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SMART BULK MODULUS SENSOR By KARTHIK BALASUBRAMANIAN 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 2003 Copyright 2003 by Karthik Balasubramanian I would like to dedicate this work to my parents for whom my education meant everything. ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Dr. Christopher Niezrecki, for assisting and guiding me in my research. This experience with him has been enriching and delightful, to say the least. I would like to express my sincere thanks to Dr. John Schueller for his invaluable suggestions and insight. I would also like to thank Dr. Carl D. Crane for serving on my supervisory committee. My thanks go out to Mr. Howard Purdy for his effort and assistance in performing all the machining operations associated with this work. Finally, I would like to thank my family and friends for being there during all those tough times. TABLE OF CONTENTS page A CK N O W LED G M EN TS ................................................................... .................... iv L IST O F T A B L E S .................................................................. .................. .. vii L IST O F FIG U R E S .............................................. ................................................. viii A B S T R A C T ............. .......... .......... ................................... .... ..... .... .. ..... ........x CHAPTER 1 IN T R O D U C T IO N ......... ................. ............................................ .................... 1.1 The Im portance of Bulk M odulus .................................... ............................... 1 1.2 Current Bulk Modulus Measurements............................................................5 1.2.1 V vibration Tester.............................. ....................6 1.2.2 A acoustic C oupler ................................................................ .................... 6 1.2.3 H olographic Interferom eter .................................................. .................... 6 1.2.4 Electronic Speckle Pattern Interferometer..................... ...................7 1.2.5 Doppler Interferometer........................ ... ........................7 1.2.6 Normal Impedance and Flow Ripple Apparatus ........................................8 1.2.7 A acoustic M methods ................................................................ .................... 9 1.2.8 O their M methods .............. ............................................ ...... .................... 10 1.3 M otivation and A approach ........................................................ .................... 11 2 THEORETICAL DEVELOPMENT....................................................................14 2.1 Piezoelectric M material ......... ................................ ......................... .................... 14 2.2 Constitutive Equations............................................................................... 16 2.3 Speed of Sound and Transfer Function Measurements........................................17 2.4 Piezoelectric Based Sensor................. .....................................................18 3 EXPERIM ENTAL SETUP .................................................................................. 23 3.1 T est E quipm ent............ .......... ...... ........................................... ....................23 3.1.1 A lum inum B lock ................................................... .................. 24 3.1.2 Transducers, Plunger and Modal Hammer...............................................24 3.1.3 Signal Conditioner and Signal Analyzer .................................................26 3.2 Experim ental Setup...............................................................26 3.3 Preexperimental Procedure........................... ...... ..... ....................28 3.4 Experim ental Procedure........................... ....... ........ .................... 29 4 R E SU L T S ........ ...... ... .... .......... ................................ ....................3 1 4.1 T heoretical R results ................................................................. .................... 3 1 4 .2 E xp erim mental R esu lts ........................................................................ ....................35 4.2.1 Tim e Dom ain M easurem ents ........................................ ................... 35 4.2.2 Transfer Function Measurements ............................................................39 4.2.3 Speed of Sound Measurements ................................... ...................43 5 SUMMARY AND CONCLUSIONS...............................................46 5.1 Sum m ary and C conclusions ..................................................... .................... 46 5.2 Future W ork .............. ........................................................................... .. .... ... 47 APPENDIX MATLAB CODES...................................................................................49 LIST O F REFEREN CES ......... ..................................... ..........................................52 BIO G RA PH ICA L SK ETCH ................................................................ ...................54 LIST OF TABLES Table Page 11. Bulk modulus of Brayco 745 at 3000 psi................................................................2 31. Test equipm ent list ............................................... ....... ............................. 23 41. Sim ulation param eters.................................................................. .................... 32 42. Speed of sound measurement values for pure hydraulic oil...................................44 43. Speed of sound measurement values for water .......................................................44 44. Speed of sound measurement values for hydraulic oil with air bubbles ................ 44 LIST OF FIGURES Figure page 11 Volume lost for different bulk modulus values ................... ... ...............3 12 Dynamic bulk modulus measurements using Doppler interferometry ..................8 13 Equipment for testing dynamic bulk modulus .................................................11 14 A pressurized flexible container filled with a mixture of liquid and air............... 12 21 Exaggerated motion of piezoelectric material ....................................................14 22 Sm all size piezoelectric stacks .............................................. .................... 15 23 Induced strain actuator using a PZT or PMN electroactive stack ........................15 24 Simplified representation of the actuatorfluid system for mathematical model... 19 31 Cross Section of the aluminum block along with the brass plugs .........................24 32 Three pressure transducers and a steel plunger..................... ....................25 33 M odal ham m er ...................................................................... .... ....25 34 Signal conditioner and signal analyzer ........................................................26 35 Experim mental setup.................................................. ................................... 27 36 Schematic diagram of the experimental setup ................... ... ...............27 37 M acro Finnpipette ............................................ ....... .... ............................. 30 41 Actuator displacement for varied fluid loading ...................... ....................32 42 Determination of bulk modulus for a given PZT actuator displacement...............33 43 Actuator displacements for varied fluid loading....................................................34 44 Time domain plots (force sensor and pressure transducers) for hydraulic oil.......35 45 Ringing pattern of the pressure response signal measured by transducer 1 for hydrau lic oil ................ .... ................................................................ .......... ... 36 46 Time domain plots (force sensor and pressure transducers) for water. .................37 47 Time domain plots (force sensor and pressure transducers) for hydraulic oil w ith bubbles.. ..... ......... ......................................................... ....... .......... ...38 48 Ringing pattern of the pressure response signal measured by transducer 1 for hydraulic oil with bubbles........................... ..... ...... .................... 38 49 Transfer function magnitude and phase plots of tests with pure hydraulic oil ......39 410 Coherence plot for tests with pure hydraulic oil........................................40 411 Transfer function and phase plot for tests with water........................................ 