Showing papers on "Multiple-scale analysis published in 1989"

The Non-Linear Response of a Slender Beam Carrying a Lumped Mass to a Principal Parametric Excitation: Theory and Experiment

Abstract: The non-linear response of a slender cantilever beam carrying a lumped mass to a principal parametric base excitation is investigated theoretically and experimentally. The Euler-Bernoulli theory for a slender beam is used to derive the governing non-linear partial differential equation for an arbitrary position of the lumped mass. The non-linear terms arising from inertia, curvature and axial displacement caused by large transverse deflections are retained up to third order. The linear eigenvalues and eigenfunctions are determined. The governing equation is discretized by Galerkin's method, and the coefficients of the temporal equation—comprised of integral representations of the eigenfunctions and their derivatives—are computed using the linear eigenfunctions. The method of multiple scales is used to determine an approximate solution of the temporal equation for the case of a single mode. Experiments were performed on metallic beams and later on composite beams because all of the metallic beams failed prematurely due to the very large response amplitudes. The results of the experiment show very good qualitative agreement with the theory.

Non-linear non-planar parametric responses of an inextensional beam☆

Abstract: The non-linear integro-differential equations of motion for an inextensional beam are used to investigate the planar and non-planar responses of a fixed-free beam to a principal parametric excitation. The beam is assumed to undergo flexure about two principal axes and torsion. The equations contain cubic non-linearities due to curvature and inertia. Two uniform beams with rectangular cross sections are considered: one has an aspect ratio near unity, and the other has an aspect ratio near 6.27. In both cases, the beam possesses a one-to-one internal resonance with one of the natural flexural frequencies in one plane being approximately equal to one of the natural flexural frequencies in the second plane. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the interaction of the two resonant modes, yielding four first-order non-linear ordinary-differential equations governing the amplitudes and phases of the modes of vibration. The results show that the non-linear inertia terms produce a softening effect and play a significant role in the planar responses of high-frequency modes. On the other hand, the non-linear geometric terms produce a hardening effect and dominate the planar responses of low-frequency modes and non-planar responses for all modes. If the non-linear geometric terms were not included in the governing equations, then non-planar responses would not be predicted. For some range of parameters, Hopf bifurcations exist and the response consists of amplitude- and phase-modulated or chaotic motions.

On the response of the non-linear vibration absorber

Abstract: The steady state vibrations of a non-linear dynamic vibration absorber are studied using the method of multiple scales, in conjunction with digital simulations. The main results are concerned with certain dynamic instabilities which can occur if the absorber is designed such that the desired operating frequency is approximately the mean of the two linearized natural frequencies of the system. A combination resonance can occur in this case, resulting in large amplitude almost-periodic vibrations. This motion destroys the effectiveness of the absorber and can coexist with the desired low-amplitude periodic response, which leads to initial condition dependent dynamics.

A generalized analysis method of auto-parametric resonances in power systems

Abstract: The envelope equation derived earlier is generalized to analyze a resonance among multiple modes by using the method of multiple scales, an excellent method for analyzing large-scale nonlinear systems. It is demonstrated through numerical simulations that the generalized envelope equation is valid for predicting the resonance phenomena themselves and also for examining the critical factors responsible for severe resonances. The factors chosen for study are the heaviness of the load, small dampings, and disturbances. The method is promising for the analysis of general multimode resonances. >

The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows

Abstract: Two perturbation methods used in weakly nonlinear stability theory, namely, the method of multiple scales and the amplitude expansion method, are examined for their equivalence through formal analyses and numerical calculation of the Landau constants. The method of multiple scales is shown to give results equivalent to those obtained from the amplitude expansion formulation for slightly supercritical states if a normalization condition is applied to the fundamental mode. The convergence of the expansion in the method of multiple scales is also discussed.

Nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales

Abstract: The method of multiple scales (MMS) in conjunction with the Galerkin method is used to analyze the nonlinear forced and damped response of a rectangular Orthotropic plate subjected to a uniformly distributed harmonic transverse loading. The effects of damping and in-plane loads are considered. The analysis considers simply supported as well as clamped plates. For each case, both movable and immovable edge conditions are considered. By using MMS, all possible resonances such as primary, subharmonic, and superharmonic reso- nances are studied. For the undamped response without in-plane loading, comparisons of the MMS results with those obtained by the finite-element method show excellent agreement. EVELOPMENT of composite materials comprising lam- inates of Orthotropic or multilayered anisotropic materi- als Recently has been receiving substantially growing research efforts. Due to the increasing demands for energy-efficient, high strength, minimum weight aircraft designs, many re- searchers believe that the use of composite materials offers promising alternatives for aircraft designs. Thin, laminated, composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher. Responses of this kind cannot be predicted by using the linear theory. Consequently, the need to study large- amplitude-deflection vibrations of composite structures is of paramount importance. The literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von Karman geometrical nonlinearity were first considered by Whitney and Leissa.1 Based on these equa- tions, different methods of analysis have been developed by several researchers. An excellent number of collections on nonlinear free and forced vibrations of composite plates cov- ering the work through 1979 can be found in the comprehen- sive book by Chia. 2 Bert3 has conducted a survey on the dynamics of composite plates for the period 1979-81. A re- view of the literature on nonlinear vibrations of plates can be found in the review paper by Sathyamoorthy4 and the book by Nayfeh and Mook.5 Large deflection analysis of symmetrically laminated rec- equations of motion are presented in terms of the lateral displacement and stress function. The equations are nondi- mensionalized following the transformation introduced by Brunelle and Oyibo.9 Though multimode analysis can be treated, the present study is focused on single-mode analysis. A deflection function representing the first mode and satisfy- ing the boundary conditions is assumed, and subsequently the stress function is found. Next, the Galerkin method is applied to obtain the modal equation, which is solved analytically by using the method of multiple scales (MMS).10 The MMS also provides solutions for subharmonic and superharmonic reso- nances. The effects of damping ratio, plate aspect ratio, and in-plane loading then are studied.

Two-mode nonlinear vibration of orthotropic plates using method of multiple scales

Abstract: Nonlinear forced oscillation of a rectangular orthotropic plate subjected to uniform harmonic excitation is solved using the method of multiple scales. The governing equations are based on the von Karman type geometrical nonlinearity, and the effect of damping is included. The general multimode solution is developed for simply supported boundary conditions, and the solution is specialized for two-symmetric modes analysis. The primary resonances and the subharmonic and superharmonic secondary resonances are studied in detail. HIN laminated composite panels subjected to transverse periodic loadings can encounter deflections of the order of panel thickness or even higher. The effect of these periodic excitations on the panel can be very severe. Responses of this kind cannot be predicted by linear theory. Consequently, the need to study large deflections using nonlinear methods of analysis is of paramount importance. The formulation of the equations governing the fundamen- tal kinematic behavior of the laminated composite plates in the presence of the von Karman geometrical nonlinearity is attributed to Whitney and Leissa.1 Based on these equations, various methods have been developed to solve nonlinear free and forced vibrations of composite panels. A good survey on mainly nonlinear free and forced vibrations of isotropic plates is given in a book by Nayfeh and Mook.2 The most compre- hensive work on geometrically nonlinear analysis of both static and dynamic behavior of the laminated panels through 1972 is collected in a book by Chia. 3 Bert4 has conducted a survey on the dynamics of composite panels for the period of 1979-81. A review of literature on linear vibrations of plates can be found in a review paper by Sathyamoorthy.5 Relatively few investigations have been reported on the nonlinear forced vibration of isotropic or composite panels under harmonic excitations. Yamaki6 presented a one-term solution for free and forced vibrations of the rectangular plates, using Galerkin's method. Lin7 studied the response of a nonlinear flat panel to periodic and randomly varying load- ings. Nonlinear forced vibrations of beams and rectangular plates were studied by Eisely8 using a single-mode Galerkin's method in conjunction with the Linstedt-Duffing perturbation technique. Free and forced response of beams and plates undergoing large-amplitude oscillations using the Ritz averag- ing method were studied by Srinivasan.9 Bennett10 studied the nonlinear vibration of simply supported angle-ply laminated plates by considering the instability regions of the response of such plates subjected to harmonic excitations. Nonlinear free and forced vibration of a circular plate with clamped bound-

