Showing papers on "Multiple-scale analysis published in 2013"

Non-linear response of buckled beams to 1:1 and 3:1 internal resonances

Abstract: The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral–partial–differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency–response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.

Nonlinear oscillations following the Rayleigh collapse of a gas bubble in a linear viscoelastic (tissue-like) medium

Abstract: In a variety of biomedical engineering applications, cavitation occurs in soft tissue, a viscoelastic medium. The present objective is to understand the basic physics of bubble dynamics in soft tissue. To gain insights into this problem, theoretical and numerical models are developed to study the Rayleigh collapse and subsequent oscillations of a gas bubble in a viscoelastic material. To account for liquid compressibility and thus accurately model large-amplitude oscillations, the Keller-Miksis equation for spherical bubble dynamics is used. The most basic linear viscoelastic model that includes stress relaxation, viscosity, and elasticity (Zener, or standard linear solid) is considered for soft tissue, thereby adding two ordinary differential equations for the stresses. The present study seeks to advance past studies on cavitation in tissue by determining the basic effects of relaxation and elasticity on the bubble dynamics for situations in which compressibility is important. Numerical solutions show a clear dependence of the oscillations on the viscoelastic properties and compressibility. The perturbation analysis (method of multiple scales) accurately predicts the bubble response given the relevant constraints and can thus be used to investigate the underlying physics. A third-order expansion of the radius is necessary to accurately represent the dynamics. Key quantities of interest such as the oscillation frequency and damping, minimum radius, and collapse time can be predicted theoretically. The damping does not always monotonically decrease with decreasing elasticity: there exists a finite non-zero elasticity for which the damping is minimum; this value falls within the range of reported tissue elasticities. Also, the oscillation period generally changes with time over the first few cycles due to the nonlinearity of the system, before reaching an equilibrium value. The analytical expressions for the key bubble dynamics quantities and insights gained from the analysis may prove valuable in the development and optimization of certain biomedical applications.

Asymptotic Analysis of Resonances in Nonlinear Vibrations of the 3-dof Pendulum

Abstract: The dynamical response of a harmonically excited three degrees-of-freedom planar physical pendulum is studied in the paper. The investigated system may be considered as a good example for several engineering applications. The asymptotic method of multiple scales (MS) has been adopted in order to carry out the analytical computations. The solutions up to the third order have been achieved. MS method allows to identify parameters of the system being dangerous due to the resonances and yields time histories for the assumed generalised co-ordinates. Three simultaneously occurring resonance conditions have been analysed. The energy transfer from one to another mode of vibrations is illustrated and discussed. The modulation equations in an autonomous form allow obtaining the frequency response functions and drawing resonance curves.

Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed

Abstract: In this paper, stability in parametric resonance of axially moving viscoelastic plates subjected to plane stresses is investigated. The plate material obeys the Kelvin–Voigt model in which the material time derivative is used. The generalized Hamilton principle is employed to obtain the governing equation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The governing equation can be regarded as a continuous gyroscopic system with small periodically parametric excitations and a damping term. The method of multiple scales is applied to the governing equation to establish the solvability conditions in principal and summation parametric resonances. The natural frequencies and modes of linear generating equation are numerically calculated based on the given boundary conditions. The necessary and sufficient condition of the stability is derived from the Routh–Hurwitz criterion. Some numerical examples are presented to demonstrate the effects of related parameters on the frequencies and the stability boundaries. The differential quadrature scheme is developed to solve numerically the linear generating system and the primitive equation model. The numerical calculations confirm the analytical results.

