Showing papers on "Multiple-scale analysis published in 1990"

Non-linear non-planar oscillations of a cantilever beam under lateral base excitations

Abstract: The non-planar responses of a cantilevered beam subject to lateral harmonic baseexcitation are investigated using two non-linear coupled integrodifferential equations of motion. 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. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the case of a one-to-one internal resonance and a primary resonance. The results show that the non-linear geometric terms are important for the responses of low-frequency modes because they produce hardening spring effects. On the other hand, the non-linear inertia terms dominate the responses of high-frequency modes. We also obtain quantitative results for non-planar motions and investigate their dynamic behavior. For different range of parameters, the non-planar motions can be steady whirling motions, whirling motions of the beating type, or chaotic motions. Furthermore, we investigate the effects of damping.

Response statistics of non-linear systems to combined deterministic and random excitations

Abstract: A second-order closure method is presented for determining the response of non-linear systems to random excitations. The excitation is taken to be the sum of a deterministic harmonic component and a random component. The latter may be white noise or harmonic with separable non-stationary random amplitude and phase. The method of multiple scales is used to determine the equations describing the modulation of the amplitude and phase. Neglecting the third-order central moments, we use these equations to determine the stationary mean and mean-square response. The effect of the system parameters on the response statistics is investigated. The presence of the nonlinearity causes multi-valued regions where more than one mean-square value of the response is possible. The local stability of the stationary mean and mean-square responses is analysed. Alternatively, assuming the random component of the response to be small compared with the mean response, we determine steady-state periodic responses to the deterministic part of the excitation. The effect of the random part of the excitation on the stable periodic responses is analysed as a perturbation and a closed-form expression for the mean-square response is obtained. Away from the transition zone separating stable and unstable periodic responses, the results of these two approaches are in good agreement. Comparisons of the results of these methods with that obtained by the method of equivalent linearization are presented.

Chaos and instability in a power system — Primary resonant case

Abstract: We investigate some of the instabilities in a single-machine quasi-infinite busbar system. The system's behavior is described by the so-called swing equation, which is a nonlinear second-order ordinary-differential equation with additive and multiplicative harmonic terms having the frequency Ω. When Ω≈ω0, where ω0 is the linear natural frequency of the machine, we use digital-computer simulations to exhibit some of the complicated responses of the machine, including period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). To predict the onset of these complicated behaviors, we use the method of multiple scales to develop an approximate first-order closed-form expression for the period-one responses of the machine. Then, we use various techniques to determine the stability of the analytical solutions. The analytically predicted period-one solutions and conditions for its instability are in good agreement with the digital-computer results.

Response of a single-degree-of-freedom system to a non-stationary principal parametric excitation

Abstract: We examine the non-stationary response of a one-degree-of-freedom non-linear system to a non-periodic parametric excitation with varying frequency. We use the method of multiple scales to obtain equations governing the stationary and non-stationary responses of the system, and we analyze the stability of the stationary responses. The response displays several phenomena, including penetration of the trivial response into the unstable trivial region, oscillation of the response about the non-trivial stationary solution, convergence of the non-stationary response to the stationary solution, lingering of the non-trivial response into the stable trivial region, and rebounding of the non-trivial response. These phenomena are affected by the sweep rate, the initial conditions, and the system parameters. Digital and analog computers are used to solve the original governing differential equation. The results of the simulations agree with each other and with those obtained by using the method of multiple scales.

Non-linear dynamic response of a rotating machine

Abstract: The dynamic response of a two-degree-of-freedom seismically mounted rotor system with cubic non-linearities is investigated. The rotating machine is subjected to internal forces caused by the eccentricity of the center of mass of the rotor. The equations of motion of the system are determined using Lagrange's equation. The method of multiple scales is then used to determined the response of the system; this is determined when the excitation frequency is near the first and second modal frequency under internal resonance conditions.

a Perturbation Method and Some Applications

Abstract: For differential equations with one fast variable, a perturbation method is introduced that transforms a solution valid over only a short time interval to a new solution composed of averaged variables plus a periodic function of the averaged variables The averaged variables are governed by a set of differential equations where the fast variable has been removed and thus can be numerically integrated quickly or solved directly This method is applied to a perturbed harmonic oscillator with a cubic perturbation, van der Pol's equation, coorbital motion in the restricted three-body problem, and to nearly circular motion of a particle near one of the primaries in the restricted three-body problem

The Effect of Nonlinearities on Flexible Structures

Abstract: : Experimental-theoretical studies have been conducted on the influence of nonlinearities on flexible structures in the presence of either an external or a parametric excitation. A single-degree-of-freedom system with quadratic and cubic nonlinearities under the influence of a harmonic parametric excitation was studied using the method of multiple scales and digital-and analog-computer simulations. A global bifurcation diagram was obtained showing the different possible attractors (point, limit cycle, chaotic attractors). For small excitation amplitudes, the perturbation results are in excellent agreement with the digital-and analog-computer simulations. For moderate to large excitation amplitudes, the accuracy of the perturbation solution is questionable and only digital -and analog- computer simulations were used. The results are in full agreement. Keywords: Nonlinear oscillations, Flexible structures, Resonances, Attractors, Bifurcations.

Application of the method of multiple scales to linear waveguide boundary value problems

Abstract: The perturbation analysis method of multiple scales is applied to the problem of linear sound propagation in a rectangular waveguide with viscous and thermal dissipation effects at the boundary. In the past, the method of multiple scales has been applied to the corresponding problem of finite level fluctuations in a waveguide. The second‐order solution (representing the first correction due to nonlinearity) cited in these investigations would however preclude the solution of the linear problem, as it grows without bound with the axial coordinate of the waveguide. In this investigation an alternate solution at second order is proposed, which is bounded throughout the spatial domain of the duct. This property of the alternate solution allows the linear problem to be solved, including the region arbitrarily close to the cutoff frequency for higher‐order modes. In addition, the uniform solution may be directly applied to the analysis of finite level fluctuations, with consequences as yet unknown.

Modulational Stability of Rotating-Disk Flow

Abstract: The problem of modulational primary stability of rotating-disk boundary layer flow for stationary modes was studied by using the method of multiple scales. The system of equations for infinitesimal disturbances is solved by linear stability theory to determine the modulated solution which represents the time and azimuthal variations of the wave pattern. The solution for the amplitude of the most critical disturbance indicates that the disturbance spreads out as the slow time τ increases and at the center it grows exponentially; after an initial decrease due to a τ-1/2 factor.