Showing papers on "Multiple-scale analysis published in 2000"

A theoretical and experimental investigation of indirectly excited roll motion in ships

Abstract: The phenomenon of indirectly exciting the roll motion of a vessel due to nonlinear couplings of the heave, pitch and roll modes is investigated theoretically and analytically. Two nonlinear mechanisms that cause large-amplitude rolling motions in a head or following sea are investigated. The first mechanism is internal or autoparametric resonance and the second is parametric resonance. The energy put into the pitch and heave modes by the wave excitations may be transferred into the roll mode by means of nonlinear coupling among these modes; thus, the roll can be indirectly excited. As a result, a ship in a head or following sea can spontaneously develop severe rolling motion. In the analytical approach, the method of multiple scales is used to determine a system of nonlinear first-order equations governing the modulation of the amplitudes and phases of the system. The fixed-point solutions of these equations are determined and their bifurcations are investigated. Hopf bifurcations are found in the case of two-to-one internal resonance. Numerical simulations are used to investigate the bifurcations of the ensuing limit cycles and how they produce chaos. Experiments are conducted with tanker and destroyer models. They demonstrate some of the nonlinear effects, such as the jump phenomenon, the subcritical instability, and the coexistence of multiple solutions. The experimental results are in good qualitative agreement with the results predicted theoretically.

A Modified Perturbation Technique Depending Upon an Artificial Parameter

Abstract: In this paper, a modified perturbation method is proposed to search for analytical solutions of nonlinear oscillators without possible small parameters. An artificial perturbation equation is carefully constructed by embedding an artificial parameter, which is used as expanding parameter. It reveals that various traditional perturbation techniques can be powerfully applied in this theory. Some examples, such as the Duffing equation and the van der Pol equation, are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weak nonlinear systems, but also for strongly nonlinear systems. In applying the new method, some special techniques have been emphasized for different problems.

Stability analysis of thin viscoelastic liquid film flowing down on a vertical wall

Abstract: This paper investigates the stability of thin viscoelastic liquid film flowing down on a vertical wall using a long-wave perturbation method to find the solution for generalized nonlinear kinematic equations with a free film interface. To begin with, a normal mode approach is employed to obtain the linear stability solution for the film flow. The linear growth rate of the amplitudes, the wave speeds and the threshold conditions are obtained subsequently as the by-products of linear solutions. The results of linear analysis indicate that the viscoelastic parameter k = k0/(ρh0*2}) destabilizes the film flow as its magnitude increases. To further investigate practical flow stability conditions, the weak nonlinear dynamics of a film flow are presented by using the method of multiple scales. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg-Landau equation. Modelling results indicate that both the subcritical instability and the supercritical stability conditions are possible in a viscoelastic film flow system. The results of nonlinear modelling further indicate that the threshold amplitude ea0 in the subcritical instability region becomes smaller as the viscoelastic parameter k increases. If the initial finite amplitude of disturbance is greater than the value of threshold amplitude, the system becomes explosively unstable. It is also interesting to note that both the the threshold amplitude and the nonlinear wave speed in the supercritical stability region increase as the value of k increases. Therefore, the flow becomes unstable when the value of k increases. The viscoelastic parameter k indeed plays a significant role in destabilizing the film flow travelling down along a vertical plate.

Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative

Abstract: Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.

On Nonlinear Vibration of Nonuniform Beam with Rectangular Cross-Section and Parabolic Thickness Variation

Abstract: We present the exact solution of differential equation in the linear case of free bending vibrations of nonuniform beam with rectangular cross-section using the factorization method. This beam with constant width and parabolic thickness is a good approximation of the gear tooth profile. It permits a nonlinear bending vibrations study (moderately large curvatures) of the gear tooth (the cantilever beam case). The case of the beam with a sharp end is considered. We use the method of multiple scales to treat the governing partial-differential equations and boundary conditions directly. In the absence of internal resonance (weakly nonlinear systems) the nonlinear modes are taken to be perturbed versions of the linear modes. We determine the nonlinear planar mode shapes and natural frequencies of a gear tooth with a sharp end variation (the cantilever beam case).

Instability of parametrically second- and third-subharmonic resonances governed by nonlinear Shrödinger equations with complex coefficients

Abstract: A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. The analysis based on the method of multiple scales. Two types of the modified nonlinear Schrodinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. Estimates of nonlinear coefficients are made. The resulting equations have an interesting in many dynamical and physical cases. Temporal modulational method is confirmed to discuss the stability behavior at both parametric second- and third-harmonic resonance cases. Furthermore, the Benjamin–Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is found.

Response of Two Quadratically-Coupled Oscillators to a Principal Parametric Excitation

Abstract: The authors study the dynamics of two oscillators coupled with quadratic nonlinearities in the case of two-to-one internal resonance when the higher mode is subjected to a principal parametric excitation. They use the method of multiple scales to obtain an approximate solution to the equations of motion and investigate theoretically its stability. Then, they verify the analysis experimentally. The authors use a cantilever steel beam and an analog second-order circuit to represent the two oscillators. The interaction between the two systems is achieved by fitting the beam with piezoceramic actuators and a strain gage and coupling the beam with the circuit through electronically generated quadratic nonlinearities. They subject the first mode of the beam to a principal parametric excitation and tune the frequency of the circuit to approximately one-half the frequency of the first mode of the beam. The theoretical and experimental results indicate that the system exhibits complicated responses, such as jumps,...

Analytical prediction of the two first period-doublings in a three-dimensional system

Abstract: Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

Nonlinear Vibration of Composite Shell Subjected to Resonant Excitations

Abstract: Analysis of the vibration of a shallow, simply supported, nonsymmetric unbalanced cross-ply laminated, circular cylindrical composite shell is presented. The subject is particularly relevant considering the widespread use of cylindrical shell structures in engineering applications. This research applies the discretized Lagrangian/method of multiple scales solution technique. The Donnell shallow shell strain-displacement relations and the single-mode displacement field from the linear eigenvalue problem are applied. The system Lagrangian is developed and integrated over the spatial domain and then substituted into Lagrange’s equation. The resulting equation of motion is a second-order temporally nonlinear ordinary differential equation in the form of the Duffing oscillator. The natural frequency, the coefficient of the cubic nonlinearity, and the strength of the nonlinearity are investigated. The method of multiple scales is applied to the nonlinear equation of motion in order to analyze the frequency response. Primary resonance, subharmonic resonance, and superharmonic resonance are analyzed.

An Alternative Example of the Method of Multiple Scales

Abstract: An alternative example of the method of multiple scales is presented. This example arises in the study of the classical heat equation with a slowly varying flux imposed at one end. The module presents introductory ideas about dimensionless variables, multiple-scale expansions, and scaling of the dependent variable. The necessarily obfuscating algebraic computation is less than that for more familiar multiple-scale examples, such as the perturbed oscillator. The results are analyzed for both their physical and mathematical importance.

Formation of Patterns in Intense Hadron Beams. The Amplitude Equation Approach

Abstract: We study the longitudinal motion of beam particles under the action of a single resonator wave induced by the beam itself. Based on the method of multiple scales we derive a system of coupled amplitude equations for the slowly varying part of the longitudinal distribution function and for the resonator wave envelope, corresponding to an arbitrary wave number. The equation governing the slow evolution of the voltage envelope is show to be of Ginzburg-Landau type.