKHVH DQDO\WLFDO WRROV WR LQYHVWLJDWH WUXH PDWHULDO UHVSRQVH EXW UDWKHU WR VWXG\ WKH UHOLDELOLW\ RI D FHUWDLQ 6+3% FRQILJXUDWLRQ 7KHVH QXPHULFDO FRGHV DUH QRW LQWHQGHG WR UHSODFH WKH H[n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n UHFWLRQV KDYH EHHQ GHULYHG WR DFFRXQW IRU WKH HIIHFWV RI UDGLDO DQG D[LDO LQHUWLD >@ DQG IULFWLRQ >@ 7KH REMHFWLYH RI WKLV LQYHVWLJDWLRQ LV WR XQGHUn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n O\ LI RQH LQFUHDVHV WKH LQSXW VWUHVV WR REWDLQ D KLJKHU VWUDLQUDWH DQG WKLV LQSXW VWUHVV LV LQ WKH IRUP RI D VWHS IXQFWLRQ WKHQ DSSDUHQWO\ WKH FRPELQDn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n SOLHG GLUHFWO\ WR GLIIHUHQW GLDPHWHU V\VWHPV )RU D LQFK GLDPHWHU EDU D OHQJWK / RYHU GLDPHWHU G /Gf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e G W f ZKHUH RE LV WKH PHDVXUHG VWUHVV VXEVFULSW V SHUWDLQV WR WKH VSHFLPHQ DQG / G fÂ§ DQG Y? fÂ§ DUH WKH D[LDO DQG UDGLDO LQHUWLD FRQWULEXWLRQV UHVSHFWLYHO\ R (TXDWLRQ f FDQ EH UHZULWWHQ LQ QRQGLPHQVLRQDO IRUP DV 6 aa 2E B G HG U 3V G U 9 2E a 2E / >G XV @ f PAGE 82 7KH OHIW KDQG VLGH LV WKH HUURU LQ WKH PHDVXUHG VWUHVV ,I RQH DVVXPHV D QRPLQDO 3RLVVRQfV UDWLR RI WKH FRQWULEXWLRQ RI WKH WHUP /G FDQ EH FDOFXODWHG UHODWLYH WR %HUWKROI DQG .DUQHV /G RI 7KLV OHDGV WR WKH FRQFOXVLRQ WKDW IRU D QRPLQDO /G WKH HUURU LQ WKH PHDVXUHG VWUHVV ZLOO LQFUHDVH E\ D IDFWRU RI 7KLV PHDQV WKDW WKH PD[LPXP VWUDLQUDWH WKH\ FDOFXODWHG VKRXOG EH UHGXFHG HYHQ IXUWKHU 7KLV ZRXOG OHDG WR XQDFFHSWDEO\ ORZ VWUDLQUDWHV 7KHUHIRUH WKH WKLQJ WR GR LV DFFHSW WKH HUURUV GXH WR LQHUWLD HIIHFWV DQG FRUUHFW WKHP XVLQJ HTXDWLRQ f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n FXODWLRQV WKDW WKH PHDVXUHG VWUHVV ZLOO EH ODUJHU WKDQ WKH WUXH VWUHVV UHVXOWn LQJ LQ DQ DUWLILFLDOO\ KLJK VWUHQJWK LQ WKH PDWHULDO EHLQJ DQDO\]HG %XW KRZ PXFK" (DUO\ ZRUN DQG VWXGLHV GRQH E\ 'DYLHV DQG +XQWHU >@ DQG E\ 6DPDQn WD >@ VKRZ WKDW LQ JHQHUDO LI WKH GLDPHWHU DQG WKH OHQJWK RI WKH VSHFLPHQ DUH RI WKH VDPH RUGHU RI PDJQLWXGH WKHQ IULFWLRQ FRQWULEXWHV WR RQO\ D IHZ SHUFHQW HUURU ,W DSSHDUV WKDW WKH LPSRUWDQW SDUDPHWHU LV WKH UDWLR RI WKH DUHD FRQVWUDLQHG E\ IULFWLRQ RYHU WKH YROXPH RI PDWHULDO LQ WKH VSHFLPHQ ,Q RWKHU ZRUGV D WKLQ VSHFLPHQ ZLOO EH PRUH DIIHFWHG E\ IULFWLRQ WKDQ D WKLFN RQH 6LQFH WKH DUHD FRQVWUDLQHG E\ IULFWLRQ LV MW G WZR IDFHVf DQG WKH YROXPH LV -W Gn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n