| Abstract: | This paper is concerned with establishing error bounds on natural frequencies of a discrete structural model using the MacNeal–Rubin Component Mode Synthesis (CMS) method. A method is proposed whereby lower bounds to natural frequencies of a finite element model with compatible elements can be obtained after upper bounds have been calculated with the CMS technique. This method uses the idea that the inertial contribution from the neglected high frequency component modes can be expressed as a function of a residual approximation frequency. Increasing the residual approximation frequency raises the effective inertia added to the system, which in turn decreases the system natural frequencies. The technique is formulated for discrete, undamped systems, and is applied to finite element models of a beam and a container ship. It is also shown that for a particular mode of the system, estimates of the residual approximation frequency can be improved using Newton's method. Thus, a frequency determined with CMS can be steadily improved until it converges to the frequency of the model. Numerical results for the finite element models indicate that convergence is fast, even when the initial error is large. |