40 412 Coherence plot for tests with water ................................... .................... 41 413 Transfer function and phase plot for hydraulic oil contaminated with air b u b b le s ............................................................................................................. 4 2 414 Coherence plot for hydraulic oil contaminated with air bubbles...........................42 415 Comparison of transfer functions measurements of pressure with respect to force for pressure transducer 1 .................................... ........................ ..... ............... 43 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 SMART BULK MODULUS SENSOR By Balasubramanian Karthik December 2003 Chair: Christopher Niezrecki Major Department: Mechanical and Aerospace Engineering One application that smart materials may have a significant impact in is the measurement of bulk modulus. Accurately knowing the bulk modulus of a fluid is very important in many hydraulic applications. The absolute bulk modulus has a major effect on the position, power delivery, response time, and stability of virtually all hydraulic systems. The focus of this research is to develop a novel sensor to measure bulk modulus of a fluid in real time. The work investigated three different strategies to determine the bulk modulus of a fluid within a system. The first approach is to develop a theoretical model to extract the bulk modulus of the fluid system by knowing the excitation voltage and measuring the strain. The results indicate that matching the stiffness of the actuator to the stiffness of the fluidic system is critical in obtaining a high sensitivity to the bulk modulus measurement. The second approach determines the frequency response functions by performing transfer function measurements using an impulse response test. In this test, the transfer function of the pressure response with respect to the applied force is measured. By doing so it is possible to extract information about the properties of a fluid. The tests are performed on three different fluids: water, hydraulic oil and hydraulic oil with bubbles. The results indicate that magnitudes of the peaks (at 1400 Hz) were larger and sharper for water compared to oil. Also, the magnitude of the peaks (at 1400 Hz) in the case of hydraulic oil with bubbles was not only reduced but also occurred at a lower frequency compared to the other two fluids. The third approach uses speed of sound measurements to determine the bulk modulus of the fluid in real time. The results indicate the theoretical values are reasonably close to the actual bulk modulus values. Also, the hydraulic oil contaminated with bubbles has a lower bulk modulus value compared with the pure hydraulic oil. CHAPTER 1 INTRODUCTION The motivation for developing an in situ bulk modulus sensor is described in this chapter. The importance of the bulk modulus in hydraulic applications, followed by a general review of the current methods of measuring bulk modulus, is also presented. Lastly, the approach adopted for determining the bulk modulus of a fluid in real time is discussed. 1.1 The Importance of Bulk Modulus The US market for pumps, cylinders, motors, valves and other fluid power components alone is $12 billion annually, exceeding the value of other wellknown industries such as machine tools and robotics. The largest consumers are the aerospace, construction equipment, heavy truck, agricultural equipment, machine tool and material handling industries that take these components and integrate them in equipment worth many billions of dollars. Yet despite the industry's importance, and the fact that fluid power imports grew at an average annual growth rate of 30% during much of the 1990's, little support has been given to the industry by the engineering research community (National Fluid Power, 2002). The bulk modulus of a fluid (or solid) is a measure of its compressibility and is given by P = AV (1.1) where 8, effective bulk modulus of the fluid V0 unpressurized fluid volume, P fluid pressure and V instantaneous fluid volume. The negative sign indicates a decrease in volume with a corresponding increase in pressure. The parameters that primarily affect the bulk modulus of a fluid are temperature, entrained air and compliance. An example of how a typical hydraulic fluid is affected by these quantities is shown in Table 11. As the temperature of the degassed fluid rises by 100 degree F, the bulk modulus is reduced by 39%. Likewise, if an amount of air representing 1% of the volume of the fluid is added to the system, the bulk modulus is reduced by 44%. If both the temperature of the fluid rises by 80 degree F and 1% air is added to the system, the bulk modulus is reduced to 40% of its original value (Magorien, not dated). Table 11. Bulk modulus of Brayco 745 at 3000 psi. Entrained Air (%) Temperature (oF) Adiabatic Bulk Modulus (psi.) 0 80 268,000 0.1 80 250,000 1 80 149,000 0 180 163,000 1 180 106,000 The output for hydraulic pumps and the position of the masterslave actuators is directly affected by the fluid bulk modulus resulting in volume loss. For a pump, the volume lost causes a loss of horsepower while for masterslave actuators the volume loss causes a loss of stroke for the slave. This effect is shown in Figure 11 (Magorien, undated). For an actuator that is stopping a moving load or requires fast load reversal, the compressibility of the fluid greatly affects the system performance. The fluid must first be compressed before the cylinder and piston can move the load to perform any useful work The power lost generally increases as the actuator size increases and the response time decreases Ifthe bulk modulus of the hydraulic fluid is low, then energy is wasted i compressing the fluid Pressure [psi] Figure 11 Apart from the reasons previously discussed, one other reason that motivates the need to determine the exact values of the bulk modulus of a fluid is its enormous variability Although it was published long ago i 1967, Hydraulc Control Systems by Hebert E Merritt is still the "bible" of dynamic hydraulic system design It remains the mostcited item in hydraulic research papers and is usually the basis for most dynamic hydraulic modeling Meritt says that "Interaction of the spring effect of a liquid and the mass of mechanical parts gives a resonance in nearly all hydraulic components. In most cases this resonance is the chief limitation to dynamic performance. The fluid spring is characterized by the value for the bulk modulus" (Merritt, 1967, pg. 14). Since the mass is commonly known in most practical systems, or can be determined by the instrumentation such as load cells or strain gages, the bulk modulus of the hydraulic fluid determines the dynamic performance capability of many hydraulic systems. If the bulk modulus of a fluid is reduced by 50% due to the introduction of a 1% volume of air, the system's natural frequency will be reduced by 30%. This greatly reduces the stability of the system. However, the bulk modulus is usually unknown. Merritt again says: In the absence of entrapped air, the effective bulk modulus would be 210,000 psi. In any practical case, it is difficult to determine the effective bulk modulus other than by direct measurement. Estimates of entrapped air in hydraulic systems run as high as 20% when the fluid is at atmospheric pressure. As pressure is increased, much of the air dissolves into the liquid and does not affect the bulk modulus. Blind use of the bulk modulus of the liquid alone without regard for entrapped air and structured elasticity can lead to gross errors in calculated resonances. Calculated resonances in hydraulic systems at best are approximate. (Merritt, 1967, pg.18) Perhaps the second most cited text is John Watton's is Fluid Power Systems. He says succinctly: Bulk modulus is a measure of the compressibility of a fluid and is inevitably required to calculate hydraulic natural frequencies in a system. It is perhaps one fluid parameter that causes most concern in its numerical evaluation due to other effects which modify it. (Watton, 1989, pg.28) As hydraulics engineering matures, the levels of sophistication and detail are greatly increasing. Yet, close examination of much hydraulics research indicates that a value of the effective bulk modulus is just roughly assumed with no high degree of 5 accuracy. These assumed values vary widely, often despite similar oils and components. For example, in recent issues of the Journal ofDynamics Systems, Measurement and Control, the values ranged from 150 kPa (Abbott et al., 2001) to 784 MPa (Hayase et al., 2000), which is equivalent to 22 to 114,000 psi. Hence, the values of bulk modulus do vary in practice and are usually unknown. Bulk modulus assumptions commonly range from 60,000 psi to 150,000 psi for common hydraulic systems (Habibi, 1999). Consequently, achievable hydraulic system performance suffers. Typically, designer's primary goal when creating hydraulic control systems is to achieve a trajectory with small error. The second goal is to design the system such that it is robust to variations in the hydraulic fluid bulk modulus (Eryilmaz and Wilson, 2001). They say, "Other control laws often employ hydraulic fluid bulk modulus a difficult to characterize quantity as a parameter." Knowing the bulk modulus of a fluid in real time will improve the performance of a wide variety of controllers. To date, there is no convenient method to measure the bulk modulus of fluid in a hydraulic system. 