Vertical Vibration of Machine Foundations

Abstract: A study is reported in which the method of multiple scales is used to obtain an approximate analytical solution of the vertical vibration of foundations on soil. The study takes into account the nonlinearity of the soil structure, radiation, hysteretic and viscous damping, and the effect of embedment. Large responses of a foundation resting on a soil occur when the excitation frequency is equal to the natural frequency of the vibratory system, or the primary resonance. These and other findings from the study are discussed.

The effect of nonlinearities on the response of a single-machine-quasi-infinite busbar system

Abstract: A single-machine quasi-infinite busbar power system is formulated taking into consideration quadratic and cubic nonlinearities. The model equation contains parametric (time-varying coefficients) and external (inhomogeneous terms) excitations. The method of multiple scales is used to approximate the response of the system to simultaneous principal parametric resonances and subharmonic resonances of order one-half. In contrast with the linear analysis, the nonlinear analysis shows that the response can exhibit: (1) limit cycles instead of infinite motions; (2) multivaluedness that can lead to jumps; (3) subcritical instabilities; and (4) constructive and destructive interference of resonances. >

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

Abstract: The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter e characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order e, the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's “Weakly Nonlinear Geometrical Optics” method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced.

Impurity localized modes of nonlinear lattice systems

Abstract: Impurity localized modes of nonlinear lattice systems are studied by applying the method of multiple scales. The solutions of the localized modes of both infinite and semi-infinite nonlinear lattice systems are obtained up to the third order of the amplitude. According to the sign of the parameter specifying the second order nonlinearity, nonlinear lattice systems expand or shrink in the vicinity of impurity. Moreover, the frequencies of the localized modes are shown to decrease or increase with their amplitude, depending on the magnitude of the parameters specifying the second and the third order nonlinearities. As an example, numerical experiments for the Toda lattice with an impurity and the semi-infinite Toda lattice with an impurity are performed, and the results are in good agreement with those obtained by the perturbation method.

Perturbation treatment of the reverse ac Josephson effect.

Abstract: An analytic study of the reverse ac Josephson effect is carried out by perturbation method. The transition from the transient state to the steady state is studied by the method of multiple scales. Our solution to the problem warrants that the energy dissipated on the resistance is identical to the energy obtained from the external source over one period of oscillation. For low-resistance junctions we show that the induced steady-state dc voltage is practically zero and for high-resistance junctions it is finite and its possible values will be discussed.

Nonlinear electrohydrodynamic instability conditions of an interface between two fluids under the effect of a normal periodic electric field. III

Abstract: A charge-free surface separating two semi-infinite dielectric fluids influenced by a normal periodic electric field is subjected to nonlinear deformations. We use the method of multiple scales in order to solve the nonlinear equations. In the first-order problem we obtained Mathieu's differential equation. For the second order, we obtain the nonhomogeneous Mathieu equation and we use the method of multiple scales to obtain a sequence of equations. In the third order we obtain the second-order differential equation of periodic coefficients. Also, we obtain a formula for surface elevation. The stability conditions are determined.

Nonlinear electrohydrodynamic instability conditions of an interface between two fluids under the influence of a tangential periodic electric field II

Abstract: A charge-free surface separating two semi-infinite dielectric fluids influenced by a tangential periodic electric field is subjected to nonlinear deformations. We use the method of multiple scales in order to solve nonlinear equations. In the first order problem, we obtain Mathieu's differential equation. In the second order problem, we obtain the nonhomogenous Mathieu equation and we use the method of multiple scales to obtain a sequence of equations. In the third order problem, we obtain a differential equation of the second order of periodic coefficients. Also, we obtain a formula for surface elevation. The stability conditions are determined.