Combination and principal parametric resonances of axially accelerating viscoelastic beams: Recognition of longitudinally varying tensions

Nonlinear parametric vibration is investigated for axially accelerating viscoelastic beams subject to parametric excitations resulting from longitudinally varying tensions and axial speed fluctuations. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite axial support rigidity is also modeled. The governing equations of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle and the viscoelastic constitutive relation. The equation is simplified into a governing equation of transverse nonlinear vibration in small but finite stretching problems. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent and space-dependent coefficients. The method of multiple scales is employed to analyze the combination and the principal parametric resonances with the focus on steady-state responses. In the difference resonance, there is only trivial zero response which is always stable. In the summation and the principal resonances, the trivial responses may become unstable and bifurcate into nontrivial responses for certain excitation frequencies. Some numerical examples indicate that the longitudinal tension variation makes the instability frequency intervals of trivial responses small and the nontrivial response amplitudes large (small) in the summation (principal) resonance. It is also found that the nontrivial responses are not sensitive to the axial support rigidity. Numerical solutions are calculated via the differential quadrature to support results via the method of multiple scales.