Shock Waves, Vibrations, and Resonance in Linearly Elastic Beams

Abstract

Equations can describe the stresses in linearly elastic simple beams, subjected to different types of suddenly applied, or dynamic, incident loads. The assumption of linearly elasticity simplifies the nonlinear aspects of structural vibrations. Simplified linear models are applicable to many vibration analyses, and provide significant insights into vibration behavior. When a load impacts a beam, there are two different stresses created. The primary bending stress is the bending stress of the beam due to perpendicular loading. The secondary bending stress is a much higher frequency bending stress that travels along the surface of the beam relative to the moving load front. In general, the primary bending stress has a much larger magnitude than the secondary bending stress. The maximum stress values for the primary and secondary bending stresses may be described by dynamic load factors (DLF’s); where the maximum stress caused by an applied load equals the DLF multiplied by the static stress; and the static stress is the equilibrium stress for the load in question applied to the beam. For example, consider the primary stresses due to a sudden pressure increase, which is applied perpendicular to a beam’s axis. The maximum stress in that beam is double the static stress, and deflection, that would be caused if that incident pressure is statically applied. That is, the DLF < 2 for a pressure that is suddenly applied over the incident surface of a beam: the maximum possible stress is twice the static stress. Now consider the secondary stress. A localized bending stress occurs when a pressure wave travels parallel to the axis of a beam, and this bending stress has a DLF < 4. That is, a high frequency bending stress travels along the beam with a maximum magnitude equal to four times the applied pressure; the maximum stress is four times the static stress. As other examples, stresses due to different types of loads may be evaluated. A gradually increasing pressure yields 1< DLF < 2, depending on the loading rates. A harmonically applied load may yield a DLF theoretically equal to infinity, but damping greatly reduces the maximum DLF. Also, these harmonic loads can significantly excite higher mode frequencies, i.e., the applied load may excite any one of a number of a structure’s natural frequencies. New theory is presented here to describe shock waves, suddenly applied loads, and resultant vibrations to coherently relate these complex quantities.