The Amplitude Incremental Variational Principle And Its Application In The Strongly Nonlinear Vibration

The nonlinear vibration theory has a wide application in engineering technology fields.In recent years, with the emergence of new materials and new structural types, the stronglynonlinear vibration theory is gradually becoming a focus of the nonlinear dynamics study.Therefore, the development and improvement of the strongly nonlinear vibration theory hasgreat significance in theoretical research and engineering application.In most nonlinear vibration analysis methods, the incremental harmonic balance method(IHB) is an effective quantitative analytical Method. It can analyze not only the weaklynonlinear vibration system，but also the strongly nonlinear vibration system. It can improvethe precision of periodic solutions. And it applies to the quantitative analysis of a wide rangeof parameter variation.The amplitude incremental variational principle of periodic motion in nonlinear elasticsystem is the theoretical fundament of the incremental harmonic balance method. Based onthe basic equations of elastodynamics, this paper uses the variational integral method to dothe systematic study on the amplitude incremental variational principle of periodic motion innonlinear system. The amplitude incremental potential energy principle and complementaryenergy principle are established. The amplitude incremental generalized potential energyprinciple and generalized complementary energy principle with two kinds of variables areestablished. And the amplitude incremental generalized potential energy principle andgeneralized complementary energy principle with three kinds of variables for nonlinearvibration are established. Thus the theoretical fundament of the incremental harmonic balancemethod is improved.On the basic of the amplitude incremental variational principle of the nonlinearvibration,the amplitude incremental variational principle of the rectangular thin plate, whichis simply supported on the four sides, is established. And the nonlinear vibration problem ofthe rectangular thin plate is solved with the incremental harmonic balance method.