Showing papers on "Multiple-scale analysis published in 2001"

Non-linear vibrations and stability of an axially moving beam with time-dependent velocity

Abstract: Non-linear vibrations of an axially moving beam are investigated. The non-linearity is introduced by including stretching effect of the beam. The beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity. Approximate solutions are sought using the method of multiple scales. Depending on the variation of velocity, three distinct cases arise: (i) frequency away from zero or two times the natural frequency, (ii) frequency close to zero, (iii) frequency close to two times the natural frequency. Amplitude-dependent non-linear frequencies are derived. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity

Abstract: In this study, the non-linear transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with variable velocity are investigated. The pipe is on fixed supports and the immovable end conditions result in the extension of the pipe during vibration and hence introduce further non-linear terms to the equation of motion. The velocity is assumed to be a harmonic function about a mean velocity. These systems experience a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved analytically by direct application of the method of multiple scales (a perturbation technique). Principal parametric resonance cases are investigated in detail. Non-linear frequencies are derived depending on amplitude. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

Nonlinear vibrations of doubly-curved cross-ply shallow shells

Abstract: We consider nonlinear forced vibrations of a doublycurved cross-ply laminated shallow shell with simply supported boundary conditions. We investigate its response to a primary resonance of its fundamental mode (i.e., fi « ^11). The nonlinear partial-differential equations governing the motion of the shell are based on the von Karman-type geometric nonlinear theory and the first-order sheardeformation theory. An approximation based on the Galerkin method is used to reduce the partialdifferential equations of motion to an infinite system of nonlinearly coupled second-order ordinarydifferential equations. These equations are solved by using the method of multiple scales. We found that symmetric modes do not have an effect on the results for the case of primary resonance of the fundamental mode of vibration. It is shown that using a single-mode approximation can lead to quantitatively and in some cases qualitatively erroneous results. A multi-mode approximation that includes as many modes as needed for convergence is used.

Global analysis for a nonlinear vibration absorber with fast and slow modes

Abstract: A two-degree-of-freedom model of a nonlinear vibration absorber is considered in this paper. Both the global bifurcations and chaotic dynamics of the nonlinear vibration absorber are investigated. The nonlinear equations of motion of this model are derived. The method of multiple scales is used to find the averaged equations. Based on the averaged equations, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple software program. The fast and slow modes may simultaneously exist in the averaged equations. On the basis of the normal form, the global bifurcation and the chaotic dynamics of the nonlinear vibration absorber are analyzed by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of this model is also found by numerical simulation.

Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions

Abstract: The non-linear response of a base-excited slender beam carrying a lumped mass subjected to principal parametric resonance is investigated. The attached mass and its location are so adjusted that the system exhibits 1:3:5 internal resonances. Method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and stability of the system. The steady-state response thus obtained is compared with those found by single- and two-mode analyses and very significant differences are observed in the bifurcation and stability of the response curves. The effects of external and internal detuning, amplitude of excitation and damping on the non-linear steady state, periodic, quasi-periodic and chaotic responses of the system are investigated. Funnel-shaped chaotic orbits, fractal orbits, cascade of period-doubling, torus doubling and intermittency routes to chaos are observed in this system. A simple illustration is given to control chaos by changing the system parameters.

Nonlinear Stability Analysis of the Thin Micropolar Liquid Film Flowing Down on a Vertical Cylinder

Abstract: We investigate the weakly nonlinear stability theory of a thin micropolar liquid film flowing down along the outside surface of a vertical cylinder The long-wave perturbation method is employed to solve for generalized nonlinear kinematic equations with free film interface The normal mode approach is first used to compute the linear stability solution for the film flow The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis

Vibrations of a Stretched Beam with Non-Ideal Boundary Conditions

Abstract: A simply supported Euler-Bernoulli beam with immovable end conditions is considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.

Control of a directly excited structural dynamic model of an F-15 tail section

Abstract: We used two simple control laws based on linear velocity and cubic velocity feedback to suppress the high-amplitude vibrations of a structural dynamic model of the twin-tail assembly of an F-15 fighter when subjected to primary resonance excitations. We developed the nonlinear differential equations of motion and obtained an approximate solution using the method of multiple scales. Then, we conducted bifurcation analyses for the open- and closed-loop responses of the system and investigated theoretically the performance of the control strategies. The theoretical findings indicate that the control laws lead to effective vibration suppression and bifurcation control. Furthermore, we conducted experiments to verify the theoretical analysis. We built a digital control system that consists of a SIMULINK modeling software and a dSPACE controller installed in a personal computer. Actuators made of piezoelectric ceramic material were used. The results show that both laws are effective at suppressing the vibrations. To compare the performance of both techniques, we calculated the power requirements for a simple system.