1.2 Current Bulk Modulus Measurements In order to measure the bulk modulus of a fluid, a sample of the fluid is typically placed in a chamber having a piston that can vary the volume of the fluid. As the fluid is loaded its compressibility is determined. However this method requires a fairly sophisticated testing apparatus. Some special techniques to determine the dynamic bulk modulus of fluids, elastomers have also been created. One such method is holographic interferometry (Holownia, 1986). Other researchers have investigated measuring bulk modulus of fluids and solids through acoustic methods (Marvin et al., 1954). This Section discusses some of these measuring techniques. 1.2.1 Vibration Tester Philipoff and Brodynan found the bulk compressibility of plastics using harmonic vibrations at very low frequencies (105 Hz 10 Hz). The apparatus consists of a hardened steel pressure chamber, a steel plunger that rests on a jack together with a vibrating testing machine. A strain gauge is used to prestress the system to a desired value after which the jack is locked in its place. The instrument is calibrated using mercury and water to determine a correction factor. The compressibility of the plastic is calculated after knowing the area of the plunger, the volume of the plastic, the mercury volume, the volume of the pressure chamber and the correction factor. They concluded that the compressibility is a complex quantity but could not determine the exact phase angle for plastics (Philippoff and Brodnyan, 1955). 1.2.2 Acoustic Coupler McKinney et al. improvised on the prior setup to determine the dynamic bulk modulus of materials over the frequency range of 50 Hz to 10,000 Hz using an acoustic coupler. In this method, the sample is placed within the cavity of a rigid pressure vessel that is equipped with two sets of pressure transducers. The cavity is filled with light oil, which is used as a transmitting medium. One set of piezoelectric transducers is used as an actuator to compress the fluid. The second set is used as a receiver whose output voltage measures the resulting pressure changes. The ratio of the output to input voltages is then used to determine the compliance of the coupler and its content, in particular, the compliance of the sample (McKinney et al., 1956). 1.2.3 Holographic Interferometer Holographic interferometry is a technique that is used to measure the static and dynamic bulk modulus of elastomers. Experimentally, this is realized by subjecting a sample placed in a pressure cell to a hydrostatic pressure of up to 20 atmospheres. The sample is then exposed to a single laser beam that captured its contraction onto a holographic plate resulting in fringe pattern from which the bulk modulus is calculated. The pressure cell is provided with static and dynamic pressure supply lines. The peakto peak dynamic pressure is on the order of 1 to 5 atmospheres. Glycerine and transformer oil are the two fluids that are used in the pressure cell. The frequency range that is covered is from 1 to 1000 Hz (Holownia, 1986). 1.2.4 Electronic Speckle Pattern Interferometer Howlonia and Rowland measured the volume changes of a rubber sample in glycerine by applying sinusoidal pressures using electronic speckle pattern interferometry (ESPI). The system comprises of three control areas: the electrooptical system, the hydraulic system and the electrical system. The electrooptical system provides a HeNe laser that is used for illuminating the rubber sample. A photodiode is used to analyze the beam modulation. The hydraulic system supplies the sinusoidal pressures to be applied on the sample through an oilglycerine interface. A piezoelectric transducer measures the applied pressure. The electrical system monitors the signal from the photodiode and the transducer on a cathoderay tube, where the pressure and phase angles are measured by observing the fringes. The advantage of this technique over holography is that the fringes obtained can be seen "live" as the sample deflects. The range of frequency covered is 50 1000 Hz (Holownia and Rowland, 1986). 1.2.5 Doppler Interferometer Guillot and Jarzynski used the laser Doppler interferometer to detect the strain resulting from the compression of a sample to determine the bulk modulus. In this method, the sample is placed at one end of a tube and a loud speaker is attached at the other end as shown in Figure 12. The loudspeaker is driven with a transient signal and the pressure inside the tube is recorded using a sound level meter. A pair of photodiodes detected the displacement signals from each side of the sample. A phaselocked loop circuit processes the output of each photodiode. After integration and calibration, the displacements on either side of the sample were determined. These displacement values are then used to calculate the bulk modulus. The frequency range for this method is from 200 Hz to 2 kHz (Guillot and Jarzynski, 2000). Loudspeaker Rgid Threads 1/4" Thick steel ube L . SSample li/2" iube) 2 5 cm Microphone 2 5 cm 180,. Figure 12. Dynamic bulk modulus measurements using Doppler interferometry (Guillot and Jarzynski, 2000). 1.2.6 Normal Impedance and Flow Ripple Apparatus Fluid bulk modulus is measured by using a normal impedance and flow ripple apparatus in accordance with the International Standards Organization (ISO 10671). The test rig contains two reservoirs in order to switch between fluids, if needed. One variable speed motor is used to drive the pump while another variable speed secondary motor is used to develop pressure pulses along the fluid line. Three pressure transducers assembled at the outlet of the pump are connected to the data acquisition system. An iterative method is used for calculating the speed of sound and the bulk modulus wherein the starting value for the effective bulk modulus is assumed and the speed of sound calculated based on this assumption. Then a correction to the speed of sound is made and the revised speed of sound is calculated. This revised speed of sound is then used for calculating the effective bulk modulus (Qatu and Dougherty, 1998). 1.2.7 Acoustic Methods The dynamic bulk modulus of polyisobutylene is calculated from the longitudinal wave and shear moduli. For a purely elastic material the longitudinal wave modulus (M) is related to the bulk modulus ( 8,) and shear modulus (G) by M ( +4G) (1.2) 3 In a viscoelastic medium the same relation is applicable, except that both f, and G become complex functions rather than constants. In this method, a signal is transmitted from one piezoelectric transducer to a sample and from the sample to another transducer that acts as a receiver. The medium used for transmitting the plane wave is ethylene glycol. The velocity is determined by measuring the phase shift introduced in the received signal on inserting the sample. The frequency of interest is from 0.9 to 7 megacycles per second (Marvin et al., 1954). Bums et al. discuss two experimental methods for measuring the dynamic bulk modulus in elastomers. In the impedance tube method the speed of a longitudinal plane wave in the material is measured. The longitudinal wave modulus is calculated from the sound speed and the density of the solid. The bulk modulus is then calculated from the shear modulus and the wave modulus using Equation 1.2. This method is most useful at relatively high frequencies, above 10 kHz. The second method involves using an acoustic coupler at low frequencies. This technique is similar to the one developed by McKinney, et al. Both the methods were used by the authors but the coupler gave them more accurate results (Burns et al., 1990). Koda et al. determined the longitudinal, shear and bulk moduli of polymeric materials by measuring the longitudinal and transverse sound velocities. Experimentally, the sound velocity is obtained from the measurement of time required for transmission through a specimen of thickness. The transmit time is determined by using the TAC (timetoamplitude converter) method in a double transducer system. Lead zirconate titanate (PZT) and quartz are used as transducers for the transverse and longitudinal sound waves, respectively. The sound waves propagate through an aluminum buffer before reaching the specimen. The TAC system starts when the transducer detects the sound wave reflected from the interface between the buffer and the specimen and stops when another transducer receives the transmitted sound wave. From the sound velocity, the longitudinal and shear moduli are obtained. The relation between the longitudinal wave modulus, shear modulus and the bulk modulus are then used to calculate the bulk modulus (Koda et al., 1993). 