Stability and Sommerfeld effect in a multi-resonant types vibrating system with isolated rigid frame driven by four exciters

Abstract

The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange’s equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It’s shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power.