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DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS By MICHAEL A. KOPLOW A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Michael A. Koplow ACKNOWLEDGMENTS I would like to express my sincere gratitude for everyone who has helped make this thesis possible. My greatest appreciation to all of my committee members for their insight and comments. Special thanks go to Dr. Mann and Dr. Sankar for their advice and confidence. I would also like to thank Raul Zapata, Abhijit Bhattacharyya, Ryan Carter, and the MTRC for their time, effort, and energy during this work. For my dad, thank you for your love and guidance. You have made this all possible. I hope my thoughts and inspirations come as free flowing for me as they did for him; for he is the spirit that guides me. Finally, I would like to thank my girlfriend, Briana, my sister, Sarah, my brother, David, and my mother for all their support during this project. Through their continued love and support, this project was a success. It is common sense to take a method and try it; if it fails, admit it frankly and try another. But above all, try something. Franklin D. Roosevelt TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ...... iii LIST OF TABLES ...................... ......... vi LIST OF FIGURES ................... ......... vii ABSTRACT ...................... ............. ix CHAPTER 1 INTRODUCTION .................... ....... 1 1.1 Introduction to the Problem .......... ....... .... 1 1.2 Machining and the Material Removal Process ............ 2 1.3 Application to Industry ................... ... 2 1.4 Literature Survey ........................... 3 2 EXPERIMENTAL MODAL TESTING ................... 7 2.1 Dynamic Response of Linear Systems ................ 7 2.2 Impact Testing Overview ........... ............ 8 2.3 Contact Sensor Mass Loading Effects ... . .... 13 3 DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS 15 3.1 Derivation of the Equation of Motion ............ 15 3.2 Dynamic Response Prediction of Uniform Beams . .... 17 3.3 Experimental Response of Uniform Beams . . ..... 20 4 DYNAMIC RESPONSE PREDICTION OF DISCONTINUOUS BEAMS 23 4.1 Receptance Derivation for Discontinuous Beams with Aligned Neu tral Axes .... .......... ............ 23 4.1.1 Discontinuous stepped beam solution for force excitation at location C . . . . .... .. 23 4.1.2 Discontinuous stepped beam solution for force excitation at location A ................ .... .. 28 4.1.3 Extension of the analytical solution for applied couples .29 4.1.4 Comparison of the analytical solution to receptance coupling 31 4.1.5 Experimental verification of the stepped beam solution .38 4.2 Receptance Derivation for Discontinuous Beams with Misaligned Neutral Axes ..... . ......... 40 4.2.1 Discontinuous misaligned beam solution for force excita tion at location C ........... ... ..... 41 4.2.2 Experimental study of the misaligned neutral axis solution 45 5 STABILITY OF LAYER REMOVAL PROCESS . . 49 5.1 Limiting Chip Width for Machining Process . . ... 49 5.2 Mode Shape Analysis as a Function of the Notch Height . 52 6 CONCLUSIONS AND FUTURE WORK ................. .. 57 REFERENCES ............. .. ............ ... 60 BIOGRAPHICAL SKETCH .............. . .. 63 LIST OF TABLES Table page 31 EulerBernoulli beam notation. ................ ..... 16 32 Boundary conditions for classical beam ends. ............ ..18 33 C'!i teristic equations for the free vibration of uniform EulerBernoulli beam s. . . . . . .. . . 19 34 Beam primary receptances. .................. .... 19 41 Notation for force excitation at position A .............. ..30 42 Discontinuous notched beam continuity conditions. .......... ..42 43 Axial vibration boundary conditions for classical beam ends. . 43 44 Notation for FRF with force excitation at position C including a mis aligned neutral axis. .................. .... 45 LIST OF FIGURES Figure page 11 Alcoa testing procedures .............. ... ... .. .. 3 21 Signal processing overview ............... .. 9 22 Comparison of different modal hammers for: (a) a force measurement in the time domain and (b) a force amplitude measurement in the frequency domain. .................. .... 11 31 Schematic of a fixedfree forced beam. ................ .. 15 32 Free body diagram of a beam element. ............... 16 33 Schematic of a uniform beam subjected to a force of amplitude F and frequency u, applied at x L. ................. 20 34 Experimental setup for FRF testing on a uniform beam. . 21 35 Comparison of experimental (solid) and analytical (dashed) FRFs for the uniform beam. .................. .... 22 41 Schematic of the stepped beam with aligned neutral axis and free bound ary conditions at locations A and C. ............... 24 42 Schematic of a stepped beam subjected to: (a) a force of amplitude F and frequency w, applied at location C, (b) a force of amplitude F and frequency w, applied at location A,(c) a couple of amplitude M and frequency w, applied at location C, and (d) a couple of am plitude M and frequency u, applied at location A. . ... 24 43 Receptance coupling components (a) and assembly (b) models for ex citation at C. . . . . . .. . 32 44 Receptance coupling components (a) and assembly (b) models for ex citation at 3. . . . . . .. . 35 45 Beam dimensions for comparison of the stepped beam analytical so lution to receptance coupling and experiment. Dimensions are in (m m ). . . . . . .. . . 36 46 FRF comparison between analytical (solid) and receptance coupling (dashed) methods when forced at position C. . . 36 47 FRF comparison between analytical (solid) and receptance coupling (dashed) methods when forced at position A. . . 37 48 Experimental setup for FRF testing on a stepped beam ....... ..38 49 Comparison of experimental (solid) and analytical (dashed) FRF when forced at position C. .................... ...... 39 410 Comparison of experimental (solid) and analytical (dashed) FRF when forced at position A. .................... ..... 40 411 Schematic of a discontinuous notch beam with a misaligned neutral axis and free boundary conditions at locations A and C. ..... ..41 412 Free body diagram of (a) forces and (b) displacements for a discon tinuous notched beam with a misaligned neutral axis. . ... 41 413 Dimensions for analytical study of beam with jump discontinuity. Di mensions are in (mm). ............... .... .... 46 414 Comparison of the analytical FRF with a misaligned neutral axis (solid) to the analytical FRF with an aligned neutral axis (dashed) when forced at position C. .................... ...... 47 415 Dimensions for experimental study of beam with a jump discontinu ity forced at the end position. Dimensions are in (mm). . 47 416 Comparison of experimental (solid) and analytical (dashed) FRF for 3 sectioned notch beam with forcing at the end location. . 48 51 Schematic of the clampedfree notched beam during machining. Di mensions are given in (mm). .............. ...... 51 52 Analytical FRF for the notched beam with fixedfree boundary con ditions. ............... ............. .. 52 53 Experimental mode shapes as a function of the notch height for Al coa testing conditions. ................ .. .... 53 54 Analytical mode shapes as a function of the notch height assuming fixedfree boundary conditions. ................ .... 54 55 Analytical mode shapes as a function of the notch height assuming compliantfree boundary conditions. ............. .. .. 55 56 Comparison of limiting chip thickness, bum, as a function of the notch depth for: (a) experiment, (b) a fixedfree model, and (c) a compliant free model. ............... ......... .. .. 56 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS By Michael A. Koplow August 2005 C'!I wi': Brian P. Mann Major Department: Mechanical and Aerospace Engineering The dynamic response of discontinuous structures is often of vital importance in the design of many engineering applications. In many cases, it is preferable to have an analytical model of the system which can reduce the amount of design, ' in. and manufacturing of products. This work grew out of the need to examine the dynamic response of a discontinuous beam used in an industrial application. As milling operations were being performed on the beam, the natural frequencies of the beam would shift, leading to unstable vibrations of the cutting process. The goal of this research was to ai 1v.. the dynamic response and characterize the stability of the discontinuous beam. The present work considers beams with two types of discontinuities. The first is that of a stepped beam with an aligned neutral axis. The second is that of the notched beam which contains a jump discontinuity and a misalignment of the indi vidual beam segments neutral axes. The discontinuous beam is modeled as separate uniform EulerBernoulli beams with continuity conditions at the discontinuity. The analytical results are compared to receptance coupling substructure analysis and experiment. Results show that the stepped beam model produces very accurate results compared to other analytical techniques and experiment. Results for the notched beam show errors due to neglecting shear and rotary inertia components of the beam segments. A stability analysis is performed considering the workpiece to be the most flexible portion of the cutting operation. Additionally, a study of the notch height is performed to analyze the change in dynamic response as a function of the material removal process. CHAPTER 1 INTRODUCTION 1.1 Introduction to the Problem Structural dynamics is widely used in research and in industry to make accurate predictions of the response of many different structures. While the modeling and dynamic response predictions for continuous structures has been well developed, there are relatively few techniques available for modeling discontinuous structures. Difficulties often arise in the modeling of structures with complex geometry; i.e. structures containing joints, connections, or notches. In many cases, these structures are either modeled with finite element packages or tested using experimental work pieces. Design using these methods are often time consuming and costly and thus it is often beneficial to have analytical solutions for structural responses. Beams provide a fundamental model for the structural elements of many engineering applications. For instance, helicopter rotor blades, spacecraft antennae, and robot arms are all examples of structures that may be modeled with beam like elements [1; 2]. The work presented in this thesis grew out of the need to examine an industrial machining process where the dynamic response of a beam like structure was the primary limiting factor. This material removal process additionally presented two unique challenges: (1) a change in the beam's dynamic response and machining stability limit as each 1,r of material was incrementally removed; and (2) a discontinuity in the beam structure which prevented direct application of conventional beam theory. The goal of this work is to present analytical solutions for the dynamic of a discontinuous beam that were developed to better understand the aforementioned industrial process. 