Nonlinear oscillations of rotor active magnetic bearings system

Abstract: In this paper, an analytical approximate solution is constructed for a rotor-AMB system that is subjected to primary resonance excitations at the presence of 1:1 internal resonance. We obtain an approximate solution applying the method of multiple scales, and then we conducted the system bifurcation analyses. The stability of the system is investigated applying Lyapunov’s first method. The effects of the different parameters on the system behavior are investigated. The analytical results showed that the rotor-AMB system exhibits a variety of nonlinear phenomena such as bifurcations, coexistence of multiple solutions, jump phenomenon, and sensitivity to initial conditions. Finally, the numerical simulations are performed to demonstrate and validate the accuracy of the approximate solutions. We found that all predictions from analytical solutions are in excellent agreement with the numerical integrations.

Nonlinear Dynamics of Torsional Vibration for Rolling Mill's Main Drive System Under Parametric Excitation

Abstract: The jointed shaft in the drivelines of the rolling mill, with its angle continuously varying in the production, has obvious impact on the stability of the main drive system Considering the effect caused by the joint angle and friction force of roller gap, the nonlinear vibration model of the main drive system which contains parametric excitation stiffness and nonlinear friction damping was established The amplitude-frequency characteristic equation and bifurcation response equation were obtained by using the method of multiple scales Depending on the bifurcation response equation, the transition set and the topology structure of bifurcation curve of the system were obtained by using the singularity theory The transition set can separate the system into seven areas, which has different bifurcation forms respectively By taking the 1780 rolling mill of Chengde Steel Co for example, the simulation and analysis were performed The amplitude-frequency curves under different joint angles, damping coefficients, and nonlinear stiffness were given The variations of these parameters have strong influences on the stability of electromechanical resonances and the characteristic of the response curves The best angle of the jointed shaft is 4 761 3° in this rolling mill

Dynamics of axially accelerating beams with multiple supports

Abstract: This study represents the transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on multiple simple supports. This is one of the examples of a system experiencing Coriolis acceleration component that renders such systems gyroscopic. A small harmonic variation with a constant mean value for the axial velocity is assumed in the problem. The immovable supports introduce nonlinear terms to the equations of motion due to stretching of neutral axis. The method of multiple scales is directly applied to the equations of motion obtained for the general case. Natural frequency equations are presented for multiple support case. Principal parametric resonances and combination resonances are discussed. Solvability conditions are presented for different cases. Stability analysis is conducted for the solutions; approximate stable and unstable regions are identified. Some numerical examples are presented to show the effects of axial speed, number of supports, and their locations.

Nonlinear vibrations of axially moving multi-supported strings having non-ideal support conditions

Abstract: In this study, nonlinear vibrations of an axially moving multi-supported string have been investigated. The main difference of this study from the others is in that there are non-ideal supports allowing minimal deflections between ideal supports at both ends of the string. Nonlinear equations of the motion and boundary conditions have been obtained using Hamilton’s Principle. Dependence of the equations of motion and boundary conditions on geometry and material of the string have been eliminated by non-dimensionalizing. Method of multiple scales, a perturbation technique, has been employed for solving the equations of motion. Axial velocity has been assumed a harmonically varying function about a constant value. Axially moving string has been investigated in three regions. Vibrations have been examined for three different cases of the velocity variation frequency. Stability has been analyzed and stability boundaries have been established for the principal parametric resonance case. Effects of the non-ideal support conditions on stability boundaries and vibration amplitudes have been investigated.

Parametric and internal resonances of an axially moving beam with time-dependent velocity

Abstract: The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beamvelocity is assumed to be comprised of a constantmean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of secondmode is approximately three times that of firstmode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

On the planar motion in the full two-body problem with inertial symmetry

Abstract: Relative motion of binary asteroids, modeled as the full two-body planar problem, is studied, taking into account the shape and mass distribution of the bodies. Using the Lagrangian approach, the equations governing the motion are derived. The resulting system of four equations is nonlinear and coupled. These equations are solved numerically. In the particular case where the bodies have inertial symmetry, these equations can be reduced to a single equation, with small nonlinearity. The method of multiple scales is used to obtain a first-order solution for the reduced nonlinear equation. The solution is shown to be sufficient when compared with the numerical solution. Numerical results are provided for different example cases, including truncated-cone-shaped and peanut-shaped bodies.

