Nonlinear dynamics of a planar beam–spring system: analytical and numerical approaches

Abstract

The multiple timescales method is applied to the exact partial differential equations of the planar motion of a hinged-simply supported beam with a linear axial spring of arbitrary stiffness. The forceddamped and free oscillations of the system around frequencies corresponding to nth natural bending mode are examined thoroughly and compared with numerical simulations as well as with already published results obtained by Lindstedt-Poincaré method. A special numerical technique using explicit finite element method to draw the frequency-response curves is appositely developed. The well-known jump phenomena between resonant and non-resonant branches, as well as superharmonic resonances, have been detected numerically.

Nonlinear vibrations of an extensional beam with tip mass in slewing motion

“…In contrast to standard procedure based on Galerkin orthogonalisation method and modal reduction, we apply directly the multiple time scales method to partial differential equations and associated boundary conditions [21]. The proposed method has been applied in [14] to solve coupled transversal-longitudinal oscillations of fixed simply supported beam with a spring at the end. According to the multiple time scale method we introduce different time scales related to small parameter…”

“…The results showed softening versus hardening dichotomy in the resonance curves and also strong interactions between flexural and longitudinal (axial) vibrations, leading to internal resonances occurring for specific stiffness of the axial spring. Those new results obtained for nonlinear beams (considered as fixed structures), the observed interactions between transversal and longitudinal vibrations and the dichotomy of the softening and hardening effect [14] motivated authors to equivalent studies of rotating structures. The nonlinear phenomena observed for the rotating inextensible beams [23,24] suggested to analyse a more general case of the extensional beam.…”

“…The importance of the definition of geometric curvature was discussed in [18]. The strict nonlinear shearable beam model was solved analytically by a direct attack of the nonlinear partial differential equations and the results were verified by a finite element method by means of the commercial Abaqus_CAEÓ software [14]. The results showed softening versus hardening dichotomy in the resonance curves and also strong interactions between flexural and longitudinal (axial) vibrations, leading to internal resonances occurring for specific stiffness of the axial spring.…”

Influence of Mechanical Couplings on the Dynamical Behavior and Energy Harvesting of a Composite Structure

“…For instance, numerical Finite Element (FE) simulations on clamped–clamped and hinged–hinged planar beams with homogeneous material properties show a hardening nature, while simply supported boundary conditions cause the softening of the frequency response for the first flexural mode [ 1 ]. This phenomenon has been investigated analytically and numerically in planar beams by means of axial–transversal coupling in nonlinear frequency amplitude response via implementing an additional axial spring [ 2 ], lumped tip mass [ 3 ], and higher-order resonances [ 4 ]. The predicted behaviors were numerically (FEM) and experimentally validated on a kinematically excited prototype with an additional mass moment of inertia caused by physical hinges [ 5 ].…”

Nonlinear Dynamics of a Planar Hinged-Supported Beam with One End Lumped Mass and Longitudinal Elastic Support

“…by finite element method and then examined to understand and optimize the way it works, without building advanced laboratory test devices or sophisticated analytical algorithms. Moreover, such numerical analysis is available to cover predictable global phenomena in a wide range as well as unexpected local effects, see [1,2]. The main disadvantages of this approach is studying only specific cases and needing huge computational time for very complex systems.…”