Multiple scales analysis of slow–fast quasi-linear systems

This article illustrates the application of multiple scales analysis to two archetypal quasi-linear systems; i.e. to systems involving fast dynamical modes, called fluctuations, that are not directly influenced by fluctuation–fluctuation nonlinearities but nevertheless are strongly coupled to a slow variable whose evolution may be fully nonlinear. In the first case, fast waves drive a slow, spatially inhomogeneous evolution of their celerity field. Multiple scales analysis confirms that, although the energyE, the angular frequencyωand the modal structure of the waves evolve, the wave actionE/ωis conserved in the absence of forcing and dissipation. In the second system, the fast modes undergo an instability that is saturated through a feedback on the slow variable. A new multi-scale analysis is developed to treat this case. The key technical point, confirmed by the analysis, is that the fluctuation energy and mode structure evolve slowly to ensure that the slow field remains in a state of near marginal stability. These two model systems appear to be generic, being representative of many if not all quasi-linear systems. In each case, numerical simulations of both the full and reduced dynamical systems are performed to highlight the accuracy and efficiency of the multiple scales approach. Python codes are provided as electronic supplementary material.

Higher-Order Multiple Scales Analysis of Weakly Nonlinear Lattices With Implications for Directional Stability

Recent focus has been given to nonlinear periodic structures for their ability to filter, guide, and block elastic and acoustic waves as a function of their amplitude. In particular, two-dimensional (2-D) nonlinear structures possess amplitude-dependent directional bandgaps. However, little attention has been given to the stability of plane waves along different directions in these structures. This study analyzes a 2-D monoatomic shear lattice composed of discrete masses, linear springs, quadratic and cubic nonlinear springs, and linear viscous dampers. A local stability analysis informed by perturbation results retained through the second order suggests that different directions become unstable at different amplitudes in an otherwise symmetrical lattice. Simulations of the lattice’s equation of motion subjected to both line and point forcing are consistent with the local stability results: waves with large amplitudes have spectral growth that differs appreciably at different angles. The results of this analysis could have implications for encryption strategies and damage detection.

Multiple Scales Analysis and Assessment of Environmental Air Quality in Tianjin

Following the development of regional economy, especially the construction of Tianjin Binhai new district, environmental pollution and ecological hazards has been resulted from the pollutant discharge and reclamation etc. The environmental air quality is very important for the health of human being, so it is very important for analyzing and evaluating the environmental air quality for guiding the environmental management and control. In this paper, the environmental air quality of recent fourteen years in Tianjin is analyzed. Multiple scales method is applied to obtain the environmental air current situation. The results of analysis and assessment show the environmental air quality is becoming worse in recent years.

Second-Order Perturbation Analysis of Low-Amplitude Traveling Waves in a Periodic Nonlinear Chain

Traveling waves in one-dimensional nonlinear periodic structures are investigated for low-amplitude oscillations using perturbation analysis. We use second-order multiple scales analysis to capture the effects of quadratic nonlinearity. Comparisons with the linear and cubical nonlinear cases are presented in the dispersion relationship, group velocity and phase velocity and their dependence on wave number and amplitude of oscillation. Quadratic nonlinearity is shown to have a significant effect on the behavior.

The Small Scale Effect on Linear Vibration of Beam-Like Nanostructures

Transverse free dynamics of a beam-like nanostructure with axial load is investigated. The effects of a small size at nano-scale unavailable in classical mechanics are presented. Explicit solutions for natural frequency, vibration mode and transverse displacement are obtained by separation of variables and multiple scales analysis. Results by two methods are in close agreement.