Thermal deformation compensation of a composite beam using piezoelectric actuators

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Smart Mater. Struct. 13 (24) 3 37 SMART MATERIALS AND STRUCTURES PII: S (4) Thermal deformation compensation of a composite beam using piezoelectric actuators GSong 1,XZhou 1 and W Binienda 2 1 Smart Materials and Structures Laboratory, Department of Mechanical Engineering, University of Akron, OH 44325, USA 2 Department of Civil Engineering, University of Akron, OH 44325, USA Received 25 February 23, in final form 2 September 23 Published 26 November 23 Online at stacks.iop.org/sms/13/3 (DOI: 1.188/ /13/1/4) Abstract Maintaining the surface shape of precision structures such as spacecraft antenna reflectors has been a challenging task. Surface errors are often introduced by thermal distortions due to temperature differences. This paper presents numerical and experimental results of active compensation of thermal deformation of a composite beam using piezoelectric ceramic actuators. To generate thermal distortion of the composite beam, two film heaters are bonded to only one side of the beam using thermally conductive materials. To correct thermal deformation caused by the film heaters, PZT (lead zirconate titanate), a type of a piezoelectric ceramic material, is used in the form of patches as actuators. These PZT patches are bonded on the other side of the beam. First, finite-element analyses are conducted with consideration of the coupled effects of structural, electric and thermal fields on the composite beam. These analyses include static coupled field modeling of the beam deformation with PZT actuation, transient modeling of the beam under thermal loading, and static coupled field modeling of the composite beam with thermal distortion and simultaneous PZT actuation to correct this distortion. Then, experiments are conducted to study the thermal effect, the PZT actuation effect and active thermal distortion compensation using PZT actuators with a proportional, integral and derivative feedback controller. Finite-element modeling and experimental results agree well and demonstrate that the proposed method can actively perform structural shape control in the presence of thermal distortion. (Some figures in this article are in colour only in the electronic version) 1. Introduction Shape control of precision structures such as aerospace antennas is receiving increasing attention. With the development of smart materials and structures, this technology has been used for shape control of structures and research in this area has been active. For example, shape memory alloy (SMA) wires and piezoelectric actuators can be bonded onto a structure to provide an active force to control the shape of the structure. Bruchet al [1] studied optimal piezoactuator locations/lengths and applied voltage for shape control of beams. Oh et al [2] studied active shape control of a double-plate structure using piezoceramics and SMA wires, and Achuthan et al [3] dealt withshape control of coupled nonlinear piezoelectric beams. In the field of shape control, thermal deformation compensation is receiving a great deal of attention. The coupled effect of elastic, electric and thermal fields makes the analysis of thermal deformation compensation more complex. Many coupled thermal piezoelectric mechanical models have been developed. Bao et al [4] investigated nonlinear static deflections, dynamic characteristics, temperature effects and control characteristics of a simply supported piezothermoelastic laminated beam with a large initial nonlinear static deflection. Shen and Kuang [5] extended Reddy s third-order shear deformation theory to encompass piezothermoelastic laminated plates. Gu et al [6] developed a higher-order temperature field theory to accurately model the /4/13+8$3. 24 IOP Publishing Ltd Printed in the UK 3

2 Thermal deformation compensation of a composite beam using piezoelectric actuators Table 1. Material properties of the composite beam. Elastic Shear moduli Thermal Heat Thermal expansion moduli (GPa) (GPa) Poisson s ratio Density, ρ conductivity, K capacity, C coeff. (µm m 1 C 1 ) E 11 = E 22 E 33 G 12 G 13 = G 23 ν 12 ν 13 = ν 23 (kg m 3 ) (W m 1 C 1 ) (J kg 1 C 1 ) α 11 = α 22 α temperature distribution in laminated structures, and Ashida and Tauchert [7] proposed a general solution procedure for plane-stress problems of circular plates constructed from piezothermoelastic material. Though both theoretical and numerical research in this area has been reported, experimental testing and verification are rarely reported. In this paper, lead zirconate titanate (PZT) patches are used as actuators to compensate for the thermal deformation of a composite beam undergoing thermal loading. Both finiteelement analyses and experiments are performed. Finiteelement analyses are carried out using the commercial finiteelement code ANSYS. First, static piezoelectric analysis is developed for the coupling of