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.

Non-linear dynamic response of a rotating machine

Abstract: The dynamic response of a two-degree-of-freedom seismically mounted rotor system with cubic non-linearities is investigated. The rotating machine is subjected to internal forces caused by the eccentricity of the center of mass of the rotor. The equations of motion of the system are determined using Lagrange's equation. The method of multiple scales is then used to determined the response of the system; this is determined when the excitation frequency is near the first and second modal frequency under internal resonance conditions.

A homogenization analysis of the field theoretic approach to the quasi-continuum method

Abstract: Using the orbital-free density functional theory as a model theory, we present an analysis of the field theoretic approach to quasi-continuum method. In particular, by perturbation method and multiple scale analysis, we provide a formal justification for the validity of numerical coarse-graining of various fields in the quasi-continuum reduction of field theories by taking the homogenization limit. Further, we derive the homogenized equations that govern the behavior of electronic fields in regions of smooth deformations. Using Fourier analysis, we determine the far-field solutions for these fields in the presence of local defects, and subsequently estimate cell-size effects in computed defect energies.

Asymptotic analysis of a vibrating cantilever with a nonlinear boundary

Abstract: Nonlinear vibration of a cantilever in a contact atomic force microscope is analyzed via an asymptotic approach. The asymptotic solution is sought for a beam equation with a nonlinear boundary condition. The steady-state responses are determined in primary resonance and subharmonic resonance. The relations between the response amplitudes and the excitation frequencies and amplitudes are derived from the solvability condition. Multivaluedness occurs in the relations as a consequence of the nonlinearity. The stability of steady-state responses is analyzed by use of the Lyapunov linearized stability theory. The stability analysis predicts the jumping phenomenon for certain parameters. The curves of the response amplitudes changing with the excitation frequencies are numerically compared with those obtained via the method of multiple scales. The calculation results demonstrate that the two methods predict the same varying tendencies while there are small quantitative differences.

Effects of a Dynamic Vibration Absorber on Nonlinear Hinged-Free Beam

Abstract: This study examined the vibrations of a hinged-free nonlinear beam placed on a nonlinear elastic foundation. The authors found that specific combinations of an elastic modulus in the elastic foundation resulted in 1:3 internal resonances in the first and second modes of the beam. This prompted adding a dynamic vibration absorber (DVA) on the elastic beam in order to prevent internal resonance and suppress vibrations. When the DVA was placed at the free end of the beam, the boundaries were time dependent. Thus, a shifting polynomial function was used to convert the nonhomogeneous boundary conditions into homogeneous boundary conditions. The authors analyzed this nonlinear system using the method of multiple scales (MOMS). Fixed-point plots were also used to facilitate the observation of internal resonance. This made it possible to study the influence of nonlinear geometry and nonlinear inertia associated with the vibration of the elastic beam. The authors also examined the combination of optimal ma...