Sensitivity Analysis of Suppressing Vibration by Inducing a Node on a Harmonically Forced Euler–Bernoulli Beam

In this paper, a properly tuned vibration absorber (spring-mass system) is used to induce a point of zero vibration, or node, anywhere along an arbitrarily supported beam for the purpose of suppressing vibration. Other benefits of enforcing a node include the ability to place a sensitive instrument near or at a point where there is little or no vibration and to keep any point along the beam stationary without using a rigid support. The assumed-modes method is used to discretize the equations of motion, which conveniently leads to the beam displacement at the node location in matrix form. Exploiting the Sherman–Morrison inverse formula, one can obtain compact, closed-form expressions for the beam deflection at the node location and the displacement of the absorber mass, explicitly in terms of the absorber parameters, attachment location, excitation frequency, forcing location and the node location. The resulting expressions can then be used to systematically perform a parameter sensitivity analysis and rapid design modifications. The proposed work contributes to the parametric study of Euler–Bernoulli beams by leveraging closed-form sensitivity expressions to rapidly account for inverse problems involving parameter tolerances and perturbations. The method introduced in this paper can also be easily extended to accommodate changes in attachment location and the tolerable deflection of the absorber mass without the need to perform a potentially expensive and time-consuming re-analysis. Numerical examples are presented to illustrate the utility of using the sensitivity expressions in designing an accurate and robust vibration absorber when slight modifications are introduced.