The Inverse Method For A Class Of Nonlinear Systems Construction And Its Applications In Engineering

The resonance phenomenon has plagued the whole engineering field for nearly300 years as the main failure factor of mechanical system,since it was discovered by Euler in 1750.The low-frequency vibration isolation has inevitably become a key common problem to be solved in many engineering fields,in order to weaken or eliminate the negative effects caused by resonance.However,different application scenarios put forward a large number of differentiated requirements for the lowfrequency vibration isolation system.Such as,the design of zero-stiffness support structure for eliminating resonance completely,the design of low-frequency active vibration isolator with displacement suppression demand,and the design of nonlinear vibration isolator with multiple degree of freedom low-frequency vibration isolation ability.Therefore,a new and modular idea for the design of low-frequency vibration isolator with different requirements in various fields can be provided,by proposing a universal inverse construction method of low-frequency vibration isolator based upon complex vibration control targets.In this paper,an archetypal model with arbitrary designable features is proposed firstly,and its potential for the construct arbitrary mechanical features is studied deeply to obtain its general structural inverse construction formula.Then take the important difficulties and challenges,faced by the current low-frequency vibration isolation engineering field,as the typical cases to provide novel and efficient solutions.The main research results of this paper are as follows:An archetypal model with arbitrary designable characteristics is proposed firstly.And a general inverse method for its structure construction is formed according to the design requirements of any performance index,by mapping the mechanical characteristics of the model to the construction of the trajectory function inside the structure.The high-order quasi-zero stiffness with high-order degradation characteristics is extended and defined by the analogy of quasi-zero stiffness,and the definition of infinite order quasi-zero stiffness(or zero-stiffness)is given by approaching the degradation times to infinity.The conditions for the construction of arbitrary order quasi-zero stiffness and zero-stiffness characteristics are deeply discussed,based upon the archetypal model and its inverse construction method.The conclusion shows that the existence of a group of curves makes the solution of the archetypal model to any finite order quasi-zero stiffness structure exist and unique;For the zero-stiffness characteristic,elliptic curve and specific geometric parameter configuration are the necessary and sufficient conditions for its construction.The zero-stiffness and its disturbed system model is established based upon a pair of elliptic trajectories,and the bifurcation characteristics of the system equilibria depended on geometric parameters are analyzed.It can be known that the variation of geometric parameters will cause the system to convert among stable linear stiffness,stable first-order quasi-zero stiffness,unstable linear stiffness,unstable first-order quasi-zero stiffness and infinite order quasi-zero stiffness(zero-stiffness).The theoretical analysis results show that the zero-stiffness system shows the mechanical suspension characteristics at any position,and the resonance phenomenon is completely eliminated.The experimental results show that it is very feasible to accurately construct a passive zero-stiffness support structure without resonance by mechanical method.Therefore,this kind of system expresses unprecedented application potential in the application fields of ground simulation in zero microgravity environment,modal experiment of large flexible structure,and ultralow frequency or even zero-frequency vibration isolation,which provides a novel design idea for the construction of passive zero-stiffness support structure.An inverse design method for the restoring force function of the support system based on the complex characteristics of loads is proposed.Taking the design o f floating-slab vibration isolator of urban rail transit as an example,the inverse design is carried out for its displacement suppression ability and low-frequency vibration isolation ability simultaneously.The energy storage capacity of the support system in the unit displacement determines the displacement suppression capacity of the system,and the equivalent stiffness near the energy concentration positions of the dynamic component determines the low-frequency vibration isolation capacity of the system.The theoretical results show that the low-frequency vibration isolation performance of the single novel vibration isolator near the positions where the vibration energy is concentrated is always much better than that of the existing linear system,under any effective loading condition.The simulation results show that the displacement suppression performance of the system is significantly improved compared with the existing linear system,and the deviation between the response and the target value in the inverse design process is tiny,which indicates the feasibility of this inverse design method and the high-accuracy of the prediction for performance targets;Simultaneously,the broadband vibration isolation performance has been promoted unprecedentedly compared with the existing steel-spring scheme,through the analysis of the vibration signals of the foundation and the tunnel wall.Therefore,this novel construction method of vibration isolator can effectively solve the problem of lowfrequency passive vibration isolator represented by the design of floating-slab vibration isolator,which widely exists in the field of active vibration isolation with the demand of displacement suppression performance.The multiple degree of freedom archetypal model with arbitrary designable characteristics and its inverse construction method based upon the target restoring force functions are proposed.The definition of multiple degree of freedom zerostiffness is proposed,and the construction conditions of the zero-stiffness system are analyzed,by analogy with the definition of single degree of freedom zero-stiffness system.The conclusion shows that there are limited solutions to the construction of multiple degree of freedom zero-stiffness characteristics based upon the archetypal model;and the zero-stiffness characteristic in any translational direction can be realized by introducing plane,sphere,ellipsoid,or hyperboloid into the system,according to the ratio of the spring stiffness in each axis.The definition of multiple degree of freedom quasi-zero stiffness system is analogized and generalized,based on the definition of single degree of freedom quasi-zero stiffness system.It is proved that there are infinite solutions for the construction of multiple degree of freedo m first-order quasi-zero stiffness system based on multiple degree of freedom archetypal model.The theoretical and numerical analysis of the system with multiple degree of freedom quasi-zero stiffness is carried out,by introducing the multiple degree o f freedom quasi-zero stiffness construction conditions of the system.The conclusion shows that the low-frequency vibration isolation performance of the system in any direction is significantly improved compared with the linear system with the same support stiffness.It provides a new inspiration of the structural design for the zerostiffness support system and the high performance low-frequency isolator with multiple degree of freedom.