Influence of flexible structure vibration on the excitation forces delivered by multiple electrodynamic shakers

Multi-shaker vibration testing has gained increasing interest in recent years as researchers have demonstrated that complicated environmental response can be accurately replicated in the laboratory using multiple electrodynamic shakers. One of the crucial issues to be addressed is the discrepancy between the actual excitation forces and the required ones due to the coupling between the structure and multiple shakers. To analyze the excitation force characteristics, a coupled multi-shaker-cantilever beam system is studied and modeled using Hamiltonian principle and Galerkin's approach, and the excitation forces exerted on the beam are derived. Comparisons of experimental and numerical results indicate that the developed model can accurately predict the excitation forces under multi-point excitation. Furthermore, the relationship between the excitation force Frequency Response Function (FRF) matrix and the dynamic characteristics of the coupled system is revealed, showing that driving point FRF F_ii under multi-point excitation differs from that under single-point excitation. If the model parameters of two shakers are not perfectly identical, there will be an apparent discrepancy in amplitude-frequency characteristic between cross-point FRF F_ij and F_ji (i ≠ j), while the phase-frequency characteristics are identical. Besides, when multiple shakers simultaneously drive the structure, the excitation forces are coupled due to structure vibration. Therefore, it is necessary to consider this coupling effect when performing the multi-point excitation for flexible structure. The coupled shaker-beam model proposed in this paper can provide the basis for designing decoupling controller.