Control of thermally induced vibrations using smart structures D.J. Inman,* R.W. Rietz," R.C. Wetherhold* "Department ofengineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA ^Delphi Harrison Thermal Systems, General Motor Corporation, 200 Upper Mountain Road, Lockport, NY 14094, USA 'Department of Mechanical and Aerospace Engineering, State University ofnew York at Buffalo, NY 14260-4400, USA Abstract Thermally induced vibrations of a smart beam are investigated. The smart beam refers to an aluminum structure with an integrated active control system consisting of piezoceramic sensors and actuators. A simply supported aluminum beam is used to illustrate the nature of thermally induced vibrations caused by a suddenly applied heat flux. A distributed piezoelectric sensor/actuator pair is used to actively suppress the motion caused by thermal disturbances. The effect of large temperature changes on the structure and actuator are taken into account in the control system design. An optimization routine is used to find a suitable placement for the sensor/actuator pair. Assuming full-state feedback, an LQR solution is obtained for vibration suppression. A classical control solution using PD feedback is also given. Simulation results show that thermally induced vibrations can be controlled using currently available smart structure technology, provided temperature effects are properly accounted for. 1 Introduction The problems associated with thermally induced vibrations in space structures are well documented.**'*'*'*'* Since the advent of space exploration, thermal disturbances have been linked to serious reductions in performance for certain spacecraft as well as the loss of spacecraft due to uncontrolled thermally induced oscillations. Unmodeled thermal effects are also believed to be the primary cause of many unexplained spacecraft failures.*^
4 Structures in Space Thus there is sufficient motivation to study the active control of thermally induced vibrations.^ The motivation for considering the use of the smart structures approach to controlling thermally induced vibrations lies in the unobtrusive nature of embedded and/or surface mounted piezoceramic elements. Compared to other choices of actuators, piezoceramic devices are relatively light weight, have reasonable power requirements^ and are spatially much smaller than traditional actuators used in vibration suppression such as proof mass or reaction mass actuators. Much literature exists on the use of piezoceramics to control a variety of structures.^ The results available include optimal actuator placement, modeling and comparison of various control strategies. Here we examine these results in the presence of thermal excitations. The temperature effects considered here neglect the direct thermoelastic coupling and thus we focus on the uncoupled heat conduction analysis. This essentially results from the assumption that temperature variations due to changes in strain are small compared to the relative magnitude of the thermal load. With this assumption, the coupling between temperature changes and the vibratory response of a structure results from the explicit dependence of the thermal boundary value problem on the structural deformation. The basic problem of interest here is illustrated in figure 1. A thermal disturbance is applied to one side cf a beam causing it to deform and vibrate. The vibration of the beam, made of a "smart structure," is sensed by an embedded or surface mounted self sensing actuator connected to an distributed actuator pair Figure 1. A smart beam with self sensing actuators located in the region 0<Li<z<L2<I/ under the influence of a thermal disturbance.