structural and electric fields. Then, transient thermal analysis is performed to see how the thermal field is going to change the shape of the composite beam, and this analysis includes the coupling of thermal and structural fields. Also, static coupled-field analysis is performed for the composite beam with both the thermal loading introduced by the film heaters and the piezoelectric actuation. In this analysis, the model includes structural, electric and thermal fields. In the experiments, to generate thermal distortion of the composite beam, two film heaters are surface-bonded to one side of the beam using thermally conductive materials. Four PZT patches are surface-bonded on the other side of the beam. First, experiments are conducted to study the PZT actuation effect and the thermal effect. Then, active compensation for thermal deformation using PZT actuators is conducted with a proportional, integral and derivative (PID) feedback controller. The numerical and experimental results agree well and demonstrate that the proposed method can actively perform structural shape control in the presence of thermal distortion. 2. Piezoelectric constitutive relations Generally, in the Cartesian coordinate system the initial polarization direction of the piezoelectric materials is chosen to be the 3-axis or Z-axis. It is assumed that the piezoelectric material is orthotropic and that it is isotropic in the plane perpendicular to the piezopolarization. The actuation strain of piezoelectric material can be modeled like thermal strain [8]. Based on the assumption that the total strain in the actuator is the sum of mechanical strain, thermal strain and controllable actuation strain due to the electric voltage, the coupled constitutive relation for a piezoelectric actuator can be written as: ε = d c E + Sσ + α T or in matrix form as ε 1 S 11 S 12 S 13 ε 2 S 12 S 11 S 13 ε 3 S = 13 S 13 S 33 γ 23 S 44 γ 31 S 44 γ 12 S 66 d 31 α 1 d 31 { } α E1 2 d + 33 α E d T E d 15 3 σ 1 σ 2 σ 3 τ 23 τ 31 τ 12 where σ, ε and S are stress (N m 2 or lb in 2 ), strain and compliance coefficients (m 2 N 1 or in 2 lb 1 ) respectively; E and d refer to applied electric field (V m 1 or V in 1 ) and piezoelectric strain coefficients (m V 1 or in V 1 ) respectively; α and T represent thermal coefficients of expansion (K 1 or F 1 )andtemperature difference (K or F), respectively. In our case, the PZT patches are bonded on the opposite side of the beam to the heaters. Therefore, we may consider T to be approximately zero. Similarly, the coupled constitutive relation for a piezoelectric sensor is written as or in matrix form as { D1 D 2 D 3 } = + D = d d σ + ee + g T [ d15 d 15 d 31 d 31 d 33 [ e11 e 11 e 33 ]{ E1 E 2 E 3 } + { g1 g 2 g 3 ] } T σ 1 σ 2 σ 3 τ 23 τ 31 τ 12 where D denotes the electric displacement (C m 2 or C in 2 ) and e is the permittivity (F m 1 or F in 1 )ofthepiezoelectric material; g denotes thermal piezoelectric coupling constants (C m 2 K 1 or C in 2 F 1 ). 3. Finite-element simulation 3.1. The composite beam with embedded PZT patches and film heaters The composite beam considered here is made of Epoxy/7781 glass fiber. Its properties are shown in table 1. Four piezoelectric patches are surface-bonded on one side of the beam and two film heaters on the other side. Figure 1 shows the front view and side view of the composite beam with 31

3 GSong et al *33*.254 (mm 3 ) PZT 5A *44.5*.32 (mm 3 ) 44.5*44.5*.254 (mm 3 ) PZT 5H Heater Film Heaters PZT Composite Beam Figure 1. Composite beam with piezoelectric actuators (mm). Figure 2. Finite-element mesh of the beam with piezoelectric actuators. Table 2. Material properties of the piezoelectric patches. Piezoelectric strain coeff. Compliance (1 12 m 2 N 1 ) (pm V 1 ) Density, ρ Electric S 11 S 33 S 44 S 66 S 12 S 13 (kg m 3 ) permittivity, e (nf m 1 ) d 31 d 33 d 15 PZT 5A PZT 5H Table 3. Material properties of the film heaters. Thermal Heat Thermal conductivity capacity expansion coeff. Density (W m 1 C 1 ) (J kg 1 C 1 ) (µm m 1 C 1 ) (kgm 3 ) embedded piezoelectric patches and film heaters. One of the four piezoelectric patches (the one near the base of the composite beam) is PZT-5A and the other three are PZT- 5H. PZT is the most commonly used type of piezoceramic. In addition, PZT acts as a capacitive load and requires very little power in static operation. Properties of the piezoelectric patches and film heaters are given intables2and3respectively. As stated before, the poling direction of the PZT patches is in the 3-direction. Finite-element analyses are performed using the commercial software ANSYS. Since 3D eight-node coupled-field element solid5 has a thermal, piezoelectric and structural field capability, and can have nonlinear piezoelectric properties, the complete system is modeled with element solid5. The 1-, 2- and 3-directions coincide with x-, y- andz-directions respectively. Figure 2 gives the front view of the total mesh of the complete system. There are 6942 nodes and 