OHVV 7KHUH LV QR RSWLRQ LQ +8// IRU PRGHOLQJ IULFWLRQf 7KH UHVXOWV WKHQ ERXQG WKH IULFWLRQ SUREOHP DV WKH ORFNHG FDVH UHSUHVHQWV D ZRUVW FDVH VFHQDULR )LJXUHV DQG VKRZ WKH WUDQVPLWWHG VWUHVV IRU D LQFK EDU ZLWK ]HUR DQG LQILQLWH IULFWLRQ UHVSHFWLYHO\ )LJXUHV DQG VKRZ WKH VDPH WKLQJ IRU D LQFK EDU $V H[SHFWHG WKH FDVHV ZLWK LQILQLWH IULFWLRQ VKRZ D KLJKHU WUDQVPLWWHG VWUHVV LPSO\LQJ DQ DUWLILFLDOO\ KLJKHU VWUHQJWK LQ WKH VSHFLPHQ ,W LV LQWHUHVWLQJ WR QRWLFH WKDW WKH HUURU GXH WR IULFWLRQ LQ WKH LQFK EDU LV DERXW b DQG LQ WKH LQFK EDU LV RQO\ b 7KLV DJUHHV ZLWK WKH FULWHULRQ GHULYHG DERYH ZKLFK VWDWHV WKDW WKH HUURU LV SURSRUWLRQDO WR / ,W DSSHDUV WKH ODUJHU 6+3% VHW XS ZLOO EH OHVV VXVFHSWLEOH WR IULFWLRQ LQ OLJKW RI WKH IDFW WKDW WKH VSHFLPHQ OHQJWK ZLOO EH LQFUHDVHG DORQJ ZLWK WKH GLDPHWHU +RZHYHU RQH VKRXOG QRW LPSO\ IURP WKLV WKDW IULFWLRQ LQWURGXFHV HUURUV LQ WKH H[LVWLQJ V\VWHPV RI WKH RUGHU RI b 7KDW ILJXUH LV DQ XSSHU ERXQG IRU D FDVH ZKHUH WKH VSHFLPHQ LV DWWDFKHG WR WKH EDUV ,Q UHDOLW\ WKH IULFWLRQ FRHIILFLHQW ZLOO EH VRPHZKHUH OHVV WKHQ HVSHFLDOO\ LI FDUH LV WDNHQ LQ OXEULFDWLQJ WKH HQGV 7KHUHIRUH WKH HUURUV GXH WR IULFWLRQ DUH QHJOLJLEOH LQ W\SLFDO 6+3%V DQG ZLOO EH VR DOVR IRU D SURSRVHG LQFK V\VWHP ,I KRZHYHU RQH ZDV WR WHVW WKLQ VSHFLPHQV WKHQ WKH GDWD UHGXFWLRQ VKRXOG WDNH IULFWLRQ HIIHFWV LQWR DFFRXQW PAGE 84 675(66 .f ( ( ( Â‘ ( f ( Â‘ fÂ§ (fÂ§ f 67$ ; < 7 06(&f )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK =HUR )ULFWLRQ %HWZHHQ 6SHFLPHQV DQG 6+3% PAGE 85 675(66 .%f H22[,2f [ 2 7,0( 06(&f )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK ,QILQLWH )ULFWLRQ %HWZHHQ 6SHFLPHQ DQG 6+3% PAGE 86 675(66 .%f )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK =HUR )ULFWLRQ PAGE 87 675(66 .%f 7,0( 06(&f )LJXUH +8// 2XWSXW LQFK %DU 7UDQVPLWWHG 6WUHVV ZLWK ,QILQLWH )ULFWLRQ PAGE 88 &+$37(5 6,;,1&+ 6+3% )($6,%,/,7< 6XPPDU\ 7KH REMHFWLYH RI WKLV HIIRUW ZDV WR DVVHVV WKH IHDVLELOLW\ RI D LQFK 6+3% 7KHUH LV DQ DEXQGDQFH RI H[SHULPHQWDO GDWD IRU H[LVWLQJ 6+3% V\Vn WHPV DQG LW ZDV XVHG LQ WKLV VWXG\ WR WHVW WKH DFFXUDF\ DQG UHOLDELOLW\ RI WKH QXPHULFDO ILQLWH GLIIHUHQFH FRGH +8// 7KH ILUVW SDUW RI WKLV ZRUN UHVHDUFKHG WKH PXOWLWXGH RI HIIRUWV FRQn GXFWHG RQ WKH VXEMHFW LQ WKH SDVW \HDUV ,Q SDUWLFXODU LW ZDV HVWDEOLVKHG WKDW LW ZDV LPSRUWDQW WR ORRN DW WKH WZRGLPHQVLRQDO HTXDWLRQV RI PRWLRQ DV WKH VLPSOLILHG RQHGLPHQVLRQDO FDVH