Internal Resonance in a Vibrating Beam: A Zoo of Nonlinear Resonance Peaks

Abstract

In oscillating mechanical systems, nonlinearity is responsible for the departure from proportionality between the forces that sustain their motion and the resulting vibration amplitude. Such effect may have both beneficial and harmful effects in a broad class of technological applications, ranging from microelectromechanical devices to edifice structures. The dependence of the oscillation frequency on the amplitude, in particular, jeopardizes the use of nonlinear oscillators in the design of time-keeping electronic components. Nonlinearity, however, can itself counteract this adverse response by triggering a resonant interaction between different oscillation modes, which transfers the excess of energy in the main oscillation to higher harmonics, and thus stabilizes its frequency. In this paper, we examine a model for internal resonance in a vibrating elastic beam clamped at its two ends. In this case, nonlinearity occurs in the form of a restoring force proportional to the cube of the oscillation amplitude, which induces resonance between modes whose frequencies are in a ratio close to 1:3. The model is based on a representation of the resonant modes as two Duffing oscillators, coupled through cubic interactions. Our focus is put on illustrating the diversity of behavior that internal resonance brings about in the dynamical response of the system, depending on the detailed form of the coupling forces. The mathematical treatment of the model is developed at several approximation levels. A qualitative comparison of our results with previous experiments and numerical calculations on elastic beams is outlined.

A multi-parametric recursive continuation method for nonlinear dynamical systems

“…DiBerardino and Dankowicz [60] showed that ISs can be created by introducing asymmetry into a nonlinear system. In [61], the presence of IS is explained analytically by analyzing the 1:3 internal resonance configuration between two Duffing oscillators for different couplings. In [62], an experiment was carried out to illustrate the IS phenomenon between a Duffing oscillator and a clamped-clamped beam at a 1:3 internal resonance configuration.…”

Effects of internal resonances in the pitch glide of Chinese gongs

“…Internal resonances are energy exchanges that occur between modes that are strongly coupled by geometrical nonlinearities. 2 They have been reported in numerous studies that concern the nonlinear behavior of beams, 38 plates, 35 shells, 13 and even other percussion instruments like large Chinese tamtams 10 and steelpans. 12 In the case of the Chinese opera gongs, internal resonances have already been demonstrated using modal active control.…”

Limit cycles and bifurcations in a nonlinear MEMS resonator with a 1:3 internal resonance

“…As the drive amplitude is further increased, the nonlinear response of the first mode exhibits features such as dips, plateaus and oscillations in its frequency response sweep, Fig. 1(b), the appearance of these features is equated with the onset of internal resonance [12,16,17]. Experimentally, such features are observed when the first mode drive amplitude exceeds 0.22 V PP (for a forward frequency sweep).…”