1.2 Machining and the Material Removal Process Machining is the most important manufacturing process in terms of time and money spent. Machining involves the process of removing material from a workpiece in the form of chips. Researchers have expended many efforts to identify the limits of stability and safe cutting conditions, depths of cut and spindle speeds, for milling operations. The goal is to prevent chatter, or undesired large vibrations. C'! ,I I i r, related to the dynamics of the structure during i, 1 i ini. will adversely affect the quality of the produced surface, and may lead to increased tool wear and tool failure. The mechanism for chatter is commonly identified as the regeneration effect [35]. In most models, chatter occurs due to the interactions between the tool and the wavy surface left on the workpiece from previous revolutions. Stability analyses of machining in literature show frequency diagrams labeling stable and unstable depths of cut as a function of various spindle speeds. The stability limits are obtained assuming that chatter occurs due to dynamics of the spindle holder and tool resonance frequencies. The tool is usually considered the most flexible part of the dynamic system. However, during milling operations on beams, the natural frequencies of the workpiece shift, causing the beam to become the most flexible part of the system. This shift may also result in chatter. This type of chatter can occur even after several successful passes of stable material removal have been performed. 1.3 Application to Industry Alcoa, a in i ii aerospace aluminum manufacturer, has implemented a 1i,,r removal process to experimentally extract the residual stress of various aluminum alloys as shown in Fig 11. The test procedure requires the removal of a 1,r of material, using a machining process called milling, and the measurement of the workpiece static displacement. Static displacement measurements are used to estimate the remaining residual stress in the material. The milling process and static measurement cycle is repeated several times to determine the residual stress at each lvr of the workpiece material. Machine Spindle Workpiece Free end Fixed end Figure 11: Alcoa testing procedures. The concern is that the large amplitude vibrations can occur during the machining process; these vibrations are likely to cause (1) a shift in the transducer zero location or create an offset from the transducer measurement; (2) the nominal depth of cut will be different than the actual depth of cut due to relative movement between the tool and the workpiece; and (3) transverse tool vibrations will either remove more or less material than anticipated (i.e. a larger/smaller slot than the one used for residual stress calculations). The uncertainty created from these factors will inevitably diminish the ability of Alcoa to correlate the initial residual stress distributions to changes in material processing. Therefore, the primary concern of this work is to reduce the large amplitude vibrations by analyzing the machining stability limits of the workpiece at various stages of the material removal process. 1.4 Literature Survey A literature survey has shown that the dynamic response for the transverse vibration of continuous EulerBernoulli beams has been well studied using both modal superposition techniques [6] and receptance techniques [7]. Modal super position requires first solving the eigenvalue problem for the free vibration of the structure without damping. As the name ii. i the forced vibration solution is obtained by assuming orthogonality of the modes and then summing up the indi vidual responses of each mode. Receptance techniques solve for the forced vibration solution by assuming a solution for the mode shape functions and by applying forces directly into the boundary conditions. Receptance techniques do not require the two step process of modal superposition, but are limited somewhat by forcing the applied forces into boundary conditions. The static behavior of EulerBernoulli beams with jump discontinuities has been studied using generalized solutions [810]. In these methods, the disconti nuities are modeled as delta functions at the point of discontinuity. The authors investigate the means by which the discontinuities can be applied to the governing differential beam equations. While these authors do a superb job at modeling the discontinuities in the static sense, they do not apply their formulations for dynamic loading. Several authors have studied the free vibration of stepped beams with aligned neutral axes. The discontinuous structures have previously been treated to find natural frequencies and mode shapes expressed as determinants equated to zero [1114]. These works analyze the boundary conditions and continuity con ditions to solve for the system frequency equations. Jang and Bert [11; 15] obtained the first exact results of the frequency equation for stepped beams with classical boundary conditions. Maurizi and Belles [12] extended the work of Jang and Bert to include the effects of elastically restrained boundary conditions. De Rosa and coworkers [13; 16; 17] analyzed the free vibration of stepped beams with elas tic supports including an intermediate support, and the effects of concentrated masses. N i.,1. waran [14; 18; 19] considered the effects of multiple beam spans, a nonsymmetrical rigid body at the discontinuity, and applied static axial forces. Tsukazan [20] studied the use of a dynamical bases for computing the beam modes. While all of these previous works have treated the free vibration case, very little work has been presented on the forced vibration case. Alternative coupling techniques, such as receptance coupling substructure synthesis [2123], can also be used to examine the dynamic behavior of discontin uous beams. Substructuring methods allow the prediction of assembly frequency response functions (FRFs) using FRFs from individual components obtained ei ther analytically or experimentally. The solution forms a two by two matrix of the primary receptances of the individual beam components for each frequency. The technique requires an inversion of the two by two matrices per frequency. For highresolution FRFs, the solution becomes computationally expensive. In this research, an analytical solution for the dynamic response of discontinu ous beams is considered. Two different discontinuities are considered: (1) a stepped beam with both aligned neutral axes and (2) and notched beam consistent with the material removal process detailed earlier. The analytical results are verified by receptance coupling methods and via experiment. One limitation of this work is that a partial differential equation is needed to obtain an assumed mode shape solution. Also, it is required to have information about the continuity conditions between individual components. The presented work can easily be extended to beams with nbeam sections and different classical boundary conditions. The organization of the thesis is as follows. C'!i lpter 2 details a background of information concerning experimental modal testing. The chapter outlines the experimental procedure, data analysis techniques to obtain frequency response measurements, and techniques to eliminate mass loading effects of experimental data due to contact sensors. ('!i lpter 3 gives a derivation of the uniform Euler Bernoulli beam as well as the procedure to obtain frequency response functions for 6 classical boundary conditions. The .111 Iiltical results are verified by experiment. ('!i lpter 4 extends the analysis of the uniform beam to include discontinuities. The cases of stepped and notched beams are considered. Results are verified using receptance coupling techniques and experiment. C'! Ilpter 5 analyzes the stability of the milling process for discontinuous beams. Additionally, the chapter examines the dynamic behavior of the beams as a function of the notch height using both experimental and analytical data. Finally, ('! Ilpter 6 summarizes the conclusions and provides recommendations for future work. CHAPTER 2 EXPERIMENTAL MODAL TESTING This chapter details the basic operations for acquiring experimental frequency response measurements. The goal of this chapter is to provide an overview of experimental modal analysis techniques. Specifically, this chapter provides infor mation about frequency response measurements, methods for obtaining time series measurements, and the data analysis techniques required to convert time domain measurements to frequency domain measurements. The discussion is followed by a brief description of mass loading effects due to contact sensors. 2.1 Dynamic Response of Linear Systems The impulse response function, h(r), can fully describe the dynamic response of a linear system. The impulse response is the output of the system due to a corresponding unit impulse applied at any time r. The output y(t), for any input x(t), is given by the convolution integral [24] y(t) h(r)x(t 7)dr. (2.1) The convolution integral of the input and impulse response is usually very difficult to solve in the time domain. Converting the time domain signal into the frequency domain allows for easy computation of Eq. (2.1). As will be shown, the convolution integral in the time domain becomes simple algebra in the frequency domain. The Fourier transform is used to convert the time domain impulse response function h(r) into the frequency domain frequency response function (FRF). For a physically realizable system, the frequency response function, given by Bendat and Piersol [24], is H(f) j= h(r)ej2fLdr, (2.2) where j is the imaginary term. By definition, the frequency response function is defined as the Fourier transform of the impulse response function. The Fourier transform is typically applied in many computational packages using the Fast Fourier Transform (FFT) which restricts the limits of the integral to a finite time interval. Taking the Fourier transform of Eq. (2.1) yields Y(f) H(f)X(f), (2.3) where the capital letters denote Fourier transforms and f denotes frequency dependence. The common notation is to use lower case letters to represent time domain signals, h(r), while capital letters are used to represent frequency domain signals, H(f). Equation (2.3) shows the relationship between the frequency response function H(f) and the input and output. In practice, the term FRF is often used interchangeably with the term transfer function. However, this is a misnomer as there is a subtle difference. Transfer functions, as typically applied in control theory, are Laplace transforms of the impulse response function. The difference is found in the integration of Eq. (2.2). Rather than integrating only the imaginary variable jf, Laplace transforms integrate a complex variable s = a + jf. Therefore Laplace transforms account for transient and steady state responses whereas the Fourier transform assumes an invariant signal. Laplace transforms can be thought of as more general because they plot the poles and zeros on the complex plane. Fourier transforms ignore the real portion and are only concerned with the jf axis of the complex plane. 