Vibration Control of a Mechanical Structure with Piezoelectric Actuators - a Comparison of Bimorph and Stack Actuators

The practical results of the vibration suppression of a cantilever beam with a low voltage piezoelectric bimorph patch and a stack actuator are discussed. After the equation of motion of the controlled system is obtained by the finite element method, a modal analysis is performed to reduce the numbers of degrees of freedom. A modal linear quadratic feedback controller is realized on a real-time digital control system to damp the first five modes of the mechanical structure. Optimal placement of the actuators is considered by the mode shapes and strain distribution. Comparison of the time history of an impulse excitation and the frequency spectrum shows the control effectiveness of both actuator types. Because of the advantageous actuator position, the stack actuator is better suited to control higher frequencies. Introduction Vibration control can be achieved by either passive or active methods. Passive vibration control approaches suffer from being ineffective at low frequencies. On the other hand, active control approaches provide numerous advantages; e.g., reduction of size and weight, programmable flexibility of design. Actuators based on the piezoelectric effect are suitable for active vibration control because of their wide frequency range and relatively high actuation forces. The aim of this study is to compare the effectiveness of two types of commonly used piezoelectric actuators. Stack actuators are mainly employed in the control of large space structures like satellites [1]. Bimorph piezoelectric patches applied on a mechanical structure like a beam or a plate proved to be very effective in vibration control [2]. But there are restrictions in positioning these actuators, so that the performance may decrease. Mathematical Model The equation of motion of the controlled system is obtained by the finite-element-method Mx Cx Kx Ef Du ( ) ( ) ( ) ( ) ( ) t t t t t + + = + . The positions and numbers of the excitation forces f(t) and the control forces u(t) are represented by the matrices E and D. To transform the physical coordinates of the mechanical system into decoupled modal coordinates q(t), a modal analysis is performed x q ( ) ( ) t t = Φ ( ) ( ) ( ) ( ) ( ) q q q Ef Du t t t t t T T + + = + diag(2 ) diag( ) i i 2 δ ω Φ Φ . To realize the controller in real-time, a reduction of the number of degrees of freedom has to be made. Therefore, the system is modeled with the first five modes, which results in a sufficiently accurate plant model. Transformation into discrete state space formulation z q q ( ) ( ) ( ) k k k =       with a sample frequency of 10kHz leads to z A z H f B u ( ) ( ) ( ) ( ) k k k k + = + + 1 Dis Dis Dis [ ] y C C qq C z 1 2 ( ) k k k =       = Φ Φ ( ) ( ) (k) . This model is updated by an identification process to verify the resonance frequencies, damping ratios and voltage-force ratios of the actuators. Piezoelectric actuators generate the required control forces or moments and strain gauges are employed as sensors to measure the bending motion of the beam. A real-time digital control system board is implemented into a Personal Computer, which interfaces with the strain gauge amplifiers via the A/D converters on one side and with the actuator amplifier via the D/A converter on the other side. A Kalman-modal state observer estimates the states of the system ( ) z k , which are fed back by a modal linear quadratic state controller u Gz ( ) ( ) k k = − . Piezoelectric Elements Piezoelectric patches are bending actuators, which can be applied on the surface of the structure with a glue layer. The free deflection of this actuator is much higher than the free deflection of the stack actuator, but the blocked force is rather low (see Tab. 1) [3]. When a bimorph actuator is bonded symmetrically on both sides of the structure, a pure bending moment at both ends of the actuator is generated, which is proportional to the applied voltage (see Fig. 1). The effectiveness of this actuator type is also proportional to the length of the actuator [4]. The low voltage piezoelectric stack is a multilayer ceramic construction, which delivers low motion and high force.