Nonlinear Forced Vibration of Curved Beam with Nonlinear Viscoelastic Ends

This article develops a mathematical formulation to investigate the nonlinear forced vibration of curved viscoelastic beam with nonlinear viscoelastic boundary conditions around buckled position numerically, for the first time. The nonlinear integro-differential equation of buckling problem and the corresponding nonlinear nonhomogeneous boundary conditions are discretized by the differential integral quadrature method (DIQM) and after that, they are solved via Newton-iterative method to compute the nonlinear static deflection paths. By employing DIQM, the linear vibration problem is converted to a linear eigenvalue problem that is solved easily. The Galerkin technique is implemented to reduce the nonlinear partial integro-differential equation governing the nonlinear dynamic problem into Duffing-type equation. The Duffing equation is discretized via periodic spectral differentiation matrix operators. Finally, the pseudo-arc length continuation algorithm is applied to solve the nonlinear eigenvalue problem resulting from the discretized duffing equation. Validation of numerical techniques implemented in the solutions is proved with previous works. Parametric studies are conducted to deliberate the influences of the amplitude of initial curvature, axial load, linear and nonlinear support parameters on the static and dynamic behaviors of straight and curved beams. It should be noted that the proposed algorithm computes both stable and unstable solutions.