Parametric Instability of Tapered Beams by Finite Element Method

Abstract

The dynamic stability behaviour of a tapered beam has been studied using a finite element analysis. The instability zones of the parametric stability diagram have been discussed for the entire ranges of static and dynamic load factors. It has been observed that at high values of static load and beyond a particular value of the dynamic load factor, the periodic solution of the Mathieu equation does not exist in the principal region. This leads to unstable behaviour due to large displacement of the beam due to increasing values of static and dynamic load factors.

Study of Second Regions of Dynamic Instability of a Uniform Beam

“…And the other equation to bound the stability region is, Equations (6a) and (6b) can include the periodic terms if the eigenvalues are obtained in terms of the disturbing frequency, W. They define the boundaries of the region and should be compared with the boundaries of the first regions of dynamic stability, with period 2T [5][6][7], for which the assumed solution was shown in equation (4). Corresponding to the solution of period 2T (the first regions of dynamic instability), the stability boundaries are obtained as,…”

“…At this stage it is important to consider the simplified case which is implicitly included in equations (6) and (7). In case of free vibration, a = 0 and b = 0.…”

“…In case of free vibration, a = 0 and b = 0. Then, from both equations (6) and 7, neglecting the trivial solutions, we obtain the universally well known free vibration problem,…”

“…To determine the regions of dynamic instability, the disturbing frequency, W, is taken as W = (W/w 1 )w 1 , where w 1 = the fundamental natural frequency, as obtained from the solution of equation (8). This formulation is also used elsewhere [6].…”

“…Thomas and Abbas [5] considered the effect of shear deformation and rotary inertia on dynamic stability. Datta and Chakraborty [6] have studied the behavior of parametric instability of tapered beams by the finite element method. Datta and Nagraj [7] also studied the dynamic instability behavior of tapered bars with flaws supported on an elastic medium.…”