Study On Nonlinear Tuned Mass Damper–simply Supported Beam Based On Incremental Harmonic Balance Method

The use of nonlinearly tuned mass dampers in bridge vibration control has the advantages of wider bandwidth,stronger robustness,and no problem of amplification beyond the target bandwidth,as compared with linearly tuned mass dampers.Therefore,the study of nonlinear tuned mass dampers is still the focus of many scholars all over the world.However,when solving strong nonlinear problems such as the vibration response of nonlinear tuned mass damper-bridge systems,the numerical calculation methods used have problems such as the calculation speed is slow,the calculation result is divergence,calculation accuracy is difficult to control,and a complete picture of the solution cannot be provided.As a result,using a more efficient calculation method to conduct the thorough research on the nonlinear tuned mass dampers has great research value.To achieve the purposes of this thesis,a numerical model for an Euler-Bernoulli simply supported beam installed with a nonlinear tuned mass damper was established,which was subjected to a harmonic excitation at a point on the beam,a periodic moving constant load on the beam,and a periodic moving mass on the beam,respectively.The kinetic energy,potential energy,and dissipated energy of the model are first formulated in the continuous expressions using the nonlinear dynamics theory.Then,the combination of the Lagrange-Euler equation and the Galerkin truncation derived the governing equations of vibration for the model.These equations were further discretized by the principle of mode superposition,which were solved by the incremental harmonic balance method（IHBM）.In the end,the semi-numerical semi-analytic solution of the vibration response was obtained from the discretization equations.The frequency-amplitude and velocity-amplitude diagrams of the vibration response of the model were obtained by means of the arc length method.To realize the influence of the nonlinear tuned mass damper on the vibration response of the model,a series of parametric studies were carried out in the numerical simulations.The simulation results show that,in a range of the lower frequencies or moving speeds of the excitations,when the nonlinear tuned mass damper has the smaller mass ratio u,the larger linear stiffness coefficient K₁,the larger cubic nonlinear stiffness coefficient K₂,or the larger damping coefficient C₁,or when the excitation has the smaller amplitude,the vibration of the beam has a better reduction.With the increasing frequency or the moving speed of the excitations,the nonlinear resonance appears in the vibration response of the model.The effect of the model parameters on the vibration of the beam becomes complex.Nonlinear dynamic phenomena such as jump discontinuity,multiple solutions,and right shift of the resonance frequency can be observed.As the frequency or the moving speed of the excitations reach maximum values in the simulations,the effect of the model parameters on the vibration of the beam diminishes and the amplitude of the vibration response of the beam tends to zero.