Journal of Sound and <ibration (2000) 238(1), 171}178 doi:10.1006/jsvi.2000.3013, available online at http://www.idealibrary.com on LETTERS TO THE EDITOR CHANGING THE PROPAGATION DIRECTION OF FLEXURAL ULTRASONIC PROGRESSIVE WAVES BY MODULATING EXCITATION FREQUENCY B.-G. LOH AND P. I. RO Precision Engineering Center, North Carolina State ;niversity, Raleigh, NC, ;.S.A. (Received 12 July 1999, and in ,nal form 24 February 2000) 1. INTRODUCTION When #exural waves propagate, particles on the surface of the vibrating beam move elliptically. Then, an object on the beam is forced to move in the opposite direction of wave propagation through frictional force [1]. Changing wave propagation directions reverse transport direction of an object. In references [2, 3], the change in wave propagation direction was made possible connecting either a voltage source or absorptive circuitry to either active or passive modules. In reference [4], modulating the relationship between phase di!erences in excitation forces allows for alteration of wave propagation direction. However, in reference [5], it was observed that changing the excitation frequency also caused the direction of wave propagation to reverse. This particular phenomenon has never been observed. Also, the dependency of the wave propagation direction on the excitation frequency has important rami"cations on the active control of object transport using ultrasonic #exural progressive waves. This paper presents changing the wave propagation direction of #exural ultrasonic progressive waves with variations in the excitation frequency. 2. PROTOTYPE AND WORKING PRINCIPLES The prototype in Figure 1 consists of a beam and modules that contain piezoelectric actuators and horns [6]. The beam and horns are made of 6061-T6 aluminum because of the excellent acoustical characteristics of this material. The beam is 11 mm wide, 3)1 mm thick and 500 mm long. The piezoelectric actuators are bolted Langevin-type transducers (BLT) manufactured by NGK Spark Plug Co. Ltd. (model no. DA2228). Their resonant frequency is 28 kHz. As shown in Figure 2, two sets of identical horn-and-actuator modules were assembled with each of the supporting structures. The horns were attached to the supporting structure at the nodes of the horns. Each set was designed to operate independently. The two modules are supplied with sinusoidal voltages of the same frequency but with a phase di!erence of 903. Using normal mode expansion, the vibration of the beam can be represented by the linear superposition of an in"nite number of natural mode shapes of vibration [7, 8]. Of particular interest are the two modes that have natural frequencies closest to the excitation frequency. The vibration amplitudes of these two modes are large, compared to other mode shapes. Therefore, the vibration can be approximated by 0022-460X/00/460171#08 $35.00/0 ( 2000 Academic Press