3196 elements. In this finite-element model, the piezoelectric patches and the composite beam are assumed to be bonded together perfectly. The displacement boundary condition is to fix all the nodes on the base, where x =. Even though for most elements the dimension in the z-direction is much smaller than the other two, this should be no problem. In fact, because the 2-direction is not important, we may consider the width of the beam to be of the samedimension as its thickness. In this way, we can refine the model and have elements of more reasonable dimensions. The results of the refined model turn out to be the same as the results obtained from the one that we are showing here Static piezoelectric analysis for the coupling of structural and electric fields First, PZT actuation analysis is performed for the composite beam with four PZT actuators. As stated before, element solid5 in ANSYS has a three-dimensional thermal, piezoelectric and structural field capability. Figure 3 shows the displacement in the 3-direction of the composite beam when 12 V is applied on the PZT actuators. It should be noted that the graph plotted in ANSYS is in units of meters. Table 4 gives the relationship between the tip displacement of the beam and the applied voltage on the embedded PZT actuators. Generally, the tip displacement of the composite beam is proportional to the voltage applied on the actuators. We can see that a voltage of 12 V can produce a tip displacement of about 5 mm on the composite beam. Obviously, it can produce more displacement for an aluminum beam Transient thermal analysis for the coupling of thermal and structural fields The transient thermal analysis conducted here includes the coupling of thermal and structural fields. To match the experimental condition, we assume that a voltage of 42 V is applied tothefilmheaters for 5 s and the electric resistance of each film heater is 32 for numerical analyses. The rate of heat generation is calculated as 42 2 /(32 VH) = (W m 3 ), where VHstands for the volume of one film heater. Automatic time stepping is used in the ANSYS program. The time step size is 1 s and stepped loads are applied. The firstand second-order transient integration parameters are chosen as.5. The reference temperature is 26 Canditis assumed 32

4 Thermal deformation compensation of a composite beam using piezoelectric actuators Table 4. Relationship between the voltage and the tip displacement. Voltage (V) Tipdisplacement (mm) Figure 4. Temperature history on both sides of the beam ( C). Figure 5. Tipdisplacement history of the beam (m). Figure 3. Beam displacement in the 3-direction when v = 12 V. that the convection coefficient to air is (W m 2 K 1 ). In fact, at room temperature, for low rates of convection, radiation may contribute up to 5% of the total heat transfer. Here, for simplicity, radiation is not considered. Figure 4 shows the temperature history on both sides of the beam. The fact that the temperature on both sides increases so fast may be due to the assumption that the film heaters and the beam are perfectly bonded without considering the thermal resistance between them. This thermal resistance may be significant. Another reason may be that radiation is ignored. Figure 5 shows the tip displacement history of the beam Static coupled-field analysis including the coupled effect of structural, electric and thermal fields Finite-element analysis is also performed to study the compensation effect of the PZT patches for the deformation introduced by thermal loading on the composite beam. This static analysis includes structural, electric and thermal fields. To consider a more general case, we fix the temperature of one side of the composite beam at 45 Candtheother side at 3 C(PZTpatches are bonded on this side). The reference temperature is still 26 C. We assume that four PZT patches are embedded on the same composite beam at the same location. First, the static beam deformation is analyzed when only the thermal loading is applied. The displacement in the 3-directon along the length of the beam is shown in figure 6. It can be seen that the tip displacement of the beam is about 11.7 mm under thermal loading. Then a voltage of 15 V is applied on the PZT actuators to compensate for the thermal deformation of the composite beam. Figure 7 shows the beam displacement under both the thermal loading and the PZT actuation. Figure 8 shows a comparison of the beam displacement under thermal loading with and without compensation, which demonstrates the compensation effect very clearly. In addition, from the displacement curve of the beam with compensation, it can be seen that the part of the beam close to the base has less 33

