Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Trajectory-based modelling of broad-band electromagnetic wavefields

Abstract: SUMMARY A trajectory-based solution of Maxwell's equations is derived using the method of multiple scales. This time-domain technique utilizes an asymptotic expansion in terms of the ratio of the wave front length-scale to the length-scale of the heterogeneity. The approach provides an alternative to an expansion in terms of frequency and allows one to model electromagnetic wave propagation over a wide frequency band. At the lower end of the frequency band, the trajectory-based solution reduces to a previously derived diffusive solution. Similarly, at higher frequencies one obtains the ‘delta-like’ solution associated with hyperbolic wave propagation. However, the solution is also valid at intermediate frequencies which cannot be characterized as either diffusive or hyperbolic. A numerical illustration demonstrates the importance of both conduction and displacement currents at frequencies between 10 and 100 MHz. The amplitudes computed using the trajectory-based approach compare well with analytic results for a homogeneous whole-space. Using the technique I am able to model observations from a broad-band (3–300 kHz) experiment at the Richmond Field Station in California. In addition, ground penetrating radar waveforms in the 5–200 MHz range, gathered at the Boise Hydrogeophysical Research Site, are matched using the results of a radar velocity tomogram.

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

Abstract: The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility) Galerkin approximation is utilized to discretize the equations of motion The obtained equation is simulated to find the natural frequency of the system In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system The experimental results verify the theoretical findings very closely It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear modelsCopyright © 2006 by ASME

Nonlinear Metamaterials with Multiple Local Mechanical Resonators: Analytical and Numerical Analyses

Abstract: This paper examines the role of stiffness nonlinearity on a periodic one-dimensional chain with multiple local resonators. The cells of the chain consist of lumped masses connected through nonlinear springs. Each cell is embedded with multiple local resonators having different parameters. In one case the local resonators are assumed to be linear and in another case they are nonlinear. The dispersion equation for the system is derived analytically by the method of multiple scales (MMS). The results are validated via comparison with those in the literature and numerically via Matlab. The nonlinearity shows enhancement in the bandgap regions, especially with increasing number of local resonators.

Single-degree-of-freedom model of displacement in vortex-induced vibrations

Abstract: In contrast to the approach of coupling a nonlinear oscillator that represents the lift force with the cylinder’s equation of motion to predict the amplitude of vortex-induced vibrations, we propose and show that the displacement can be directly predicted by a nonlinear oscillator without a need for a force model. The advantages of the latter approach include reducing the number of equations and, subsequently, the number of coefficients to be identified to predict displacements associated with vortex-induced vibrations. The implemented single-equation model is based on phenomenological representation of different components of the transverse force as required to initiate the vibrations and to limit their amplitude. Three different representations for specific flow and cylinder parameters yielding synchronization for Reynolds numbers between 104 and 114 are considered. The method of multiple scales is combined with data from direct numerical simulations to identify the parameters of the proposed models. The variations in these parameters with the Reynolds number, reduced velocity or force coefficient over the synchronization regime are determined. The predicted steady-state amplitudes are validated against those obtained from high-fidelity numerical simulations. The capability of the proposed models in assessing the performance of linear feedback control strategy in reducing the vibrations amplitude is validated with performance as determined from numerical simulations.