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Research Papers

# Spatial and Temporal Excitation to Generate Traveling Waves in Structures

[+] Author and Article Information
Ran Gabai

Dynamics Laboratory, Faculty of Mechanical Engineering, Technion IIT, Technion City, Haifa, 32000, Israelrang@technion.ac.il

Izhak Bucher

Head of the Dynamics Laboratory, Faculty of Mechanical Engineering, Technion IIT, Technion City, Haifa, 32000, Israelbucher@technion.ac.il

J. Appl. Mech 77(2), 021010 (Dec 11, 2009) (11 pages) doi:10.1115/1.3176999 History: Received December 14, 2008; Revised March 26, 2009; Published December 11, 2009; Online December 11, 2009

## 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.

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## Figures

Figure 2

Spatial distribution of forces and moments applied to a beam to achieve a pure traveling wave (λ=0.4 m) (left) using all the modes to calculate the forces, (right) using only the first ten modes of the beam, (top) force distribution, and (bottom) moment distribution

Figure 1

Modal coordinates describing a pure traveling wave having different wavelengths. (Left) λ=0.4 m and (right) λ=0.1 m. Small windows show an enlarged portion of most contributing coefficients.

Figure 3

Illustration of the beam response in time and space when using only the first ten modes to describe the desired traveling wave response

Figure 8

Membrane response and power-flow obtained by exciting only the ten most significant modes from Fig. 6

Figure 9

Force distribution over the membrane calculated to excite the ten most significant modes from Fig. 6

Figure 10

Force and moment distribution along a beam with an abrupt change in cross section. (Top) force distribution and (bottom) moment distribution.

Figure 11

Traveling wave response of a beam with an abrupt change in cross section

Figure 12

Distribution of forces generating a pure traveling wave on a membrane with a localized mass

Figure 4

Beam’s spatial response (a) and (c), and complex amplitude (b) and (d). The numbers indicate the measured points and the corresponding locations on the complex amplitude. (a) and (b) The response to the optimal excitation (a pure traveling wave). (c) and (d) The response to the forces only (without moments).

Figure 5

Nine initial modes of the membrane in the example: (a) ϕ11, (b) ϕ21, (c) ϕ31, (d) ϕ12, (e) ϕ22, (f) ϕ14, (g) ϕ32, (h) ϕ13, and (i) ϕ41

Figure 6

Modal coefficients for different wavelengths and wave directions: (a) λ=0.4γ=25 deg, (b) λ=0.4γ=50 deg, (c) λ=0.2γ=35 deg, and (d) λ=0.7γ=75 deg

Figure 7

(Top images) Membrane’s vibration and the corresponding mean power-flow (marked by arrows) and the matching force distribution (bottom images). (a) and (b) λ=0.4γ=25 deg, and (c) and (d) λ=0.2γ=35 deg.

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