Dynamic Stability of Viscoelastic Bars under Pulsating Axial Loads

Recent applications require the extension of stability theory to axially loaded viscoelastic bars. For elastic bars it is known that a pulsating load leads to a Mathieu equation. Meaning, there are some instable regions before the static buckling load and some stable regions beyond, depending on the excitation (amplitude, frequency, offset). To fill this gap we follow along the lines of Weidenhammer and adopt the classical Euler‐Bernoulli beam kinematics for coupled longitudinal and bending vibrations. Modeling viscoelasticity by the Standard Linear Solid model introduces an internal variable. Hence, there are auxiliary transversal and longitudinal displacements corresponding to the internal variable in addition to the physical transversal and longitudinal displacements of the bar centerline. The modeling further assumes only longitudinal vibrations in the stable regime, thus there are no bending vibrations and the description simplifies to a system of linear partial differential equations. Further, a one‐sided coupling is assumed for the stability analysis, i.e. the longitudinal vibrations are prescribed and induce bending vibrations. Still, the time‐variant coefficients of the transversal dynamics make it difficult to find an analytical solution. Consequently, an approximation is obtained by Ritz method from Hamilton's principle. Applying the eigenforms of elastic buckling as trial functions finally leads to a system of rheolinear ordinary differential equations. Its stability chart with respect to static and harmonic load components is calculated numerically by Floquet theory.