PD\ QRW EH DGHTXDWH IRU D ODUJHU V\VWHP 7KH 3RFKKDPPHU&KUHH VROXWLRQ IRU DQ LQILQLWHO\ ORQJ EDU ZDV XVHG DV WKH EDVHOLQH IRU WKH PDWKHPDWLFDO DQDO\VLV (DUO\ SDSHUV RQ LQHUWLD DQG IULFWLRQ HIIHFWV ZHUH VWXGLHG DQG WKH PHWKRGRORJ\ H[WHQGHG WR WKH LQFK EDU V\Vn WHP SULPDULO\ WR KDYH D IHHO IRU WKH PDJQLWXGH RI WKH HUURUV LQWURGXFHG E\ WKRVH IDFWRUV 7DNLQJ D FORVHU ORRN DW WKH JRYHUQLQJ HTXDWLRQV DQG DW WKH 3RFKKDPn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n SODFHPHQWV DQG VWUHVVHV DORQJ WKH UDGLDO GLUHFWLRQ FDQ DOVR EH FDOFXODWHG LI WKH ZDYHVSHHGV RI WKH )RXULHU FRPSRQHQWV DUH NQRZQ ,W ZDV VKRZQ KRZ WKLV PHWKRGRORJ\ JLYHV VXUSULVLQJO\ JRRG UHVXOWV FRQVLGHULQJ WKH PDQ\ RUn GHUV RI PDJQLWXGH GLIIHUHQFH LQ FRPSXWLQJ WLPH FRPSDUHG WR WKH ILQLWH GLIIHUHQFH WHFKQLTXH $OO WKUHH PHWKRGRORJLHV UHDIILUPHG WKH LQLWLDO VWDWHPHQW RI VFDOH LQn YDULDQFH RI DQ\ 6+3% ,W ZDV VKRZQ KRZ LI WKH LQSXW SUHVVXUH SXOVH KDV D ZDYHOHQJWK RI PLFURVHFRQGV D LQFK 6+3% ZLOO EHKDYH DV D RQHGLPHQn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n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n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n FDO 5HSRUW $005& 75 $UP\ 0DWHULDOV DQG 0HFKDQLFV 5Hn VHDUFK &HQWHU :DWHUWRZQ 0$ 1RYHPEHU .ROVN\ + 6WUHVV :DYH LQ 6ROLGV 'RYHU 3XEOLFDWLRQV 1HZ PAGE 93 6DPDQWD 6. '\QDPLF 'HIRUPDWLRQ RI $OXPLQXP DQG &RSSHU DW (OHYDWHG 7HPSHUDWXUHV -RXUQDO RI WKH 0HFKDQLFV DQG 3K\VLFV RI 6ROLGV 9RO SS -DKVPDQ :( 5HH[DPLQDWLRQ RI WKH .ROVN\ 7HFKQLTXH IRU 0HDVXULQJ '\QDPLF 0DWHULDO %HKDYLRU -RXUQDO RI $SSOLHG 0HFKDQLFV 9RO SS &KLX 66 DQG 1HXEHUW 9+ 'LIIHUHQFH 0HWKRG IRU :DYH $QDO\VLV RI WKH 6SOLW +RSNLQVRQ 3UHVVXUH %DU ZLWK D 9LVFRHODVWLF 6SHFLPHQ -RXUn QDO RI WKH 0HFKDQLFV DQG 3K\VLFV RI 6ROLGV 9RO SS PAGE 94 5DMHQGUDQ $0 DQG %OHVV 6+LJK 6WUDLQ 5DWH 0DWHULDO %HKDYLRU 7HFKQLFDO 5HSRUW $):$/75 $):$/ .LUWODQG $)% 10 'HFHPEHU 5RVV &$ 6SOLW +RSNLQVRQ 3UHVVXUH %DU 7HVWV )LQDO 7HFKQLFDO 5HSRUW (6/75 $)(6& 7\QGDOO $)% )/ 0DUFK )ROODQVEHH 36 DQG )UDQW] & :DYH 3URSDJDWLRQ LQ WKH 6SOLW +RSNLQn VRQ 3UHVVXUH %DU $60( -RXUQDO RI (QJLQHHULQJ 0DWHULDOV DQG 7HFKn QRORJ\ PAGE 95 *RQJ -& 0DOYHUQ /( DQG -HQNLQV '$ 'LVSHUVLRQ ,QYHVWLJDWLRQ LQ WKH 6SOLW +RSNLQVRQ 3UHVVXUH %DU -RXUQDO RI (QJLQHHULQJ 0DWHULDOV DQG 7HFKQRO RJ\ 9RO SS -XO\ -HURPH (/ )HDVLELOLW\ RI D LQFK 6SOLW +RSNLQVRQ 3UHVVXUH %DU 6+3%f )LQDO 7HFKQLFDO 5HSRUW (6/75 $)(6& 7\QGDOO $)% )/ $XJXVW +RSNLQVRQ % $ 0HWKRG RI 0HDVXULQJ WKH 3UHVVXUH 3URGXFHG LQ WKH 'HWRQDWLRQ RI +LJK ([SORVLYHV RU E\ WKH ,PSDFW RI %XOOHWV 3KLORVRSKLn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n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n WHULVWLF LPSHGDQFHV RI WKH PDWHULDOV 7KH XVHU LV DVNHG WR HQWHU WKH SXOVH GXUDWLRQ QXPEHU RI WLPH VWHSV HDFK H VHFRQGV ORQJf WKH GLDPHWHU RI WKH EDU DQG WKH GLPHQVLRQV RI WKH VSHFLPHQ 7KH LQSXW SXOVH LV UHSUHVHQWHG YLD D )RXULHU VHULHV WHUPV DUH UHWDLQHG $PSOLWXGH RI WKH LQFLGHQW SXOVH LV QRUPDOL]HG WR (DFK FRPn SRQHQW KDV D XQLTXH SKDVH DQJOH IUHTXHQF\ DQG DPSOLWXGH DQG WKH\ DUH DOO LQ SKDVH DW WKH RQVHW 7KH 3RFKKDPPHU&KUHH VROXWLRQ IRU PRGH 0 O LV XVHG WR FDOFXODWH WKH ZDYH VSHHG RI HDFK )RXULHU FRPSRQHQW LQ WKH EDU PDWHULDO DV ZHOO DV LQ WKH VSHFLPHQ PDWHULDO (DFK SXOVH LV WKHQ VWDUWHG DW D GLVWDQFH RI PHWHUV IURP WKH VSHFLPHQ DQG LV IROORZHG WR WKH LQFLGHQW EDUVSHFLPHQ LQWHUIDFH WKURXJK WKH PXOWLSOH UHIOHFWLRQV DQG WUDQVPLVVLRQV WKDW RFFXU DW WKDW SRLQW DQG LV IROn ORZHG LQWR WKH VSHFLPHQ WR WKH VSHFLPHQWUDQVPLWWHU EDU LQWHUIDFH ZKHUH PRUH UHIOHFWLRQV DQG WUDQVPLVVLRQV WDNH SODFH 6LQFH WKH VSHHG RI HDFK PAGE 97 FRPSRQHQW RI WKH SXOVH LV NQRZQ DQG VLQFH WKH FKDUDFWHULVWLF LPSHGDQFHV DUH NQRZQ WKH SURJUDP FDOFXODWHV WKH DPRXQW WKDW JHWV UHIOHFWHG DQG WUDQVn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n VVHO IXQFWLRQV DUH HYDOXDWHG QXPHULFDOO\ )RU HDFK UDGLDO YDOXH WKH WHUPV PDNLQJ XS WKH WUDYHOLQJ ZDYH DUH DGGHG IRU HDFK WLPH LQFUHPHQW 7KH SRUn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n PLWWHG SXOVHV DUH VKRZQ PAGE 98 5()/(&7(' 675(66 5()/(&7(' 675(66(6 $7 5$',$/ /2&$7,21 LQFK 6+5 0&546(&21'6 7,0( PLF5L%DFDQGF 5()/(&7(' 675(66 $7 5$',$/ /2&$7,216 )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV 5DGLDO 9DULDWLRQV RI 5HIOHFWHG 6WUHVVHV LQFK 6+3% 7RSf LQFK 6+3% %RWWRPf PAGE 99 75$160,77(' 675(66 75$160,7 675(66(6 $7 5$',$/ /2&$7,21 ,QFK 6+3% 62 0&5*6((41'6 7,0( PLFURVHFRQGV 7,0( PLFURVHFRQGV 75$160,77(' 675(66 $7 5$',$/ /2&6 RR V )LJXUH 0RGLILHG 3RFKKDPPHU&KUHH 5HVXOWV 5DGLDO 9DULDWLRQV RI 7UDQVPLWWHG 6WUHVVHV LQFK 6+3% 7RSf LQFK 6+3% %RWWRPf PAGE 100 QRQ RQ QQQQQQQQQQQQQQQQQQQQ 7+,6 352*5$0 86(6 32&++$00(5 $1' &+5(( )5(48(1&< 62/87,21 2) 7+( 7:2 ',0(16,21$/ (48$7,216 2) 027,21 ,1 & PAGE 101 R R R QRRQ R R QR R QR QRQR &$/&8/$7( 7+( $03/,78'(6 2) ($&+ 38/6( $6 7+(< &5266 7+( ,17(5)$&(6 %$&. $1' )257+ $7 67$7,21 21 7+( ,1&,'(17 %$5 GMOf DrOfrrQHQrrfrVLQQrSLrVf GMfUHIrGMOf 7+,6 1(;7 3,(&( ,6 7+( ),567 5()/(&7,21 %$&. :+,&+ :,// $'' 72 7+( 5()/(&7(' 675(66 GMf WUrUHIrWUrGM f G Mf GMfrUHIrr $7 67$7,21 21 7+( 75$160,77(5 %$5 GMf WUrWUrGMf 3/86 :+$7 %281&(6 %$&. $1' )257+ ,1 7+( 63(&,0(1 $1' $''6 72 7+( 75$160,77(' 675(66 GMf WUrUHIrUHIrWUrGMf GMf GMfrUHIrr FRQWLQXH 68%5287,1( $1*/( &$/&8/$7(6 3+$6( &+$1*(6 9(/2&,7,(6 $1' :$9(/(1*7+6 )25 ($&+ 7(50 ,1 7+( )285,(5 6(5,(6 FDO DQJOHGLDFROFRZRPUDSKLSKFGOf 68%5287,1( &5266 &$/&8/$7(6 7+( &5266 6(&7,21$/ 9$5,$7,21 2) 7+( ',63/$&(0(176 $1' 675(66(6 21/< /21*,78',1$/ 675(66(6 $5( %528*+7 %$&. ,172 7+( 0$,1 352*5$0 6,*0$ 55 $1' 6,*0$ == $5( 287387(' 72 ),/( )25'$7 FDOO FURVVGLDFGOPPVLJf &$/&8/$7( 7+( 7,0(6 ($&+ 38/6( 7$.(6 72 5($&+ 7+( 675$,1 *$*(6 QfÂ§ GR M OPP QQ WLPHQOf ]OrFROfSKOQff WLPHQf WLPHQOfr]OrFROfSKOQff WLPHQf WLPHQf]OrFRfSKQffr WLPHQf WLPHQOf]rrFROfSKOQff ]OrFRfSKQff WLPHQf WLPHQfr]OrFRfSKQff WLPHQf r]OrSKQfrOHf FRQWLQXH PAGE 102 & N 5(35(6(176 7+( 5$',$/ 9$5,$7,21 2) 7+( 675(66(6 & N O ,6 $7 7+( &(17(5 2) 7+( %$5 & N ,6 $7 7+( 685)$&( 2) 7+( %$5 GR N & QQ ,6 7+( 180%(5 2) 32,176 0$.,1* 83 7+( 38/6( GR L OQQ F WKLV LV WLPH fÂ§ GHOWD W LV KDOI D PLFURVHFRQG W LrrOH ILf IULf rUHI XLf rWUrWU IUULf rWUrUHIrWU XWLf rWUrUHIrUHIrWU IUUULf IUULfrUHIrr XWWLf XWLf rUHIrr Q O GR M OPP QQ SKL WWLPHQOf ILf ILfVLJMNfrGMOfrFRVQrZRrSKLf SKLU WWLPHQf SKL U }WWLPH Q f IULf IULfVLJMNfrGMfrFRVQrYURrSKLUf IUU L f IUU L fVLJ M NfrG M frFRVQOUZRrSKLUf IUUULf IUUULfVLJMNfrGMfrFRVQrZRrSKLUf SKLW WWLPHQf SKLW WWLPHQf XLf XLfVLJMNfrGMfrFRVQrZRrSKLWf XWLLLLf XWLLLLfVLJMNfrG MfrFRVQrZRrSKLWf XWW LM M M f rXWW LM M M fVLJ M Nf rG M f rFRV QrZRrSKL Wf FRQWLQXH PAGE 103 L LL MM WLPHOf MMM MMr GR M MMQQ LL LIMJHMMMfLL LLO LIM,WMMMf[ IUMf IUMfIUULf[rIUUULLf X M f rX M fXW L f[rXWW LLf FRQWLQXH & 287387 7+( 7+5(( 38/6(6 72 ),/(6 $1' GR M OQQ ZULWHNfIMfIUMfXMf FRQWLQXH FRQWLQXH IRUPDWIOf VWRS HQG PAGE 104 VXEURXWLQH DQJOH GFROFRZRPPSKOSKFOGOf GLPHQVLRQ FfSKLfSKfFOfGOfGOOf FSYf rYrrrYrYrYrYrY r rYrrOff GR QFRXQW O FR FRO LIQFRXQWHTfFR FR SL F ] P PPr ZGrZRFR Y L LL [ LrZ YY [S rSLrFSYfrY LI[S[f FLf FSYf GOOLf Y LILPf GR L OP LIQFRXQWHTfJR WR GOLfGOOLf FO LffÂ§FLf SKOLfOFLfrFRf SKLf OFLfrFRf FRQWLQXH FRQWLQXH UHWXUQ HQG PAGE 105 VXEURXWLQH FURVVGLDFGOPPVLJQRf GLPHQVLRQ FfGOfVLJfVLJQRf GLPHQVLRQ VLJUfVLJQUfVLJU]f GLPHQVLRQ VLQU]f UHDOr