Structures in Space 5 active control circuit. The vibrations are then canceled by the command given by the closed loop circuit to the piezoceramic self sensing actuator system. The result is a structure which can withstand a thermal load without excessive vibration. 2 Models The thermal elastic effects considered here are limited to a linear theory. This requires the following basic assumptions: direct thermal-mechanical coupling is negligible small strain the structure posses constant (temperature-independent) properties The first of these assumptions implies that the temperature distribution in the body can be determined independently from the structural vibration and that the temperature of the structure does not change as a result of its deformation. Small strain is assumed, along with the usual assumptions of an Euler-Bernoulli beam. The last assumption implies that the equations of equilibrium are satisfied throughout, linear elastic stress-strain relations hold and that body continuity is maintained. This last assumption also requires that temperature variation and stresses remain small and that the structure returns to its original equilibrium shape when applied loads are reduced to zero. In particular then, the following does not apply at high temperatures and/or for high stress loadings. Here direct thermal elastic coupling refers to strain variations that are directly coupled to the structure temperature variations. This leads to an irreversible thermoelastic dissipation which we will not consider. Rather we use the uncoupled heat conduction analysis as described in Boley and Weiner/ which is referred to as thermal-structural coupling. Thermalstructural coupling occurs through boundary conditions such that structural vibrations or deformations affect the heat transfer to the structure and vice versa. Figure 2 illustrates this situation,, While a linearization of a seriously nonlinear problem, this model does capture the behavior of structural deformation growing without bound because of a thermal input (Foster and Thornton?). This occurs even with the assumption that the thermal and structural responses can be determined independently. 2.1 The Heat Conduction Model The heat conduction model used here is fairly simple and straightforward. Let T(x, y, t) indicate the temperature as a function of elapsed time t, depth of the beam y and the length of the beam x. Then for the "top" surface of the beam exposed to a heat flux q and the bottom surface insulated, the governing relationship is
6 Structures in Space dt(x,y,t) pc\ dy* ) dt dt 0 ^(x,y U AC,f) ay'*--* dy '%/= T(x,y,0) = Here, y is the depth of the beam running between h/2 < y < A/2, h being the beam thickness, A; is the thermal conductivity of the structure, <7o is a constant magnitude (uniform) heat flux, p is the mass density, c is the specific heat of the beam and d denote the usual partial derivatives. The coefficient K = k/pc is the thermal diffusivity. Following Boley and Weiner,* the temperature distribution T(x,y,t) gives rise to a moment applied to the beam. This thermal moment, denoted by MT(X, t), is distributed along the length of the beam, and is defined to be (1) MT(X, t) = Ea&T(x, y, t)yda (2) where AT(z, %/, t) denotes the change in temperature distribution, A is the cross sectional area of differential definition da = dxdy, E is the beam's modulus of elasticity and a is the coefficient of thermal expansion for the beam. As will become clear in the following sections, because of the initial assumptions, the thermal moment will only affect the boundary conditions of the Euler-Bernoulli beam equation. 2.2 Self-Sensing Actuators The "smart" structure aspect of the system proposed here refers simply to the use of a piezoceramic sensor actuator system and corresponding active control system. The piezoceramic system is described in this section and the control law is described in section 4. The piezoelectric sensor and deformed beam undeformed beam Figure 2. Structural deformation causes a change in the angle of incidence, 0, of the heat flux q. Here 6 changes </> as the beam deflects.
Structures in Space 7 actuator system is described by Dosch et al.^, and is modeled as applying a bending moment, Ma(z,t), described by where Va(t) is the voltage (command) applied to the piezoceramic selfsensing actuator as determined by a given feedback control law (described in the following section). Here h(x) denotes the Heaviside step function used to model the placement of the piezoceramics along the beam and Ka = bd$\ep(ta + 4) is an actuator constant, characteristic of the piezoceramic. Here, 6 is the width of the piezoelectric patch, d^\ is the relevant piezoelectric constant, Ep is the elastic modulus of the piezoceramic, and Zo and tb are the thickness of the piezoceramic actuator and the beam respectively. The details and derivation of this distributed moment can be found in Fanson and Caughey^ and its limitations are discussed in Crawley and Anderson^. The sensor equation associated with the self sensing actuator system can be characterized by the voltage Va(t) developed across the piezoceramic and is %,(*) = #,[w*(z2,z) - w=(zi,z)] (4) where tu% denotes the partial derivative of w with respect to x and is the slope of the beam at the specified point, and the sensor constant Kg is defined by (3) K. = (5) Here C* is the capacitance of the piezoceramic material and T/C is the perpendicular distance from the beam's neutral axis. In active control, the sensor voltage Vg(t) is used to determine the vibration of the beam. This signal is used in the development of the control law of section 4 to provide feedback to determine the voltage Va(t) to apply the beam to suppress unwanted vibration. The self-sensing actuation circuit (Dosch et al.