2.2 Impact Testing Overview The following section describes a method for obtaining frequency response functions from a physical system. The discussion is limited to impulse inputs from impact hammers and output responses measured by accelerometers. In this method, both the input and output responses are measured. From Eq. (2.3), the frequency response function is simply the response output divided by the force input in the frequency domain. Extensions for other types of excitation and measurement can be found in literature [25; 26]. Figure 21 shows an illustration of the process to obtain the frequency re sponse function from time series data. The steps are broken down into a sequence for the input (hammer/force measurement) and the output (accelerometer mea surement). An overview of the modal testing process is listed below. A detailed explanation of each step follows. Time domain measurements are obtained using a data acquisition system and modal testing equipment. Windows are applied to clean the data and avoid leakage. The fast Fourier transform is used to convert the input and output data into the frequency domain. The FRF is obtained as the output over the input. The results are averaged over multiple impacts to ensure good coherence. TIME DOMAIN ANTIALIASING WINDOWING APPLY FFT AVERAGING REAL AND DATA FILTER IMAG FRF I^L / \p I I_ 'r I I Figure 21: Signal processing overview. The first step is to acquire time series measurements using a data acquisition system, a modal hammer, and a transducer. For the purposes of this discussion, it is assumed to have only one input and one output, but multi input and output systems are possible. During the analog to digital conversion process, an anti aliasing filter is applied to remove any high frequency signals that may exist in the data. The Nyquist frequency states that the highest possible observable frequency is equal to half of the sampling frequency. Therefore, the sampling frequency must be greater than twice the maximum frequency of interest present in the signal. If the data are sampled at too low of a rate, the signal will be aliased and the correct frequency content will not be observable. For reconstructing the true signal, a general rule of thumb is to sample at 510 times the highest frequency of interest. If the signal has aliased, it is not possible to reconstruct the true signal and the signal is unusable. The impact hammer is a popular excitation system because it is easy to implement. The energy applied to the structure is directly related to the hammer mass. Hammers range in size from a few ounces to several pounds with varying contact tip materials. The frequency content depends on the hammer mass and the contact stiffness. The choice of modal hammer depends on the application and desired frequency range. Figure 22 shows a comparison of the effects of various modal hammers. Softer tips will excite a larger amplitude of motion, but will contain a smaller frequency bandwidth. Stiffer tips typically excite a larger frequency range, but will contain less amplitude. Additionally, larger hammers will provide more energy and will thus excite for longer time. Smaller hammers will show a more refined impulse that dissipates more rapidly. Using a modal hammer, the input should ideally show a single impulse. How ever, double hits (or multiple impacts) can occur and are sometimes unavoidable. The double hit problem can often be minimized by selecting the smallest possible hammer. The output should ideally show a damped response with transients that decay to zero. Depending on the sampling rate, boundary conditions, and number F soft tip/ large mass stiff tip time frequency (a) (b) Figure 22: Comparison of different modal hammers for: (a) a force measurement in the time domain and (b) a force amplitude measurement in the frequency domain. of samples taken, this may or may not be the case. To prevent leakage in the data, it is alvx preferable to allow the system to naturally damp out. Next, windows are applied to remove static and random noise from the signal. For the input signal, as stated above, the force should be a perfect impulse at one instant and equal to zero everywhere else. Impulses are finite in duration and the FFT of the impulse provides the input frequency spectrum. Therefore any value present, excluding the impulse, can be regarded as noise and will be eliminated. It is important to retain any double hits because they are integral to the system and should not be erased from the signal. To remove input noise, multiply the signal by a square wave filter (force window) that is equal to one during the impulse and zero everywhere else. Because only noise components are removed, the force window will not add any artificial effects. For the output signal, the signal should begin at zero (a number of precursor scans taken before the impact) and end at zero after the impact has damped out. The Fourier transform requires that a signal be periodic in order to obtain the FFT. While accelerometer tests are not periodic, forcing the signal to zero at the ends will result in an accurate transformation. If the signal has not naturally attenuated to zero, the Fourier transform will distort the frequency domain signal. The distortion occurs in the form of leakage which is a smearing of the frequency content over a wide range [25]. To prevent leakage, the sample should be allowed enough time for the signal to naturally attenuate to zero. In several instances it becomes impractical to allow for naturally system decay due to file size limitations or time constraints. In these cases, an exponential window can be applied to force the signal to zero. Although it is sometimes necessary to use an exponential window to prevent FFT distortion, the exponential window should be used with caution as it will add artificial damping into the system. Once the signals have been processed to reduce noise and leakage, the next step is to convert the time domain signal into the frequency domain using the FFT transform algorithm. The FFT provides the complex (real and imaginary) valued linear Fourier spectrum of the input and output signals. The ratio of the output over input spectrums at this point is called the accelerance function given by J.A The accelerance function has resulted because the output was measured using an accelerometer. Converting the accelerance function to a receptance function requires the following. Consider a sinusoidal input given by F(t)= Rsint, (2.4) where R is the amplitude of excitation and a is the frequency term in the units of rad/s. From linear system theory, a sinusoidal input will result in a sinusoidal output of the same frequency. Therefore, the output will have a response of the same form as Eq. (2.4). Differentiating the response twice yields W2F. Converting the system from accelerance to receptance requires dividing by C2. The measured FRF is the ratio of output over input divided by c2, Y(f) A(f)/F(f) (2.5) F(f) 2 The results are averaged over multiple impacts to improve accuracy and reduce random noise. The system response can then be plotted in terms of the real and imaginary responses or similarly magnitude and phase plots. The final step is to determine the accuracy of the experiments. The coherence function [24] is one measure that can be used to gage the effectiveness of the impact tests. The coherence function is a real valued quantity which provides a measure of the linear dependence of two impact tests as a function of frequency. For a linear system, the coherence function, 72, is given by Bendat and Piersol [24] as 2 IGy(f)12 (2.6) G7 x(f)Gyy(f) where Gxy is the cross power spectrum between the input and output signals, Gxx is the input power spectrum, and Giii is the output power spectrum. The value of the coherence function will be between 0 and 1, whereby a value of 0 corresponds no relationship between two signals and a value of 1 corresponds to a perfectly linear relationship. There are several causes of poor coherence functions. The coherence will drop as a result of nonlinearities in the system and due to poor signal to noise ratios for antiresonance frequencies. Furthermore, the coherence will be poor if the user does not impact the same location with the same force. It is important to note that the coherence must be calculated using data containing multiple averages. A test with only one impact will misleadingly show perfectly linear data because the coherence will measure the single test onto itself. 