Analytical and Experimental Investigation of Modal Disparity

This paper describes an analytical and experimental investigation on the property of modal disparity in a vibrating structure. For a given structure, the concept of modal disparity describes its ability to generate significant changes in the mode shapes by some type of on-the-fly structural modification. In the present study we consider a vibrating beam that experiences a controlled stiffness change, induced by the activation and deactivation of an electromagnetic brake. The beam has two stiffness states corresponding to the two states of the brake. The mechanisms of braking and brake release are mathematically modeled using the principle of impulse and momentum and a finite element model of the beam that adequately describes both stiffness states is developed. The transformation matrix which maps the modal displacements and velocities between the two stiffness states is derived and it is shown that this matrix is a measure of modal disparity, i.e., how well energy can be redistributed between the modes. Experiments were conducted to validate energy redistribution between modes through stiffness variation. A clamped-clamped aluminum beam with a hinge and built-in electromagnetic brake was used for the experiments. A piezoelectric strain sensor was used for sensing and a pair of piezoelectric transducers were used for excitation. It is shown that significant amounts of energy can be shifted back and forth between different sets of modes in a systematic and predictable manner. This confirmation of modal disparity for a specific structural system opens the door for a number of potential applications in passive and active control of vibrating structures.