To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string,the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string.To derive the governing equation,Newton's second law,Lagrangean strain,and Kelvin's model are respectively used to account the dynamical relation,geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms,closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance.The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance.Some numerical examples are presented to show the effects of the mean transport speed,the amplitude and the frequency of speed variation.
This paper shows that first integrals of discrete equation of motion for Birkhoff systems can be determined explicitly by investigating the invariance properties of the discrete Pfaffian. The result obtained is a discrete analogue of theorem of Noether in the calculus of variations. An example is given to illustrate the application of the results.
