Piezo patch sensor/actuator control of the vibrations of a cantilever under axial load

The effect of axial force in the vibration control of beams is studied by means of an integral equation formulation, which facilitates the numerical solution of the problem of finding the eigenfrequencies and eigenfunctions of a freely vibrating beam controlled by piezo patch sensors and actuators. The basic formulation of the problem is given in terms of a differential equation which is converted to an equivalent integral equation formulation by introducing an explicit Green’s function. After approximating the kernel of the integral equation in terms of Fourier series based on the eigenfrequencies of the host structure, a method of solution is outlined which consists of expressing the integral equation as an infinite series of linear equations and using a finite of number of these equations in the numerical solution. Numerical results are obtained for different values of the gain, the axial load and for three patch sizes. The first three vibration frequencies of the controlled and uncontrolled beam are given. The tip deflections of the beam for different axial loads are also plotted to compare transient behaviour.