Nonlinear Stability, Thermoelastic Contact, and the Barber Condition

Abstract: The behavior of a one-dimensional thermoelastic rod is modeled and analyzed. The rod is held fixed and at constant temperature at one end, while at the other end it is free to separate from or make contact with a rigid wall. At this free end we impose a pressure and gap-dependent thermal boundary condition. This condition, known as the Barber condition, couples the thermal and elastic problems. Such systems have previously been shown to undergo a bifurcation from a unique linearly stable steady-state solution to multiple steady-state solutions with alternating stability. Here, the system is studied using the asymptotic matching techniques of boundary layer theory to derive short-time, longtime, and uniform expansions. In this manner, the analysis is extended into the nonlinear regime and dynamic information about the history dependence and temporal evolution of the solution is obtained. @DOI: 10.1115/1.1345699# The analysis of thermal contact problems has revealed a wealth of interesting phenomena. Beginning with J. R. Barber in 1978 ~@1#!, who pointed out that the solution of such problems poses certain difficulties, and continuing to this day, numerous researchers have turned their attention to these problems. Barber observed that the classical assumption of perfect insulation during a separated phase and perfect thermal contact during contact led to models with solutions which were unacceptable on physical grounds. Introducing a pressure and temperature-dependent boundary condition, which would subsequently become known as the Barber condition, he allowed for a smooth transition between the insulated and perfect thermal contact states. Studying a linearized version of a thermal contact problem which included the Barber condition, he showed that the paradoxes inherent in simpler models could be avoided and physically relevant solutions recovered. In 1980, Barber, Dundurs, and Comninou @2# investigated a thermal contact problem using the Barber condition in a onedimensional model of a thermoelastic rod. Imposing a temperature gradient across the rod, they demonstrated that the system underwent a bifurcation from a unique linearly stable steady-state solution to multiple solutions with alternating stability as the magnitude of the thermal gradient was varied. Since that time, various authors have explored the Barber condition and its implications for thermal contact problems ~@3,4#!. While such analyses have been extended to multiple materials ~@3,5,6#!, various geometries ~@7,8#!, and to numerical simulations ~@4#!, most theoretical work to date has relied upon linear stability theory. In a recent article ~@9#! we developed a nonlinear theory which described the history dependence and dynamics of solutions near the bifurcation point for a simplified model of a onedimensional thermoelastic rod. Our model did not, however, include the Barber condition. Since the Barber condition is much more physically realistic than the boundary condition used in ~@9#!, it is desirable to have a nonlinear theory for a model which incorporates the Barber condition. We carry out such an analysis here. While the model studied here differs from the model studied in ~@9#!, only in the use of the Barber condition, the method of analysis differs significantly. In particular, here we use the asymptotic matching techniques of boundary layer theory to derive shorttime, long-time, and uniform asymptotic expansions of the solution. In our prior analysis we used the method of multiple scales, or two-timing, to accomplish similar goals. The switch in techniques is not merely a matter of taste. Rather, any attempt to apply multiple scale techniques to the model considered herein will soon encounter algebraic difficulties. That is, such an attempt becomes analytically intractable. However, as is shown, boundary layer theory may be applied with little difficulty. This not only allows us to carry out the analysis for the one-dimensional rod model with the Barber condition, but gives us hope that similar techniques will yield a nonlinear stability theory for more complicated multidimensional problems. We begin in Section 2 by formulating the governing equations for our model. We make the standard assumption that quasi-static uncoupled thermoplasticity is valid and use the Signorini contact condition to capture periods of separation and contact. We impose the Barber condition on the thermal part of the problem, leaving the contact resistance function unspecified. A solution is constructed for the elastic problem and the system of governing equations is reduced to a nonlocal and nonlinear heat conduction problem. In Section 3, we impose physically realistic constraints on the contact resistance and develop a linear theory. We review the analysis due to Barber @2#, and show that the system studied undergoes a bifurcation from a single linearly stable steady-state solution to multiple steady-state solutions. Finally, in Section 4, we study the behavior of our system near the bifurcation point. That is, we inquire as to what happens when the system is started nearby the now linearly unstable steady-state solution. Using asymptotic matching techniques, we incorporate the effect of stabilizing nonlinear terms into our theory and obtain information about the dynamics and history dependence of the solution. We show that as conjectured, the solution does indeed approach one of the stable solutions obtained in the linear theory.