M2UMOUMODM2NUMONUMONDKDDN SL U GR LL U UGLD EHWD DmGLD DPX QfÂ§ GR M OPP QQ [FQfrrr JDP rSLrGOQfGLD LI JDPOH22fJR WR KDSLrGOQfrVTUWDEVEHWDr[ff DN SLrGOQfrVTUWDEVr[ff KK EHWDr[O DNN r[O LI KKOW22fJR WR MOU EHVVMOKDrUDf M2U W}HVVMRKDrUDf MODEHVVMOKDf UDWLROM2UMOD JR WR MOU EHVVLOKDrUDf M2UEHVVLRKDrUDf MODEHVVLOKDf UDWLROM2UMOD LI DNNOW22fJR WR MONU EHVVMODNrUDf M2NU EHVVMRDNrUDf MONDEHVVMODNf UDWLRM 2N UMOND JR WR MONU EHVVLODNrUDf M2NU W!HVVLRDNrUDf MONDEHVVLODNf UDWLR MNUMOND VLJMLLf rDPXrJDPrrrMODKDffr [EHWDr[frKDfrUDWLROEHWDr[fr DNfrUDWLR[ff VLJQRMLLf VLJMLLfVLJMOf PAGE 106 R R VLJU MLLf rDPXrJDLQrrrMODKDf fr [frKDfrUDWLROEHWDr[fr DNfrUDWLR[2fEHWDr[O2frDUfrMOUMOD EHWDr[O2frDUfrMONUMONDf2[ff VLJQUMLLf VLJUMLLfVLJMf VLJU]MLLf rDPXrJDPrKDDfrMODrMOUMODfMONUMONDff VLQU]M LL+VLJU]MLLfVLJMf FRQWLQXH FRQWLQXH GR L OPP ZULWHfVLJQRLLLfLL Of FRQWLQXH GR L OPP ZULWHfVLQU]LLLfLL Of ZULWHfVLJQULLLfLL Of FRQWLQXH IRUPDWHf UHWXUQ HQG PAGE 107 R 7+,6 &$/&8/$7(6 7+( %(66(/ )81&7,21 2) 25'(5 IXQFWLRQ EHVVMR[f UHDOr \SLSSSSTOTTTTUOUUUUU fII VO V V V V GDWD SLSSSSOnG2GG r GGTOTTTTGO r GGGG GDWD UOUUUUUG2G2 r GGGG r VLVVVVVGG2 r GGGG2 LIDEV[f,WfWKHQ \ [rr EHVVMR UO\rU\rU\rU\rU\rUfffff r VO\rV\rV\rV\rV\rVfffff HOVH D[ DEV[f ] D[ \}]rr [[ D[ EHVVMR VTUWD[frFRV[[frSO\rS\rS\rS\ r rSfff f]rVLQ[[f0TO\rT\rT\rT\rTfffff HQGLI UHWXUQ HQG 7+,6 &$/&8/$7(6 7+( %(66(/ )81&7,21 2) 25'(5 r IXQFWLRQ EHVVMO[f UHDOr \SOSSSSTOTTTTUOUUUUU VOVVVVV GDWD UOUUUUUGG2 OG2GGG VLVVVVVGG2 GGGOG GDWD SOSSSSOGGGG GTOTTTTGG GGG LIDEV[f,WfWKHQ HOVH Y}[rr EHVVMO [rUO\rU\rU\rU\rU\rUfffff VO\rV\rV PAGE 108 R Q R R QRQ 7+,6 &$/&8/$7(6 7+( 02',),(' %(66(/ )81&7,21 2) 25'(5 IXQFWLRQ IFLHVVLR[f UHDOr \SOSSSSSS r TOTTTTTTTT GDWD SLSSSSSSO2G2GG r GGGOG GDWD TOTTTTTTTTG2GO r GGGGO r GOGOG LIDEV[f,WfWKHQ \ [frr EHVVLR SO\rS\rS\rS PAGE 109 6$03/( 287387 )520 02',),(' 32&++$00(5&+5(( 352*5$0 ,1&+ 6+3% 0,&526(&21'6 38/6( ,/(6 )252'$7 $1' )25'$7 675(66 9$/8(6 $/21* 7+( &5266 6(&7,21 /21*,78',1$/ 675(66 5 &(17(5f )285,(5 7(506 ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( (((( ( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( O2((((( ((((( ((((( ((((( 5 $ 685)$&(f ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( ((((( (((( ( ((((( PAGE 110 (((((((( (( ((((((&(( (( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( f(( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( f(( (((((((( (( ((((+(((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( Â‘(( (((((((( f(( (((((((( Â‘(( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( (( (((((((( (( 