^) describes the details of this arrangement which forms a perfectly collocated control law. 2.3 Structural Model Following the standard Hamilton's principle, the equations of motion and admissible boundary conditions for an Euler-Bernoulli beam with piezoceramic actuator and applied thermal moments are derived. A Lagrangian containing the thermal moment and piezoceramic actuator is formulated. Under the standard geometric assumptions the kinetic energy of the system (neglecting the actuator mass) is '*
8 Structures in Space where w(x, t) is the beam deflection in the ^/-direction. Using a constitutive law that includes the change in temperature, the potential energy becomes The effect of the piezoceramic actuator is modeled by taking the variation of the nonconservative work term where A(x) = h(x x\) h(x x%) denotes the location of the actuator. Substitution of these last three expressions into the Lagrangian results in the following equation of motion and corresponding admissible boundary conditions: (8) with boundary conditions either and either r\2 f\ EI-^ = -Mr - MaA(x) or^ = 0 (10) ox* ox The boundary conditions are, as usual, a condition on moment or slope, or on shear or displacement. Equations (9), (10) and (11) are those governing the behavior of an uncoupled thermal-structural analysis of a beam. Under the assumptions stated above the thermal moment is calculated independent of the beam equation using equations (1) and (2). The actuator moment is calculated based on the sensor signal and control law described in the following sections. 3 Thermal effects on the sensor The major difficulty in using a smart structure approach to control thermally induced structural vibration under the limiting assumptions stated above is accounting for the effect of changes in temperature on the piezoceramic. There is a pyroelectric coupling such that a temperature change produces an electric displacement. In lead zirconate titanates (used in the simulation to follow), the charge generated by a temperature change is approximately 2 x 10* coulombs/m^/f and is linear in our range of interest (-50 C <T< 200 C, typically experienced by orbiting satellites). As
Structures in Space 9 long as the maximum temperature is below the Curie temperature (temperature at which the crystal structure changes and becomes symmetric, which is about 350 C) the voltage induced across the piezoceramic is (2x (12) If the temperature is measured independently, the thermally induced electric displacement (producing the voltage Vp across the piezoceramic wafer) can be subtracted out. Figure 3 is a plot of the piezoelectric voltage versus temperature (using a 20 C reference point) for the particular PZT sensor used in the control simulations that follow. 100 CO SENSOR ACTUATION Figure 3. Piezoelectric voltage versus temperature and the variation with temperature The other affect present is the possibility of the d$\ piezoelectric constant changing with temperature. This is known to change ±4% in the temperature range of interest: -50 < T < 200 C, which turns out not to affect the control design or the closed loop behavior. However, the temperature affect on the voltage must be accounted for in any closed loop design because if it is not accounted for, the control system will see the temperature induced change in voltage as a structural motion and attempt to compensate for it. 4 Control formulation For the closed loop control formulation a simply supported beam is considered which is initially at rest at 0 C. A constant uniform heat flux is suddenly applied to the top surface of the beam, assuming the bottom surface is insulated. This scenario is motivated by satellites moving from dark to direct sunlight. The equation of motion as described above is nondimensionalized in time and length by introducing r = # and = f
10 Structures in Space An assumed modes method is used by writing the solution as afinitesum of modes of the simply supported beam (0j(f) = \/2sin j?r, j = 1,2,...oo). Using thefirstfivemodes of equation (9) and transforming the system into modal coordinates yields (see table 1 for physical parameter values) Pi(t) + B*kuH(t) = mr9i + fai (13) where the overdot represents differentiation with respect to the nondimensional time variable r, B* = ( ) (^j), *, = (ur)* and 9i = 7^J3 ((-1) - 1)' /<>«= /, &(0/a(& r)df (14) Here /a(,t) = S^(Ma(r)A(0) (15) where A is the nondimensional version of A giving the actuator location by A(f) = h( - i)-h( -&). Here (i = ^J- and& = 7*- The nondimensional moment supplied by the actuator is given by *** -«) d6) The nondimensional thermal moment my calculated using equation (1), is found to be The nondimensionalized sensor voltage from the self-sensing actuator is Va(r) = ks ( ** ) {^-(6) - <t>'i(f.i)\pi(t) - (6 - (18) An artificial damping term is added to the beam equation of the form D = 0.01M +.OOOltf. Using thefirstfivemodes, the equation of motion becomes where '? + Dp + Kp = gmr(r) -h ^mo(r) (19) ^ = (B7r)*dtap(l, 16,81,256,625), g = [-0.09122 0-0.003379 0-0.0007298]^ (20) and the vector f«is defined by fi = F*i7r\/2(cosi7rf2 -cosurfi) z = 1,2,3,4,5 (21)