1.2.8 Other Methods Holownia and James used hydraulic pressure change rather than volume change to compress an elastomeric specimen to measure dynamic bulk modulus from 100 Hz up to 1200 Hz. In this method, two identical chambers are filled with liquid and a rubber sample placed in one of them as shown in Figure 13. A stepped piston, attached to a vibrator, is then used to pressurize both the chambers. The resulting pressure, which will depend on the volume and compressibility of the rubber sample, is then measured using piezoelectric transducers (Holownia and James, 1993). Piezoelectric pressure transducers Amplifier ual beam osciloscope Rubber sample  00< 1 Static pressure l pressurization chamber and some means to change pressure. A deformation jacket, a fluid Stepped piston Figure 13. Equipment for testing dynamic bulk modulus (Holownia and James, 1993) Fishman and Machmer discuss three different methods to determine bulk modulus. Their methods use a steel cell, capable of sustaining 70 MPa internal pressures as a pressurization chamber and some means to change pressure. A deformation jacket, a fluid displacement volume change measurement device and load ram displacement are used to measure the volume changes. The test specimen used in all the cases was Adprene (Fishman and Machmer, 1994). 1.3 Motivation and Approach In all of these tests, the fluid and elastomer properties are measured by using some type of sophisticated test apparatus. None of these test procedures consider the hoses and fittings that would normally contain the hydraulic fluids during operation. However, these hoses and fittings always have some compliance that will greatly affect the effective bulk modulus of the system. Likewise, these tests do not consider the entrained air or contaminants within an actual system. In order to determine the effective bulk modulus, the compliance of the container and the entrained air needs to be considered, as shown in Figure 14 (Manning, 2001). The effective bulk modulus is also given by 1 ~J v, Ve Va 1 1 Ve a+A v,/3 fl P (1.2) where p, effective bulk modulus p bulk modulus of the liquid a bulk modulus of entrained air P bulk modulus of container V effective volume that undergoes deformation Measurements of bulk modulus that are not performed on an actual system are approximate at best. Fluid sampling is a difficult task and when fluid sampling can be performed, it does not maintain the actual conditions within the hydraulic system. As a result, not considering the effective bulk modulus and how it changes with temperature, entrained air, contaminants and container compliance will lead to hydraulic controllers with substandard performance. S L  T Expanded Volume, V + V5 Figure 14. A pressurized flexible container filled with a mixture of liquid and air (Manning, 2001) The objective of this work is to develop a system to measure the bulk modulus of a fluid in real time. The following chapter will discuss the creation of a mathematical I . , 13 model of a fluidic system response to a quasistatic excitation that incorporates the constitutive Equations of a piezoelectric actuator. The specific test apparatus used to measure the effective bulk modulus and its variation with entrained air, two approaches (speed of sound measurements and frequency response functions using a modal hammer test) to compute bulk modulus, and an analysis of the results using these two methods are also presented. Conclusions will be drawn and future work will be discussed. CHAPTER 2 THEORETICAL DEVELOPMENT Within this chapter a brief review of piezoelectric actuation and its constitutive Equations is presented The piezoelectric actuator is the heart of the bulk modulus sensor being investigated Additionally bulk modulus variability on speed of sound is described along with transfer function measurements Finally a mathematical formulation to extract the bulk modulus of a fluid using piezoelectric actuator is derived 2.1 Piezoelectric Material Piezoelectric material comes in variety of forms, ranging from rectangular patches, thin discs and tubes, to very complex shapes fabricated using solid freeform fabrication or injection molding Due to its crystalline nature, a piezoelectric maternal expands and contracts when an electric field is applied as shown in Figure 21 Typical free strains induced in these elements are on the order of 0 1%to 0 2% Because of the free strain or displacement (in plane d31, out of plane d33) of these piezoceramics is so small, they typically cannot be used as actuators in their raw form, rather, amplification is required As a result numerous, novel and ingenious mechanisms to amplify the actuator motion have been developed One such development is the use of piezoelectric stacks 2 L L Unactivated Activated Figure 21 Exaggerated motion of piezoelectric material (Niezrecki et al, 2001) Piezoelectc stacks consist of many layers of electroactive matenals (PZT or PMN) alternatively connected to the positive and negative terminals of a voltage source as shown i Figure 22 These electro active materials, when activated, expand and produce output strain the range of7501200[nm/m Figure 22 Small sze piezoelectic stacks(Giurgiutu, et a, 2000) The piezoelectic stacks are constructed using two methods Method 1 produces stacks of lower stiffness by mechanically assembling and gluing together the layers of active maternal and the electrodes as shown in Figure 23 Method 2 produces stacks of higher stiffniess by assembling together ceramic layers and electrodes and then cofinng them Stacks of high density can be produced if the second method is subject to high isostatic pressure I d of acte I V [Ie a. ed eleaozes Figure 23 Induced strain actuator usng a PZT or PMN electroactive stack (Giurgiutu, etal, 2000) 2.2 Constitutive Equations The piezoelectric materials exhibit electromechanical coupling, i.e., mechanical stress produces an electrical response and vice versa. The electromechanical properties of these materials are related to the electric dipoles that exist in the molecular structure. Poling the material produces an alignment in the electric dipoles. Application of an external field or mechanical stress will produce motion in the electric dipoles. This motion of the dipoles gives piezoelectric materials their electromechanical properties. The total strain in a piezoelectric material is the summation of the strain due to external forces and the strain due to the applied electric field and is given by, S, = s, T +dk, E/ (2.1) where S mechanical strain, s elastic compliance, T mechanical stress, d piezoelectric strain coefficient, E electric field i, j, k orientation of the piezoelectric crystal as shown in Figure 2.1. The total electric displacement in a piezoelectric material is the summation of the electric displacement due to mechanical stress and the applied field and is given by, D =d /= Tk +s E (2.2) where D electric displacement s dielectric permittivity. Equation 2.1 and 2.2 are combined to yield, IS =[s d]ITI (2.3) D d s E Equation 2.3 clearly indicates that the mechanical and electrical domains are coupled through the piezoelectric strain coefficient, d. For piezoelectric stacked actuators, the threedimensional tensor Equation can be reduced to a onedimensional Equation in which the induced deformation of the actuator is dependent on the electric field (or voltage) and the mechanical loading applied on the actuator. The reduced constitutive relation is then given by: S3 =d33 E3 + s T (2.4) Equation 2.4 can be rewritten in terms of the piezoelectric strain, s the applied voltage, v, the thickness of the piezoelectric stack, t, the Young's modulus of the piezoelectric material, E and the piezoelectric stress, op. The above Equation then becomes, v 1 = d33+(r (2.5) t EP The first term in Equation 2.5 is the induced strain under stress free conditions. This Equation is used in the development of the theoretical model that will be discussed later in this Section. 