5$',$/ 675(66 (((((((((( (((((((((( (OO(OO(((((((( (((((((((( (((((((((( PAGE 111 (( (( (( O2(22( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( (( ((22 6+($5 675(66 (( (( (( (( 8(( (( (( (( (( (( (( (( (((((((( (((((((( (((((((( (((((((( ((((((( ( (((((((( (((((((( (((((((( (((((((( ((((((( ( (((((((( ((((((( ( (((((((( ((((((( ( (((((((( ((((((( ( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((( (((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((( (( ((( ((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( ( ((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((( (((((((&( (((((((22(22 (((((((&( (((((((22(22 PAGE 112 (((((((((22(22 (( (((((((22(22 (((( (((((22(22 ( ( ( ( ( ( ( ( ( 22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 ((((&(((6((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 PAGE 113 (((O2((((((22(22 (((((((((( (((((((((( (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 (((((((((22(22 PAGE 114 6$03/( 287387 )520 32&++$00(5&+5(( 352*5$0 ),/( )25'$7 ILUVW SDJH RQO\f LQFLGHQW UHIOHFWHG WUDQVPLWWHG VWUHVVHV PAGE 115 $33(1',; % $03/,78'(6 $1' 5$',86 29(5 :$9(/(1*7+ 9$/8(6 PAGE 116 5DWLR RI )RXULHU &RHIILFLHQWV IRU *LYHQ D$ RI %DU 'LDPHWHUV $1' ,1&+ %$56 'XUDWLRQ fÂ§! XVHH 86HF QVHF %DU 'LDPHWHU fÂ§! LQFK LQFK LQFK 'Q$R D Q PAGE 117 $33(1',; & 0,6&(//$1(286 +8// 3/276 PAGE 118 " L LRDFmRR f RmrDR } & } 6m, L f Â‘M 67 $ -2 m ,:-2O 0$,! } 0r46.r ? n M 1Y 6387 +23.,1621 %$5 Â‘ &0 =21,1* 3URZHP 6327 +23.,1621 %$5 &0 =21,1* 3URW;mIQ } } m} , 2LO Â‘ A f} \ r @L 6Wr .7 r!Am 3 I r 2+ 26r 21 L 06&&, 6387 +23.,1621 %$5 &0 =21,1* 3URWrrIQ @ I \ 68 } 72 : 8! f n0r2' mfLm2 n\ZYn7 fÂ§ 67 .7M 6387 +23.,1621 %$3 O &0 =21,1* 3URWQRQU V D } f } } RF } M rfÂ§ ?$?$$Y ?$ n 6387 +23.,1621 %$5 )LJXUH +8// 2XWSXW LQFK 6+3% 5DGLDO DQG /RQJLWXGLQDO 9HORFLWLHV DW ,QFLGHQW %DU 6SHFLPHQ ,QWHUIDFH %DU 6XUIDFH 7RS 0LGGOH RI WKH %DU %RWWRP %DU &HQWHU 5LJKWf PAGE 119 675(66 .%f 7 06(&f )LJXUH +8// 2XWSXW LQFK 6+3% $OO 7UDQVPLWWHG 6WUHVVHV DW %DU 6XUIDFH PAGE 120 675(66 .f )LJXUH +8// 2XWSXW LQFK 6+3% $OO 7UDQVPLWWHG 6WUHVVHV DW %DU &HQWHU &RPSDUH ZLWK )LJXUH PAGE 121 675(66 .