Critical Criterion And Mechanism Of Amplitude Death In Coupled Non-autonomous Systems With Multi-frequency Excitations

Interconnected network systems are very common in nature.They can be divided into autonomous systems and non-autonomous systems.Autonomous system means a kind of dynamic system without external excitation while non-autonomous system is a kind of dynamic system affected by external force.This paper mainly studies the amplitude death phenomenon of non-autonomous network systems.Amplitude death is a special dynamic behavior,that is,each subsystem of a non-autonomous system can collectively present a suppressed state of slight oscillation.Studying the mechanism of amplitude death is of great significance to the stability design and control of the system.Existing research on amplitude death in non-autonomous systems under single frequency excitation has been very sufficient,but for the case of multi-frequency excitation,the research about amplitude death is still blank.In order to fill the gap,this thesis focuses on the amplitude death of non-autonomous network system under multi-frequency excitation.Based on the observation that the response of the system under the state of amplitude death always corresponds to the harmonic function of the external excitation frequency,a fundamental solution for the amplitude death state is assumed.By using harmonic balance method,the fundamental solution can be determined by solving the coefficients of a set of nonlinear algebra equations.The analytical form of the fundamental solution is important because it provides the distribution of the amplitude death state in a parameter design space,very useful to the stability design of systems.Based on the basic solution and implicit function theorem,the parameter boundary of the amplitude death can be deduced,which provides theoretical basis for the stability design.Finally,a method for calculating the critical condition of amplitude death under multi-frequency excitation is proposed,and the correctness of the method is verified by the examples of single freedom Duffing system and multi-degree of freedom coupled Duffing system under multi-frequency excitation.Based on the application background of multi-module floating platform,this paper calculates the dead state parameter boundary of floating platform under multi-frequency excitation.Firstly,the network model of multi-module floating body system under multi-frequency excitation is established,including single floating body dynamic model and connector mechanical model;then the response characteristics of floating platform under multi-frequency excitation are studied,and it is found that the multi-module floating body system still has the following two response features under multi-frequency excitation:1)the response frequency of the system is equal to the external excitation frequency when the system enters the amplitude dead state;2)the amplitude death phenomenon arise from the jump between different solution branches.These two points determine whether the fundamental solution can be used to obtain the critical criterion of amplitude death.Finally,using the implicit function theory,the amplitude death area in the(ks-ξ)parameter plane is obtained and verified by numerical scanning.By assuming the fundamental solution and applying approximate analytical method,the amplitude death parameter domain of the system can get calculated quickly,which can save a lot of calculation time compared with the numerical method.At the same time,the amplitude death can provide theoretical guidance for the design and optimization of stability parameters in engineering,which has important theoretical significance and application value.