Active Control of Damped Sandwiched Beams

A governing equation of motion for three layer sandwiched beams is derived. The three layer sandwiched beam is fixed at one end and is excited by a harmonic force at the other end. The boundary conditions are expressed in terms of transverse displacement. The developed governing equation is a sixth order differential equation in terms of transverse displacement. The six complex roots of the complex characteristic equation are obtained by satisfying the six homogeneous boundary conditions. In this analysis displacements are computed at five equally spaced points on the beam. The frequency responses at these points are used for extracting the modal parameters. The resonant peaks in the frequency response are assumed to represents the response of a single mode. Nyquist diagram representing frequency responses are plotted. It is observed that perfect circles are obtained for all modes. The natural frequency is observed at the lowest point on the circle. Modal frequencies and the loss-factors are determined by using half power point technique. Computation indicates that damping of vibrations in the beam is possible only up to certain extent. Higher loss factors requires [sic] special materials with very high material loss-factor, which may not be available. The modal parameters extracted from the frequency at each mode are used for identifying the system and also for developing a mathematical model (transfer function) for the system. The transfer function is used. to evaluate the dynamic behavior of the system as well as to control the vibrations actively at the specified points. An Integral controller is designed according to the specified designed values. A proportional plus feedback controller is used to establish a good stability for the transient response of the system. The computed results showed that with the help of active control the possibility of excessive vibration is avoided with a good stability. The frequency response of the passively damped and actively controlled sandwiched beams for typical cases are presented