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.

Internal resonance in parametric vibrations of axially accelerating viscoelastic plates

Abstract: In this paper, instability boundaries of axially accelerating plates with internal resonance are investigated for the first time. The relation between the acceleration and the longitudinally varying tensions are introduced. The governing equation and the corresponding boundary conditions are derived from the generalized Hamilton principle. The effects of internal resonances and the nonhomogeneous boundary conditions on the instability boundaries are highlighted. By the method of multiple scales, the modified solvability conditions in principal parametric and internal resonances are established. The Routh-Hurwitz criterion is introduced to determine the instability boundaries. The effects of the viscoelastic coefficient and the viscous damping coefficient on the instability boundaries are examined. Abnormal instability boundaries are detected when the internal resonance is introduced. The phenomenon of local zigzag and V-shape boundaries are explained from the viewpoint of modal interactions. The numerical calculations of the differential quadrature schemes about the first four complex frequencies, the first four complex modes, and the stability boundaries are used to confirm the results of the analytical method.

Principal Parametric Resonance of Axially Accelerating Viscoelastic Strings

Abstract: The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) nontrivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial speed fluctuation amplitude, and the viscoelastic parameters.Copyright © 2005 by ASME

Experimental nonlinear identification of a single structural mode

Abstract: We describe a procedure for the identification of the nonlinear parameters of a single mode of a structure possessing quadratic and cubic geometric and inertia nonlinearities and linear and quadratic damping We use this procedure to identify the parameters of the first mode of a portal frame consisting of three beams and two masses The generalized coordinate of this mode is modeled by a second-order ordinary-differential equation possessing quadratic and cubic geometric and inertia nonlinearities, linear viscous and quadratic damping (airflow drag), and parametric and external excitation terms The linear natural frequency and damping coefficient are estimated using linear tests Then, the structure is excited by a harmonic force having a frequency that is approximately twice the natural frequency of the first mode, thereby producing a combination of a principal parametric resonance and a subharmonic resonance of order one-half We use the method of multiple scales to determine a second-order uniform expansion of the model equation and hence the frame response to such an excitation We estimate the nonlinear parameters by regressive fits between the theoretically obtained response relations and the experimental results We report deviations and agreements between model and experiment