2.3 Speed of Sound and Transfer Function Measurements Apart from the model presented is Section 2.4, the speed of sound and transfer function measurements may also be used to determine the bulk modulus of the fluid in real time. As a signal is applied to a piezoelectric actuator, the deflection of the actuator will generate a propagating acoustic wave that will travel from the actuator through the fluid and will reach a pressure sensor. Because the pressure sensor is located some distance from the actuator, there will be a time delay between the induced actuator pulse and the measured pressure sensor response. From the measured time delay and known distance, the wave speed and fluid bulk modulus can be determined. The speed of sound of a wave traveling in a fluid is given by (Streeter and Wylie, 1975): c= (2.6) where c wave speed p fluid mass density Another approach that will be investigated involves performing transfer function measurements. Transfer function measurements are routinely used in structural dynamic analysis to characterize the mechanical properties (natural frequency, damping, stiffness, etc.) of structures. These same experimental techniques can be applied to hydraulic fluids to determine their properties by knowing the frequency response functions (FRFs). The frequency response functions for a fluid can be obtained by using an impulse response test. In this test, the transfer function of the pressure response with respect to the applied force is measured. By doing so it is possible to extract information about the properties (stiffness, bulk modulus, etc.) of a fluid. 2.4 Piezoelectric Based Sensor Within this Section an expression relating a piezoelectric (PZT) actuator input voltage to its displacement for a given fluidic system (containing entrained air and mechanical compliance) is derived. The model can be used to extract the bulk modulus of the fluid in real time. 19 The excitation is caused by a PZT actuator, which drives the piston as shown i Figure 24 The simplified mathematical model treats the fluid within the container as spnng with stiffness, kj, which is m series with PZT actuator The internal stiffness of the actuator is treated as another spring with stiffness, k, as shown i Figure 2 4 PZT Actuator (a) Fluid &Air I1 111111111111111 Free Displacement Excted Achlator Compressed Fluiad Figure 24 Simplified representation of the actuatorfluid system for mathematical model A) Actuator fluidic system B) Equivalent model C) Actuator n excited state The boundary condition at the interface between the actuator and the fluidic spring dictates that the forces acting on the actuator and the spring are equivalent and are given by F=kx=k% ,y x) where x,, free displacement of the actuator x displacement of the actuator for a given applied voltage (loaded by the fluid) F the force induced by the actuator 20 During expansion, the active material works to compress the fluid and to expand the internal stiffness of the piezoelectric material. Rearranging Equation 2.8 and solving for the actuator displacement yields: kp fee kf + k Equation 1.1 can also be arranged in terms of the change in pressure (AP): AV AP=? A (2.1( V, The change in volume AV, is related to the area of the fluid, A f, and change in actuator displacement, Ax. Likewise, the induced force is equivalent to the product of induced pressure and the area of the actuator. Using the relationship in Equation 2.10 yields: AF P, (Af Ax) A (2.1 AP V Rearranging Equation 2.11 and expressing it in terms of the applied force induced by the actuator, AF , AF= A Ax aF= The stiffness of the fluid can be expressed in terms of the bulk modulus by rearranging Equation 2.12 AFT A Ax V0 / Af p, Af f if if where if is the length of the fluid as described in Figure 2.4. The axial stiffness of the PZT actuator can be expressed as: (2.12) where E modulus of elasticity of the piezoelectric material, I length of the actuator and A crossSectional area of the actuator. The displacement of the actuator can be expressed in terms of the bulk modulus of the fluid by substituting the expressions for kf and k, (Equation 2.13 and 2.14) into Equation 2.9.The expression can then be solved for the actuator displacement and is given by: AE xfle (8, Af/,f +A, E, /1, (2.15) The free displacement of a piezoelectric actuator is directly related to the dielectric coefficient, d33, the applied electric field and the length of the actuator. The free displacement for a stress free state is the product of the stress free strain and the length of v the piezoelectric actuator. From Equation 2.5 the induced strain is d33 and hence the t free displacement is given by: free d 33 tp t where v applied voltage and t thickness of the actuator. Substituting the expression for the free displacement (Equation 2.16) into Equation 2.15 produces a solution for the actuator displacement in terms of the effective bulk modulus and other known quantities, and is given by: Ap E d33 v x=  p p 33 / (2.17) t( A,/ 1, +A, E, /1 Equation 2.17 can be rearranged to obtain an expression for the effective bulk modulus of the fluid, and is given by: A, E 1If d33 v 1 fle =; xt < Xfe (2.18) Af xt 1I The expression derived in Equation 2.18 assumes that the displacement of the actuator is a known quantity. In order to utilize this result, the displacement of the actuator needs to be measured. This can be accomplished placing a strain sensor on the PZT actuator to measure the displacement. Assuming the geometry, physical properties of the actuator, applied voltage, and the displacement are known, Equation 2.18 can be used to extract the effective bulk modulus of the fluid in real time. CHAPTER 3 EXPERIMENTAL SETUP One objective of this experiment is to determine the transfer function of the pressure response with respect to applied force. By obtaining these frequency response functions (FRFs), it may be possible to correlate this data to the stiffness (or bulk modulus) of the fluid. Another objective is to compute the bulk modulus of the fluid in real time using speed of sound measurements. This chapter will describe how the experiments are setup and run, to achieve these objectives. 3.1 Test Equipment The primary test equipment consists of an aluminum block with a cavity, a plunger, three pressure transducers and a modal hammer. A comprehensive list of all the test equipment and their quantities is shown in Table 31. This is followed by a brief description of the essential components. Table 31. Test equipment list Equipment Quantity Aluminum block (highly rigid) with a 1 cavity Pressure transducers 3 Aluminum plunger 1 Modal hammer 1 Brass plugs 2 Signal conditioner 1 DSP siglab 1 BNC cables 3 3.1.1 Aluminum Block The aluminum block is essentially rigid in order to eliminate the effects of compliance The presence of compliance i the equipment can significantly affect the measured values of the bulk modulus The block is 0 411m [16 18 in] length, has a width of 0 069 m [2 72 in] and is 0 112 m [4 41 in] i height It has a0 012 m [047 m] hole drilled along its length and is situated at a distance of 0 067 m [2 64 in] from the top of the block The hole (cavity) has been designed to house a volume of fluid (hydraulic, water, air etc), representing a closed hydraulic system Two brass plugs are used at each end to seal the chamber The plugs maintain the pressure within the block by preventing any leakage (air or fluid) Four additional holes have been drilled vertically to allow the propagating wave to travel from the plunger to the three pressure transducers A drawing of the block along with the brass plugs, transducers and the plunger is shown in Figure 3 1 Figure 31 Cross Section of the aluminum block along with the brass plugs 3.1.2 Transducers, Plunger and Modal Hammer Three pressure transducers (PCB ICP pressure sensors, 1000 psi, 5 mV/psi) are used in the aluminum block and are shown in Figure 32 Transducers one and two are model 101A04 type while the third transducer is model 113A24 type The transducers are taped withTeflon before they are fit tothe block m orderto prevent ayleaks andto ensure atigl fit The almlnum or steel plunger is fit at one ad of the block and the stem of the plunger has two grooves that accommodate w o nngs (see Figure 32) The purpose of the onngss to prevent ayleakage The plunger acts as aston hands used to generate a pressure pulse ihn the fld Grease is appledto the stm to facilitate easy movement of the pluger Figure 32 Three pressuretranscers and steel plunger The type of modal hammer usedfor the expementisthe086B04 PCB modal hammer as shownmFi e 33 The hammer usedhas a rang from 0 to 1000 lbs anda setivity of 