2.3 Contact Sensor Mass Loading Effects Discrepancies between measured and theoretical FRFs are partly due to mass loading effects due to the added inertia of the accelerometer. Several authors [25; 27; 28] have shown that measured and predicted FRFs may be compensated to include the additional dynamics of the sensor. The correction for a driving point FRF, or direct impact FRF, from Ashory [27], is given by Am At 1 (2.7) 1 1 MA7 where A11 represents a direct impact accelerance FRF, the super script m rep resents the measured accelerance, the super script t represents the theoretical accelerance function without the additional inertia, and M is the extra mass of the accelerometer in (kg). This form of correction is known as mass cancelation. As shown in Eq. (2.7), the mass loading effect is frequency dependent. Equation (2.7) is used to remove the effects of the additional inertia added by the attached sen sor. In this formulation, the experimental results are altered to resemble the true vibrations of the beam if the sensor were not attached. Due to limited bandwidth and noise in the measured signal, applying Eq. (2.7) will distort some modes in the experimental data. Although the signal will shift to the correct frequency response, the damping ratio is incorrectly shifted and will distort some modes in the FRF. To avoid these problems, it becomes easier to shift the theoretical response to the experimental response for comparison. In this formulation, the theoretical response is shifted to resemble the vibrations of the beam as if the additional dynamics were included. The theoretical result can be compensated to include the accelerometer mass by [27] At A 11 (2.8) An 1 + MA\, '4 where the terms are the same as defined above. Because theoretical models contain no noise and can be set to include a very fine resolution, the modes will shift to the correct locations without distortion. For transfer FRFs, those that are measured in a different location than sensed, the correction is given by A4 A77j 1 At (2.9) 12 1 + MAt' where A12 represents a transfer accelerance FRF measured at one location and forced at another. As discussed previously, the accelerance result can be trans formed into receptance by dividing by L2. CHAPTER 3 DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS This chapter develops the equations of motion for continuous, uniform beams and reviews the method to acquire receptance functions for the case of excitation at the boundary conditions. For this analysis, the system will be modeled using the Euler Bernoulli beam theory, neglecting shear and rotary interia. The beams are modeled using receptance techniques whereby external forces are applied directly to the boundary conditions. The discontinuous beam formulation will be shown in later chapters to be an expansion of the uniform beam. 3.1 Derivation of the Equation of Motion This section derives the equation of motion for an EulerBernoulli beam. Figure 31 shows a typical fixedfree beam with an applied transverse force. Figure 32 shows the free body diagram for a differential beam element with a constant cross sectional area, where the beam notation is defined in Table 31. Sf(x,t) dxt  Figure 31: Schematic of a fixedfree forced beam. The shear force and bending moments illustrated in Fig 32 show the positive sign convention for the beam element. Positive shear and bending moments are assumed to produce upward displacements and rotations. f(x,t) t (x+dx) M(x) Q(x+dx) Q \ dx Figure 32: Free body diagram of a beam element. Table 31: EulerBernoulli beam notation. f(x,t) = Applied transverse force as a function of space (x) and time (t) Q = Shear force acting on the cross section M = Internal bending moment p = Mass density (kg/m3) A = Cross sectional area (m2) E = Young's Modulus (Pa) I = Area moment of inertia about the neutral axis (m4) For the EulerBernoulli beam analysis, there are 2 underlying assumptions. The first assumption is that the beam is long and slender. The length of the beam is assumed to be much greater than the height, such that the shear force is dominated by the bending stresses. As a result, the shear stresses and rotatory inertia terms are considered negligible. Therefore Q = OM/8t. The second assumption is a small slope of deflection curvature of the beam. This is the assumption of small angles. For small angles, it can be shown that 0 = 9v/ax. In practice, the EulerBernoulli approximation is valid when the beam length is at a minimum of 5 to 10 times its height. In these cases, the EulerBernoulli beam shows very accurate results for the lowest modes. Errors will begin to acrue for higher modes. However, the a i1,J ,i i; presented in this work is most interested in the fundamental mode vibration because the small amplitude of the higher modes is less important. Summing the forces on the differential element and using Newton's second law, it follows that Fy = ma, => Q dx + f(x,t)dx = pAdx (3.1) O1 x at2 Application of the first assumption of slender beams to Eq. (3.1) results in 82M /1 82(x. (.) x + f(x, t) p A t2 (3.2) From bending theory, recall M = EI Applying the second assumption for small deflections, it follows that 02 02vU 02v(, t) 2 (EI82) + fa) (x,) pA N (3.3) Assuming only uniform beams, Eq. (3.3) may be rewritten as a2v(x,t) a4v(X,t) pA + El f( xt) (3.4) A t2 + X2 Equation (3.4) represents the equation of motion for transverse vibration of a uniform Euler Bernoulli beam. The result is a fourth order partial differential equation dependant on space and time. 3.2 Dynamic Response Prediction of Uniform Beams The solution to Eq. (3.4), when subjected to an input of frequency w (rad/s), can be separated into a solution in space and time v X(x) sin ut. (3.5) Substitution of Eq. (3.5) into Eq. (3.4) yields dependence upon the spatial quantity alone 4( 34X(x) = 0, (3.6) OX4 where 34 w2pA (3 EI(1 + i) ' is the solution to the eigenvalue problem and r] is a nondimensional structural damping factor. The general mode shape solution to X(x) is X(x) = asin 3x + bcos x + c sinh 3x + dcosh 3x, (3.8) where a, b, c, and d are constants determined by suitable boundary conditions. The free vibration solution is written as a 4 by 4 determinant obtained by applying 4 boundary conditions to Eq. (3.8). The boundary conditions for classical conditions are listed in Table 32. The frequency equation solution becomes a transcendental function where 3 is the unknown quantity. Values for 3 are determined by roots of the transcendental equation. Because the equation is transcendental in nature, the roots are not easily obtained. The values may also be found by the zero crossings in a plot of the equation as a function of 3. The natural frequencies are then determined solving for w in Eq.(3.7). The characteristic equations for the free vibration problem for fixedfree and freefree beams are given by Balachandran and Magrab [29] and are listed in Table 33 Table 32: Boundary conditions for classical beam ends. v = 0 for a fixed end, El D = E17 0 for a free end, v =EI s = 0 for a pinned end, and H1 El 0 for a sliding end. The forced vibration solution [7] is obtained by equating applied forces into the boundary conditions. Applied forces are equated to the shear force while Table 33: C'!i o .:teristic equations for the free vibration of uniform Euler Bernoulli beams. FixedFree Beam: cos 3L cosh 3L + 1 = 0 FreeFree Beam: cos f3L cosh fL 1 = 0 applied couples are equated to the bending moment. The signs on the forces are determined by the positive sign convention as shown in Fig 32. The FRF for a uniform beam is obtained by solving a set of four equations with four variables. Note that for any position of a uniform beam, it is possible to model the response as a 2 by 2 matrix of its primary receptances. These receptances form the transfer functions of the beam, as listed in Table 34. Table 34: Beam primary receptances. 1 Translation due to an applied force v/F 2 Translation due to an applied moment v/ \ 3 Bending due to an applied force 0/F 4 Bending due to an applied moment 0/ \! Consider the freefree uniform beam shown in Fig 33. The desired FRF is the direct receptance at the location x = L due to an applied force. The boundary conditions for this case are a2v(0) At x = 0 El = 0, (3.9a) ox2 El 0, (3.9b) ax3 At x = L EI =L 0, (3.9c) aX2 03 v(L) E = F sin t. (3.9d) odx Applying the boundary conditions given in Eq. (3.9) to Eq. (3.8) results in the following FRF solution [7] v sin fL cosh fL cos /L sinh f3L (3.10) F EI(1 + i)/33(cos 3L cosh fL 1) ' where the denominator, cos 3L cosh 3L = 1, forms the frequency equation whose roots determine the natural frequencies of the system. Fsin(o)t) t L x=O x=L Figure 33: Schematic of a uniform beam subjected to a force of amplitude F and frequency a/, applied at x=L. 3.3 Experimental Response of Uniform Beams This section provides experimental verification for the uniform beam recep tance functions. The experiment consists of the uniform beam of aluminum 7051 which is 393.7 [mm] long with a cross section that is 25.4 [mm] wide and 19 [mm] tall. Freefree boundary conditions were obtained by hanging the beam with a taut nylon 1 ii. rigidly attached to the end of the beam via a thin piece of plexiglass as shown in 34. The free boundary conditions were applied because they provide very accurate and repeatable results. Fixed boundary conditions are very difficult to experimentally obtain because there is .l1. i, some measure of compliance in the connection. Experiments were conducted by forcing the beam with a modal hammer and obtaining the response with a low mass accelerometer mounted onto the beam. Mass loading effects due to the contact sensor were corrected using Eq. (2.8). The accelerometer mass was measured to be m = 0.8 grams. By com parison, the total mass of the beam is 0.54 [kg]. The material has a density of p = 2830 [kg/m3] and a Young's modulus of E = 71 [GPa]. Structural damping was obtained as r = 0.0003 via a best fit approximation to the experimental data. FreeFree beam impact I I so hammer (U  Figure 34: Experimental setup for FRF testing on a uniform beam. Figure 35 shows the results for the experiment for the first 2 modes for direct FRFs at location x = L. The data show modes at 630 Hz and 1706 Hz for the experimental test. As the data show, the experimental results are in excellent agreement with the analytical predictions. Results show that analytical predictions have higher natural frequencies than the experimental measurements. Because damping is fit to the entire structure, it does not perfectly match for each mode. In this case, the length to height ratio was 21:1 and the results show that the EulerBernoulli beam approximation is capable of modeling the response of the first 2 modes. It is expected that higher modes would show greater errors. x 103 Total Response 1 5 5 1 500 1000 1500 Freq (Hz) 5 5 0 5 500 1000 Freq (Hz) 1500 x 103 Mode 1 J 600 620 640 660 Freq (Hz) x 10 5 0 5 10 I 15 600 620 640 660 Freq (Hz) and analytical (dashed) FRFs for Figure 35: Comparison of experimental (solid) the uniform beam. CHAPTER 4 RESPONSE PREDICTION OF DISCONTINUOUS BEAMS This chapter develops the receptance functions for the dynamic response of discontinuous beams. For brevity, the derivation will applied to the case of free boundary conditions at the end locations with one change in cross section. The discontinuity is treated by assuming two separate uniform EulerBernoulli beams coupled with continuity conditions at the joint between beams. The problem is solved as a boundary value problem with 8 unknown constants. This method can be easily expanded for beams containing different boundary conditions or additional uniform sections. The case of a notched beam with an unaligned neutral axis is treated with a coupling of the transverse bending and axial vibrations. 