Structures in Space 11 Here the f% and 2 are determined by the position of and length of the actuator. Next the sensor voltage is used to generate closed loop control studies for the response of the pinned-pinned beam to a thermal load as described here. The load is a step thermal input, and the nondimensional initial conditions are a zero initial displacement and a specified initial velocity given by p(0)=w2[-l/16 0-1/432 0-1/2000]? to account for the change of variables used to transform equation (9) to an equation with homogeneous boundary condtitions (details may be found in Rietz^, p. 54). Table 1 Physical parameters of the system used for simulation (6061-T6 Aluminum) (0 C < T < 100 C) Modulus of Elasticity Density Thermal Conductivity Thermal Diffusivity Coefficient of Thermal Expansion E (Nm~*) k (Wnf'K-*) K (mv*) a (K~*) 69 x 10* 2700 170 6.4 x 24 x 10-6 Beam Parameters 6(m) 0.02 h(m) 0.002 L(m) 0.5 B 0.8542 Piezoceramic Parameters p = 0.275 x 10-*F =.02m =.05m = 0.225m = 2.54 x lo-^m Eg = 6.3 x 10* N/m* 5 Closed loop results The above formulation can be used to discuss a variety of control strategies. A standard PD control is implemented which feeds back the position and the derivative. Likewise a standard LQR control is implemented. As expected, LQR control produces a faster response, bringing the system to rest in its thermally loaded static rest position in just under 0.9 sec. This response is illustrated in figure 4. Of course LQR requires full state feed-
12 Structures in Space back. The more practical case of PD control requires a little over 1 second to reach steady state as shown in figure 5.. 0.4 Oj 0.6 0.7 0.8 0.9 Time (sec) Figure 4 Deflection of the center of a simply supported beam versus time m the presence of a thermal load. The dashed line is uncontrolled, the solid line is the closed loop response with an LQR controller 0.7._. 0.6 I 05 a 0.4 3 03.1 & 8i 0.2 o.i * 0-0.1 0.1 0.2 0.3 0.4 OJ 0.6 0.7 0.8 0.9 Time (sec) Figure 5 Deflection of the center of a simply supported beam versus time «1WI L^T f \^ *1 I*"*. The dashed line is uncontrolled, the solid line is the closed loop response with a PD controller
Structures in Space 13 Even though LQR gives better performance we continue discussing PD control, as LQR would require five actuators, or five piezoceramic patches, to implement in this case when PD control requires only one. Next consider the temperature effects on the closed loop system. This is described pictorially by the block diagram of figure 6 and the response curves of figure 7. In figure 6 the block diagram of a closed loop, PD controller with temperature compensator is illustrated. In the temperature compensator block the pyroelectric effect is subtracted from the sensor output voltage. As a result, the output of the temperature compensator block is a conditioned sensor signal which is related to the motion of the beam. Figure 7 indicates the result of compensating for the temperature effects on the piezoceramics. The plot indicates clearly that without compensation for the piezoelectric effect in the piezoceramic sensor circuit, the closed loop response would seek the wrong steady state value. Effectively, the uncompensated closed loop system produces an unnecessary static deflection. thermal disturbance controlled variable feedback Figure 6 Block diagram of the temperature compensated, PD control system 6 Summary The analysis and simulation of the response of a simple "smart structure", consisting of a beam with piezoceramic sensing and actuator system, subject to a thermal disturbance has been presented. A linear model is assumed with only a partially coupled thermal elastic mechanical model. Namely, the heat flux is assumed to affect the structural model through a boundary condition only, and the temperature may be calculated independently of the structural response. In this case the closed loop control
14 Structures in Space r i closed loop without compensator 0.1 0.2 03 0.4 OJ 0.6 0.7 0.8 0.9 1 Figure 7 Time response to a thermal disturbance for open loop (dashed), PD closed loop without compensator (solid), and PD control with temperature compensation (dot/dash). system must compensate for the pyroelectric voltage generated. The temperatureeffect on the piezoceramic constant appearing in the control actuator did not significantly affect the closed loop response and could be neglected in the case presented here. An effective method for controlling temperature induced structural vibrations have been formulated which compares well with, but is more practical than those previously published.^ Acknowledgment Thefirstauthor gratefully acknowledges the support of AFOSR grant number F49620-95-1-0280 which was used to support Dr. R. W. Rietz's graduate studies. References 1. Boley, B and Weiner, J. H., 1960, Theory of Thermal Stresses, John Wiley and Sons, N Y.
