E. Esmailzadeh et N. Jalili, PARAMETRIC RESPONSE OF CANTILEVER TIMOSHENKO BEAMS WITH TIP MASS UNDER HARMONIC SUPPORT MOTION, International journal of non-linear mechanics, 33(5), 1998, pp. 765-781
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.