5 GSong et al Power Amplifier for Piezo Patches Power Supply for Laser Sensor Film Heater (not showing) Thermal Couple (not showing) Piezo Patches Thermal Couple Reference Thermal Couple meter Signal Conditioners for Thermal Couples Current Amplifier for film Heater Laser Dot Laser Range Sensor Figure 6. Beam displacement in the 3-direction (m). Figure 9. Actual experimental set-up. Figure 1. Block diagramof thesystem. Figure 7. Beam displacement in the 3-direction with compensation (m). Displacement in 3-direction (mm) Without Compensation With Compensation Beam Length (mm) Figure 8. Beam displacement along the length in the 3-direction. displacement, which is due to the fact that the PZT patches are bonded close to the base. Of course, if we want to keep thewhole beam roughly straight, we may increase the applied voltage or patch more PZT actuators on the beam. 4. Experimental set-up Experiments are conducted to study the PZT actuation effect, thermal effect and active compensation of thermal deformation using PZT actuators. In ordertogenerate thermal distortion to the composite beam in the experiments, two film heaters are surface-bonded to only one side of the composite beam using thermally conductive materials. Four PZT patches are bonded on the other side of the beam as stated in section 3. Two thermocouples are used to measure the temperature. One is on the surface of the film heaters and the other is on the surface of the beam on the opposite side to the heaters. Figure 9 depicts the actual experimental set-up and figure 1 illustrates the operating block diagram of the same system. The cantilevered composite beam is highly under-damped. For its dominant first mode at 2.6 Hz, its damping ratio is.1. One of the concerns in active shape control using PZT actuators is to avoid excitation of this flexible beam. A laser range sensor is employed to detect the tip displacement of the beam PZT actuation First, experiments are conducted to study the PZT actuation effect. Voltages are applied to the four PZT patches through four power amplifiers. The tip displacement of the beam is measured by the laser range sensor. When the voltage is applied in or opposite to the poling direction, the beam is going to bend in or out, as illustrated in figure 11. Experiments are performed to bend the beam in both directions. The actuating voltage in the experiments varies from to 12 V in increments of 2 V. Figure 12 shows the experimental results in comparison with the numerical results. Two more sets of experiments are conducted to reveal the effect of hysteresis of the beam with PZT actuators. The applied voltage varies from to 12 V and then from 12 to V. Figure 13 clearly shows the hysteresis phenomenon that was not modeled 34

6 Thermal deformation compensation of a composite beam using piezoelectric actuators Figure 11. Relationship between the applied voltage and the bending direction. 7 Tip Displacement (mm) Bending out Bending in ANSYS Results Voltage applied on PZTs (Volts) Figure 12. PZT actuation effect. 6 Tip Displacement (mm) Bending out Bending in Figure 14. Temperature of the thermocouples Voltage Applied on PZTs (Volts) Figure 13. Hysteresis phenomenon. with the finite-element method. Another nonlinearity observed is that the beam does not behave the same in both bending directions Thermal effect Open-loop experiments are conducted to study the thermal effects generated by the film heaters on the composite beam. First, a voltage of 42 V is applied to the heaters for 5 s, starting at 6 s. Figure 14 shows the temperature history measured by the thermocouples. It can be seen that the temperature on the side of the heaters increases more quickly than that on the opposite side. It reaches its peak at 56 s when the power for the heaters is removed. From 6 s to 56 s the temperature difference between the two thermocouples increases up to 11 C. Figure 15 shows the tip displacement of the beam measured by the laser sensor. At first, the tip moves very quickly, which is due to rapid changing of the temperature difference. Then it moves slowly and comes backward when the power applied on the film heaters is removed at 56 s. At 9 sthe tip of the beam does not reach its equilibrium position due to the fact that a temperature difference (about 3 C) still exists. Figure 15. Tipdisplacement of the beam. For the second thermal actuation test, a voltage of 5 V is applied to the heaters for 5 s, also starting from 6 s. Similarly, figure 16 depicts the temperature curve measured by the thermocouples and figure 17 is the tip displacement of the beam measured by the laser sensor. During this experiment, the beam performs in the same way as intheprevious one Active shape control In this active shape control experiment, the PZT actuators under feedback control are used to actively compensate the thermal distortion caused by the temperature difference on the beam. A PID feedback controller is used to control the PZT actuators. The block diagram of the feedback controller is shown in figure

