Spatial and Temporal Excitation to Generate Traveling Waves in Structures

Abstract

The problem in the creation of traveling waves is approached here from an unconventional angle. The formulation makes use of normal vibration modes, which are standing waves, to express both traveling waves and the required force distribution. It is shown that a localized force is required at any discontinuity along the structure to absorb reflected waves. This convention is demonstrated for one- and two-dimensional structures modeled as continua, and as discretized numerical approximation of the mass and stiffness matrices. Harmonic vibrations can be characterized as standing or traveling waves or as a combination of both. By applying forces that have been specially designed for the purpose, the vibratory response can become a pure traveling wave. The force distribution is important for the design of ultrasonic motors and in control applications, attempting to absorb and create outgoing and incoming waves.

Vector control method applied to a traveling wave in a finite beam

“…In this paper, we will consider that if the two modes a and b have the same amplitude and are phase shifted by 90° (W a = jW b ), a traveling wave in the beam results. However, Gabai and Bucher [8] point out that perfect traveling wave is obtained when a large number of modes is taken into account; then, we expect to have an imperfect traveling wave in our case. This point will be quantified by experimental tests.…”

“…Finally, the vibration amplitude for the two modes, and their phase shift, can be calculated using (8) and 9…”

“…Hence, (6), (8), and (9) describe the evolution of the vibration amplitude for the two modes a and b. These equations yield a model which can be represented by way of a causal ordering graph (coG), as shown in Fig.…”

“…In that case, the tuning of the exciters-their supply voltage, frequency and phase shift-becomes complex to obtain the traveling wave. The analytical solutions for a traveling wave in a string, a beam, and a membrane are given in [8]; the authors calculate the analytical expression of the applied forces to obtain a pure traveling wave in an undamped medium. They also show that it is sufficient to excite ten modes only; the other modes' contribution can be neglected.…”

“…These methods can be off-line [8], [9], which means that the excitation is calculated before the experimental runs, or on-line [10] and can thus adapt themselves to variation in the operating conditions. However, their tuning procedure is based on steady states, and dynamic performances are not studied.…”