Input shaping using chebyshev polynomials to reduce vibration of flexible structures

Abstract

Residual vibration of the system, induced by its point-to-point motion, limits the performance of a flexible mechanical system to do next job. In this study a modified input shaping method using impulse sequence with variable amplitude and`` switching-time is proposed through the impulse response concept and modal filtering technique. To get the input shaping parameters of the proposed method, the information of the state transition matrix on the duration of the motion is required. A simple and efficient numerical scheme for the estimation of the state transition matrix is also presented using Chebyshev polynomials. Through an example of a linear time varying system with greatly varying parameters it is verified that the proposed method is more effective in reducing residual vibration of time varying system than the TVIS method as well as the conventional input shaping method. To demonstrate another usefulness of the proposed method, two nonlinear systems are controlled. Obtaining a linear time-varying model by perturbing the nonlinear dynamic equations about the nominal states, the proposed method can be implemented. Various error types are investigated to check the robustness of the method. From the examples it is obvious that the proposed method is effective in broader classes of flexible systems and the estimation method is efficient and useful.