Vibration Analysis And Control Of 1-dimensional Elastic Structures Subjected To Nonlinear Boundaries

The nonlinear boundary value problem and the nonlinear boundary control for flexible structures are investigated in the current work.For the first time,two new analytical methods,the multi-scale method together with modal correction and the harmonic balance method together with modal correction,are proposed to overcome strong nonlinear boundary value problem.Based on these methods,the nonlinear boundary control is studied.It indicates that the boundary control can restrain the vibration of a flexible structure in a wide region.Besides,the boundary control works not only for a static flexible structure,but also for a gyroscopic system.In general,the partial differential governing equation(PDE)of a flexible structure is solved via the method of modal expansion.However,the nonlinear and the non-homogenous boundary make it impossible for that the modal functions are ruined.To overcome the non-homogenous boundary value problem,the method of modal correction is employed.According to this method,the solution of the PDE contains two parts: the first one satisfies the linear homogenous boundary while the second part is a correction.As the first part satisfies the linear homogenous boundary,it is expressed by the modal expansion.The second part helps the solution to satisfy the initial boundary.In this way,the non-homogenous boundary can be translated into a homogenous boundary value problem.The multi-scale method(MSM)can translate a nonlinear equation into linear equations with different time-scales.Referring to this idea,the nonlinear boundary can also be rescaled into linear non-homogenous boundaries with different time-scales.After that,the linear non-homogenous boundary value problems could be solved out one by one together with the method of modal correction.During this processing,only the primary resonance is considered.The analytical solution may lose its accuracy while the nonlinearity in the boundary is strong,as higher-order harmonics and non-resonant responses are neglected.To improve the analytical solution,the higher-order harmonics and non-resonant responses are introduced into the solvability condition,or namely,an iteration is employed.This makes MSM can deal with strong nonlinear boundaries.The harmonic balance method(HBM)is good for strong nonlinear ordinary differential equations(ODE).A PDE should be discretized into ODEs first before HBM is carried out.However,the mode is ruined by the nonlinear boundary as mentioned above.In this paper,a smart way is proposed for the first time to enable HBM to deal with nonlinear boundary problems.It treats the boundary as generalized governing equations.In this way,some corresponding generalized coordinates are introduced into the initial system.With the help of the modal correction,the PDE of the flexible structure is projected into modal space.Hence,a set of ODEs are produced.Together with those generalized governing equations,the new augmented system can be solved out by HBM.Physically,the boundary produces reflected wave to the flexible structure.They will cause the transverse vibration and may yield standing waves.In other words,the standing wave will be ruined or restrained by changing the boundary.According to this,two types of nonlinear boundary isolations are introduced.The first one employs pure nonlinearity into the initial boundary.It will maintain the linear characters of the main structure while the resonance is restrained.The second one changes the initial boundary to the quasi-zero stiffness support.It eliminates the two resonances in low frequency region and can restrain resonances in higher frequency region.Except the nonlinear boundary isolation,a new kind of nonlinear boundary absorber is also introduced.It absorbs the vibration energy via the rotation angle at the end caused by the transverse motion.Hence,the absorber can work in a wide frequency region,regardless of the resonant frequency.The method of multi-scale method together with modal correction is good for solving resonance.The harmonic balance method together with modal correction can obtain the response with wide frequency band.Both of them can deal with strong nonlinear boundary problems.The investigation indicates that the transverse vibration can be restrained with a good efficiency by the nonlinear boundary control.This paper produces a productive way for the vibration control in engineering.