4.1 Receptance Derivation for Discontinuous Beams with Aligned Neutral Axes This section develops the receptance functions for a discontinuous beam with an aligned neutral axis and the case of free boundary conditions at the end locations with one step change in cross section as shown in Fig 41. The solution can be viewed as expansion of the uniform beam receptance derivation whereby the individual sections are modeled as separate beams with continuity conditions applied at the joints. The following section will solve for the cases of force and couple excitation as shown in Fig 42. The results are compared to an alternative solution using receptance coupling and to experiment. 4.1.1 Discontinuous stepped beam solution for force excitation at location C This section develops the frequency response function for force excitation at position C as shown in Fig 42(a). The solution for the first beam section (AB) is given by < L1  L > A B C Figure 41: Schematic of the stepped beam with aligned neutral axis and free boundary conditions at locations A and C. F sin(cot) I F sin(o t) I C A Msin(ot) A B C Msin(ot) C Figure 42: Schematic of a stepped beam subjected to: (a) a force of amplitude F and frequency w, applied at location C, (b) a force of amplitude F and frequency w, applied at location A,(c) a couple of amplitude M and frequency w, applied at location C, and (d) a couple of amplitude M and frequency w, applied at location Xi(xl) = c1 sin ixl + c2 cos/j3ll C 3 sinh/ Pix + c4 cosh/ Pix , (4.1) where the subscript 1 refers to the (AB) beam section. The (AB) beam sectional properties are given by El, II, pi, A1, and 31. As with the uniform beam, /1 is Ell(l+i ). Applying the free boundary condition at location A written as p04 requires 2v1 (0) a03V(0) Iax (4.2) Substituting Eq. (4.2) into Eq. (4.1) yields cl c3 and c2 = c4. The resulting expression becomes Xl(xi) = ci (sin oixl sinh 3lxl) + c2 (cos 0 cosh Plxl) The solution for the second beam section (BC) is given by X2 (2) = C5 sin 32x2 + c6 cos 2x2 + c7 sinh 32x2 + cs cosh 32x2, (4.3) (4.4) where the subscript 2 refers to the (BC) beam section. The (BC) beam sectional properties are given by E2, 2, p2, A2, and 32, which is given by 23 2= 22(1+* is understood that 31 and 32 are functions of frequency and the explicit notation has been left out. The continuity conditions at location B for the given case of a colinear neutral axis state that the deflection, slope, bending moment, and shear force are equal for the opposite sides of the joint. The analytical expressions for the continuity conditions are vl(L1) dvl(L1) dxl EllI dx d21 ) d3v1(Li) Ell dx V2(0), v(Oo), dv2 (0) dx2 d',_(O) E212 (0) d '(0) dE2 2 X Applying the continuity equations yields (4.5a) (4.5b) (4.5c) (4.5d) FicI + F3C2 0 1 0 F3C1 F2C2 21121 0 /321121 FicI + F4C2 0 3021121 0 F4c1 + FC2 321 0 3 P21 where the undefined terms in the above matrix are sin 31L1 sin 31L1 cos i$L1 cos PiLi + sinh 3iL ,  sinh/3iL , + cosh3iL1 ,  cosh/3iL , 1 0 02 0 (4.6) (4.7a) (4.7b) (4.7c) (4.7d) E212 $32 121 = and 21 0 (4.8) ElI1 P1 Constants c5, C6, C7, and cs are eliminated by solving Eq. (4.6). The solution for X2(x) may now be expressed in terms of the remaining unknown constants cl and X2 (x2) =C1 (T1 sin 32x2 + T2 cos /2x2 + T3 sinh /322 + T4 cosh /322) + C2 (VI sin /322 + V2 cos 32X2 + V3 sinh /322 + V4 cosh 32X2) (4.9) where the undefined terms are F4 F3 T =2F + 3 (4.10a) 21212321 2021 F2 F1 T2 = 2+ (4.10b) 21210221 2 F4 F3 T3 F + (4.10c) 2121p, 2021l F2 F1 T4 1 + (4.10d) 2I21, 1 2 F1 F2 Vl 22 1 =2 (4.11a) 7212,231 2021 F4 F3 V2 1 + T3 (4.11b) 212P221 2 ' F1 F2 V3 2I F1 2 (4.11c) 2212,231 2021 F4 F3 V4  2+ (4.11d) 12 02 1 2 2 Constants cl and c2 are determined by the boundary conditions at location C. The boundary conditions at location C require c _(L2) = 0, (4. 12a) 2 ax E212 *( = F sin uLt. (4.12b) The boundary conditions state that the bending moment is equal to zero while the shear force is equal to the applied impulse load. Applying the conditions of Eq. (4.12) to Eq. (4.9) yields cIZ1 + c2Z2 0, (4.13a) F c1Z3 + c2Z4 = 1 (4.13b) E2 2 where the relationships for Z1 to Z4 are Zi Ti, T2/3 T3/3 T4 sin 2L2 Z2 Vi2 V2 V3 sinh P2L2 (4.14) Z3 T13 T2/33 T33 T4 3 cos 32L2 Z4 2 V V V303 V403 COSh 2 L2 Z4 V3 V2/ V/3 V/3 cosh032L2 Solving Eq. (4.13) yields the frequency response solution v 1 [Z2 (T1 sin 0x2 + T2 cos 02X + T3 sinh 02X2 F (1 + iq/)E212(ZiZ4 Z2Z3) +T4 cosh /2X2) Z1 (Vi sin /2x2 + V2 cos 2x2 + V3 sinh 02x2 + V4 cosh 22)] , (4.15) where the compound beam is forced at position C, x2 represents the spatial output location, and r represents the structural damping factor. The denominator Z1Z4 Z2Z3 = 0 forms the so called frequency equation whose roots are the natural frequencies of the system. 4.1.2 Discontinuous stepped beam solution for force excitation at location A This section develops the frequency response for force excitation at position A as shown in Fig 42(b). The continuity conditions are the same as discussed above, however the boundary conditions at location A require 20v1(0) a2V 0) (4.16a) E V (0) Fsin t (4.16b) The sign change on the forcing term is due to the free body sign convention as shown in Fig 32. The boundary conditions at location C now require 2t, (L2) 0' ,(L2) (2 = 0. (4.17) Using the same procedure as outlined before, the response of the compound beam to force excitation is obtained. However, for loading at position A, the order of the procedure is reversed. In this case, the boundary conditions at location C are applied first, then the continuity conditions at location B, and then finally the boundary conditions at location A. Using the method as outlined before, the solution becomes v 1 S( i)E (ZZ ZZ7) [Zs (V5 sin 3iAx + V6 cos 31 + V7 sinh P3ax F (1 + iM)Elli(Z5Z8 Z6Z7) +Vs cosh pixi) Z6 (T5 sin x, + T6 cos 11 + T7 sinh pix + Ts cosh pixi)] (4.18) where the compound beam is loaded at position A. Additional terms are applied to reduce notation. The constants are defined in Table 44, where 321 and I21 are the same as above. 4.1.3 Extension of the analytical solution for applied couples This section examines the case of applied couples as shown in Fig 42(c) and Fig 42(d). For both systems, the continuity conditions are the same as discussed above. For excitation at location C, the boundary conditions at position A are Table 41: Notation for force excitation at position A 25 T212 4T Z7 T/3f + T4/3. Zs = Vi2/3 + V4/3 F5 = sin 32L2 sinh/32L2 + cosh 32L2 cos/32L2. F6 = sin 32L2 cosh /32L2 + sinh /32L2 cos 32L2. F7 = sinh /32L2 cos 32L2 + sin 32L2 cosh /32L2. Fs = sin 32L2 sinh 32L2 + cosh 32L2 cos 32L2. Ts sin/31Li1 1F6 + +cos/1Li (F 1) + 21 (F5 + 1). T6 sin/1L1 (1 F) (Fs + 1) +cos/3iLi 2F6+ ). T sinh/3iLi ( F6 ) +cosh/3IL1 (F61 (1 Fs) + (F5 + 1) . si2312 31 321 21 121/321 2 8 +F6 ). T6 = sinh 1 2L (F ) 1) 2 (F5 + 1) + cosh1 AL ( 2 F6 + 2 V5 = sin/1 (2L1 (F8 1) + (F8 + 1) + cos/i3iL( F7 + F~1 17) V6 sin/31L1 ( I1 F7+ 7) + cos/3iLi ( (8 1) +2 ( + 1)). V7 = sinh/31L1 (21 (Fs 1) 2 (F + 1)) +cosh/3lL1 ( 7 22 F7) 23 3 Vs sinh 1L1 ('21 F7 F7) + cosh Li (+2 (1 F8)+ (8 + 1)). <21(0) 301(0) o 0. (4.19) ax: axz Due to the free body sign convention, the boundary conditions at position C now become 2 Vl (L2 ) E2s 2 = Msin wt (4.20a) ax 8"',_(L2) 0 () 0. (4.20b) For excitation at position A, the boundary conditions at location C require ?12 (L2) _(L2) ax ax (4.21) while the boundary conditions at location A require 8201(0) El 21 02V Msinwct, (4.22a) ax2 03U1(0) 3v 0. (4.22b) axI The system FRFs are obtained using the same procedure as outlined before. Boundary conditions at the unforced end are applied first, then the continuity conditions, and then the boundary conditions at the point of excitation. 4.1.4 Comparison of the analytical solution to receptance coupling This section compares the responses given by the analytical results of sec tions 4.1.1 and 4.1.2 to receptance coupling substructure synthesis. Receptance coupling is an alternative method capable of of predicting the dynamic response of a stepped beam. The receptance coupling method involves coupling the receptances for uniform beams obtained analytically or experimentally at the discontinuity using compatibility and equilibrium conditions. The end result is a 2 by 2 matrix for each frequency consisting of the primary receptances given by the individual beam components. The component matrices are written as hi 1 1 vI vI R =11 f m ,1 (4.23) 1 P01 01 where the subscripts indicate either direct or cross receptances due to applied component forces and moments. The individual beam receptances, h, 1, n, and p, are frequency dependent vectors such that the size of the total matrix is 2 by 2 by N where N is the length of the frequency vector. The entries into the matrices are found from a model of the uniform beams given by Bishop [7]. The analysis that follows uses lowercase h, 1, n, and p to correspond to component receptances and capital H, L, N, and P for assembly receptances. R is used to denote the component receptance matrix and G is used to identify the assembly receptance matrix. The relationship between input forces and output responses can be written as vi h 111 f(4.24) O1 ll P (4.24) Using reduced notation, the displacement vector (v, 0) can written as the gen eralized coordinate u and the force vector (f, m) can written as the generalized coordinate q such that u = Rlqi (4.25) for a direct measurement at location 1. The goal of the process is to determine relationships between the assembly and component models to obtain the desired assembly FRF. To examine the solution process, consider the case of solving for the direct FRF due to force excitation at location 1 as shown in Fig 43. Note that location 1 corresponds to location C in the previous section. q2b I lq 2 1 Q Q1 S1q2 11 u2 Ul A "2b B U2 U (a) (b) Figure 43: Receptance coupling components (a) and assembly (b) models for excitation at C. The solution for the case of excitation at location C, given by receptance coupling [2123], is obtained first by analyzing the component displacements and rotations. The component displacements and rotations show the internal reaction forces due to an applied force or moment on the total system. For body A, the component displacement and rotations are Slb Ribib Rlb2b 0 4 I b (4.26) I' _ R2blb R2b2b q2b '. . = R2b2bq2b Similarly, for Body B ( l R11 R12 q (4.27) 2 R21 R22 92 ul = R11qi + R12q2 U2 R21iq + R22q2 Next, consider the assembly displacements and rotations. The assembly displacements and rotations show the applied external forces on the total system. In this case, the applied force is located at position 1. The force at assembly position 2 is "turned off' such that there are only reaction forces at this location. The assembly displacements and rotations can be written as ) Gil C1 12 Qi (4.28) U2 G21 G22 0 U. = GllQ . The relationship between the components and the assembly is defined by the compatibility