7 GSong et al Figure 16. Temperature of the thermocouples. Figure 17. Tipdisplacement of the beam. In this experiment, a voltage of 42 V is applied to the two film heaters for a period of 5 s, starting from 6 s. The temperatures measured by the thermocouples are shown in figure 19, which is almost the same as figure 14, as expected. Only three PZT 5H actuators are activated, and figure 2 shows the actuating voltage applied to the PZT patches. For the PID controller, P gain =.6, D gain =.25 and I gain =.1. Though no active vibration control using a PZT actuator is in place, an effort has been made to limit any rapid change in control voltage applied to the PZT actuators. This effort includes employing two low-pass filters, one used to process the laser range sensor output and the other to process the PID control output. The side-effect of these low-pass filters is a slow response of the closed-loop system. As the thermal loading is added to the beam at 6 s, deformation of the beam is initially observed as in figure 21 up to.52 mm in terms of the tip displacement. This deformation is caused by the slow response of the PZT actuation closed-loop system to avoid causing excessive vibration. Compared with the displacement in the open-loop experiment, which is also shown in figure 21, this deformation is much smaller. Starting from 12 s, the beam bends towards its desired position under control from the PZT actuators. Since the beam is highly under-damped, minor oscillations are seen from 18 to 28 s while the beam converges to its desired position. From 28 to 56 s the beam reaches its final position with an error of.2 mm in the tip displacement. During this period, the closed-loop system experiences no vibrations. This clearly demonstrates the effectiveness of the method of using PZT actuators for active compensation of thermal distortion to the beam. With the removal of the power to the film heaters at 56 s, an initial error of.58 mm is observed. Once again, this error is due to the slow response of the closed-loop system. At 85 s a steady-state position is reached with an error of.1 mm. Again, the effectiveness of the shape control method is demonstrated. 5. Conclusions In this paper, PZT patches are used as actuators to compensate for the thermal deformation of a composite beam undergoing thermal loading. Both finite-element analyses and experiments are performed. First, static piezoelectric analysis is developed for the coupling of structural and electric fields. Then, transient thermal analysis is performed to see how the thermal field is going to change the shape of the composite beam, and this analysis includes the coupling of thermal and structural fields. Also, static coupled-field analysis is performed for the composite beam with both the thermal loading introduced by the film heaters and the piezoelectric actuation. In this analysis, the model includes structural, electric and thermal fields. In the experiments, to generate thermal distortion to the composite beam, two film heaters are bonded to only one side of the beam using thermally conductive materials.to correct thermal deformation caused by the film heaters, PZT patches are used asactuators. First, experiments are conducted to study the PZT actuation effect and the thermal effect. Then, active compensation of thermal deformation using PZT actuators is conducted with a PID feedback controller. The numerical results and experimental results agree well and demonstrate Command error PID Low Pass Filter Amplifier Film Heater PZT Disturbance Beam Tip Position Low Pass Filter Sensor Figure 18. Block diagram of the feedback controller. 36

8 Thermal deformation compensation of a composite beam using piezoelectric actuators Figure 19. Temperature of the thermocouple. Figure 21. Tipdisplacement of the beam with and without control. References Figure 2. Actuating voltage of the PZTs. that the proposed method can actively perform structural shape control in the presence of thermal distortion. Acknowledgments The authors would like to acknowledge the support provided by NSFviaaCAREER grant and NASA via a cooperative grant. [1] Bruch J C Jr, Sloss J M, Adali S and Sadek I S 2 Optimal piezo-actuator locations/lengths and applied voltage for shape control of beams Smart Mater. Struct [2] Oh J T, Park H C and Hwang W 21 Active shape control of a double-plate structure using piezoceramics and SMA wires Smart Mater. Struct [3] Achuthan A, Keng A K and Ming W C 21 Shape control of coupled nonlinear piezoelectric beams Smart Mater. Struct [4] Bao Y, Tzou H S and Venkayya V B 1998 Analysis of non-linear piezothermoelastic laminated beams with electric and temperature effects J. Sound Vib [5] Shen S and Kuang Z-B 1999 An active control model of laminated piezothermoelastic plates Int. J. Solids Struct [6] Gu H, Chattopadhyay A, Li J and Zhou X 2 A higher order temperature field theory for coupled thermo-piezoelectric-mechanical modeling of smart composites Int. J. Solids Struct [7] Ashida F and Tauchert T R 21 A general plane-stress solution in cylindrical coordinates for a piezothermoelastic plate Int. J. Solids Struct [8] Chopra I 2 Smart Structures Theory (College Park, MD: University of Maryland) pp