Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Numerical Study of the Influence of Waving Bottom on the Fluid Surface Wave

The effect of waving bottom on the fluid surface wave was investigated. Starting from the basic equations of potential flow theory and boundary conditions, we used the multiple scales perturbation method to deduce fluid surface waves satisfy the first-order approximate equation and second-order approximate equation. Under the second-order approximation, the fluid surface waveform was simulated with the Matlab in the presence of different waving bottom form. The results show that there are three solitary waves on the surface of the fluid. With the development of time, the amplitude of each solitary wave has not changed. It seems that they are not affected each other and propagate independently. So it suggests that the waving bottom is effective for maintaining surface wave energy balance income and expenditure in spreading process.

Modifications to the response of a parametrically excited cantilever beam by means of smart active elements

This article is concerned with applying active smart material elements for modifying parametric vibration in a flexible composite beam structure. The glass epoxy beam is bonded to two theoretically prestrained shape memory alloy (SMA) strips and fitted with a lumped end mass. In this study, the components of the recovery force generated during the SMA activation are derived with respect to a three-dimensional frame when the structure is undergoing combined bending and torsional motions. In order to employ Lagrangian dynamics, the generalized forces are formulated and the equations of motion are then derived. Three different parametric resonances for the structure are predicted by using the multiple scales perturbation method. In addition, the effects of the SMA strips on the natural frequencies, the mode shapes, and the instability regions of the structure are all investigated. It is shown that the different thresholds of instability for parametric resonances within a composite structure of this sort may be influenced by smart active elements.

Internal Resonance of Finite Beams on Nonlinear Foundations Traversed by a Moving Load

Vibration analysis of beams traversed by moving load is an old and well known topic in structural mechanics, and has been of great interest for researchers of different fields, such as mechanical, railway and civil engineering. Many researchers have conducted different investigations in this field. In the present research, the nonlinear vibration of the system is studied and consequently the response of the system to a moving load is determined as a closed form solution. Furthermore, the effects of load amplitude on the response of the system are investigated. Galerkin’s method is first utilized to truncate the governing equation of motion and then MMS (Method of Multiple Scales) perturbation method is applied to study the nonlinear vibration of the system, in the presence of the internal resonance. Effects of damping of the foundation as well as magnitude of the moving load on the frequency responses are investigated. The proposed methodology and obtained results can be used to investigate the behavior several systems among which railway system shows a good compatibility.

The Parametric Resonance Instability in a Drilling Process

This study investigates dynamic instability in a high-speed drilling process. A pretwisted beam is used to simulate the drill. The time-dependent nature of the thrust force and the drilling depth is considered in the equation of motion of the drill. A moving Winkler-type elastic foundation assumption is applied to the drill tip to approximate the time-varying boundary conditions in the drilling process. Galerkin’s method is used to formulate the characteristic equation in a discrete form. The variation of the instability regions of the drill system is solved and analyzed by employing the multiple-scales perturbation method. The numerical results indicate that the unstable regions suddenly enlarge and shift toward a lower frequency when the drill first contacts the work piece. The effects of the rotational speed, pretwisted angle, and thrust force of the drill on the variation of the dynamic instability in high-speed drilling are also studied and are found to be highly influential.

ANALYTICS OF HOMOCLINIC BIFURCATIONS IN THREE-DIMENSIONAL SYSTEMS

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion is based on the collision between the periodic and the hyperbolic fixed point involved in the bifurcation. The second uses the infinity condition of the period of the periodic orbit. For illustration a specific system is investigated.

Dynamic Stability in a Cracked Turbo Blade-Disk

Analysis of the stability in a cracked blade-disk system is proposed. The effect of modal localization on the stability in a rotating blade-disk was studied. A crack near the root of a blade is regarded as a local disorder in this periodically coupled blade system. Hamilton’s principle and Galerkin’s method were used to formulate the equations of motion for the cracked blade-disk. The instability regions of this cracked blade-disk system were specified by employing the multiple scales perturbation method. Numerical results indicate that the rotation speed, shroud stiffness and crack depth in the blades affect the stability regions of this mistuned system significantly.