This study is concerned with the influence of axial inertia upon the elastic bending motion of initially slightly curved columns acted on by time-dependent axial forces. The equations of motion include both axial inertia and nonlinear strain terms. Numerical solutions were obtained for a similar problem previously studied by Hoff [1] but in which axial-inertia effects were neglected; i.e., the problem of a simply supported column initially bent in the shape of a half sine wave and loaded by displacing one end axially at a constant rate. The range of solutions pertains to conventional structural compression members (slenderness ratios less than 150), and to minimum rates of loading compatible with elastic response of common engineering materials. This study suggests that axial-inertia effects are of secondary importance in so far as the gross elastic response of conventional structural columns is concerned.
The free motion of an undamped pendulum-type vibration absorber is studied on the basis of approximate nonlinear equations of motion. It is shown that this type of mechanical system exhibits the phenomenon of auto parametric excitation; a type of “instability” which cannot be accounted for on the basis of the linearized system. Complete energy transfer between modes is shown to occur when the beam frequency is twice the simple pendulum frequency. On the basis of a numerical solution, approximately 150 cycles of the beam oscillation take place during a single cycle of energy interchange.
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