Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam with Finite Deformation

The motion equations governing the dynamical behavior of a viscoelasticTimoshenko beam with finite deformation are derived and simplified byGalerkin method. The viscoelastic material is assumed to obey thethree-dimensional fractional derivative constitutive relation. Thedynamical behaviors of the simplified systems with order 1 and order 2are numerically computed and compared by using the computational methodpresented by the authors. The dynamical behaviors of the systems areuniform qualitatively, but there is a little deviation quantitatively.And the truncated system with order 1 is safer than the one of order 2.It is also shown that the lower order system is reasonable. Theinfluences of the load parameter and the fractional derivative parameter(material parameter) on the deflection of the beam are consideredrespectively. The numerical methods in nonlinear dynamics, such as phasediagram, and Poincare section, are applied to reveal dynamical behaviorsof the nonlinear viscoelastic Timoshenko beam. There are plenty ofdynamical behaviors, such as periodicity, bifurcation, quasi-periodicityand chaos in the dynamical system.