Dynamic Response Prediction And Application Of Large Amplitude Vibrations Of Cantilever Tapered Beams

Tapered beam is widely used in building,mechanical and aviation fields because of its excellent mechanical properties.However vibration in these applications is inevitable.This paper is focused on the study of the large amplitude nonlinear vibration behavior of single and double tapered cantilever beams on the nonlinear foundation.An alternative analytical method is presented to solve the issue.Unlike other literatures,the admissible lateral displacement function satisfying the geometric boundary conditions of a single or double tapered cantilever beam is derived by using Rayleigh-Ritz method.It is briefer in expression than that of Bessel functions.The linear natural frequency obtained by this paper method what is compared with the linear natural frequency obtained by known literature and finite element analysis software ANSYS verify the correctness of deflection function derived by using Rayleigh Ritz method.Besides,Analytical approximate solutions in closed and explicit form was obtain with combining Galerkin and Newton linearization method to solve motion equation.No matter how the amplitude is large or small,these approximate solutions show excellent consistency with the numeric method obtained by shooting method.Finally,the value of frequency under different amplitudes is discussed.The influence of the first approximation to frequency and the second approximation to frequency under the different amplitudes on the accuracy are aslo discussed.For?=,2.0?=2.0,?=.0,4?=.05,?=.0,6?=8.0,the analytical phase trajectories of a single tapered beam and a double tapered beam are compared with the exact phase trajectories of the numerical integration formula.The exact consistency between the approximate phase trajectory and the exact solution is obtained.Furthermore,The frequency of a single tapered beam and a double tapered beam under three different cones of?=0.2,?=0.4,?=0.8are analyzed and the influence of the basic stiffness parameters of the nonlinear vibration on the vibration response are also analyzed.More specifically,at the elastic foundation coefficients nonlinear foundation stiffness isk_NL=0,shear foundation stiffness isk_S=0,linear foundation stiffness isk_L=0,5,10,50,200,the frequency of the beam is analyzed respectively;linear foundation stiffness isk_L=0,shear foundation stiffness isk_s=0,nonlinear foundation stiffness isk_NL=,050,100,150,200respectively,the frequency of the beam is analyzed;linear foundation stiffness isk_L=0,nonlinear foundation stiffness isk_NL=0,shear foundation stiffness isk_S=0.1,8.0,6.0,2.0,0,the frequency of the beam is analyzed respectively.The dynamic response of the tapered beams in the actual projects are calculated.The frequency are obstained under different amplitude.Besides,the three-dimensional diagram of the actual engineering vibration is drawn.Moreover,due to brevity of expressions,the present analytical approximate solutions are convenient to investigate effects of various parameters on the lar ge amplitude vibration response of tapered beams.