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.

Contact and pressure balance structures in two-fluid cosmic-ray hydrodynamics

Abstract: The role of cosmic-ray-modified contact discontinuities and pressure balance structures in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the two-fluid cosmic-ray hydrodynamic equations in the long-wavelength limit consist of the backward and forward propagating cosmic-ray-modified sound waves, with sound speed dependent on both the cosmic-ray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum ofthe cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmic-ray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero perturbations in the thermal gas, whereas the cosmic-ray-modified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, N-wave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the short-wavelength entropy wave and the thermal gas sound waves in a slowly varying, large-scale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of squeezing instability for short-wavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmic-ray-modified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks and compression and rarefraction waves with the solar wind termination shock are briefly discussed.

Dynamic Response of Rotor-Bearing Systems With Time-Dependent Spin Rates

Abstract: This paper presents an investigation into the vibration of rotor-bearing systems with time-dependent spin rates. Due to this spin rate, parametric instability may take place in certain situations. In this work, the Galerkin method is used to eliminate the dependence on the spatial coordinates, and then the method of multiple scales is applied to derive periodic solutions and expressions for the boundaries of unstable regions analytically. Numerical results are given for the case where the spin rate is characterized as a small, harmonic perturbation superimposed on a constant rate. The effects of system parameters on the changes of the boundaries of unstable regions are shown.

Effect of fast parametric excitation on the instability behaviour of a spinning disc bounded in a compressible fluid-filled enclosure

Abstract: This paper focusses on investigating how high-frequency (fast) excitations affect the aeroelastic instability of a spinning disc immersed in a compressible fluid-filled enclosure. The method of multiple scales is used to recast the governing equations of motion into separate equations for fast and slow motions. The slow motions represent the dynamic behaviour of the non-autonomous coupled system. The stability boundaries of the coupled system are investigated through a bifurcation analysis with respect to the mean disc rotation speed and the forcing amplitude. The study reveals that an increase in the forcing amplitude of the fast parametric excitation results in a corresponding increase in the frequencies and the critical speeds of the coupled disc modes associated with the slow motions. Moreover, it is shown that the aeroelastic stability can be postponed or suppressed due to the stiffening and gyroscopic effects induced by the fast excitations.

Time-delay dynamics of the MR damper–cable system with one-to-one internal resonances

Abstract: In this study, we investigate the nonlinear resonant response of a magnetorheological (MR) damper–stay cable system with time delay, and the one-to-one internal resonance is considered. Based on Hamilton’s principle, the motion equations of the MR damper–cable system are derived, and the Galerkin method is applied to obtain the discrete model. Then, the method of multiple scales is applied to determine the modulation equations and the second-order solution of the nonlinear response of the MR damper–cable system. Following, the equilibrium solution and dynamic solution of the modulation equations are examined via the Newton–Raphson method and shooting method. The results show that the equilibrium solution may undergo Hopf bifurcation, resulting in the periodic solution. Moreover, the effects of the time delay and the inclination angle on the resonant response of the MR damper–cable system are investigated as well as those of the damper position. The numerical results show that the time delay increases the amplitudes of in-plane and out-of-plane modes and results in the more remarkable hardening behavior and relatively poor mitigation performance of the MR damper. However, the large time delay may suppress the complex chaotic modulation motion of the MR damper–cable system.

Internal Resonance of Axially Moving Beams with Masses

Abstract: Transverse vibrations of axially moving beams with multiple concentrated masses have been investigated. It is assumed that the beam is of Euler–Bernoulli type, and both ends have simply supports. Concentrated masses are equally distributed on the beam. This system is formulated mathematically and then sought to find out approximate solutions. In case of three-to-one internal resonance, analytical solutions are derived by means of method of multiple scales (a perturbation method). It is assumed that axial velocity of the beam is harmonically varying around a mean-constant velocity. Steady-state vibration characteristics are investigated from the amplitude-phase modulation equations. Then, the effects of both magnitude and number of the concentrated masses on nonlinear vibrations are investigated numerically in detail.