943 NN The purpose ofthe hammer is to provide the impacttothe plunger Sorderto generate pressure wave nthe fluidic cavty Figure33 Modal ammer 3.1.3 Signal Conditioner and Signal Analyzer The PCB signal conditioner (model 482 A16) used for the setup is shown in Figure 34. It is a low noise ICP sensor signal conditioner with a 4channel configuration. Unity gain is chosen for all the experiments. A 2042 model DSP technology SigLab analyzer (Figure 3.4) that has 4inputs, 2 ouputs and a 20 kHz BW is used in the experimental setup to analyze the pressure and force signals. , Figure 34. Signal conditioner and signal analyzer 3.2 Experimental Setup A diagrammatic representation of the experimental setup is shown in Figure 35. Figure 36 provides a schematic representation of the setup. The modal hammer and the three transducers are connected to the inputs of the signal conditioner using four BNC cables. Four additional cables are used to connect the output terminals of the signal conditioner to the input terminals of the DSPT signal analyzer. The signal analyzer is then connected to a personal computer. ATO Block l withbras phig preigei arm tlee tailducet Figure35 Experimental setup ~m1Coltuarr Figure 36 Schematic diagram of the experimental setup The modal hammer is used to provide an impact force on the plunger that acts like a piston. This causes a pressure pulse to travel through the cavity. The transducers then measure the pressure pulse. The delivery of the pressure pulse can be ultimately provided by a piezoelectric actuator. Once the force and pressure are known, the transfer function is determined between the pressure and applied force. By knowing this it is possible to extract information about the properties (stiffness, bulk modulus, etc.) of a fluid. In order to measure the speed of sound in the fluid, the same experimental setup is used as shown in Figure 3.6. The wave speed is determined by measuring a time delay between induced actuator pulse (provided by the modal hammer) and the measured pressure sensor response. From the measured time delay and known distance between the bottom of the plunger and each of the pressure sensors through the fluid (0.17 m, 0.29 m and 0.41 m), the wave speed and fluid bulk modulus can be determined using the formula stated in Section 2.3. 3.3 Preexperimental Procedure All the measurements are conducted using three different fluids within the cavity: water, Mystik AW/AL (antiwear/antileak) ISO 32 hydraulic oil and oil contaminated with a specified quantity of air. Before performing the measurements for a given fluid the following procedure is meticulously performed to ensure that the system is free of air bubbles: 1. The cavity is cleaned using a solution of water and liquid detergent (when using oil). The surface of the block is then cleaned using a dry cloth. The block is oven dried at temperature of 200'C to remove any water particles. It is then allowed to cool to room temperature. 2. The cavity at one end of the aluminum block is fitted with a brass plug. The three transducers and the plunger are then fitted into their respective holes. Teflon tape is used while fitting the transducers and the brass plugs to ensure a tight fit. Grease is also applied on the plunger to facilitate easy movement. 3. The block is then kept in the upright position such that the other end of the cavity is at the top. Water or hydraulic oil is slowly filled into the cavity using a test tube. Now, the block is rested against a wall in a 45degree position and maintained in that position for one whole day. This is to ensure that the air bubbles rise up and the cavity is filled only with the fluid. 4. The cavity is repeatedly checked to see if it is filled with the fluid to its brim during the course of the day. If not, more fluid is poured into the cavity. Once the cavity is completely filled with fluid (or devoid of bubbles), the other brass plug is fit into the cavity. To further ensure that there are no air bubbles, the plunger is pressed down by providing some force. If the system is full of water (without any bubbles) then the plunger will provide firm resistance (upward) to the downward force that is applied. 3.4 Experimental Procedure The experimental procedure consists of performing tests on three different fluids: water, hydraulic oil and hydraulic oil with known quantity of air bubbles. The experimental setup shown in Figure 3.6 is used for all the three tests. The pre experimental procedure (from step 1 to step 4) is performed before switching to a different fluid. The transfer function measurements and speed of sound measurements for water and hydraulic oil follow the same procedure described in Section 2.3. In the case of hydraulic oil with air bubbles a slight modification is involved. In order to determine a known quantity of air bubbles an instrument called Finnpipette (macro) is used as shown in Figure 37. It has range of 0.55ml with a capability to increase every 0.01 ml for accurate measurement. The aluminum block is filled with water up to its brim and then the Finnpipette is preset to the quantity needed to pipette. The white tip is inserted into the water and the red top is pressed down partially to pipette the required quantity of fluid into the instrument. The red top is then pressed completely to eject the fluid. For the experiments that used oil as a fluid and an air bubble, the bubble volume was estimated to be approximately 0.6 ml. The plug is then screwed onto the 30 block. The presence of an air bubble allowed the plunger to have a softer response as compared to when the cavity was filled completely with fluid. Tip Preset scale Figure 37. Macro Finnpipette Red top Figure 37. Macro Finnpipette Red top CHAPTER 4 RESULTS This chapter presents the results obtained from analyzing the theoretical model that can be used to extract the bulk modulus of a fluid in real time. It also discusses the experimental results obtained from the transfer function and speed of sound measurements using the three fluids: water, hydraulic oil and hydraulic oil with bubbles. 4.1 Theoretical Results Within this Section a test case for a fluidic system is simulated using the expressions found in Equations 2.17 and 2.18 (which have been restated below as Equations 4.1 and 4.2). A, E, d33 v x=( A/l+aE /) (4.1) t(8,A, 1, +APE/l f8e = ;A f xxt7J 1 (4.2) A, xt 1 The simulation parameters chosen are shown in Table 41. It is assumed that the piezoelectric actuator is cylindrical along with the cavity containing the fluid. The piezoelectric actuator material chosen for this simulation is a piezoelectric material manufactured by Piezo Systems, Inc. (PSI5HS4ENH). The thickness of the actuator is chosen such that at 250 V, the electric field applied at its maximum level (3.0 x 105 V/m) Table4 Simulationparameters Variable Value 1 01m If 05m AA 785e7nm (Diameter 0001m) A1 785e5 m (Diameter =001 m) E 5GPa t 8 33e4 meters d, (PZT5H) 650e12 mV Given a known applied voltage and an effective bulk modulus,Equation 4 1 can be used to calculate the displacement of the actuator The appendix contains the code used for calculating and plotting the results When the effective bulk modulus is zero, this represents the free expansion of the piezoelectric actuator As the bulk modulus increases, the displacement of the actuator decreases as shown in Figure 41 10 1 5 2' 0 X 10 Effective Bulk Modulus [Pa] Figure 41 Actuator displacement for varied flud loading ~CI Likewise, knowing the applied voltage and actuator d splacement allows the effective bulk modulus to be calculated using Equation 4 2, as shown in Fig 42 It should be noted that the expression in Equation 4 2 will generate negative values if the assigned displacement is higher than the PZT actuator free displacement, implying that the fluid is pulling on the PZT actuator and elongating it Therefore, the calculated values in this range have been set to zero Likewise if the displacement approaches zero, the calculated bulk modulus will reach infinity These values have been artificially limited to 2 2 Pa (319,000 psi, the bulk modulus for water) in the l lsimulaton X 10 22 2 x 10 1 8 2 14 1 22 x 10 1 58 12 10 0 200 04 0 100 02 10 I50 2 0 Voltage [V] Actuator Displacement [m] Figure 42 Determination of bulk modulus for a given PZT actuator displacement In order to practically utilize the formulation derive din Equation 4 2, it is imperative that the displacement of the actuator has a strong sensitivity to the effective bulk modulus The simulation has shown that the stiffness of the actuator plays an important role in its sensitivity to changes in bulk modulus This effect is seen in Fig 4 3 in which the diameter of the actuator is changed and the displacementresponse is computed for a variety of fluid bulk modulus vluues Changes the diameter of the actuator are directly related to changes in the internal stiffness of the acdtator If the actuator's authority is too high (large diameter), it will expand without significant sensitivity to the fluid This is seen he te p portion of Figure 4 3 in which the actuator displacement is essentially constant as the fluid bulk modulus changes If the actuators authority is too low (small diameter), it will not be able to expand a significant amount to be accurately measured over a useful range This is seen at the bottom portion of Figure 43 Matching the stiffness of the acatorto stiffness of the fluid should yield the highest sensitivity to changes in bulk modulus and is an important design consideration 2 E1 E 05 0 6 05 S10 20 Diameter [m] Bulk Modulus [Pa] Figure 43 Actuator displaceents for varied fluid loading 4.2 Experimental Results Within this Section the time domain and frequency response for three fluids (hydraulic oil, water and hydraulic oil with bubbles) is analyzed and compared. This is followed by speed of sound measurement calculations for the three fluids. 