A NUMERICAL TREATMENT OF THE PERIODIC SOLUTIONS OF NON-LINEAR VIBRATION SYSTEMS

摘要/Abstract

摘要： Direct numerical integration can be used to find the periodic solutions for the equations of motion of nonlinear vibration systems. The initial conditions are iterated so that they coincide with the terminal conditions. The time interval of the integration (i.e.,the period) and certain parameters of the equations of motion can be included in the iterations.The integration method has a variable steplength.This shooting method can produce periodic solutions with a shorter computer time. The only error occurs in the numerical integration and it can therefore be estimated and made small enough. Using this method one can treat a variety of vibration problems, such as free conservative, forced, parameter-excited and self-sustained vibrations with one or several degrees-of-freedom. Unstable solutions and those which are sensitive to parameter changes can also be calculated.The stability of the solutions is investigated based on the theory of differential equations with periodic coefficients. The extrapolation method and the procedure of automatic steplength control are used to estimate the initial values of iterations by determining the resonance curve and other vibration characteristics.Some examples have been calculated to illustrate the applicability of the method. The non-linearity way be expressed by an analytical function or any other functions, such as a piecewise linear function. Several remarkable features of nonlinear vibrations are presented through the periodic solutions obtained. Finally, some results are compared with those obtained by other approximation methods and experiments.

关键词: plate bending, membrane-plate similarity, convergence, finite element

Abstract: Direct numerical integration can be used to find the periodic solutions for the equations of motion of nonlinear vibration systems. The initial conditions are iterated so that they coincide with the terminal conditions. The time interval of the integration (i.e.,the period) and certain parameters of the equations of motion can be included in the iterations.The integration method has a variable steplength.This shooting method can produce periodic solutions with a shorter computer time. The only error occurs in the numerical integration and it can therefore be estimated and made small enough. Using this method one can treat a variety of vibration problems, such as free conservative, forced, parameter-excited and self-sustained vibrations with one or several degrees-of-freedom. Unstable solutions and those which are sensitive to parameter changes can also be calculated.The stability of the solutions is investigated based on the theory of differential equations with periodic coefficients. The extrapolation method and the procedure of automatic steplength control are used to estimate the initial values of iterations by determining the resonance curve and other vibration characteristics.Some examples have been calculated to illustrate the applicability of the method. The non-linearity way be expressed by an analytical function or any other functions, such as a piecewise linear function. Several remarkable features of nonlinear vibrations are presented through the periodic solutions obtained. Finally, some results are compared with those obtained by other approximation methods and experiments.

Key words: plate bending, membrane-plate similarity, convergence, finite element

凌复华. A NUMERICAL TREATMENT OF THE PERIODIC SOLUTIONS OF NON-LINEAR VIBRATION SYSTEMS[J]. Applied Mathematics and Mechanics (English Edition), 1983, 4(4): 525-546.

Ling Fu-hua. A NUMERICAL TREATMENT OF THE PERIODIC SOLUTIONS OF NON-LINEAR VIBRATION SYSTEMS[J]. Applied Mathematics and Mechanics (English Edition), 1983, 4(4): 525-546.

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