Theoretical modeling on the combination resonance of size-dependent microbeams

The combination resonance of size-dependent microbeams is investigated. Two harmonic forces act on the microbeam, and combination resonance is observed while the excitation frequencies differ from the resonant frequency. Microbeams with two different sources of nonlinearities including three kinds of boundary conditions, clamped-free (nonlinearity comes from large curvature and nonlinear inertial), clamped-clamped, and hinged-hinged (nonlinearity originates from mid-plane stretching-bending coupling), are taken into consideration to have a deep understanding of this phenomenon. A traveling load acting on the microbeam is presented as a special case of combination resonance. The modal discretization technique is applied to discretize the equations of motion, and then the Lindstedt–Poincare method, a perturbation approach, is employed to solve the resultant equations. The conditions for combination resonance are presented, and frequency-response curves and time histories at the resonance point are obtained for microbeams of each boundary condition. Results reveal that different sources of nonlinearities result in different performances of combination resonance. The free vibration part constitutes a large percentage of the final response. Furthermore, the situation of coexistence of combination resonance and superharmonic (or subharmonic) resonance is determined. The special case demonstrates a higher amplitude than the common combination resonance for all the boundary conditions. Parametric studies are then carried out to discuss the effects of the length scale parameter, excitation force as well as its position, and damping on the performance of the microbeam.

Combination resonance analysis of imperfect functionally graded conical shell resting on nonlinear viscoelastic foundation in thermal environment under multi-excitation

In this work, nonlinear forced vibrations of truncated conical shells are presented using a semi-analytical method. The material properties are varied along the thickness direction as a power law distribution. The functionally graded truncated conical shells are exposed to external harmonic load and placed in the thermal environment and have an initial imperfection. Furthermore, the functionally graded truncated conical shells rests on generalized nonlinear viscoelastic foundations which consisted of a Winkler and Pasternak foundation parameters augmented by a Kelvin–Voigt viscoelastic model and a nonlinear cubic stiffness. The fundamental equations are extracted using first-order shear deformation theory in conjunction with nonlinear von Kármán relationships. The partial differential equations of truncated conical shells are reduced through Galerkin’s method, and the result is extracted using the multiple scales method. To analyze the resonance analyses, a two-term external excitation is considered. In this regard, various secondary resonances are investigated, and finally, the analyses about combination resonances are represented. To investigate the presented approach, a comparison study is performed with those addressed by other researchers. To analyze the nonlinear combination resonance behavior of truncated conical shells, the effect of geometrical characteristics, material properties, power law index, thermal effects, external load amplitude, and initial imperfection are examined. Finally, the steady-state responses of the nonlinear system are analyzed. As one of the most interesting results, the softening behavior of truncated conical shells with inverse quadratic distribution is the most, and for the quadratic distribution is the least.

Investigation of instability in ultrasonic welding under parametric excitation

This paper develops a time-varying model for battery tabs based on the parametric excitation of Euler-Bernoulli beams. The instability caused by combination resonance under a high-frequency longitudinal load is considered. A Galerkin procedure is used to discretize the time-dependent problem into the Mathieu equation. The critical axial load is obtained from the transition curve of combination resonance. The effectiveness of the stability analysis was verified by numerical simulations involving longitudinal and bending loads.

Asynchronous parametric excitation: validation of theoretical results by electronic circuit simulation

A validation of recent theoretical results on the stability effects of asynchronous parametric excitation is presented. In particular, the coexistence of both resonance and anti-resonance at each combination resonance frequency is to be confirmed on a close-to-experiment simulation model. The simulation model reproduces the experimental setup developed by Schmieg in 1976, remaining the only experimental study on asynchronous excitation to this day. The model consists of two oscillating electronic circuits with feedback-free coupling through parametric excitation. In contrast to a mechanical system, the phase relations of the parametric excitation terms in an electronic system can be easily adjusted. The implementation of the simulation model is performed in the electronic circuit simulation software LTspice. The electronic model itself is first validated against the experimental results obtained by Schmieg and is then used to confirm the theoretical findings. The results of the electronic circuit simulation show excellent qualitative and quantitative agreement with analytical approximations confirming the coexistence of resonance and anti-resonance effects near a combination resonance frequency.

Simultaneous Resonance and Anti-Resonance in Dynamical Systems Under Asynchronous Parametric Excitation

Since the discovery of parametric anti-resonance, parametric excitation has also become more prominent for its stabilizing properties. While resonance and anti-resonance are mostly studied individually, there are systems where both effects appear simultaneously at each combination resonance frequency. With a steep transition between them and a high sensitivity of their relative positions, there is a need for a concurrent study of resonance and anti-resonance. The semi-analytical method of normal forms is used to derive approximate analytical expressions describing the magnitude of the stability impact as well as the precise locations of stabilized and destabilized areas. The results reveal that the separate appearance of resonance and anti-resonance is only a special case occurring for synchronous parametric excitation. In particular, in circulatory systems the simultaneous appearance is expected to be much more common.

Investigation of the favorable conditions to apply the combination resonances approach for crack detection purposes

Various structural health monitoring methods were recently applied to fault diagnosis of rotating machinery, in which vibration response–based techniques demonstrated to be well adapted for different scenarios. However, only severe cracks are detected when most of such techniques are applied. Thus, innovative methods are proposed to identify the existence of incipient cracks in rotating shafts. In a previous contribution, the combination resonance approach for crack identification purposes was presented. This method is based on the application of additional forces at combination frequencies along with a heuristic optimization method to identify crack signatures in the vibration responses of the rotor. In the present work, the best conditions to apply the combination resonance approach are investigated in terms of the amplitude and frequency of the so-called diagnostic force. The vibration responses and phase angles of a cracked rotor were determined by using the harmonic balance technique. The FLEX approach is used to model the crack. Numerical simulations were performed using the finite element model of a horizontal rotor.

Simultaneous Resonance and Anti-Resonance in Dynamical Systems Under Asynchronous Parametric Excitation

Since the discovery of parametric anti-resonance, parametric excitation has also become more prominent for its stabilizing properties. While resonance and anti-resonance are mostly studied individually, there are systems where both effects appear simultaneously at each combination resonance frequency. With a steep transition between them and a high sensitivity of their relative positions, there is a need for a concurrent study of resonance and anti-resonance. The semi-analytical method of normal forms is used to derive approximate analytical expressions describing the magnitude of the stability impact as well as the precise locations of stabilized and destabilized areas. The results reveal that the separate appearance of resonance and anti-resonance is only a special case with synchronous parametric excitation. In particular, in circulatory systems the simultaneous appearance is expected to be much more common.