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The effect of modal truncation on the damping ratio and their derivatives with respect to an added mass is investigated for a simply supported, multispan beam with a linear quadratic Gaussian control system. It is found that both the damping ratios and derivatives converge slowly, but the derivatives converge more slowly than the damping ratios. However, it is shown that when Ritz vectors corresponding to static displacements due to actuator forces are added to the reduced model, the convergence of both the damping ratios and their derivatives is accelerated. It is also shown that the accuracy of the damping ratio predicted by a reduced-model control design can be improved significantly if the Ritz vectors are included in the design of the control system. Thus, it appears that Ritz vectors added to the reduced model of flexible structures can improve greatly the accuracy of both the design and analysis of the control system.

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.

The sensitivity of control system performance to structural changes is calculated for a multi-span beam with direct-rate feedback vibration control. 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. The convergence of the derivatives of the damping ratios with increasing number of modes is shown to be slower than the convergence of the damping ratios themselves. In particular, in some cases the convergence of finite-element approximations to the derivatives is much faster than the convergence of the modal approximations. The results indicate that the use of reduced models based on natural vibration modes may be ill-advised for calculating the sensitivity of control system performance to changes in the controlled structure.