Showing papers on "Multiple-scale analysis published in 1993"

Nonlinear response of a parametrically excited buckled beam

Abstract: A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.

Nonlinear stability of surface waves in magnetic fluids: effect of a periodic tangential magnetic field

Abstract: Nonlinear wave propagation on the surface between two superposed magnetic fluids stressed by a tangential periodic magnetic field is investigated using the method of multiple scales. A stability analysis reveals the existence of both nonresonant and resonant cases. From the solvability conditions, three types of nonlinear Schrodinger equation are obtained. The necessary and sufficient conditions for stability are obtained in each case. Formulae for the surface elevation are also obtained in both the non-resonant and the resonant cases. It is found from the numerical calculation that the tangential periodic magnetic field plays a dual role in the stability criterion, while the field frequency has a destabilizing influence.

The dynamic stability and nonlinear resonance of a flexible connecting rod: Continuous parameter model

Abstract: The transverse vibrations of a flexible connecting rod in an otherwise rigid slider-crank mechanism are considered. An analytical approach using the method of multiple scales is adopted and particular emphasis is placed on nonlinear effects which arise from finite deformations. Several nonlinear resonances and instabilities are investigated, and the influences of important system parameters on these resonances are examined in detail.

Whirling of a forced cantilevered beam with static deflection. I: Primary resonance

Abstract: The response of a slender, clastic, cantilevered beam to a transverse, vertical, harmonic excitation is investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Previous work often has neglected the static deflection caused by the weight of the beam, which adds quadratic terms in the governing equations of motion. Galerkin's method is used with three modes and approximate solutions of the temporal equations are obtained by the method of multiple scales. Primary resonance is treated here, and out-of-plane motion is possible in the first and second modes when the principal moments of inertia of the beam cross-section are approximately equal. In Parts II and III, secondary resonances and nonstationary passages through various resonances are considered.

Stability of fluttered panels subjected to in-plane harmonic forces

Abstract: This paper presents an investigation into the stability of a fluttered panel acted on by an in-plane harmonic force. The in-plane force is assumed to be slow varying and small in magnitude compared to the aerodynamic force. Because of this small harmonic force, the system may become stable although the aerodynamic force exceeds its critical value. In this work, the finite element formulation is applied to obtain the discretized system equations. The autonomous terms in the system equations are then uncoupled by transforming these terms into Jordan canonical forms. Finally, the method of multiple scales is used to solve for analytical solutions of the system. The effects of system parameters on the changes of stability boundaries are studied numerically.

Whirling of a forced cantilevered beam with static deflection. III: Passage through resonance

Abstract: Nonstationary excitations of slender, elastic, cantilevered beams with equal principal moments of inertia are considered. The excitation frequency is slowly increased or decreased through a resonance of the first mode at a constant rate. Three resonances are investigated: primary resonance, superharmonic resonance of order two and subharmonic resonance of order two. After application of Galerkin's method with three modes, the nonlinear, nonstationary response of the first mode of the beam is determined by two methods: integration of the modulation equations obtained from the method of multiple scales, and direct numerical integration of the temporal equations of motion. Time histories are presented and the effects of excitation amplitude, rate of acceleration or deceleration through resonance, damping and initial conditions of the disturbance on the maximum response are studied. The effect of a persistent random disturbance is also examined. Although the excitation acts in the vertical plane, whirling occurs if the beam is subjected to out-of-plane disturbances.

Normal mode sound propagation in an ocean with random narrow‐band surface waves

Abstract: Normal mode sound propagation in an isovelocity ocean with random narrow-band surface waves is considered, assuming the root-mean-square wave height to be small compared to the acoustic wavelength. Nonresonant interaction among the normal modes is studied straightforward perturbation technique. The more interesting case of resonant interaction is investigated using the method of multiple scales to obtain a pair of stochastic coupled amplitude equations which are solved using the Peano-Baker expansion technique. Equations for the spatial evolution of the first and second moments of the mode amplitudes are also derived and solved. It is shown that, irrespective of the initial conditions, the mean values of the mode amplitudes tend to zero asymptotically with increasing range, the mean-square amplitudes tend towards a state of equipartition of energy, and the total energy of the modes is conserved.

Dynamic Response of Rotor-Bearing Systems With Time-Dependent Spin Rates

Abstract: This paper presents an investigation into the vibration of rotor-bearing systems with time-dependent spin rates. Due to this spin rate, parametric instability may take place in certain situations. In this work, the Galerkin method is used to eliminate the dependence on the spatial coordinates, and then the method of multiple scales is applied to derive periodic solutions and expressions for the boundaries of unstable regions analytically. Numerical results are given for the case where the spin rate is characterized as a small, harmonic perturbation superimposed on a constant rate. The effects of system parameters on the changes of the boundaries of unstable regions are shown.

Post-flutter analysis of beams subjected to subtangential forces

Abstract: This paper presents an analytical study of the post-flutter behavior of beams subjected to both distributed and concentrated subtangential forces. First, Hamilton's principle is used to derive the variational equation of motion. Next, the Ritz-Galerkin method is applied to yield a discretized system of equations. The linear terms in the system of equations are then uncoupled by transforming the system into its quasi-canonical form before the method of multiple scales is utilized to solve for analytical solutions. It is observed that for ideal beams without damping, the post-flutter behavior depends on the initial conditions of beams.

Problem of the impact interaction of an elastic rod with a Uflyand-Mindlin plate

Abstract: A problem on the impact of an elastic rod of circular cross-section onto an Uflyand-Mindlin isotropic elastic plate is considered. The radial method, which allows construction of an approximate solution behind fronts of discontinuity surfaces up to the boundary of the contact region by means of segments of power series with variable coefficients, is used as a method of solution. It is, however, established that segments of radial series are not uniformly suitable throughout the region of existence of wave motion. A method of multiple scales is used for their regularization. The method, based on asymtotics of the radial series, permits construction of a uniformly suitable expansion containing a smaller number of terms than segments of radial series, but equal to them in accuracy.

Response of a Parametrically-Excited System to a Nonstationary Excitation

Abstract: The response of a single-degree-of-freedom system to a nonstationary excitation is investigated by using the method of multiple scales as well as analog- and digital-computer simulations. The unexcited system has one focus and two saddle points. The system can be used to model rolling of ships in head or follower seas. The method of multiple scales is used to derive equations governing the modulation of the amplitude and phase of the response. The modulation equations are used to find the stationary solutions and their stability. The response to nonstationary excitations is found by integrating the original governing equation as well as the modulation equations. There is good agreement between the results of both approaches. For some frequency and amplitude sweeps, the nonstationary response found from integrating the original governing equation exhibits behaviors that are analogous to symmetry-breaking bifurcations, period-doubling bi furcations, chaos, and unboundedness present in the stationary case. T...