PARAMETRIC RESPONSE OF CANTILEVER TIMOSHENKO BEAMS WITH TIP MASS UNDER HARMONIC SUPPORT MOTION

The parametric response of a thick cantilever beam with a tip mass sub jected to harmonic axial support motion is investigated. The Timoshenk o beam theory is used to assess the effects of rotary inertia and shea r deformation for the beam. In this regard, different modal amplitudes for transverse displacement and angle of rotation of the cross-sectio n are considered. This yields a more accurate description of the dynam ic model. The governing equations of motion are then derived for an ar bitrary 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 analy sis, the general governing equations are truncated to only the first m ode of vibration. Using Green's function and Schauder's fixed point th eorem, the necessary and sufficient conditions for the existence of pe riodic 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 a mplitude and frequency in the stable and unstable regions, the solutio n exhibits many shapes besides the transition periodic shapes. A numer ical example assessing the role of slenderness ratio of the beam, is p resented to demonstrate the effectiveness of the proposed study. Resul ts indicate that for a given beam system with a known excitation, incr easing the tip mass would almost always reduce the stable periodic reg ion. The effect of the beam model assumption on the periodic domain is also studied. Results show that using purely flexural or even the Eul er-Bernoulli model rather than Timoshenko, would produce an incorrect periodic region. (C) 1998 Elsevier Science Ltd. All rights reserved.