Multi-frequency Synchronization Of The Mechanical System Driven By Dual And Multiple Exciters

With the continuous development of science and technology,the requirement of machine performance is becoming higher and higher.In order to improve efficiency,it is necessary to install two or more motors to work together at the same time,it is,therefore,of great significance to study the theory of vibratory synchronization.Since the emergence of self-synchronous vibrating machines driven by two motors,many researchers have studied the coupling mechanism of exciters to achieve synchronization,and published some relevant papers.However,most of these research achievements are based on the same driving frequency,and few theoretical studies on the multi-frequency synchronization of exciters.So,this paper will extend the fundamental frequency synchronization to multi-frequency synchronization(double-frequency and triple-frequency synchronization),which achieves the purpose of dual frequency excitation for the system.Four typical dynamical models of mechanical system are proposed,the multi-frequency synchronization mechanisms of two or multiple exciters with different distribution modes,different rotational directions and other conditions are studied,which can provide a reference for the design of new dual-frequency driving vibration equipment in engineering.Specific research contents are as follows:(1)The mechanisms of double-frequency and triple-frequency synchronization for two exciters in the same direction under the far super-resonance condition are revealed.The differential equations of the system are obtained based on Lagrange equation.Using the asymptotic method yields the expression of phase relationship between two exciters,the synchronization condition of two exciters is deduced when the speed ratio is 1:2 or 1:3,respectively.And the stability condition is obtained,which is in accordance with Routh-Hurwith criterion.The dynamic characteristics of the system are numerically analyzed,the stability index is defined and its relationship with the dimensionless parameters is analyzed.The simulations are performed by applying Runge-Kutta routine,and the curves of motor speed,phase difference,displacement and plane motion trajectory at the steady-state of the rigid frame are obtained.Finally,the validity of the theoretical results is verified by experiments.(2)The double-frequency synchronization of three exciters in the same direction in the far super-resonant state is studied.Theoretically,the motion differential equation of a rigid frame dynamical model of the mechanical system driven by three exciters with linear distribution is obtained.Combining the asymptotic method and the averaging method yields the synchronization criterion and stability criterion of exciters under the double-frequency condition,the stability index is also defined.Numerically,the stability index and other characteristic curves are given,and the stability ability of the system is discussed.The comparisons and analyses of simulation and experimental results verify the validity of the theoretical results and the feasibility of the theoretical analysis methods used,which provide a theoretical guidance for the functional design of complex frequency vibration screening equipment in engineering.(3)The theory of the double-frequency and triple-frequency synchronization for the mechanical system with a rigid frame driven by four exciters rotating in reversed directions is analyzed.By Lagrange equation,the motion differential equations of the system are derived.Applying the asymptotic method yields the expressions of phase relationships among four exciters under the conditions of double-frequency and triple-frequency,and the analytical expressions of the synchronization and stability conditions for achieving synchronization among exciters are given.The dynamic characteristics of the system are numerically analyzed,which reveals the influences of structural parameters on stability of the system,and the simulation and experimental results are given to verify and modify the theoretical model.The research can provide a reference for the design of new shaker and vibration forming equipment.(4)The vibratory synchronization mechanisms under the double-frequency and triple-frequency condition of the four-motor-driven mechanical system with two rigid frames under vibration isolation are revealed.Using the asymptotic method deduces the synchronization criterion of four exciters to realize multi-frequency synchronization.The stability criterion in the synchronous state is proposed,and the results are in accordance with Routh-Hurwith criterion.Simultaneously,the stability index is defined,the stability ability of the system is discussed,and the stable phase differences among exciters are obtained.The Runge-Kutta routine is applied to simulate the system,and the correctness of the theoretical results is verified,which can provide a theoretical basis for the design of new vibration compaction equipment in engineering.