Vibrations of a Slightly Curved Microbeam Resting on an Elastic Foundation with Nonideal Boundary Conditions

Abstract: An investigation into the dynamic behavior of a slightly curved resonant microbeam having nonideal boundary conditions is presented. The model accounts for midplane stretching, an applied axial load, and a small AC harmonic force. The ends of the curved microbeam are on immovable simple supports and the microbeam is resting on a nonlinear elastic foundation. The forced vibration response of curved microbeam due to the small AC load is obtained analytically by means of direct application of the method of multiple scales (a perturbation method). The effects of the nonlinear elastic foundation as well as the effect of curvature on the vibrations of the microbeam are examined. It is found that the effect of curvature is of softening type. For sufficiently high values of the coefficients, the elastic foundation and the axial load may suppress the softening behavior resulting in hardening behavior of the nonlinearity. The frequencies and mode shapes obtained are compared with the ideal boundary conditions case and the differences between them are contrasted on frequency-response curves. The frequency response and nonlinear frequency curves obtained may provide a reference for the choice of reasonable resonant conditions, design, and industrial applications of such systems. Results may be beneficial for future experimental and theoretical works on MEMS.

Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations

Abstract: The effect of the narrow-band random excitation on the non-linear response of sandwich plates with an incompressible viscoelastic core is investigated To model the core, both the transverse shear strains and rotations are assumed to be moderate and the displacement field in the thickness direction is assumed to be linear for the in-plane components and quadratic for the out-of-plane components In connection to the moderate shear strains considered for the core, a non-linear single-integral viscoelastic model is also used for constitutive modeling of the core The fifth-order perturbation method is used together with the Galerkin method to transform the nine partial differential equations to a single ordinary integro-differential equation Converting the lower-order viscoelastic integral term to the differential form, the fifth-order method of multiple scale is applied together with the method of reconstitution to obtain the stochastic phase-amplitude equations The Fokker–Planck–Kolmogorov equation corresponding to these equations is then solved by the finite difference method, to determine the probability density of the response The variation of root mean square and marginal probability density of the response amplitude with excitation deterministic frequency and magnitudes are investigated and the bimodal distribution is recognized in narrow ranges of excitation frequency and magnitude

From Newton's Cradle to the Discrete $p$-Schrödinger Equation

Abstract: We investigate the dynamics of a chain of oscillators coupled by fully nonlinear interaction potentials This class of models includes Newton's cradle with Hertzian contact interactions between neighbors By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times The envelope equation leading to approximate solutions is a discrete $p$-Schrodinger equation Our results include the existence of long-lived breather solutions to the original model For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times

Nonlinear Wave Propagation in Bubbly Liquids

Abstract: Weakly nonlinear wave equations for pressure waves in bubbly liquids are derived in a general and systematic way based on the asymptotic expansion method of multiple scales. The derivation procedure is explained in detail with a special attention to scaling relations between physical parameters characterizing the wave motions concerned. In the framework of the present theory, one can systematically deal with various weakly nonlinear wave motions for various systems of governing equations of bubbly liquids, thereby deriving such as the Korteweg–de Vries–Burgers equation, the nonlinear Schrodinger equation, and the Khokhlov–Zabolotskaya–Kuznetsov equation. In this sense, the method may be called a unified theory of weakly nonlinear waves in bubbly liquids.

Stability analysis and vibration reduction for a two-dimensional nonlinear system

Abstract: This study examines how the vibration absorbers influence the stability of nonlinear flow-solid interaction systems. A novel approach is proposed for the analysis of dynamic stability of the two-dimensional nonlinear system using the internal resonance contour plot (IRCP) and flutter speed contour plot (FSCP). The system considered is a planar rigid-body with plunge and pitch vibrations. The two ends of the body are supported by cubic nonlinear springs with quadratic damping. The vibration absorber attached beneath is also considered as a rigid body, with the mass and position of the absorber adjusted for optimization of vibration reduction. The method of multiple scales (MOMS) is employed to obtain a fixed point solution. The P-method is also adopted to obtain the eigen plot and Poincare Map for comparison. Finally, both the IRCP with FSCP are cross-referenced to provide guide-lines for identifying the optimal location for the absorber with regard to stability and vibration reduction without the need to alter the framework of the main body.