A Second-Order Approximation of Multi-Modal Interactions in Externally Excited Circular Cylindrical Shells

Abstract: We investigate the nonlinear response of an infinitely long, circularcylindrical shell to a primary-resonance excitation of one of itsflexural modes, which is involved in a one-to-two internal resonancewith the breathing mode The excited flexural mode is involved in aone-to-one internal resonance with its orthogonal flexural mode Thereare two simultaneous internal (autoparametric) resonances: two-to-oneand one-to-one The method of multiple scales is directly applied to thepartial-differential equations to obtain a system of six first-ordernonlinear ordinary-differential equations governing modulation of theamplitudes and phases of the three interacting modes In the absence ofdamping, the modulation equations are derivable from a Lagrangian,reflecting the conservative nature of the system The modulationequations are used to study the equilibrium and dynamic solutions andtheir stability and hence their bifurcations The response may be eithera two-mode or a three-mode solution For certain excitation parameters,the equilibrium three-mode solutions undergo Hopf bifurcations Acombination of a shooting technique and Floquet theory is used tocalculate limit cycles and their stability, and hence theirbifurcations

Instability Mechanisms of a Centrally Clamped Rotating Circular Disk Under a Space-Fixed Spring-Mass-Dashpot System

Abstract: The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed when the disk is subject to a space-fixed spring-mass-dashpot system. The disk is clamped at the center and free at the periphery. Using the method of multiple scales, the authors determine a set of four nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of two interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. They are reduced to an equivalent set of two first-order equations whose equilibrium solutions are determined analytically. The stability characteristics of these solutions is studied; the qualitative behavior of the response is independent of the mode being considered. Considering the case of a spring moving periodically along the radius of the disk, the authors show how its frequency can be coupled to the rotational speed of the disk and sub...

Transient Coupling of Launch Vehicle Bending Responses with Aerodynamic Flow State Variations

Abstract: Current analytic formulations for the self-sustained coupling of launch vehicle bending responses with aerodynamic e ow statevariations are restricted to prediction of the limit cycle state and associated stiffness and damping values. The theory is extended to include evolution of the structural response during transonic e ight to the limit cycle state. The method of multiple scales is used to analyze the nonlinear equation of motion. The resulting analytic expressions agree well with histories from a rigorous numerical solution for the idealized force ‐response coupling relationship. Itis shown that the transientresponseforlaunchvehicleaeroelastic coupling depends on the initial conditions, the structural frequency and damping, and the time lag associated with e ow state changes. It is also shown that the maximum response during the transonic regime can be signie cantly smaller than that for the limit cycle state, depending on the combination of these variables. In particular, the short time period that launch vehicles typically spend in the transonic regime precludes convergence of the bending mode response to the limit cycle state when the structural frequency and damping are low. The structure of the solution space is described, and it is shown that a subcritical Hopf bifurcation exists when the e ow state changes at 0 deg.

Combination Resonance of a Centrally Clamped Rotating Circular Disk

Abstract: The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed when the disk is subject to a space-fixed linear spring. Combination resonances are shown to occur at rotational speeds different from the classical critical speeds at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, the authors determine a set of eight nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of four interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. Using convenient variable transformations, the authors obtain an equivalent set of equations, which they use to prove that no single- mode solution is possible for this case. The stability characteristics of the trivial solution for all of the possible interactions among four of the lowest mod...

Analysis of global dynamics in a parametrically excited thin plate

Abstract: The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation.

Use of linear and nonlinear vibration absorbers for buffet alleviation of twin-tailed fighter aircraft

Abstract: We investigate theoretically and experimentally the performance of linear and nonlinear vibration absorbers to suppress high-amplitude vibrations of twin-tailed fighter aircraft when subjected to a primary resonance excitation. The tail section used in the experiments is a 1/16 dynamically scaled model fo the F-15 tail assembly. Both techniques (linear and nonlinear) are based on introducing an absorber and coupling it with the tails through a sensor and an actuator, where the control signals ae either linear or quadratic. For both cases, we develop the equations governing the response of the closed-loop system and use the method of multiple scales to obtain an approximate solution. We investigated both control strategies by studying their steady-state characteristics. In addition, we compare the power requirements of both techniques and show that the linear tuned vibration absorber uses less power than the nonlinear absorber.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Nonlinear stability analysis of the thin pseudoplastic liquid film flowing down along a vertical wall

Abstract: This article investigates the weakly nonlinear stability theory of a thin pseudoplastic liquid film flowing down on a vertical wall. The long-wave perturbation method is employed to solve for generalized nonlinear kinematic equation with free film interface. The normal mode approach is used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg–Landau equation. The modeling results indicate that both subcritical instability and supercritical stability conditions are possible to occur in a pseudoplastic film flow system. The results also reveal that the pseudoplastic liquid film flows are less stable than Newtonian’s as traveling down along the vertical wall. The degree of instability in the film flow is further intensified by decreasing the flow index n.

