Nonlinear dynamics of elastic rods using the Cosserat theory: Modelling and simulation

The parametric response of a thick cantilever beam with a tip mass subjected to harmonic axial support motion is investigated. The Timoshenko beam theory is used to assess the effects of rotary inertia and shear deformation for the beam. In this regard, different modal amplitudes for transverse displacement and angle of rotation of the cross-section are considered. This yields a more accurate description of the dynamic model. The governing equations of motion are then derived for an arbitrary axial support motion which provide the flexibility of choosing the number of characteristic modes of the beam. To formulate a simple, physically correct dynamic model for stability and periodicity analysis, the general governing equations are truncated to only the first mode of vibration. Using Green’s function and Schauder’s fixed point theorem, the necessary and sufficient conditions for the existence of periodic oscillatory behavior of the beam are established. Consequently, the phase domains of periodicity and stability for various values of the physical characteristics of the beam-mass system and harmonic base excitation are presented. Depending on the values of the excitation amplitude and frequency in the stable and unstable regions, the solution exhibits many shapes besides the transition periodic shapes. A numerical example assessing the role of slenderness ratio of the beam, is presented to demonstrate the effectiveness of the proposed study. Results indicate that for a given beam system with a known excitation, increasing the tip mass would almost always reduce the stable periodic region. The effect of the beam model assumption on the periodic domain is also studied. Results show that using purely flexural or even the Euler–Bernoulli model rather than Timoshenko, would produce an incorrect periodic region.

Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion

Natural frequencies and ope–loop responses of an elastic beam fixed on a moving cart and carrying an intermediate lumped mass

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

The dynamic response of a nonuniform beam with time-dependent elastic boundary conditions is studied by generalizing the method of Mindlin-Goodman and utilizing the exact solutions of general elastically restrained nonuniform beams given by Lee and Kuo. The time-dependent elastic boundary conditions for the beam are formulated. A general form of change of dependent variable is introduced and the shifting polynomials of the third-order degree, instead of the fifth-order degree polynomials taken by Mindlin-Goodman, are selected. The physical meaning of these shifting polynomial functions are explored. Finally, the limiting cases are discussed and several examples are given to illustrate the analysis.

D.A. Grant

Papers

Beam vibrations with time-dependent boundary conditions

The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass