Tailoring circumferential traveling waves in a damped circular plate coupled to a spring-mass oscillator using phased force excitations

Abstract

We develop an analytical approach to produce circumferential traveling waves in a lightly damped circular plate under the cyclic symmetry-breaking scenario caused by a local spring-mass oscillator, using three-point actuators distributed either uniformly or nonuniformly along a circumferential path. Modal analysis and forced vibration of the lightly damped plate-oscillator system are examined. The addition of the spring-mass oscillator leads to separation of the degenerate modes and to associated frequency curve-veering phenomena. The plate-oscillator system exhibits two adjacent modes with nearly identical natural frequencies, which closely resemble the degenerate cosine and sine modes of an axisymmetric pure plate. A typical degenerate mode pair of a pure (uniform) plate is first selected, and the actuators’ positions, amplitudes and phases are tuned based on wave kinematics to achieve continuous rotation of the plate mode. Next, an approximately degenerate mode pair resembling the pure plate modes is identified, and the excitation frequency is optimized using a standing-wave ratio and a traveling-wave index based on complex orthogonal decomposition to generate circumferential traveling waves in the plate-oscillator system. It is found that standing waves occur at the resonance frequency; however, by tuning the excitation frequency to a value halfway between the natural frequencies of the two adjacent modes, the effects of dynamical imperfections can be mitigated, leading to traveling waves. Adding external damping can result in partially modulated standing waves, which is much more significant in nonuniform configurations. The wave dynamics are verified through finite element analysis. This work will be helpful for practical application of particle transport on two-dimensional surfaces by robust circumferential traveling waves.