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.

Investigation of Dynamic Behavior of a Cable-Stayed Cantilever Beam under Two-Frequency Excitations

Abstract: Many civil structures and facilities can be modeled using cable-stayed cantilever beams. This study is to investigate the nonlinear dynamic response and dynamic behavior of a cable-stayed cantilever beam subjected to two different external excitations through theoretical analyses. First, the equations of motion of the cable and the beam are established. Then, based on the Galerkin method, dynamic structural responses are expressed into the superimposition of mode shapes, with the generalized time coordinates as unknown coefficients. To obtain the unknown coefficients, modulation equations governing the amplitude and phase are derived by using the method of multiple scales. Four representative cases of simultaneous resonances (four representative excitation cases) are considered. Based on the derived analytical solutions, for each case, the frequency response and amplitude response of the system are obtained through parametric studies and nonlinear dynamic behavior of the system are explored. The obtained results demonstrate: (1) both the beam and the cable can behave the harden spring properties and the soften spring property in the frequency response; and the cable experiences larger response than the beam although excitations are applied on the beam; (2) the effect of the amplitude variation of secondary resonance on the responses of the beam and the cable is smaller than the primary resonance; and (3) the addition of a secondary resonance, such as Order 1/2 and 1/3 sub-harmonic resonance and Order 2 and 3 super-harmonic resonance, to the primary resonance can suppress the response of the beam or the cable to a certain extent.

On the calculation of the transmission line parameters for long tubes using the method of multiple scales.

Abstract: The present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls. The perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first-order solutions and, in addition, the second-order solutions representing the correction due to boundary layer attenuation. The propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first-order corrections to the classical attenuation rates of the different modes come into play as well. Starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct, the analysis is then extended to the case where both of these quantities are of the same order of magnitude. Furthermore, the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated.

Electroviscoelastic Instability of a Kelvin Fluid Layer Influenced by a Periodic Electric Force

Abstract: The electroviscoelastic stability of a Kelvin fluid layer is discussed in the presence of the field periodicity. The surface elevations are governed by two transcendental coupled equations of Mathieu type which have not been attempted before. Analysis for the surface waves in axisymmetric modes and antisymmetric deformation which are governed by a single transcendental Mathieu equation is considered. The method of multiple scales expansion is applied to the stability analysis. The solution and the characteristic curves are obtained analytically. It is shown that the region between the two branches of the characteristic curves is unstable, whereas all points which lie outside the characteristic curves are stable. The special case of large viscosity is introduced for numerical calculations. It is found that the increase of kinematic viscosity, field frequency, and the elasticity parameter possesses a dual role in a damping nature. The phenomena of the coupled resonance is observed. The resonance region and the resonance points are functions of viscosity, elasticity, and field frequency, with nonlinear relations in the wavenumber.