Research On Nonlinear Dynamics Of Axially Moving Free-Free Beams

The slender missile is excited by kinds of external loads on high-speed flight,that will result in vibration and unstability of missiles.Dynamic analysis of missiles with large aspect ratio has theoretical significance and engineering value for design and manufacture.Moving missiles were usually modeled as free-free beams in the past literature,however a few literature about the nonlinear dynamics analysis have been published.Nonlinear dynamic equations are solved analytically and numerically.Based on Hamilton principle,equations of motion of moving free-free beam are derived.Considering velocity and materials stiffness of moving beam,modal and stability analysis are carried out by Galerkin method and state space method.Complex modal method is used to research the modal traveling wave result from the gyroscopic term in equations.Nonlinear forced vibration equations of moving beam are builded and decomposed into two ordinary differential equations by principal coordinate.Multi-scale method and incremental harmonic balance method are used to solve these equations,meanwhile that of equations are numerically sovled by Runge Kutta method.Analysis of amplitude frequency and dynamic response are studied for the moving beam.Maximum magnitudes of the first principal coordinate are given in the case of moving speed,exciting frequency and amplitude.Curves of amplitude frequency and eigenvalue trajectory are given.Principal resonance result in multiple equilibrium points are researched in the case of 1:3 internal resonance.Maximum magnitudes caused by exciting amplitude,moving speed and value of nonlinear parameter are calculated.Results show that multivalued solutions occur in that curves.Different moving speed will result in different amplitude frequency curves.Increasing the value of the nonlinear parameter will decreases the maximum magnitudes and offset resonance peak.Bifurcation characteristics of periodic motion are researched in case of moving speed,exciting amplitude and frequency by Poincarè mapping,periodic point mapping and largest Lyapunov exponent.Considering Kelvin viscoelastic constitutive,nonlinear parametric vibration equations are given.Equations are numerically solved by Galerkin method and Runge Kutta method.Modal and stability analysis are given in case of viscoelastic parameters and follower force.Conditions of parametric resonance are given.Effects of average and pulsating velocity on bifurcation of periodic motion are reserched by Poincarè mapping,periodic point mapping and largest Lyapunov exponent.LQR control and sliding mode control theory are used to study the vibration suppression of moving beam equiped with piezoelectric laminated actuator.Results show that maximum magnitudes are small when the actuator is installed in the middle of the beam in case of slower moving speed,and maximum magnitudes are small when the actuator is located at position distance from either of ends 1/3 of the length of the beam in case of faster moving speed.Sliding mode control has short control response time than that of LQR control.Sliding mode control is an ideal control strategy for vibration suppression of moving beams.