Stability and Bifurcation Analysis of an Axially Accelerating Beam

Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

“…The range of instability related to Hopf bifurcation in both trivial and nontrivial solution curves increases in the present case compared to single frequency excitations cases. Further, like principal parametric resonance of first mode [50], the two frequency excitation case has two mode trivial and nontrivial solution curves but unlike principal parametric resonance of second mode [51], it does not have a single mode solution.…”

“…4, [50]) and principal parametric resonance of second mode (Fig. 2, [51]), some qualitative and quantitative similarities and dissimilarities appear in stability, bifurcation and type of solution curves. The range of instability related to Hopf bifurcation in both trivial and nontrivial solution curves increases in the present case compared to single frequency excitations cases.…”

“…The influence of increasing fluctuating velocity component on stability and bifurcation of equilibrium solution curves of two frequency parametric excitation as well as single frequency parametric excitations [50,51] has some common effects like increase in amplitude of first mode and broadening of instability zone in both trivial and nontrivial solution curves. But the increase in number of Hopf bifurcation points; vanishing of saddle node bifurcation point (SN) and increase in maximum amplitude of second mode as illustrated in Fig.…”

“…The numerical results was used to verify the approximate solutions. The authors of the paper [50,51] have made the stability, bifurcation and dynamic response study for a traveling beam under single-frequency parametric excitation in presence of 3:1 internal resonance using direct method of multiple scales.…”

Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam

“…In this study, the effects of mass numbers and mass ratios on the nonlinear natural frequency and amplitude of the system have been investigated. The authors of the paper [23,24] have performed the stability and bifurcation analysis for a traveling beam under single-frequency parametric excitation in presence of 3:1 internal resonance using direct method of multiple scales.…”