An Improved Efficient Galerkin Averaging-Incremental Harmonic Balance Method And Its Application To High-dimensional And Strongly Nonlinear Dynamic Systems

Nonlinear phenomeona of mechanical systems occurs in the geometry,structure,coupling,and many other aspects.Compared with linear systems,nonlinear systems show rich and complex behaviors,such as jump-phenomena,super and sub-harmonic resonance,bifurcation,chaos,etc.There are great intrests in seeking periodic solutions of nonlinear vibrations to ensuring a long-term stable operation of the system.The semi-analytical methods,such as the well known incremental harmonic balance(IHB)method,can be used to solve systems with strong nonlinearities,they have clear mathematical concepts and excellent accuracy,thus are widely used in both theroical analysis and enginerring applications.Unfortunatly,the application scope of various semi-analytical methods is limited to relatively lower-dimensional,simple dynamic systems.The limitation mainly occur in three aspects: Firstly,the process of traditional semianalytical methods is cumbersome and highly coupled with the equations to be solved,so the computational procedures lack generality,hence are difficult to be used to analyze complex high-dimensional models in enginerring applications;secondly,the computational speed of trditional semi-analytical methods in the literature are generally slow,thus they may not be able to obtain periodic responses of a high-dimensional nonlinear system in a reasonable time.Thirdly,for some complex dynamic models such as multibody dynamic systems,it is difficult to derive their ordinary differential equations(ODEs),on the contrary,it is much simpler to obtain their differential algebraic equations(DAEs)that cannot be solved by traditional semi-analytical methods,which also limited the types of problems they can analyze.In this work,an efficient Galerkin averaging-incremental harmonic balance(EGA-IHB)method based on tensor contraction and fast Fourier transform(FFT)is introduced to obtain steady-state periodic responses of generalized ODE systems effectively,and to perform stability analysis.Based on the advances,an extended EGA-IHB method,and the related stability analysis method is also introduced to obtains periodic responses of index-3 DAE systems,which are obtained by using multibody dynamics modeling theory,and perform stability analysis.The content of this paper is organized as follows:Several key issues in the use of Discrete Fourier Transform(DFT)to perform harmonic balance procedure is disscussed.The Galerkin integrals and DFT procedure used to conduct harmonic balance are proved to be theoretically equivalent when spectrum aliasing does not occur.Based on the equivalence,the minimal sampling rate is obtained for systems with polynomial nonlinearities,and the sampling rate selection strategy for non-polynomial systems is also disscussed.Results of the disscusion is valid for both the alternating frequency/time-domain transform(AFT)method,and the modified harmonic balance methods based on fast Fourier transform(FFT).The correctness of the sampling rate selection strategy is verified by numeircal examples,and the computational performance of the existing semi-analytical methods is compared.Results show that the analytical harmonic balance procedure can be replaced with a equivalent but much faster numerical process.This work has laid a theoretical foundation for the design of an efficient and universal semi-analytical solver based on the incremental harmonic balance method.To improve generality and computational efficiency of the incremental harmonic balance method to use it as a universal solver and nonlinear analyzer for generalized ODEs,the efficient Galerkin averaging – incremental harmonic balance(EGA-IHB)method is proposed.The residual terms of the equations and their derivatives are expressed using truncated Fourier series,and Fourier coefficient vectors are calculated using FFT.The time-domain series used to perform FFT is obtained by an independent sampling procedure.Integrals in the Galerkin averaging procedure are calculated by tensor contraction.Note that during the computational process,the tensors remain constant,which are related only to the number of truncated harmonic terms,hence are only need to be calculated once.In this way,efficiency of the EGA-IHB method is greatly increased.Numerical examples show that,generality of the EGA-IHB is much similar to a numerical integrator,it can be used to solve majorities of ODEs with periodic responses without re-derivation.It also shows significant advantage in computaitonal speed when comapring with other semi-analytical methods in the litrature.To obtains steady-state periodic responses of multibody dynamical systems modeled by index-3 DAEs,the extended EGA-IHB method is purposed.In the procedrue of the proposed algorithm,both the generalized coordinates and Lagrange multipliers are expanded into truncated Fourier series,and the harmonic balance is conducted by using the tensor contraction and FFT based EGA procedure.The DAEs are then transformed into nonlinear algebraic equations,which are then solved by using Newton-Raphson method.To automatically generate parameter-response curves of DAE,the improved arc-length continuation method based on scaling stragegy is introduced.Based on the theoretical advances,a generalized DAE solver framework is introduced.Numerical examples show that,the extended EGA-IHB method can be used to obtains periodic responses of high-dimensional multibody systems effectively,and generate parameter-response curves.Results from the extended EGA-IHB method are also compared with those from the numerical integrators,which shows excellent agreements,which varifies the accuracy of the proposed algorithm.To perform stability analysis on the steady-state periodic response of DAE systems,an improved stability analysis method is introduced.The new method perturbs both the generalized coordinates and Lagrange multipliers to linearize the original DAEs around a steady-state periodic response,which is followed by an linear transform procedure to transfer the perturbed DAE to an equvalent linear ODE with periodic coeffecients at each specific time step.Then,ordinary procedure based on the Floquet theory for determine stability can be conducted to generate monodromy matrix,the eigenvalues of the matrix,which are refered as the “Floquet multipliers”,can be used to determine stability of the original DAE system.The precise Hsu’s method is used to generate the monodromy matrix.Numerical examples show that,the proposed stability analysis method can determines stability of the periodic responses of the DAE systems with both single degree of freedom and multi degree of freedoms,and tracks bifurcations.