Numerical Simulations of Cantilever Beam Response with Saturation Control and Full Piezoelectric Coupling

Numerical simulations of the response of a uniform, cantilever beam subjected to a base excitation are performed. A saturation absorber is implemented to control the beam response. As compared to previous investigations of similar configurations, this investigation fully includes the coupling between beam modes due to the presence of the piezoelectric actuators. It is shown that the coupling has a significant effect on the beam response, which is not present when modal coupling is neglected. Symbols C Beam structural damping Co Actuator curvature cij Damping matrix d31 Dielectric material constant E Young’s modulus m F Modal amplitude of excitation force hb Cantilever beam thickness hp Piezoceramic thickness I Cross-section moment of inertia kij Stiffness matrix L Beam length * Ph.D. candidate, Department of Aerospace Engineering † Associate Professor, Department of Aerospace Engineering, Associate Fellow AIAA ‡ Graduate student, Department of Engineering Science and Mechanics Copyright © 2002 by Donald L. Kunz. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. mij Inertia matrix Ma Moment induced by the actuator n Number of beam modes for calculation N Number of excitation frequencies Pj Generalized force vector S Cross-sectional area t Time vm Modal coordinates of control voltage Va Piezoelectric applied voltage wi Beam generalized coordinate W Beam displacement x Beamwise coordinate x1 Inboard end of the actuator x2 Outboard end of the actuator x3 Sensor location α First-mode nonlinear controller parameter m α Modal nonlinear controller parameters γ First-mode nonlinear plant parameter m γ Modal nonlinear plant parameters δij Dirac delta function e Strain φi Beam modeshape functions λm Controller natural frequency ρ Density μij Beam viscous damping ratio matrix m ζ Controller damping ratio Subscripts b Cantilever beam p Piezoceramic actuators Introduction In 1972, Haxton and Barr developed a passive control concept for autoparametric dynamic systems, which was based on creating an internal resonance. In their application, a damped spring-mass system was subjected to a periodic force. The response of the 2 American Institute of Aeronautics and Astronautics system was controlled by attaching a weightless cantilever beam with a tip mass to the system. Nonlinear coupling between the spring-mass system and the tip-weighted cantilever beam acted as an energy bridge that transferred energy from the spring-mass system to the beam. Other investigators have used this concept to control one-degree-of-freedom systems by implementing both mechanical and electronic controllers. While studying the pitch and roll motions of ships, Nayfeh, Mook, and Marshall discovered the saturation phenomenon and, using perturbation methods, were able to develop a theoretical explanation for the phenomenon. Since that time, the saturation phenomenon has been applied as a vibration control methodology for various one-degree-of-freedom systems, and also multi-degree-of-freedom systems. In References 4 and 5, while demonstrating the effectiveness of saturation control as a means to reduce the vibration response of a cantilever beam, the mass and stiffness of the piezoceramic actuators are neglected in the derivation of the governing equations for the coupled beam-controller system. The beam deflections are represented as an infinite series of trial functions, which are chosen to be the natural modeshapes of a uniform cantilever beam. Since the mass and stiffness of the actuators has been neglected, the trial functions are also eigenfunctions of the beam, and the equations for the deflection generalized coordinates can be uncoupled. In reality, the presence of the piezoceramic actuators introduces a nonuniformity in the beam. The question that one must ask is whether it is reasonable to neglect the mass and stiffness of the piezoceramic actuators. For an aluminum beam, such as the one considered herein and in References 4 and 5, the piezoceramic material is nearly three times as dense as aluminum, but contributes less that 3% of the total volume. The Young’s moduli of the piezoceramic and aluminum are nearly equal, but the length of the actuator is small compared to the length of the beam. However, in order to be effective, the actuator is placed where maximum curvature is expected. If the mass and Young’s modulus of the actuators are included in the derivation of the governing equations, the mass or stiffness matrices of the beam are no longer diagonal, and the deflection generalized coordinates are not uncoupled. In order to evaluate the influence of the actuators on the beam response, this investigation includes both the mass and Young’s modulus of the actuator in the governing equations. The values of the two saturation parameters, which control the nonlinear coupling between the plant and controller, are varied to determine the effectiveness of the controller when the actuator properties are included. The performance of the controller for systems that include the coupling terms are compared to one other using one-mode, twomode, and three-mode approximate solutions. Finally, results both with and without coupling are compared to determine the impact of including an accurate model of the actuators. Saturation Control Saturation control uses a combination of two secondorder nonhomogeneous systems. One can be either single or multi-degree-of-freedom system considered as a plant. Another is a singleor multi-degree-offreedom systems considered as a controller. To achieve saturation control, there are two requirements. The system must be made autoparametric by setting the natural frequency of the controller to a half of the plant natural frequency, and excited the plant close to its natural frequency. Hence, when the plant is excited sufficiently, energy from the plant can be transferred to the controller without spilling back. The plant vibration almost dies out because the nonlinear coupling terms of those systems set up an energy bridge between the plant and controller. Furthermore, saturation control has been applied to reduce the vibration of continuous beams. The basic model is a cantilever beam using a pair of piezoelectric patches as an actuator. In this analysis, the plant and controller are coupled on the right-hand side of the equations. On the left-hand side, however, the offdiagonal terms are neglected, uncoupling the beam mode by the reason of considering the piezoelectric patches weightless. This paper also shows that plant response does not spill over to other modes when the mass and Young’s modulus of the piezoelectric patches are neglected, and only one controller is applied. Controller L x1 x2 x hp 2hb Actuator