Integral equation approach for piezo patch vibration control of beams with various types of damping

The Euler–Bernoulli model of transverse vibrations of a cantilever beam is extended to include viscous and Kelvin–Voigt (strain rate) damping to study active vibration control under damping. The controls are of feedback type with displacement and velocity feedback controls provided by full length or patch piezoelectric actuators and sensors bonded to the top and the bottom of the beam. The resulting partial differential equation which contains discontinuities in a differential equation setting due to patch actuators is solved by formulating the problem in an integral equation setting which eliminates the singularities. The solution is reduced to the solution of a linear system of equations by using an orthonormal set of the eigenfunctions of the freely vibrating beam which provide an accurate approximation to the controlled vibration case. Control effectiveness is investigated in terms of changes in the damped natural frequencies for different gains, damping coefficients, and patch locations. The effect of the displacement and velocity feedback controls and problem parameters on the real and complex parts of the fundamental, second and third order frequencies is discussed with a view towards establishing the damped and oscillatory behaviour of the piezo controlled beams.