and equilibrium conditions. For the aligned neutral axis model, the compatibility conditions are U2 U2b = 0, (4.29a) tu = Ul, and (4.29b) u2 U2. (4.29c) The compatibility conditions state that the coordinates in the component models are in the same spatial location as the coordinates in the assembly model. Although true in this application, it is not necessary that the components and assembly are in the same spatial location for the receptance coupling method. Finally, consider the equilibrium conditions. The end result for this exercise is to obtain the solution at position 1 and therefore Q2 is not active. Also, for this model, there is a rigid connection between components A and B, such that q2 + 2b 0, (4.30a) qi = Qi. (4.30b) At this point, all of the tools necessary to solve for the transfer function are available. Recall that the goal is to find h11. This is the real v/F direct FRF at position 1. This is located in G11. Recall G11 1 (4.31) [21 P22 Therefore it is necessary to solve for Gil. Recall from the assembly displacements and rotations, the relationship for Gl is U1 Ul = GCQi = G11 (4.32) Q1 This result can be refined by utilizing the compatibility conditions and the compo nent displacements and rotations for Body B u = U = Rlql + R12q2 (4.33) Plugging this result into the expression for GC1 results in Rilqi + R12q2 Gl l (4.34) Qi Recall from the equilibrium conditions, Qi = ql, therefore R12q2 G = Rl + (4.35) Qi Because an explicit function for q2/Q1 is not readily available, the ratio is de termined as a function of the individual component receptances. The result is determined from the component displacements and compatibility conditions as follows q3 q2b q 2 Q3 Q 21 u2 u3 A "2b B U3 U2 (a) (b) Figure 44: Receptance coupling components (a) and assembly (b) models for excitation at 3. U2b R2b2bq2b = U2 R2b2bq2b = R21l + R22q2 Recall : q2b= q2, and qi= Qi R2b2bq2 R22q2 R21Q1 q2 (R22 + R2b2b)1R21Q Therefore, G11 Rl1 R12(R22 + R2b2b)1R21 , (4.37) where the desired deflection FRF due to an applied force is the first entry in the matrix for each frequency. The solution for the case of excitation at location A is found in a similar manner using Fig 44. The FRF for excitation at location A (or position 3 in Fig 44) is written as G33 R33 R32b (R22 + R2b2b) R2b3 (4.38) 19.05 < 254 S25.4 / 5.49 Vtf Figure 45: Beam dimensions for comparison of the stepped beam analytical solu tion to receptance coupling and experiment. Dimensions are in (mm). x 103 Predicted FRFs for Measurement at Location C 200 400 600 800 1000 1200 1400 1600 1800 Freq (Hz) x 104 E 10  5 200 400 600 800 1000 1200 1400 1600 1800 Freq (Hz) Figure 46: FRF comparison between analytical (solid) and receptance coupling (dashed) methods when forced at position C. 140  C /) S105 Predicted FRFs for Measurement at Location A 4 2 0I o  ...... ........  2 4 200 400 600 800 1000 1200 1400 1600 1800 Freq (Hz) x 10 1  E2 3 4 200 400 600 800 1000 1200 1400 1600 1800 Freq (Hz) Figure 47: FRF comparison between analytical (solid) and receptance coupling (dashed) methods when forced at position A. To compare the proposed stepped beam analytical solution to the receptance coupling result, consider the model given in Fig 45. The model consists of a stepped beam with a rectangular cross section and colinear neutral axis. The material is 7051 aluminum with a density of p = 2; ;[i. .//m3], a Young's modulus of E = 71 [GPa], and structural damping factor of q = 0.02. Both beam sections consist of the same material with the same damping factor. Figures 46 and 47 show a comparison of the real and imaginary portions of the FRFs obtained via sections 4.1.1 and 4.1.2 to the receptance coupling solution. The results show that the solutions are identical. The advantage of the proposed solution method is in the processing power required to obtain the solutions. As shown in Eqs. (4.37) and (4.38), the receptance coupling solution requires inverting the matrices for each frequency of interest. As the frequency vector becomes large, either due to an increased frequency resolution or larger frequency bandwidth, the function becomes very costly to perform. The proposed solution, however, requires far less computing power for large frequency vectors. freefree bean accelerometer^ cato Sensor acquisition impact hannerI Figure 48: Experimental setup for FRF testing on a stepped beam. 4.1.5 Experimental verification of the stepped beam solution This section provides experimental verification for the analyses of sections 4.1.1 and 4.1.2. The experiment consists of a stepped beam of 7051 aluminum with dimensions given in Fig 45. The material has a density of p = 2830 [kg/m3] and a Young's modulus of E = 71 [GPa]. Structural damping was obtained as a best fit approximation to the data. For excitation at location C, a damping value T = 0.003 was obtained. For excitation at location A, the damping value was determined to be r = 0.001. The freefree boundary conditions were obtained in the same manner as the uniform beam and are shown in Fig. 48. Experiments were conducted by impacting the beam with a modal hammer and obtaining the response with a low mass accelerometer mounted onto the beam. Figures 49 and 410 show the results for the experiments for the first 3 modes for direct FRFs at locations C and A respectively. The accelerometer mass was measured to be m = 0.7 grams. S103 Mode 1 500 1000 1500 Freq (Hz) x103 1 n 1 2 3 4 5 6 Figure 49: Compa forced at position C E m E 500 1000 1500 Freq (Hz) rison of experimental (solid) and 5 6 260 280 300 320 Freq (Hz) analytical (dashed) FRF when The data show experimental modes at 286 Hz, 1159 Hz, and 1759 Hz for the experimental test measured at location C. Experimental modes for the test at location A were found to be located at 291 Hz, 1165 Hz, and 1771 Hz. The differences are due to additional relative inertia of the accelerometer when placed on the thinner cross section. As the data show, the experimental results are in excellent agreement with the analytical predictions. As with the uniform beam, slight errors in magnitude occur due to the use of a structural damping factor rather than a damping factor per mode. The predicted results show modes with higher natural frequencies than the experimental measurements. For both the analytical model as well as the exper imental measurements, the test at location C shows almost all of the vibration occurring in the fundamental mode. The test at location A shows a greater distri bution of energy across the higher modes in both the experiment and analytical x 103 1 Total Response 1 n x104 Total Response 104 Mode 1 3 2 I z1 I 500 1000 1500 Freq (Hz) x104 X 10 1: 0 I 1  E E )3 3 4 E 4 5 5 6 6 500 1000 1500 260 280 300 320 Freq (Hz) Freq (Hz) Figure 410: Comparison of experimental (solid) and analytical (dashed) FRF when forced at position A. result. The magnitude of the first mode was of the order of 103 [m/N] for the test at location C and 104 [m/N] for the test at location A. The smaller motion of the test at position A is due to larger inertia of the thicker cross section as well as the additional damping caused by the proximity of position A to the nylon string. 4.2 Receptance Derivation for Discontinuous Beams with Misaligned Neutral Axes This section develops the receptance functions for a discontinuous beam with a nonuniform neutral axis and the case of free boundary conditions at the end locations with one change in cross section as shown in Fig 411. The solution is an expansion of the stepped beam receptance derivation; however this case includes a coupling of the transverse bending and axial vibrations. The results are compared to a three sectioned beam used in the lr removal process by Alcoa. v E I I 2  SL L2 A B C Figure 411: Schematic of a discontinuous notch beam with a misaligned neutral axis and free boundary conditions at locations A and C. 4.2.1 Discontinuous misaligned beam solution for force excitation at location C This section develops the frequency response for force excitation at position C. The continuity conditions of the beam are determined from the free body diagrams shown in Fig 412. In the figures, q represents the transverse shear force, m represents the bending moment, p represents the axial force, 0 is the slope, A is the relative displacement between beam sections in the x direction, and e is the distance between beam neutral axes. The equations are coupled by the axial deflection and bending moments. From the displacement figure, one can see that the relationship between the slope and axial displacement between beams is tan = . For small angles, it can be shown that tan 0 9 0. Therefore the relative axial displacement between beams is A = cO, where 0 is the slope written as d. The moments are related by the axial force such that m = m2 p26. p (L) 2(0) m (L ) q m2(O) (a) (b) Figure 412: Free body diagram of (a) forces and (b) displacements for a discontin uous notched beam with a misaligned neutral axis. The continuity conditions are defined as a set of forces and a set of displace ments. The displacement continuity conditions state that the transverse deflection, slope, and axial force are equal for the opposite sides of the joint. The force con tinuity conditions state that the bending moment, shear force, and axial force are equal. The continuity conditions are di,i'l ,1 in Table 42. Note that as the individual beam sections neutral axes are aligned, the equations become uncoupled and the continuity conditions would yield the same result as in the aligned case. Therefore it can be viewed that the aligned neutral axis condition is a special case of the more general discontinuous notched beam. Table 42: Discontinuous notched beam continuity conditions. Displacement Continuity Conditions transverse displacement vi(Li) = v2(0) slope 01(L1) 02(L1) axial displacement ui(Li) = u2(0) Oe Force Continuity Conditions shear force si(Li) = 82(0) bending moment mi(Li) = m2(L) p2(0) axial force pl(Li) = p2(0) To solve the coupled system, it becomes important to consider the axial vibration of uniform beams. The equation of motion for the axial vibration of a uniform EulerBernoulli beam is given by reference [30] as E2u(x,t) 2u(x,t) (4.39) S 2 2 (4.39) where E is the Young's modulus and p is the material density as previously defined. The solution to Eq. (4.39), when subjected to an input of frequency w (rad/s), can be separated into a solution in space and time u(x, t) = Z(x) sin ut . (4.40) Substitution of Eq. (4.40) into Eq. (4.39) yields dependence upon the spatial quantity alone a2Z(X) 2Z(x) 0, (4.41) aX2 where a = w/~. The general mode shape solution to Z(x) is Z(x) Hsin ax+ Jcosax, (4.42) where H and J are constants determined by suitable