4.2.1 Time Domain Measurements The time domain plot for hydraulic oil is shown in Figure 44. 1500  0 Time [sec] x 104 1000  S500 Transducer 0 I \ Pressure Transducer2 0 2 4 6 8 10 12 14 16 18 20 Time [sec] x 104 Figure 44. Time domain plots (force sensor and pressure transducers) for hydraulic oil. The single hit of the modal hammer on the plunger is characterized by the dominant peak force of 1756 Newtons at approximately 0 sec in the force signal plot. The plunger then generates a pressure wave within the fluid. The pressure pulse is detected by each of the three transducers and is shown in Figure 44. The pressure wave reaches transducer 1, 0.00012 seconds after impact, transducer 2, 0.00022 seconds after impact, and reaches transducer 3, 0.00030 seconds after impact. The pressure signals have a "ringing" pattern as shown in Figure 45 for transducer 1. This pattern is attributed to the compression and rarefaction of the fluid that decays with time. 100  50  0 2 4 6 8 10 12 14 Time [sec] x 10 Figure 45. Ringing pattern of the pressure response signal measured by transducer 1 for hydraulic oil The time domain plots for the tests with water are shown in Figure 46. These plots are similar to those for hydraulic oil. The single hit of the modal hammer on the plunger is characterized by the single peak force of 2211 Newtons at approximately 0 sec in the force signal plot. The pressure wave reaches transducer 1, 0.00012 seconds after impact, transducer 2, 0.00023 seconds after impact, and reaches transducer 3, 0.00033 seconds after impact. The similar ringing pattern is also observed with water. 2000 S1500 z 1000 2 500 0 LL 0 0 0 2 4 6 8 10 12 14 16 18 20 Time [sec] x 104 200 S100\/ '' 2 0 /  Pressure Transducer1 S200 Pressure Transducer2 300 Pressure Transducer3 0 2 4 6 8 10 12 14 16 18 20 Time [sec] 104 Figure 46. Time domain plots (force sensor and pressure transducers) for water. The time domain plot for hydraulic oil with bubbles is shown in Figure 47. A known amount of air was added to the hydraulic oil using the Finnpipette. The estimated bubble volume is approximately 0.6 ml. It should be noted that the plunger provides a softer response because of the presence of an air bubble compared to when the cavity was filled completely with fluid. The time domain plots for hydraulic oil with water have a different response from those of the other two fluids. This differing pattern of response is shown in Figure 48. The single hit of the modal hammer on the plunger is characterized by the single peak force of 1203 Newtons at approximately 0 sec in the force signal plot. The pressure wave reaches transducer 1, 0.00014 seconds after impact, transducer 2, 0.00025 seconds after impact, and reaches transducer 3, 0.00035 seconds after impact. The similar ringing pattern is also observed with water. 400 0 2 4 6 8 10 12 14 16 18 20 Time [sec] x 104 ' Pressure Transducerl Pressure Transducer2 Pressure Transducer3 0 2 4 6 8 10 12 14 16 18 20 Time [sec] In4 Figure 47. Time domain plots (force sensor and pressure transducers) for hydraulic oil with bubbles. 300 250  200  150  0 2 4 6 8 10 12 14 Time [sec] x 103 Figure 48. Ringing pattern of the pressure response signal measured by transducer 1 for hydraulic oil with bubbles 4.2.2 Transfer Function Measurements The transfer function and phase plots of pressure with respect to force for hydraulic oil is shown in Figure 49. 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency [Hz] 4500 5000 5500 6000 0 500 1000 1500 2000 2500 3000 3500 Frequency [Hz] 4000 4500 5000 5500 6000 Figure 49. Transfer function magnitude and phase plots of tests with pure hydraulic oil The coherence plot for pure hydraulic oil is given in Figure 410. These plots clearly indicate that there is reasonably high coherence from the signals.  Pressure Transducer 1  Pressure Transducer 2  Pressure Transducer 3 // \\ S Pressure Transducer 1  Pressure transducer 2 SPressure Transducer 3 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Frequency [Hz] Figure 410. Coherence plot for tests with pure hydraulic oil The transfer function and phase plot for water is shown in Figure 411. 'E 20 S Pressure Transducer 1 Pressure Transducer 2 W z 15 d S 1  Pressure Transducer 3 2 i 0 10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Frequency [Hz] Pressure Transducer 1 Pressure transducer 2 Pressure Transducer 3 Figure 411. Transfer function and phase plot for tests with water 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Frequency [Hz] It is clear that the response for water is not only larger in magnitude, but is also sharper. This is indicative of that fact that water has a bulk modulus that is much higher than hydraulic oil (f8wat,, 310 ksi, 8,,, 200 ksi). The coherence plots for each of the transducers in the case water are shown in Figure 412. These plots clearly indicate that there is reasonably high coherence from the signals. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Frequency [Hz] Figure 412. Coherence plot for tests with water The transfer function and phase plot for hydraulic oil with 0.6 ml bubbles is shown in Figure 413. Pressure Transducer 1 S Pressure Transducer 2 Pressure Transducer 3 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Frequency [Hz] 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Frequency [Hz] Figure 413. Transfer function and phase plot for hydraulic oil contaminated with air bubbles The magnitude of the coherence plot is shown below in Figure 414. 0 500 1000 1500 2000 2500 3000 3500 Frequency [Hz] 4000 4500 5000 5500 6000 Figure 414. Coherence plot for hydraulic oil contaminated with air bubbles A comparison of the magnitude peaks of hydraulic oil with air bubbles and the other two fluids for pressure transducer 1 is shown in Figure 4.15. This clearly shows that when oilfilled cavity is contaminated with 0.6 ml of air (by volume), the magnitude of the transfer function at the first cavity resonance is not only reduced, but its frequency has been reduced to 380 Hz. 102 Hydraulic Oil Oil with 1% Air SWater 10 ' C o o 10 n oI / ir t 102 10 10 103 10 3Frequency [Hz] Figure 415. Comparison of transfer functions measurements of pressure with respect to force for pressure transducer 1 4.2.3 Speed of Sound Measurements The fluid bulk modulus and wave speed were calculated for all the three fluids using the formula stated in Section 2.3. The values for pure hydraulic fluid are stated in Table 42. Table 42. Speed of sound measurement values for pure hydraulic oil Transducer Distance [m] Time [sec] Wave speed Actual [ fe in [ c in m/s] N/m2] 1 0.1668 .00012 1390 1.68 x 10 2 0.2945 .00022 1338 1.55 x 109 3 0.4174 .00030 1391 1.68 x 10 The density of hydraulic oil is assumed to be 870 kg/m3. The actual bulk modulus values were found to be reasonably close to the theoretical bulk modulus value of hydraulic oil, which is approximately 1.72 GPa. The speed of sound measurement values for water is shown in Table 43. Table 43. Speed of sound measurement values for water Transducer Distance [m] Time [sec] Wave speed Actual [ PF in [ c in m/s] N/m2] 1 0.1668 .00012 1283 1.64 x 10 2 0.2945 .00023 1280 1.63 x 109 3 0.4174 .00033 1265 1.60 x 10 The density of water is assumed to be 1000 kg/m3. The relative error between the average actual bulk modulus values and theoretical bulk modulus value (2.2 GPa) is 0.2636 and the percentage error is 26.36%. The values for hydraulic fluid with air bubbles are stated in Table 44. Table 44. Speed of sound measurement values for hydraulic oil with