In this paper, an approximate method for detertmining the unstable regions of dynamic
stability of thin-walled beams is given. The beam is assumed to be underth the actions of concentrated longitudinal forces at both ends and of the type where P。=const, Pt(t) a periodic 2 and small parameter. The end conditions are arbitrary.By using trigonometric series or Galerkin's method satisfying the end conditions, the fundamental equations, based on Vlasof's theory, are reduced to a system of three ordinary linear differential equations (4) or (7) of 2nd order with periodic coefficients. Moreover, they can easily be transformed to canonical form
Therefore, their characteristic equations are reciprocal equations, with characteristic roots symmetrioally distributed with respect to the real axis and unit circle in a complex plane. The condition of boundary lines between stable and unstsable regions is taken as, that all of the characteristie roots have unit modulus (absolute value), but there exist equal roots. Expanding the characteristic exponentials in series of the small parameter ,this condition is represented by the following equation: where Wn/Wnk represent different frequencics of n-mode vibrations of the beam under the of a constant force P。, and W is the frequency of P1(t). When u-0,(27) and (28) become Hence, dynamic unstability would take place at the neighbourhoods of these critieal ratios, expresse By (29) and (30).
When the unstable regions of dynamic stability are desired, we use pertubation method to determine
For Practical use, it is sufficient to determine
only. The boundary lines can then approximately determined by the following equations: For illustrating this method, a simply supported beam of narrow rectangular cross-section、under the action of varying end moments (fig. 2) is considered.
The fundamental unstable regions, corresponding
to bending, torsional and “mixed” type of dynamic unstability
are calculated and shown in figs. 4, 3, 5.
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