Research On Efficient Numerical Methods For Dynamic Problems Of Piecewise Linear Systems

Vibration of piecewise linear systems is a kind of typical non-smooth nonlinear problem,which can exhibit very complex vibrational behavior.Because the dynamic properties of the piecewise linear system are conducive to engineering design and application,the piecewise linear system is widely applied to engineering practice.In addition to the structures and systems of artificial design,the production and processing error of the structural equipment,the installation error of the device,and the wear and damage caused by the operation may lead to the existence of piecewise linear vibration.Therefore,it has theoretical significance and very important practical value to study the vibration problem of piecewise linear systems.For free vibration of a simple piecewise linear system,its dynamic responses can be accurately obtained by combination of piecewise analytic integrations.However,for its forced vibrations,only can using approximate analytical methods or numerical methods to compute its dynamic responses.Approximate analytical methods are usually used to find possible periodic solutions for weakly nonlinear systems with a few degrees of freedom.Therefore,numerical methods are more and more used for researches on piecewise linear vibration problems,especially for a piecewise linear system with a large number of degrees of freedom and strong nonlinearities.Because the piecewise linear system has different mechanical properties in different states,it is particularly important for dynamic analyses of piecewise linear systems to establish an appropriate mathematical analysis model,which can accurately determining the stress state of piecewise linear structure,and to develop a stable and efficient time-integration method.This doctoral dissertation aims at efficient numerical methods for dynamic problems of piecewise linear systems,develops efficiently numerical algorithms for dynamic analysis establishes appropriate for a tensegrity structure,a periodic piecewise linear structure and a moving vibration system problem,respectively.The main research contents are as follows:(1)By combing the precise integration method(PIM)and the Newton-Raphson scheme,an accurate and efficient numerical integration method is presented for determining the dynamic responses of a tensegrity structure.The Newton-Raphson scheme is used to determine the exact time at which the cables undergo slackening or tightening;then,the integral interval can be divided into a series of time steps without any cable slackening or tightening.Thus,the tensegrity structure can be regarded as a linear structure within a time step,which can be solved accurately via PIM.Base on the proposed method,the dynamic behaviors of the steady state motion of the tensegrity structure under a harmonic excitation are also discussed in detail,including table periodic motion,quasi-periodic motion,chaos and bifurcation behaviors.Numerical results show the accuracy and efficiency of the proposed method.(2)An efficient numerical integration method based on the parametric variational principle(PVP)is proposed for computing the dynamic response of a periodic piecewise linear system with large number of gap-activated springs.Through describing gap-activated springs by PVP,the complex nonlinear dynamic problem is transformed to a linear dynamic problem and a standard linear complementarity problem.The linear complementarity problem can be solved by using mature algorithms of quadratic programming.This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the gap-activated springs.Based on the periodicity of the structure and the finiteness of the velocity of energy propagation,an efficient PIM is developed to obtain the dynamic responses of the system.This method indicates that there are a large number of same elements and zero elements in the matrix exponential of a periodic structure,and saves calculation quantities and computer storage by avoiding repeated calculation and storage of these elements.By comparing with the Runge-Kutta method,the correctness and efficiency of the proposed method are verified.(3)PVP for the dynamic contact and nonlinear analysis of the coupled system consisting of a moving vibration system and a periodic piecewise linear structure are established,and an efficient and accurate numerical simulation method for dynamic analysis is presented.Based on PVP,formulation for dynamic contact between the moving vibration system and the periodic piecewise linear structure is established,and a corresponding algorithm is developed.This algorithm can avoid using the contact stiffness,and can accurately determine the contact state and compute the contact force between the moving vibration and the periodic piecewise linear structure.PVP for the periodic piecewise linear structure is established,and the corresponding algorithm of quadratic programming is developed.This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the piecewise linear structures.Based on the periodicity of the structure,a numerical time-integration method is developed on the basic of PIM for computing the dynamic responses of the the periodic structure under a moving force.This method is proposed on the basic of PIM,having the characteristics of good stability and high accuracy.And taking the advantage of the periodicity of the structure,the matrix exponential of only one unit cell of the periodic piecewise linear structure is computed,which greatly reduces the calculation scale,thus improves the computational efficiency and reduces the computer storage requirements.(4)A semi-anlytical method for computing the steady state interaction forces between a periodic structure and a moving vibration system with a constant speed is presented.Based on the periodicity of the steady state interactions,it is expanded into a Fourier series with undetermined coefficients.The vibration problem of the coupled system is transformed to compute steady state responses of vibration system and periodic structure under moving harmonic loads.Based on the periodicity of the structure,energy band strucure theory is combined with Fourier transform and mode superposition method,two semi-analytical algorithms are presented for determining the steady state responses of a periodic structure under a harmonic excitation moving at a constant speed.This method can directly obtain steady state interaction forces between a periodic structure and a moving vibration system with a constant speed,and avoid performing integration over a very long time.Based on the band theory of periodic structure,the natural frequencies and modes of a complete periodic structure can be obtained by using the stiffness matrix and the mass matrix of one unit cell,which improves the calculation efficiency.