Non-planar Dynamics of a CFRP Cable with an External 1/3-order Subharmonic Resonance

Based on the non-planar vibration equations of a cable made of carbon fiber reinforced polymer (CFRP), the nonlinear behaviors of the cable are studied. The one-to-one internal resonance of the lowest in-plane and out-of-plane modes of the cable is investigated. Three different cases, namely, 1/3-order subharmonic resonance of the in-plane mode, 1/3-order subharmonic resonance of the out-of-plane mode and 1/3-order subharmonic resonance of both of the in-plane and out-of-plane modes are examined. The vibration equations of the cable are discretized by using Galerkin’s method. In this way, the ordinary differential equations (ODEs) are obtained. The multiple time scale method is employed to solve the ODEs and the corresponding modulation equations are derived. Then, the response curves of the cable are obtained by using Newton-Raphson method and pseudo arclength algorithm. Meanwhile, the response curves of the CFRP cable are compared with those of the steel cable to explore the differences in nonlinear behaviors of the cables made of different materials. The results show that the CFRP cable has potential advantages over the steel cable from a nonlinear point of view.

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

Dynamics of a rotor composed of a flexible beam attached to a slewing rigid hub is presented in the paper. Dynamics of the structure is studied for a slender beam model, based on extended Bernoulli–Euler theory, which takes into account a nonlinear curvature, coupled transversal and longitudinal oscillations and non-constant angular velocity of the hub. Moreover, to demonstrate a general case for dynamical boundary conditions, lumped mass fixed at the beam tip is added. The partial differential equations (PDEs) are derived from Hamilton principle of the least action. The analytical solutions of the PDEs are obtained by the multiple time scale method applied directly to PDEs. Forced vibrations around selected resonance zones are studied and the influence of beam rotation, preset angle, hub radius, tip mass is presented. Hardening and softening phenomena, respectively for the first and the second mode, are obtained for various angular velocity values.

Isochronous Beams by an Inclined Roller Support

The paper addresses the problem of isochronous beams, namely those that oscillate with a frequency that is independent of the amplitude also in the nonlinear regime. The mechanism adopted to obtain this goal is that of having, as a boundary condition, a roller that can slide on a given path. A geometrically exact Euler–Bernoulli formulation is considered, and the nonlinear analysis is done by the multiple time scale method, that is applied directly to the partial differential equations governing the motion without an a priori spatial reduction. The analytical expression of the backbone curve is obtained, up to the third-order, and its dependence on the roller path is addressed. Conditions for having a straight backbone curve, i.e., the isochronous beam, are determined explicitly. As a by-product of the main result, the free and forced nonlinear oscillations of a beam with an inclined support sliding on an arbitrary path have been investigated.

Parametric Resonance of a Self-Excited System Under External Force and Time Delay Influence

Vibrations of a nonlinear self-excited system driven by parametric excitation are presented in the paper. The considered model with one DOF includes a self-excitation term represented by a nonlinear Rayleigh function and also a periodically varied stiffness coefficient which represents parametric excitation. The influence of the external force or/and time delay, treated as a control signal, is demonstrated. Nonlinear parametric resonance is determined numerically and analytically by the multiple time scale method. The influence of time delay on the resonance zones and the frequency locking phenomenon is analysed.

Temperature Influence on Nonlinear Responses of Timoshenko Beam

The goal of this paper is to study large amplitude vibrations of a Timoshenko beam under an influence of the elevated temperature. It is assumed that the beam gets the elevated temperature instantly and the temperature is uniformly distributed along the beam’s length and cross-section. The mathematical model represented by a set of partial differential equations is derived taking into account boundary conditions for a simply supported beam in the both ends. Next, the problem is reduced by the Galerkin method by means of free vibration modes. The influence of the temperature on a resonance localisation and nonlinear oscillations is studied numerically and analytically by the multiple time scale method.