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Nonlinear Problems in Dynamics by the Finite Element Method in Time Domain

Exact solutions of nonlinear dynamic equations are very rare and almost all of the methods for solving nonlinear differential equations are only approximate. A general solution method of systems with complex and strong nonlinearity is the numerical time integration. Numerical integrations are usually very time-consuming and may encounter numerical difficulties when the nonlinearity becomes very strong. A very efficient method for solving nonlinear differential equations in the frequency domain is the harmonic balance method. When the assumption of dominance of primary resonance in the response is satisfied, the harmonic balance method is very accurate and numerically reliable method. If the influence of higher harmonics in the response is significant, this method becomes very unreliable. The incremental harmonic balance method is more convenient for determining frequency response characteristics, because a new solution can be sought, with the previous solution used as a very good approximation. Another approach for solving nonlinear problems in dynamics is the finite element in time method, which is based on a weak form of Hamilton principle of varying action [1]. Similar to the standard finite element technique, the time interval is divided into a finite number of time elements. The solution for all the spatial degrees of freedom at all time steps within a given time interval is divided into a finite number of time elements. The solution for all the spatial degrees of freedom at all time steps within a given time interval is sought through a set of algebraic equations. Furthermore, the straightforward determination of the stability of solution is the second advantageous feature of this method. The stability of periodic solution is investigated considering small perturbations of the solution. According to Floquet-Liapunov theorem the stability can be determined by the eigenvalues of the transition matrix that relates the initial and final perturbations. The model of a three-degree-of-freedom semi-definite system with clearances under harmonic excitations is used to study the feasibility of the finite element in time method. The stability of solutions is analysed using Floquet-Liapunov theorem. Close agreement is found between obtained results and the published findings of a harmonic balance method [2].

Nonlinear Vibrations; Finite Element in Time; Clearance; Stability

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