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.

Nonlinear dynamic behavior of a rotor active magnetic bearing

Abstract: The nonlinear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects The vibration of the rotor is modeled by a coupled second order nonlinear ordinary differential equations with quadratic and cubic nonlinearities Their approximate solutions are sought applying the method of multiple scales in the case of primary resonance The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency response curves Choosing the Hopf bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained At the same time, the Floquet theory is used to determine the stability of the periodic solutions A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented The three types of primary Hopf bifurcations are found for the first time in the rotor-AMB system It is shown that the limit cycles undergo cyclic fold, period doubling bifurcations, and intermittent chaotic attractor, whereas the chaotic attractors undergo explosive bifurcation and boundary crises In the regime of multiple coexisting solutions, multiple stable equilibriums, periodic solutions and chaotic solutions are the most interesting phenomena observed

An analytical approximated solution and numerical simulations of a non-ideal system with saturation phenomenon

Abstract: Nowadays, researches about non-ideal problems have been increased considerably in technical-scientific community. Nonlinear problems have been widely studied due to their particularities in real problems, mainly when these nonlinearities induce interactions between a dynamical system with its excitation source, these kind of systems are called non-ideal systems. One of the excitation sources, which is of non-ideal kind, is an unbalanced DC motor with limited power supply. When it is coupled to a dynamical system, this system is subjected to Sommerfeld effect, where jump phenomena may occur. However, when the same dynamical system is of two-degrees-of-freedom and possesses 2:1 internal resonance, saturation phenomenon occurs. Therefore, in this work, the dynamical behavior of a non-ideal three-degrees-of-freedom weakly coupled system associated with quadratic nonlinearities in the equations of motion is investigated. The full system consists of two nonlinear mechanical oscillators coupled through quadratic nonlinearities and which produces a 2:1 internal resonance between their translational movements. In the bigger oscillator, an unbalanced DC motor is used as an excitation source. Under these conditions, equations of motion of the system were obtained using Lagrange’s method, and the method of multiple scales was applied to find an analytical approximated solution of the equations of motion. In addition, numerical simulations of the equations of motion were carried out to analyze the response of the non-ideal system by varying the torque of the motor. It is shown that when the excitation frequency is near to second natural frequency of the main system, saturation and jump phenomena occur. Furthermore, this work investigates the ranges of some torques of the motor, which causes the phenomena, and explores the possibility to harvest energy from high-amplitudes of vibration in a future work.

Nonlinear gravitational stability of streaming in an electrified viscous flow through porous media

Abstract: Weakly nonlinear Kelvin–Helmholtz instability for two viscous fluids streaming through porous media is investigated. The electro-gravitational stability of the horizontal plane interface is examined. A vertical or a horizontal electric field stresses the system. The linear form of equation of motion is solved in the light of the nonlinear boundary conditions. The present boundary value problem leads to construct nonlinear characteristic equation. This nonlinear characteristic equation has complex coefficients for the elevation function. The nonlinearity is kept to the third order. The method of multiple scales, in both space and time, is used. The use of the Taylor expansion through the multiple scale scheme leads to the derivation of the well-known nonlinear Schrodinger equation with complex coefficients from the nonlinear characteristic equation. This equation describes the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrodinger equation that occurs in the non-resonance case. The relation between the stratified kinematic viscosity and the porous permeability is performed in order to control the marginal state representation. This marginality is utilised in order to relax the complexity of the linear dispersion relation. Stability conditions are discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. It is found that the porosity of the media increases the destabilizing influence for the fluid density. In nonlinear scope, a destabilizing influence for the upper porous permeability is recorded, while a stabilizing influence is found for the lower porous permeability. Both the horizontal and vertical electric fields are still playing the same roles in linear and nonlinear examinations as in the non-porous media. Two opposite roles are presented for the variation of the stratified fluid velocity V . A stabilizing influence for V ⩽1 and a destabilizing effect for V >1 are illustrated in this examination. A dual role in the nonlinear examination is recorded for values of the wave-frequency.

Principal response of van der pol-duffing oscillator under combined deterministic and random parametric excitation

Abstract: The principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied. Jumps were shown to occur under some conditions. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analysis were verified by numerical results.