Dynamic stability of a viscoelastic beam with frequency-dependent modulus

Effect of an axial pre-load on the flexural vibrations of viscoelastic beams

“…It is known (Cheli and Diana, 2015), indeed, that when an elastic beam is subjected to a static pre-load, its resonances move toward higher or lower frequencies, in case of an applied traction or compression, respectively. Many studies have been also carried out which investigate the effects of some dynamical axial pre-loads on both the flexural (Shih and Yeh, 2005) and the axial (Ebrahimi-Mamaghani et al, 2021) responses of the viscoelastic beams. However, to the author's knowledge, there are no specific studies in the literature that analyze how the transversal response of the viscoelastic beam changes in frequency because of static axial actions, with particular reference to the possible enhancement or suppression of one or more resonances.…”

Dynamic Stability of the Periodic and Aperiodic Structures of the Bernoulli-Euler Beams

“…Parametric resonance can be observed in mechanical systems such as, for example, by applying an axial load to a cantilever beam [3,6], or it can even result from changes in stiffness [7]. Parametric resonance can be analyzed by reducing the equation of motion of the system to the form of the Mathieu equation [8][9][10][11][12][13]. Periodic solutions of the Mathieu equation allow to designate areas of stable and unstable solutions [14].…”

Numerical analysis of static and dynamic stabilities of viscoelastic columns

“…There is an amount of literatures concerned with this issue. Stevens discussed the stability of a simply supported perfect viscoelastic column subject to a harmonic axial load ; by using complex modulus and perturbation methods, he also investigated the influence of imperfections such as initial curvature ; Dost presented a dynamic analysis of simply supported perfect viscoelastic columns under a constant axial load and gave some numerical results for a set of selected viscoelastic parameters ; Szyszkowski utilized the Lyapunov stability concept to obtain an approximate closed‐form expression of viscoelastic critical load of a perfect column ; Gurgoze considered a simply supported beam subjected to a constant axial load and a periodic displacement excitation and discussed the dynamic viscoelastic stability of the lateral vibrations ; Cederbaum made use of the concept of multi‐scales to present a solution for the stability analysis of a viscoelastic column under a periodic force ; Fung investigated dynamic stability of a simple supported viscoelastic beam subjected to harmonic and parametric excitations simultaneously ; Shirahatti discussed the difference of stability behaviors between constant and periodic loadings for simply supported perfect viscoelastic columns ; Shih considered a viscoelastic beam subjected to an axially harmonic load and investigated the dynamic stability in the case where modulus of the beam was frequency‐dependent ; the dynamic stability of moxing viscoelastic beams was reported by and ; Elfelsoufi presented a mathematical model based on integral equations for numerical investigations of stability analysis of damped beams subjected to subtangential follower forces ; Hilton depicted a typical relationship between axial viscoelastic column loads and column lifetimes and found that long‐time survival of viscoelastic column can only be achieved at small fractions of the Euler loads ; Chen investigated dynamic stability of an axially accelerating viscoelastic Timoshenko beam undergoing parametric resonance based on multiple scales method, which is applied to the equations to establish the solvability conditions in summation and principal parametric resonances ; numerical calculations given by for two‐end simply supported viscoelastic column have confirmed that the dynamic behavior of one‐order and two‐order Galerkin truncated models are qualitatively the same, but there are certain errors if the quantitative comparisons are concerned; Hamed gave the creep response of flexural beams and beam‐columns made with functionally graded materials using time stepping numerical integration, which yields an exponential algorithm via the expansion of the relaxation function into a Dirichlet series ; Ghayesh analytically investigated result of the free and forced vibrations of a Kelvin–Voigt viscoelastic beam supported by nonlinear spring using multiple timescales ; Luongo presented a post‐critical analysis of nonlinear viscoelastic beams using the multiple scale method, based on fractional series expansions in the perturbation parameter ; Deng investigated the stability of a viscoelastic column under the excitation of stochastic axial compressive load based on the stochastic averaging method…”