A Study On The Effects Of Nonlinear Boundaries On The Dynamic Characteristics Of Slender Rod And Beam Structures

In power machinery engineering,slender continuous rod/beam structures are widely used,such as the engine’s connecting rod and the frame of the automobile chassis.When the dynamic characteristics of these engineering structures are analyzed,the complex boundary conditions(such as bolted joints,riveted joints,etc.)are usually simplified to linear boundaries.However,the nonlinear factors at the actual structural joints,such as gap and large deformation,have an essential effect on the dynamic characteristics of the structures,which reduce the real service life of the power machinery and even cause severe mechanical failures.In this paper,the dynamic characteristics of a slender rod/beam structure with nonlinear boundaries are studied,and the influence of linear and cubic stiffness nonlinear elastic boundary is analyzed.For the continuous rod structure,the traveling wave method,the separated variable method,and the complex exponential method are used to analyze the phase-frequency relationship of the system.The connection and difference between the three methods are investigated.The relationships between the dimensionless frequency,displacement,and phase are revealed.The backbone curves calculated by the phase closure principle and the separation variable method are compared.The influence of the small damping of the structure on the forced vibration is analyzed.For the continuous beam structure,the reflection of the propagating wave and the nearfield wave incident on the nonlinear elastic boundary are discussed separately.The relationships between the dimensionless frequency,displacement,reflection coefficient,and phase are obtained by using the harmonic balance method.The backbone curves calculated by using the phase closure principle and the separation variable method are compared.The frequency response curves of the beam structure under harmonic excitation are obtained.The finite element method is used to verify the theoretical analysis results of the slender rod/beam with nonlinear boundaries.The simulated frequency response curves are obtained by using the positive and negative frequency sweep methods,which are compared well with those obtained using the theoretical approaches.Taking turbine blades as an example,the theoretical methods in this paper are simply applied in engineering.The study finds that a slender rod/beam structure with nonlinear boundaries has a jumping phenomenon that the linear system does not have.Within a specific range of linear stiffness,the hardening stiffness boundary increases the jump frequency of the beam structure with the increase of the degree of nonlinearity,which has a negligible effect on the phase of the reflected wave.The effect of the softening stiffness boundaries is opposite to that of the hardening stiffness boundaries,and the effect is more obvious.For the softened stiffness boundaries,when the nonlinear stiffness gradually increases to the same as the linear stiffness,the system becomes unstable.The jump phenomena appear in the reflection coefficient and phase.Because the phase closure principle neglects the influence of the nearfield wave,compared with the separation variable method,the phase closure principle induces certain errors when calculating the backbone curve of a beam structure with nonlinear stiffness boundaries.