Structures in Space 15 2. Bruch, J. C., Adalis, Sadek, I. S. and Sloss, J. M., 1993, "Structural Control of Thermoelastic Beams for Vibration Suppression," Journal of Thermal Stresses, Vol. 16, No. 3, pp. 249-263. 3. Crawley, E. F. and Anderson, 1990, "Details and Models of Piezoceramic Actuator of Beams," (also AIAA paper no. 89-1388) Journal of Intelligent Material Systems and Structures, Vol. 1, No. 1, pp. 4-25. 4. Dosch, J. J., Inman, D. J. and Mayne, R. W., 1994, U.S. Patent Number 5347870. 5. Dosch, J. J., Inman, D. J. and Garcia, E., 1992, "A Self Sensing Piezoelectric Actuator for Collocated Control," Journal of Intelligent Material Systems and Structures, Vol. 3, pp. 166-185. 6. Fansen, J. L. and Caughey, T. K., 1987, "Positive Position Feedback Control for Large Space Structures," Proceedings of the 28th AIAA Structures, Structural Dynamics and Materials Conference, pp. 588-598. 7. Foster, R. S. and Thornton, E. A., 1994, "An Experimental Investigation of Thermally Induced Vibration of Spacecraft Structures," Proceedings of the 35th AIAA Structures, Structural Dynamics and Materials Conference, AIAA paper # 94-1380. 8. Leo, D. J. and Inman, D. J., 1993 "Modeling and Control Simulations of a Slewing Frame Containing Active Members," Smart Materials and Structures, Vol. 2, pp. 82-95. 9. Rao, S. S. and Sunar, M., 1994, "Piezoelectricity and Its Use in Disturbance Sensing and Control of Flexible Structures: A Survey," Applied Mechanics Review, Vol. 47, No. 4, pp. 113-121. 10. Rietz, R., 1995, "Dynamics and Control of Thermally Induced Vibration," PhD Dissertation, State University of New York at Buffalo. 11. Sharkey, J. P., Nurre, G. S., Beals, G. A., and Nelson, J. D., 1992, "A Chronology of the On-Orbit Pointing Control System Changes on the Hubble Space Telescope and Associated Pointing Improvements," AIAA Paper No. 92-4618-CP, Proceedings of the 33rd Structures, Structural Dynamics, and Materials Conference, pp. 1418-1433. 12. Thornton, E. A., "Thermal Structures: Four Decades of Progress," Journal of Aircraft, Vol. 29, No., 3, pp. 485-498.
16 Structures in Space 13. Thornton, E. A., and Kim, Y. A., 1997, "Thermally Induced Bending Vibrations of a Flexible Rolled-Up Solar Array," Journal of Spacecraft and Rockets, Vol. 30, No. 4, July-August, pp. 438-448.