Accuracy of eigenvalue derivatives from reduced-order structural models

It is well known that the Fourier series of discontinuous functions converge slowly and that the derivatives of the series may not converge at all. Since modal expansion of structural response is a generalization of the Fourier series, we may expect slow convergence of modal expansion when the applied loads exhibit discontinuities in time or space. Thus, in a structure controlled by point actuators, we may expect slow convergence of derivatives of structural response with respect to system parameters. To demonstrate this, the sensitivity of the closed-loop response to structural changes is calculated for a multispan beam with direct-rate feedback using colocated velocity sensors and point actuators. Reduced models based on the natural modes of the structure are formed, and derivatives of the damping ratios of the closed-loop eigenvalues are calculated. As expected, the convergence of the derivatives of the damping ratios with an increasing number of modes is slower than the convergence of the damping ratios themselves. The convergence is unproved when distributed actuators replace the point actuators. In transient response problems, it is known that complementing the vibration modes with a mode representing static response to the loads can improve convergence. Indeed, for the example studied, when Ritz vectors corresponding to static responses caused by unit loads at the actuators are added to the basis vectors, the convergence of the reduced-model derivatives is greatly enhanced.