Non-Linear Vibrations of a Microbeam Resting on an Elastic Foundation

Abstract: In this study, response of a microbeam bonded to a non-linear elastic foundation is investigated. The model accounts for mid-plane stretching, an applied axial load, and an AC harmonic force. The microbeam is resting on a non-linear elastic foundation which introduces a cubic non-linear term to the equations of motion. Immovable end conditions introduce integral type nonlinearity. The integro-differential equations of motion are solved analytically by means of direct application of the method of multiple scales (a perturbation method). The amplitude and phase modulation equations are derived for the case of primary resonances. Frequency-response curves and non-linear frequencies are analyzed with respect to the effective physical parameters. Influence of elastic foundation coefficients, coefficients related to dielectric constants, and mid-plane stretching on the vibrations of the microbeam are investigated.

Linear vibrations of continuum with fractional derivatives

Abstract: In this paper, linear vibrations of axially moving systems which are modelled by a fractional derivative are considered The approximate analytical solution is obtained by applying the method of multiple scales Including stability analysis, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted by graphs It is determined that the external excitation force acting on the system has an effect on the stiffness of the system Moreover, the general algorithm developed can be applied to many problems for linear vibrations of continuum

Nonlinear vibrations of beams with spring and damping delayed feedback control

Abstract: The primary, subharmonic, and superharmonic resonances of an Euler–Bernoulli beam subjected to harmonic excitations are studied with damping and spring delayed-feedback controllers. By method of multiple scales, the non-linear governing partial differential equation is transformed into linear differential equations directly. Effects of the feedback gains and time-delays on the steady state responses are investigated. The velocity and displacement delayed-feedback controllers are employed to suppress the primary and superharmonic resonances of the forced nonlinear oscillator. The stable vibration regions of the feedback gains and time-delays are worked out based on stablility conditions of the resonances. It is found that proper selection of feedback gains and time-delays can enhance the control performance of beam’s nonlinear vibration. Position of the bifurcation point can be changed or the bifurcation can be eliminated.

Control of the nonlinear oscillator bifurcation under a superharmonic resonance

Abstract: A weakly nonlinear oscillator is modeled by a differential equation. A superharmonic resonance system can have a saddle-node bifurcation, with a jumping transition from one state to another. To control the jumping phenomena and the unstable region of the nonlinear oscillator, a combination of feedback controllers is designed. Bifurcation control equations are derived by using the method of multiple scales. Furthermore, by performing numerical simulations and by comparing the responses of the uncontrolled system and the controlled system, we clarify that a good controller can be obtained by changing the feedback control gain. Also, it is found that the linear feedback gain can delay the occurrence of saddle-node bifurcations, while the nonlinear feedback gain can eliminate saddle-node bifurcations. Feasible ways of further research of saddle-node bifurcations are provided. Finally, we show that an appropriate nonlinear feedback control gain can suppress the amplitude of the steady-state response.

Optimal Control of Nonlinear Resonances for Vehicle Suspension using Linear and Nonlinear Control

Abstract: The dynamic equations of the strongly nonlinear vibration of vehicle suspension with linear and nonlinear feedback controllers are developed. The strongly nonlinear vibration is transformed into the weakly nonlinear vibration by the nonlinear controller. The forced vibration of vehicle suspension is studied by the method of multiple scales. The regions of feedback gains obtained from the stability conditions of eigenvalue equation quantitatively are presented. Taking attenuation ratio and energy function as the objective functions, the control parameters of velocity and displacement are calculated by the minimum optimal method. Illustrative examples are given to show the effectiveness of vibration control.