Primary resonance of traveling viscoelastic beam under internal resonance

Abstract

Under the 3:1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The undetermined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.

High-static-low-dynamic-stiffness vibration isolation enhanced by damping nonlinearity

“…The analytical and the numerical results perfectly agree in the stable portions for both heavy damping and light damping. To determine whether the method of harmonic balance correctly captures the dynamic behavior of the parameters chosen, the frequency response curves (FRC) of the two masses are plotted in Figure 6(a) and (b) with the numerical results via the Runge-Kutta scheme [27][28][29][30][31]. The figure shows that there is reasonable agreement, and the method of harmonic balance can be used for further investigation of the dynamic behavior.…”

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

“…( 29)-(31) into Eqs. ( 25)-( 27) and ignoring the effect of inertia in the inplane direction, and subsequently applying the Galerkin's method [22][23][24][25] , the ordinary differential equations for the variables W A 1,n and W B 1,n can be obtained,…”

Nonlinear Dynamic Analysis of Axially Moving Laminated Shape Memory Alloy Beam with 1:3 Internal Resonance

“…By the method of multiple scales, they established the modified solvability conditions in principal parametric and internal resonances. Hu et al [ 23 , 24 ] explored the nonlinear dynamics of a super critically moving beam and a traveling viscoelastic beam under the 3:1 internal resonance condition. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes.…”