air bubbles Transducer Distance [m] Time [sec] Wave speed Actual [ PF in [ c in m/s] N/m2] 1 0.1668 .00014 1191 1.23 x 109 2 0.2945 .00025 1178 1.20 x 10 3 0.4174 .00035 1192 1.24 x 10 The presence of entrained air has clearly resulted in decreasing the bulk modulus values. Thus, the speed of sound measurement test can be used to calculate the bulk modulus of hydraulic fluids in real time with reasonable accuracy. Precise results can be achieved if the possible sources of error (26.36%) are identified and removed. Some possible sources of error for this experiment include the following: Mass of the plunger: Ideally, all the force that is provided by the impact hammer on the plunger should be transmitted to the fluid. However, some of this force is spent in accelerating the plunger. The resulting transfer functions are likely distorted due to the force signal being affected by the inertia of the plunger. Different results may be obtained if the mass of the plunger is reduced. Friction of the plunger: Some of the impact force from the modal hammer is also spent in overcoming the friction that exists between the surface of the plunger and the aluminum block. Grease reduces the force spent in overcoming the friction but does not eliminate it. This could provide another source of error. Bubble removing technique: The method adopted to ensure that the system contains pure fluid (water or hydraulic oil) is not foolproof. The approach does not provide a definite way to ensure that the system is free of all the bubbles and hence could result in causing errors in the bulk modulus value. CHAPTER 5 SUMMARY AND CONCLUSIONS This work presents a novel sensing technique to determine the bulk modulus of a fluid or hydraulic system. Within this Section, a summary of the work and the conclusions drawn from it are presented. Improvements and modifications to the existing experiment are suggested. 5.1 Summary and Conclusions The work investigated three different strategies to determine the bulk modulus of a fluid within a system. The first approach was to develop a theoretical model to extract the bulk modulus of the fluid system by knowing the excitation voltage and measuring the strain. The results indicate that matching the stiffness of the actuator to the stiffness of the fluidic system is critical in obtaining a high sensitivity to the bulk modulus measurement. The second approach determines the frequency response functions by performing transfer function measurements using an impulse response test. In this test, the transfer function of the pressure response with respect to the applied force is measured. By doing so it is possible to extract information about the property (i.e. bulk modulus) of a fluid. The tests were performed on three different fluids: water, hydraulic oil and hydraulic oil with bubbles. The results indicate that magnitudes of the peaks (at 1400 Hz) were larger and sharper for water compared to oil. Also, the magnitude of the peaks (at 1400 Hz) in the case of hydraulic oil with bubbles was not only reduced but they also occurred at a lower frequency compared to the other two fluids. The third approach uses speed of sound measurements to determine the bulk modulus of the fluid in real time. The results indicate the theoretical values are reasonably close to the actual bulk modulus values. Also, the hydraulic oil with bubbles has a lower bulk modulus value compared with the pure hydraulic oil. 5.2 Future Work One improvement to the present system would be to implement a new design for the test apparatus to remove air from the chamber prior to filling the block with fluid. This would ensure the system is perfectly free of air bubbles except when it is chosen to artificially introduce bubbles for testing. Also, the new design could make use of valves and pipes as would be found in a typical hydraulic system. Another improvement would be to replace the plunger with an actual actuator/sensor. The deformation of the actuator in response to the applied voltage can be measured by attaching a fiberoptic strain sensor to the piezoelectric stack. The resulting volume change of the system can therefore be determined. The pressure change can be measured by pressure transducers. For the experiment performed, there are several potential sources of error. The mass of the plunger used to transmit pressure to the fluid moves during the impact. The impact provided by the modal hammer is measured by the force transducer. The force signal is affected by the inertial force that the plunger provides to the force transducer and the frictional force that the Orings induce on the plunger. These factors can influence the force measurement and the resulting transfer functions. One possible way to reduce these effects is to reduce the mass and friction of the plunger. Another possible source of error is the imperfect removal of bubbles in cavity. It should be possible to 48 eliminate any remaining gas by evacuating the cavity prior to testing. Modifications to the test apparatus should improve the result. APPENDIX MATLAB CODES clear clc % Program that computes the displacement of the actuator for the ISAFluid model %Variables for ISA Lp 0.1; Dp =0.001; Ap = (pi Dp2)/4; Df 0.01; Af (pi D22)/4; Lf 0.5; (Induced Strain Actuator) % Length of the PZT, m % Diameter of the PZT, m % CrossSectional area ofISA, m2 % Diameter of the Fluid, m % CrossSectional area offluid, m2 % Length ofthefluid, m %Physicalproperties for PSI5HS4ENH from Piezo Systems Inc. % http://www.piezo.com/enus/dept_10 O.html Ep =5e0; d33 650e12; Depoling Field = 3e5; t = 250/DepolingField; Voltage =0:10:250; % Modulus ofElasticity of the PZT, Pa % Dielectric coefficient m/v = C/N % V/m % Thickness of the PZT, m %V %Variables for the fluid Eflud= 0:0.1e9:2e9; %Program begins for i = :length (Voltage); forj = :length (Efluid); x(i,j)= (Ap*Ep*d33*Voltage(1,i)/t)/(Efluld(l,j)*Af/Lf + Ap*Ep/Lp); end end Figure (1) Surf (Efluid, Voltage, x) xlabel ('Effective Bulk Modulus [Pa]') ylabel ('Voltage [V]') zlabel ('PZT Actuator Displacement [m]') view ([14,32]) clear clc % Program that computes bulk modulus for the ISAfluid model % Variables for ISA Lp =0.1; % Length of the PZT, m Dp = 0.001; % Diameter of the PZT, m Ap = (pi DpA2)/4; % CrossSectional area of SA, m2 Df 0.01; % Diameter of the Fluid, m Af = (pi DfA2)/4; % CrossSectional area offluid, m2 Lf= 0.5; % Length ofthefluid, m %Physicalproperties for PSISHS4ENH from Piezo Systems Inc. % http://www.piezo.com/enus/dept_10 .html Ep = 5e10; %Modulus ofElasticity of the PZT, Pa d33 = 650e12; %Dielectric coefficient m/v = C/N VoltageM,= 250; % V depolingField= 3e5; %V/m t = VoltageMx/depolingField; %Thickness of the PZT, m Voltage= 0: 5: VoltageMax; steps= 40; Bmx 2.2e9; xf= d33*VoltageMax*Lp/t xs =xf/200; deltaLisA= Xs:(xfXs)/steps:xf; %Program begins for i = :length(Voltage); forj = :length(deltaLisA); B2(ij)(Lf*Ap*Ep/Af)*( d33*Voltage(1,i)/(deltaLsA(1,j)*t) 1/Lp ); % Make the bulk modulus values equal to zero if they are negative. if B2(ij)<0 B2(ij)=0; elseif B2(ij)>Bmx B2(i,j)=Bmx; end end end Figure (2) surf (deltaLisA, Voltage,B2) ylabel ('Voltage [V]') xlabel ('Actuator Displacement [m]') zlabel ('Effective Bulk Modulus [Pa]') colorbar clear clc % Program that computes the displacement response of a PZT actuator % Fluid loading for a variety of diameters. Diam= le3*[0.05:.05:2 2:.2:5]; %Diameter of the actuator, m Df 0.01; % Diameter of the fluid, m Af= pi*D(^2/4; % CrossSectional area, m2 Ap pi*Dp^2/4; %CrossSectional Area Lp =0.1; % Length of the PZT, m Lf = 0.5; % Length ofthe fluid, m %Physicalproperties for PSI5HS4ENH from Piezo Systems Inc. % http://www.piezo.com/enus/dept_10 O.html Ep = 5e10; OModulus ofElasticity of the PZT, Pa d33 = 650e12; %Dielectric coefficient m/v = C/N Voltage= 250; %V depolingField= 3e5; %V/m t = Voltage/depolingField; %Thickness of the PZT B= 0:.05e9:2e9; %N/m2 ii 1:1:length(Diam) Dp=Diam(ii); num=Ap*Ep*d33*Voltage; j= l:length(B); x(j,ii)=num/(t*(B(j)*Af/Lf+ Ap*Ep/Lp)); Figure(3) surf(B,Diam,x) ylabel ('Diameter [m]') xlabel ('Bulk Modulus [Pa]') zlabel ('PZT Actuator Displacement [m]') view([37,20]) LIST OF REFERENCES Abbott, R. D., McLain, T. W., and Beard, R. 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Watton, J., 1989, Fluid Power Systems Modehng, Simulation, Analog and Microcomputer Control, Prentice Hall, New York, pp. 28. BIOGRAPHICAL SKETCH The author was born in 1979 in Bangalore, India. He graduated with a Bachelor of Science in mechanical engineering degree in May 2000 from the University of Madras, Chennai, India. He then obtained Master of Science degrees in mechanical engineering and management in December 2003 from the University of Florida. 