Pseudo-arclength Continuation Technique Research Articles

Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance

The aim of the study described in this paper is to investigate the forced dynamics of an axially moving viscoelastic beam. The governing equation of motion is obtained via Newton's second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin–Voigt, a two-parameter rheological model, energy dissipation mechanism, in which material, not partial, time derivative is employed in the viscoelastic constitutive relation. The dimensionless partial differential equation of motion is discretized using Galerkin's scheme with hinged–hinged beam eigenfunctions as the basis functions. The resulting set of nonlinear ordinary differential equations is then solved using the pseudo-arclength continuation technique and a direct time integration. For the system with the axial speed in the sub-critical regime, the response of the system is examined when possessing an internal resonance and when not. By employing a direct time integration, it is shown how the bifurcation diagrams of the system are modified by the presence of the dissipation terms—i.e. by both the time-dependant and steady (due the simultaneous presence of the axial speed and the energy dissipation mechanism) energy dissipation terms. Moreover, the amplitude–frequency responses and bifurcation diagrams of Poincaré maps are presented for several values of the system parameters.

Superelongation in heligal muscles

In this paper the nonlinear planar dynamics of a fluid-conveying cantilevered pipe is investigated. The centreline of the pipe is considered to be extensible; i.e., coupled longitudinal and transverse displacements are considered. The extended version of the Lagrange equations for systems containing non-material volumes is employed to derive the equations of motion, resulting directly in a set of coupled nonlinear ordinary differential equations. The pseudo-arclength continuation technique along with direct time integration are used to solve these equations. Bifurcation diagrams of the system are constructed as the flow velocity is increased; these diagrams are supplemented by time traces, phase-plane portraits, and fast Fourier transforms for some sets of system parameters. As opposed to the case of an inextensible pipe, an extensible pipe elongates in the axial direction as the flow velocity is increased from zero; depending on the system parameters, this static elongation can be considerable. At the critical flow velocity, the system loses stability via a supercritical Hopf bifurcation, emerging from the trivial solution for the transverse displacement and leading to a flutter.