DANISH CENTER FOR APPLIED MATHEMATICS AND MECHANICS

Report No. 728 March 2007 Effective properties of mechanical systems under high-frequency excitation at multiple frequencies Jon Juel Thomsen Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Building 404, DK-2800 Lyngby, Denmark; (E-mail: jjt@mek.dtu.dk) Abstract Effects of strong high-frequency excitation at multiple frequencies (multi-HFE) are analyzed for a class of generally nonlinear systems. The effects are illustrated for a simple pendulum system with a vibrating support, and for a parametrically excited flexible beam. For the latter, theoretical predictions are supported by experimental observations, providing good agreement for a wide range of excitation conditions. The main effect of strong multi-HFE is to change the effective or apparent stiffness in a manner similar to that of mono-HFE, provided the HFE frequencies are well separated and non-resonant. Then the change in effective stiffness is proportional to the sum of squared excitation velocities, and the corresponding changes in equilibria, equilibrium stability, and natural frequencies can be computed as for the mono-HFE case. When there are two or more close excitation frequencies, an additional contribution of slowly oscillating stiffness appears. This may cause strong parametrical resonance at conditions that might not appear obvious, i.e. when the difference in two HFEfrequencies is near twice an effective system natural frequency, which due to the HFE itself is shifted away from the natural frequency without HFE. Also, it is shown that strong multi-HFE can stabilize otherwise unstable equilibria, but only if the frequencies are well separated; thus continuous broadband and random HFE does not have a stabilizing effect paralleling that of mono-HFE, or multi-HFE with non-close frequencies. The general results may be used to investigate or utilize general effects, or as a shortcut to calculate effective properties for specific systems, or to calculate averaged equations of motion that may be much faster to simulate numerically.Effects of strong high-frequency excitation at multiple frequencies (multi-HFE) are analyzed for a class of generally nonlinear systems. The effects are illustrated for a simple pendulum system with a vibrating support, and for a parametrically excited flexible beam. For the latter, theoretical predictions are supported by experimental observations, providing good agreement for a wide range of excitation conditions. The main effect of strong multi-HFE is to change the effective or apparent stiffness in a manner similar to that of mono-HFE, provided the HFE frequencies are well separated and non-resonant. Then the change in effective stiffness is proportional to the sum of squared excitation velocities, and the corresponding changes in equilibria, equilibrium stability, and natural frequencies can be computed as for the mono-HFE case. When there are two or more close excitation frequencies, an additional contribution of slowly oscillating stiffness appears. This may cause strong parametrical resonance at conditions that might not appear obvious, i.e. when the difference in two HFEfrequencies is near twice an effective system natural frequency, which due to the HFE itself is shifted away from the natural frequency without HFE. Also, it is shown that strong multi-HFE can stabilize otherwise unstable equilibria, but only if the frequencies are well separated; thus continuous broadband and random HFE does not have a stabilizing effect paralleling that of mono-HFE, or multi-HFE with non-close frequencies. The general results may be used to investigate or utilize general effects, or as a shortcut to calculate effective properties for specific systems, or to calculate averaged equations of motion that may be much faster to simulate numerically.