Vibration Suppression of an Elastically Supported Beam With Closely Spaced Natural Frequencies

Vibration suppression of distributed parameter systems has a wide range of applications. With the help of dynamic vibration absorber (DVA), the dynamic performance of a distributed system under external excitation can be improved by tuning the absorber parameters. However, a single degree of freedom DVA is not capable of handling multiple resonant frequencies at a time, which could occur for relatively complicated mechanical systems. Although widely used in civil engineering, adoption of multiple absorbers becomes less effective for mechanical systems due to limited installation space and additional inertia. The second issue for the vibration suppression of a distributed system is the coupling between the modes with close frequencies, which prevents designers from simplifying the primary system into a single degree of freedom model and employing classic methods like the fixed-point theory. Being viable alternatively, the multivariable optimization method can cause convergence problems and become tedious. Although vibration absorber has been studied for decades, the vibration suppression of a distributed system with closely spaced frequencies has not been well addressed. Therefore, it is essentially important to develop an absorber that is capable of suppressing resonant peaks appearing in clusters.

To resolve the above issues, this paper considers a benchmark problem of a uniform Euler-Bernoulli beam with elastically supported boundary conditions. The system parameters are appropriately designed so that the first two natural frequencies of the system are closely spaced where the associated modes are more likely to be excited simultaneously. In order to achieve desired vibration reduction, a two-degree-of-freedom spring-mass-damper absorber is connected to the primary beam structure through a pair of spring and dashpot. Different from conventional methods of parameter optimization, the parameters of the absorber are determined by the approach of dynamic characteristic assignment. With the given desired performance (amplitude of receptance), one can determine the relationship between the coupling stiffness and frequencies of fixed points. By introducing a virtual spring connecting the absorber and the ground, the rest parameters of the absorber are computed by the fixed-point frequencies via assignment of characteristics. Eliminating the virtual stiffness by choosing appropriate coupling stiffness, all the parameters of the two-degree-of-freedom absorber can be determined accordingly. The proposed method is computationally efficient because no multivariable optimization is required, and all the parameters of the absorber are solved via a set of nonlinear equations through characteristic assignment. Numerical examples verify that a two-degree-of-freedom absorber is effective in suppressing double resonant peaks in a distributed system.