boundary conditions. The axial force, p, can be written as p = AE (. The boundary conditions for classical conditions listed in Table 43. Table 43: Axial vibration boundary conditions for classical beam ends. u = 0 for a fixed end, AEa = 0 for a free end The solution process for system FRFs is the same as for the stepped beam. The system FRFs are obtained by enforcing boundary conditions at the unforced end, then the continuity conditions, and then boundary conditions at the point of excitation. For excitation at location C, the free boundary conditions at A require a2v1(0) a3V1(0) u1i(0) x2; 9x9x 03 (4.43) where the extra boundary condition comes from the axial vibration mode shape. The continuity conditions are listed in Table 42. The forced boundary conditions at C become (L 0 (4.44a) 8", (L2) E2 2 F sin wt, (4.44b) 2(L2 2(L2) 0. (4.44c) ax2 The result is a set of 12 equations with 12 unknown constants. Solving these equations yields a solution of v 1 F (1 + iq)E212(ZsZ5Z4 + ZsZ6Z2 + Z5Z3Z9 + ZIZ7Z4 Z6ZIZ9 Z3Z7Z2) (Z7Z2 Z5Z9) (Ti sin 0X2 + T2 cos 0x2 + T3 sinh 02X + T4 cosh 2X2) + (ZZt8 ZIZ7) (VI sin 32x2 + V2 cos 322 + V3 sinh 32x2 + V4 cosh X2) + (ZiZ9 Z8Z2) (W1 cos/22 + 2 cosh P2)] , (4.45) where the compound beam is loaded at position C. Additional terms are applied to reduce notation. The constants are defined in Table 44, where /21, 121, and F, F2, F3, F4 are the same as above. An additional constant is required, E21 z The denominator forms the characteristic equation whose roots are the natural frequencies of the system. Figure 414 shows a comparison of the analytical result for the stepped beam shown in Fig 45 to the analytical result for a similarly dimensioned misaligned beam shown in Fig 413. The distance between the neutral axes, e, is c = (h2  h1)/2 6.78 [mm]. As the results show, the shifted and aligned solutions show very similar results. The first mode has negligible differences. Higher modes are show slight differences on the order of 2 Hz. Table 44: Notation for FRF with force excitation at position C including a mis aligned neutral axis. T1 1 F4 1 + 3 T, 1 1, 4 2 21 T2 22E2121312 + 1 T3 2E21 213 F4 + 27 2F3 2E21I2123,11 2j421F2. T4 1 2 1F, V2 2E21213 1 + 21 V2 1 E7  1 3  2E211211 1 2 1 F2 V24 1 4 1 S = F, sinaiL . S3 2E21 21 3P21 V4 E21 F4 + F3 i2  sin L .1 S2 COS olLL. S3 c1F3. S4 2cIF2. Z1 TI2 sin f2L2 T2 cos f22 + T32L sinh f22 + T432 cosh 2L2 Z2 Vl30 sin f2L2 V2/3 cos f2L2 + V3/23 sinh f2L2 + V4023 cosh f2L2. Z3 Ti23 cos f2L2 + T2 03 sin f32L2 + T303 cosh f2L2 + T4323 sinh f2L2. Z4 = Vij3 cos 0f2L2 + V2j3 sin f2L2 + V3f3 cosh f2L2 + V4 3 sinh 32L2. 5 = Wl23 cos 32L2 + W2/32 cosh 32L2. Z WI3 sin /32L2 + W2/3 sinh /32L2. Z7 = a2S COS 1 2L2 a2S2 sin a2L2. Zs a2S3 sin c2L2. Z9 = 2S4 sin a2L2. 4.2.2 Experimental study of the misaligned neutral axis solution This section provides an experimental study for a three sectioned beam used in the 1.vr removal process by Alcoa as shown in Fig 415. The experiment consists of a notched beam of 7051 aluminum with dimensions given. The material has a density of p = 2830 [kg/m3] and a Young's modulus of E = 71 [GPa]. Structural damping was obtained as a best fit approximation to the data. A damping value r = 0.0008 was used in the analysis. As with the other experimental tests, the freefree boundary conditions were obtained with the schematic shown in Fig 4 8. Figure 416 shows the experimental results for the first 2 modes for direct < 254 > 140  A B C Figure 413: Dimensions for analytical study of beam with jump discontinuity. Dimensions are in (mm). excitation at location D. The accelerometer mass was measured to be m = 0.7 grams. The data shows experimental modes at 162 Hz and 1385 Hz. Analytical predictions show an error of 13 Hz, or 7 percent, with respect to the first mode. Results show that analytical predictions are higher than the experimental mea surements. Errors are most likely due to neglecting shear and rotary inertia in the analysis for the individual beam sections. The smaller middle section is much more flexible than the larger outer sections which causes shear at the discontinuities. The analysis would be more accurate if the beam sections were modeled with the more complex Timoshenko beam model [30] which includes the effects of shear and rotatory inertia. In this model, transverse and rotational modes are coupled making it difficult to obtain closed form analytical solutions. Many researchers use finite element packages to solve the coupled equations [22; 31; 32]. Other possible sources of error include the material properties and beam dimensions. Material properties are assumed to be equal to the average values for the particular material. However, these properties can vary slightly between batches of the same material. In any machining process, dimensions contain a certain amount of geometric tolerances. The dimensions listed are an average for the experimental workpiece, however local variations do occur. x 10 200 400 600 800 1000 1200 Freq (Hz) x 103 1400 1600 1800 200 400 600 800 1000 1200 1400 1600 1800 Freq (Hz) Figure 414: Comparison of the analytical FRF with a misaligned neutral axis (solid) to the analytical FRF with an aligned neutral axis (dashed) when forced at position C. A B C D Figure 415: Dimensions for experimental study of beam with a jump discontinuity forced at the end position. Dimensions are in (mm). x 103 Total Response 2 1 2 I 200 400 600 800 1000 1200 1400 Freq (Hz) x 103 2 II 2 I 4 6 8 200 400 600 800 1000 1200 1400 Freq (Hz) x 103 Mode 1 3 I 2 0 1 2 3 120 140 160 180 200 220 Freq (Hz) x 103 0 2 4 120 140 160 180 200 220 Freq (Hz) Figure 416: Comparison of experimental (solid) and analytical (dashed) FRF for 3 sectioned notch beam with forcing at the end location. CHAPTER 5 STABILITY OF LAYER REMOVAL PROCESS This chapter details the stability for machining discontinuous beams assuming slotting cutting conditions. Stability analyses of machining in literature show frequency diagrams labeling stable and unstable depths of cut as a function of various spindle speeds [35]. The goal is to obtain a width of cut, or chip width (b), for which chatter will not occur at any chosen spindle speed. C'! I I r is defined as a selfexcited vibration that will adversely affect the quality of the produced surface, and may lead to increased tool wear and tool failure. C'!i II 1, causes vibrations to grow in amplitude at a rapid rate causing the tool to leave the workpiece. The mechanism for chatter is the regeneration effect. In this model, a wavy surface is formed on the workpiece due to the motion of the tool during previous passages. As the tool enters and leaves the cut, the wavy surface is continually regenerated. Stability limits are obtained assuming that chatter occurs due to dynamics of the spindle holder and tool resonance frequencies. The tool is usually considered the most flexible part of the dynamic system. However, during milling operations on the discontinuous beam, the beam will vibrate transversely. As the material is being removed, the natural frequencies of the workpiece shift and the beam vibrations will start to grow. After a certain number of passes, the beam becomes more flexible than the tool and chatter will occur due to regeneration of the beam vibrations. 5.1 Limiting Chip Width for Machining Process This section identifies the limiting chip thickness, bum for which the cutting process is stable. In most high speed applications, the goal is to obtain stability lobes which detail stable cutting conditions for different spindle speeds. In this cutting process, the cutting is performed at modest speeds [1350 rpm] which inhibits the advantages gained by high speed machining. There is no need to obtain stability lobes for this particular application. Rather, the goal of this section is to obtain the limiting chip thickness for which the cutting process is stable for all speeds. The limiting value of chip width is given by T!ili, [3] 1 blimp (5. 1) 2K,Re(G)mnin where b is the chip width [mm], Ks is the specific cutting coefficient [N/mm2], and Re(G)min is the minimum real portion of the frequency response function. The chip width is albv, a positive number and therefore Re(G),mi must be the most negative portion of the FRF for a conservative estimate. The cutting coefficient is not a material dependent quantity, but is rather process dependent. The limiting chip width is defined as the most negative portion of the FRF because that corresponds to a phase shift of 270 degrees between passages. As the phase shift approaches 270 degrees, there is the largest phase difference between subsequent passages or the largest amount of amplitude variation per cycle. The minimum real of the FRF is obtained using the analytical calculations of the previous section. Consider the fixedfree cutting process shown in Fig 5 1. The goal is to find the most conservative estimate for bli,. The analysis is performed with two assumptions in the cutting process; (1) the amplitude of vibrations are small enough to not pinch the tool during i i 11 iiii:i. and (2) the most compliant element of the system is the beam in its final cut configuration. The tool is relieved such that the first assumption can be held. In this case, the point of largest vibration occurs at the end of the notch cross section farthest from the fixed end. This is the point that will be considered the point of excitation for the FRF. Figure 51: Schematic of the clampedfree notched beam during machining. Di mensions are given in (mm). Using the analysis of the previous chapter, the system is modeled as a fixed free beam with 3 uniform sections. The force boundary condition is applied to the shear continuity condition for the second discontinuity. The system is modeled using 7051 aluminum with dimensions given in Fig 51. To obtain a conservative estimate of the vibrations, it is assumed that the largest vibrations will occur after machining such that the notch is in its lowest height. The material has a density of p = 2830[kg/m3] and a Young's modulus of E = 71 [GPa]. Structural damping is assumed to be ] = 0.02. Figure 52 shows a plot of the real response of the notched beam after machining. The limiting chip width is calculated using Eq. (5.1). The specific cutting coefficient is found to be K = 850 [N/mm2] for the tangential component. This case uses the radial component of the cutting force which is equal to 200 [N/mm2]. The minimum real portion of the FRF is determined to be 1.25 x 105 [m/N]. Using these values, the limiting chip width is 0.2 [mm] [.008 in]. As expected, the structure is so flexible that an extremely small chip width leads to instability. Most cutting processes have limiting chip widths of orders of magnitude greater than those recorded here. Chip widths can range from 2 to 30 [mm]. For this particular test, cuts were performed in steps of 0.254 [mm] which led to chatter. x 10 Total Response Eo 1 0 2 0 500 1000 1500 2000 2500 3000 Freq (Hz) x105 Mode 1 2 Eo 1 0 30 35 40 45 50 55 60 Freq (Hz) Figure 5 2: Analytical FRF for the notched beam with fixedfree boundary condi tions. 