%f )LJXUH +8// 2XWSXW LQFK 6+3% LQFK $OXPLQXP 6SHFLPHQ ,QFLGHQW DQG 5HIOHFWHG 6WUHVVHV PAGE 122 ( ( f fÂ§ (fÂ§ fÂ§ (fÂ§ ( Â‘ ( ( Â‘ ( Â‘ ( Â‘ fÂ§ ( U 7 06(&f )LJXUH +8// 2XWSXW LQFK 6+3% LQFK $OXPLQXP 6SHFLPHQ $OO 6WUHVVHV :LWKLQ WKH 6SHFLPHQ PAGE 123 675(66 '<1(&0 )LJXUH +8// 2XWSXW LQFK 6+3% LQFK $OXPLQXP 6SHFLPHQ $OO 6WUHVVHV LQ WKH 7UDQVPLWWHU %DU PAGE 124 $33(1',; +8// 6$03/( ,1387 ),/(6 PAGE 125 .((/ /$*5$1*( 352% ,10(0/ ;/%2: +($'(5 63/,7 +23.,1621 %$5 &0 =21,1* 10 )( $/ 1/5(* 5(*,21 1;, 1<17 ),;; 1; 1< 17 )5(( 1; 1< 17 0$67(5 1; 1< 17 )5(( 1; 1< 17 35(6 1; 1< 0$7 1; 1< 1648$' 1; 1< ;O ; ; ; PAGE 126 +8// /$*5$1*( 352% &<&/( 135(6 3 ( 7 3 ( 7 ( 3 7 ( 376723 ( PAGE 127 $33(1',; ( )2575$1 /,67,1* )25 ,1(57,$ &255(&7,21 PAGE 128 RQ R RQ RQRR r/,1(6 :,7+ rr $5( 7+26( $''(' 72 &255(&7 )25 ,1(57,$r 7+( 3+$6( &255(&7,216 +$9( %((1 '21( 7+( 86(5 +$6 12: 7+( 237,21 72 6723 7+( 352*5$0 25 72 &217,18( 72 &$/&8/$7( 675(66(6 675$,16 (7& ZULWHrrfn7+( &255(&7,216 21nNV n 6(76f 2) '$7$ +$9(n ZULWH rrfn%((1 &$/&8/$7(' '2 <28 :,6+ 72 352&('( :,7+n ZULWHrrfn7+( 675(66 $1' 675$,1 &$/&8/$7,216" <1n UHDGrf3 LI23(4n1nRURSHTnQnf*2 72 LIRSQHn PAGE 129 R Q Q RQ LGVWV LGnVVRGDWn LGVWQ LGnVQRGDWn LGHQJ LGnHQRGDWn XVLQJ FRUUHFWHG VHW RI GDWD HOVH LGLI LGnLIGDWn LGUI LGnUIGDWn LGWI LGnWIGDWn RXSXW ILOHV LGVWV LGnVVIGDWn LGVWQ LGnVQIGDWn LGHQJ LGnHQIGDWn HQGLI F F LGLVWDWXV nROGnf LGLIVWDWXV nROGnf GGUVWDWXV nROGnf LGUIVWDWXV nROGnf LGWVWDWXV nROGnf LGWIVWDWXV nROGnf LIOHT2fRSHQOILOH LIOHTOfRSHQOILOH GR L OQL UHDGfELQFLf FRQWLQXH FORVHGf LIOHTfRSHQOILOH LIOHTOfRSHQOILOH GR L OQL UHDGOfEUHILf FRQWLQXH FORVHGf LIOHTfRSHQOILOHr LI OHTOfRSHQGILOH GR L OQL UHDGOfEWUDLf FRQWLQXH FORVHOf IODJ O FRU URrVOHQrrfrGLDrrffrH ZULWH r rfn '2 <28 :$17 72 &255(&7 )25 ,1(57,$" <1n UHDGrfRS LIRSHTn\nRURSHTn PAGE 130 R R R R R ZULWHfVWUVLQVWUVRWVWUHVV FRQWLQXH FORVHOOf ZULWHrrfLGVWV LIIODJHTf JR WR rr ZULWHrrfn'2 <28 :$17 72 &$/&8/$7( 675$,16 $1' 9(/2&,7,(6" <1n UHDGrfRS LIRSHTn1nRURSHTnQnf JR WR LIRSQHn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n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Â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n OHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW &O :LQIUHG 0 3KLOOLSV 'HDQ &ROOHJH RI (QJLQHHULQJ 0DGHO\Q 0 /RFNKDUW 'HDQ *UDGXDWH 6FKRRO PAGE 134 81,9(56,7< 2) )/25,'$ xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EPEBLJTX8_KSNZBC INGEST_TIME 2011-07-29T21:18:57Z PACKAGE AA00003301_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |