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.

Resonance behavior of a rotor-active magnetic bearing with time-varying stiffness

Abstract: The non-linear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The rotor-AMB system is subjected to a periodically time-varying stiffness. The simultaneous primary resonance case is considered and examined. The vibration of the rotor is modeled by a coupled second-order non-linear ordinary differential equations with quadratic and cubic non-linearities. Their approximate solutions are sought applying the method of multiple scales. The steady-state response and the stability of the system at the simultaneous primary resonance case for various parameters are studied numerically, applying the frequency response function method. It is found that different shapes of chaotic motion exist, which are determined using phase-plane method. It is also shown that the system parameters have different effects on the non-linear response of the rotor. For steady-state response, however, multiple-valued solutions, jump phenomenon, hardening and softening non-linearity occur. Results are compared to previously published work.

Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions

Abstract: In this paper, parametric and 3:1 internal resonance of axially moving viscoelastic beams on elastic foundation are analytically and numerically investigated. The beam is restricted by viscous damping force. The beam’s material obeys the Kelvin model in which the material time derivative is used. The governing equations of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle. The effects of the nonhomogeneous boundary conditions due to the viscoelasticity are highlighted, while the boundary conditions are assumed to be homogeneous in previous studies. In small but finite stretching problems, the equation is simplified into a governing equation of transverse nonlinear vibration. It is a nonlinear integro-partial differential equation with time-dependent and space-dependent coefficients. The dependence of the tension on the finite axial support rigidity is also modeled. The method of multiple scales is directly applied to establish the solvability conditions. The nonlinear steady-state oscillating response along with the stability and bifurcation of the beam is investigated. A detailed study is carried out to determine the influence of the viscoelastic coefficient and the viscous damping coefficient on dynamic behavior of the system. The numerical calculations by the differential quadrature scheme confirm the approximate analytical results.

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Abstract: Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.

Steady-state response of a fluid-conveying pipe with 3:1 internal resonance in supercritical regime

Abstract: The forced vibration response of the pipe conveying fluid, with 3:1 internal resonance, is studied here for the first time. The straight equilibrium configuration becomes bent while the velocity of the fluid exceeds the critical value. As a result, the original mono-stable system transforms to a bi-stable system. Critical excitation which can cause global responses is solved out from the potential equation of the unperturbed system. The condition of 3:1 internal resonance is established after the partial differential equation is discretized. Global bifurcations are studied in simulation ways. By the method of multiple scales, local responses around the bent configuration are investigated. The analytical results are verified by simulations. Responses at the second mode bifurcate out another branch near the resonance frequency. It is very different with the triply harmonic responses without internal resonance. The triply harmonic response is a resonant excitation to the second mode. Responses will change largely with the detuning relationship between these two modes. Influences of the excited amplitude are also studied. Based on the analytical method, critical excited conditions of jumping and hysteretic phenomena are determined. The responses will have up- and down-bifurcations in the special region.