5.2 Mode Shape Ai, .! ,i as a Function of the Notch Height This section compares the measured and theoretical mode shapes as a function of the notch height with fixedfree boundary conditions. Because all real systems contain some measure of compliance in the connection, the fixed boundary condi tion is modeled in two v, The first model assumes a perfectly rigid connection. The second model contains constraints against translation and rotation by use of translational and rotational springs. The spring connections are applied to the shear and bending moment boundary conditions respectively. The compliant boundary conditions are given as [30] d2Ul(0) dUi(0) El I = (5.2a) dx3 53 where kt and k, and the coefficients of the translation and rotational springs. For this analysis, the spring constants were obtained as a best fit approximation to the experimental complaintfree uniform beam data. The results were compared to the uniform beam configuration because the beam best represents the optimal Euler Bernoulli beam before any machining has been completed. Using this method, the springs constants were found to be kt = 5 x 106 [N/m] and k, = 1 x 105 [Nm/rad]. First Mode Second Mode x 10 x 10 5 SLL 0 5 1z U 10 1 10 a) 10 c 15l 1 5 5 _'_j 50 600 100150 200 250 300 0 800 1000 0 Freq (Hz) Notch depth (mm) Freq (Hz) Notch depth (mm) x 105 Measured Real FRF for Uniform and Notch beam 2 I I II1 Uniform Beam Discont Beam 1U  1 100 200 300 400 500 600 700 800 Freq (Hz) Figure 53: Experimental mode shapes as a function of the notch height for Alcoa testing conditions. Figures 53, 54, and 55 show plots of the measured and theoretical mode shapes for the FRFs with fixedfree boundary conditions as a function of the notch height. Results were obtained by taking FRF measurements at the notch in between lvr removal sequences. Each figure shows a plot of the first and second mode shapes as well as a comparison of the uniform beam and the notched beam in its final configuration. As the figures show, the notch causes a downward shift of 54 the natural frequencies. All of the plots show that the second mode shape increases as a function of the notch height. They also all show the same relative magnitudes for both modes. First Mode x 106 10 5 LL 100 200 300 400 Freq (Hz) x 10 Second Mode x 106 5 LL 0 LL 10 g / 5 5 0 700 800 Notch depth (mm) Freq (Hz) Real FRF for Uniform and Notch beam Uniform Beam Discont Beam 1  0 1 i  1 I III 100 200 300 400 500 Freq (Hz) 600 700 800 900 1000 Figure 54: Analytical mode shapes as a function of the notch height assuming fixedfree boundary conditions. The figures show differences in the first mode. The experiment shows a decrease in the magnitude of the first mode as a function of the notch height. The theoretical models show a decrease of the mode up to 10 [mm] of notch depth, and then a sharp rise in the mode size. This result is most likely due to a lack of sensor response for low frequencies in the experimental measurement. For large notch heights, the natural frequency shifts to below 40 [Hz]. The sensors used in the experiment are not accurate below 50 [Hz] and as a result, some of the data was clipped from the results. I 10 Nc 5 0 Notch depth (mm) Second Mode x 10. ............ ... . . . xO . . ' S.. ...... LL rl ^Y 0 .5 ....... .............. .0 .0.5 10 l5 5 60 H0 0 800 0 otch depth (mm) oFreqh (He 800 otch depth (mm) Notch depth (mm) Real FRF for Uniform and Notch beam 100 200 300 400 500 Freq (Hz) 600 700 800 900 1000 Figure 55: Analytical mode shapes as a function of the notch height assuming compliantfree boundary conditions. The figures also show differences in the natural frequencies for both modes for both the fixedfree model and the compliantfree model. The fixedfree model shows larger errors for the second mode than the first. As the compliantfree model shows, most of the differences occur in the second mode. In fact, both theoretical models show very similar results for the first mode. Errors in the natural frequency are due to two reasons. First, compliant spring constants were obtained as a best fit approximation. A more accurate model of the clamping force would provide a better FRF estimate. The second reason is due to the modeling of EulerBernoulli beams. Recall that an underlying assumption for EulerBernoulli beams is that of long, slender beams. For small values of the notch depth, the middle beam section becomes short and tall as opposed to long and slender. Therefore the Euler Bernoulli approximation is best held for large notch depths. However, as discussed First Mode 5 x10 :. " E 0.5  0 L 5 0.5 n, 1 Freq (Hz) 510 x10 O 0 g LL n 1 2 I N in the previous chapter, large notch depths also contain shear deformation at the section boundaries which is neglected in this analysis. The results promotes the use of more complicated Timoshenko beam models for the experimental system. Figure 56 shows a comparison of the limiting chip thickness as a function of notch depth for the experiment and both the fixedfree and compliantfree models. All of the models show a nonlinear relationship between a stable chip thickness and the notch depth. The general response of each model is an increasing allowable chip thickness with increasing notch depth until a depth of 12 [mm], followed by a sharp drop. C'!i I.I r most likely occurs for high notch depths because of the sharp drop in stable cutting conditions. The fixedfree model over estimates the bum values relative to the experimental data while the compliantfree model is more conservative. Figure 5: Compason of limiting chip thia ness, bum, as a fu depth for: (a) experiment, (b) a fixedfree model, and (c) a compliantfree model. Fixedfree m odel mng chp thcknesso f d l h 035 _0 035 025 0 025 015 0 15 Notch depth (ram) Notch depth (rm) (b) (c) Figure 56: Comparison of limiting chip thickness, bli,, as a function of the notch depth for: (a) experiment, (b) a fixedfree model, and (c) a compliantfree model. CHAPTER 6 CONCLUSIONS AND FUTURE WORK The objective of this work was to examine the dynamic response of discontin uous beams. This thesis examined two different discontinuities, one of a stepped beam, and one of a notched beam. The study analyzed the beams using Euler Bernoulli beam theory, neglecting shear and rotary inertia. The study examined the processes necessary to perform successful modal testing experiments. Various modal testing complications were discussed including the use of window functions and mass loading effects due to contact sensors. The dynamic response of stepped beams was found to be an extension of the solution for uniform beams. The solution process involved solving a boundary value problem for the mode shape solution of beam segments using boundary conditions and continuity conditions. A comparison to receptance coupling has shown this result to be extremely accurate for beams with long, slender uniform sections. The advantage of this formulation as compared to receptance coupling is in the computational time required to obtain solutions. Receptance coupling requires the use of matrix inversion per frequency of interest in the solution. As the frequency vector becomes large, either due to increased bandwidth or frequency resolution, the receptance coupling solution can become very costly. This result is especially true when considering a structure with many beam segments. Results were also compared to an experiment of a beam with one step change and freefree boundary conditions. The results were found to compare very favorably with the experimental results with minimal errors. The dynamic response of notched beams was investigated as an extension of stepped beams. This thesis examined the result as the dynamic coupling between axial and transverse mode shapes. It was shown that the notched beam and the stepped beam will show identical results when the distance between neutral axes is zero. In other words, the stepped beam is a special case of the more general notched beam case. Results compared to experiment have shown that this formulation is inaccurate for the applications studied in this work. The fundamental mode shape showed a 7 percent error between the experimental and analytical results. It was determined that the errors were due to the beam model used in the analysis. It was found that the effects of shear at the discontinuities cannot be ignored. The stability of beam milling was analyzed to determined the limiting chip thickness to avoid chatter. It was determined that chatter was being caused by the flexibility of the workpiece in its notched configuration. The flexibility caused regeneration of a wavy surface on the workpiece which led to greater vibration and ultimately the loss of stability. An analysis of the cutting configuration was performed to determine the analytical and experimental mode shapes as a function of the notch height. The fixed boundary condition was created with two analytical models; one of a perfectly rigid boundary condition, and the other of a compliant boundary. The second mode in the compliantfree model was shown to be more accurate than that of the fixedfree model. It was also shown that the natural frequencies shift downward as a result of increased notch height. Qualitatively, the second mode increased in size with increased notch height. The fundamental mode was shown to increase in size in the analytical models with the notch height, however it was shown to decrease in experimental tests. These results were most likely due to a problem of sensor response in the experimental tests. Some of the data in the signal was clipped because the accelerometers used in the test were unable to measure response at frequencies less than 50 Hz. As the fundamental mode shifted to a lower natural frequency in the experimental test, its response went below the 50 Hz threshold. 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Alvermann, Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams, Journal of Sound and Vibration 276 (2004) 807836. BIOGRAPHICAL SKETCH Michael Koplow was born on Nov. 1, 1980 in Boston, MA. He received a bachelor's degree with honors in mechanical engineering from the University of Florida in August, 2004. After receiving his MS in mechanical engineering from the University of Florida in August, 2005, Michael plans to pursue PhD studies at the University of California at Berkeley. 