The method of multiple scales applied to the nonlinear stability problem of a truncated shallow spherical shell of variable thickness with the large geometrical parameter

Abstract: Using the modified method of multiple scales, the nonlinear stability of a truncated shallow spherical shell of variable thickness with a nondeformable rigid body at the center under compound loads is investigated. When the geometrical parameter k is larger, the uniformly valid asymptotic solutions of this problem are obtained and the remainder terms are estimated.

The method of multiple scales applied to the nonlinear stability problem of a truncated shallow spherical shell of variable thickness with the large geometrical parameter

Abstract: Using the modified method of multiple scales, the nonlinear stability of a truncated shallow spherical shell of variable thickness with a nondeformable rigid body at the center under compound loads is investigated. When the geometrical parameter k is larger,the uniformly valid asymptotic solutions of this problem are obtained and the remainder terms are estimated.

Non-linear modal interactions in the oscillations of a liquid drop in a gravitational field

Abstract: Non-linear modal interactions in the dynamics of a vibrating drop are examined. The partial di!erential equations governing the drop vibrations are formulated assuming potential #ow and incompressibility. The solution is expressed in terms of the eigenfunctions of the (linearized) Laplace operator in spherical coordinates. A small parameter e is introduced to scale the (small) deformation of the drop surface from its position of equilibrium. A 2 : 1 internal resonance is then imposed between the second and third modes of the resulting discretized system, and the ensuing non-linear modal interactions are studied using the method of multiple scales. A bifurcation in the slow dynamics of the system is detected that leads to amplitude modulations of the drop oscillations. The method employed in this work is general and can be used to study other types of non-linear interactions involving two or more drop modes. ( 2001 Elsevier Science Ltd. All rights reserved.

Solitons in superlattices : multiple scales method

Abstract: Negative differential conductivity is one of the basic characteristics of superlattices. A self-trapped potential is used to describe the formation of electric field domains resulting from negative differential conductivity. The method of multiple scales, in which the electric field profile is separated into fast and slow spatial components, shows that the slowly varying component satisfies the nonlinear Schrodinger equation. The well-known soliton solutions of this equation provide a theoretical description of the electric field domains. The soliton solutions imply that solitons are observed as envelopes of the linearized wave functions that correspond to the electric field domains in semiconductor superlattices.

Nonlinear Dynamics and Local Bifurcations in the Flexible Beam

Abstract: The analysis of nonlinear dynamics and local bifurcations of a simply supported flexible beam subjected to harmonic axial excitation is presented. The equation of motion with quintic nonlinear term under the parametric excitation of the simply supported flexible beam is derived. The parametrically excited system is first transformed to the averaged equations using the method of multiple scales. The analysis of stability for the zero solution of the averaged equations is given. It is found that the zero solution is of a double zero eigenvalues and codimension-3 degenerate bifurcations can occur in the averaged equations. Numerical simulations are also given to find the bifurcation response curves.

Nonlinear Dynamics of Two-Degrees-of-Freedom Fluid Conveying Elasto-Visco-Plastic Model System

Abstract: The local and global nonlinear dynamics of a two-degrees-of-freedom model system conveying fluid is studied. The undeflected model consistsof an inverted T formed by three rigid rods, with the tips of the twohorizontal rods resting on the viscous foundation. The foundationexhibits a visco-elasto-plastic response, including the Bauschingereffect. The vertical rigid rod of an annular cross-section is subjectedto a conservative (dead) force. Also, it conveys fluid having bothstatic and pulsation components. First, the method of multiple scales isused for the analysis of the local dynamics of the system withvisco-elastic response. Attention is focused on modal interactionphenomena in weak excitation at primary resonance and on hardsub-harmonic excitation. Two different asymptotic expansions areutilised to get a structural response for typical ranges of excitationparameters. Numerical integration of the governing equations is thenperformed to validate the results of asymptotic analysis in each case. Afull global nonlinear dynamics analysis of the visco-elasto-plasticsystem is performed. The role of plastic deformations in thedestabilisation of the system is discussed. Large-amplitude nonlinearoscillations of the visco-elasto-plastic system are studied, includingthe influence of material hardening and of static and periodiccomponents of pulsating fluid. Chaotic regimes of motion with andwithout plastic effects are considered. The results of the analysis maybe used in devices composed of a rather short tube connected to a notcompletely fixed foundation resting on the soil exhibitingelasto-plastic behaviour.

The method of multiple scales applied to the nonlinear stability problem of a truncated shallow spherical shell of variable thickness with the large geometrical parameter

Abstract: Using the modified method of multiple scales, the nonlinear stability of a truncated shallow spherical shell of variable thickness with a nondeformable rigid body at the center under compound loads is investigated. When the geometrical parameter k is larger, the uniformly valid asymptotic solutions of this problem are obtained and the remainder terms are estimated.