Time Scale Perturbation Method Research Articles

On the multiple time-scales perturbation method for differential-delay equations

On the multiple time-scales perturbation method for differential-delay equations

On Stability Analysis of Linear Axially Moving Damped Beam with Elastic Foundation

This study investigates the dynamical behavior of an axially moving beam with elastic foundation under viscous damping. The beam is taken to be simply supported at the ends. Axial speed of beam is assumed to be time‐dependent function about a relatively low mean constant speed. The motion of axially moving beam in the term of transverse vibrations is expressed in the fourth‐order linear and homogeneous partial differential equations with variable coefficients. The governing equations of motion are solved by using the two timescales perturbation method together with the Fourier expansion method. The sum and difference‐type resonances are taken to examine the stability of the system. The influence of the parameters such as damping and stiffness of foundation on the transverse amplitude response of the system is seen. It is found out that the system is stable with the increase in damping.

On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.

Eliminate the Nonlinear Oscillations of the Modified duffing Equation by using the Nonlinear Integrated Positive Position Feedback

In this article, the nonlinear integrated positive position feedback (NIPPF) control adds to a nonlinear dynamical system modeled as the well known Duffng oscillators. This control is proposed to mitigate system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow- ow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. Before and after (NIPPF) control the nonlinear systems' steady-state amplitude are examined, the comparison is made to validate the closeness between the numerical solution and the analytical perturbative one at time-history and frequency response curves.

Analysis of longitudinal oscillations in a vertically moving cable subject to nonclassical boundary conditions

In this paper, we study a model of a flexible hoisting system, in which external disturbances exerted on the boundary can induce large vibrations, and so damage to the performance of the system. The dynamics is described by a wave equation on a slow time-varying spatial domain with a small harmonic boundary excitation at one end of the cable, and a moving mass at the other end. Due to the slow variation of the cable length, a singular perturbation problem arises. By using an averaging method, and an interior layer analysis, many resonance manifolds are detected. Further, a three time-scales perturbation method is used to construct formal asymptotic approximations of the solutions. It turns out that for a given boundary disturbance frequency, many oscillation modes jump up from order ε amplitudes to order ε amplitudes, where ε is a small parameter with 0<ε<<1. Finally, numerical simulations are presented to verify the obtained analytical results.

On resonances in transversally vibrating strings induced by an external force and a time-dependent coefficient in a Robin boundary condition

In this paper an initial–boundary value problem on a bounded, fixed interval is considered for a one-dimensional and forced string equation subjected to a Dirichlet boundary condition at one end of the string and a Robin boundary condition with a slowly varying time-dependent coefficient at the other end of the string. This problem may serve as a simplified model describing transverse or longitudinal vibrations as well as resonances in axially moving cables for which the length changes in time. By introducing an adapted version of the method of separation of variables, by using averaging and singular perturbation techniques, and by finally using a three time-scales perturbation method, resonances in the problem are detected and accurate, analytical approximations of the solutions of the problem are constructed. It will turn out that small order ɛ excitations can lead to order ɛ responses when the frequency of the external force satisfies certain conditions. Finally, numerical simulations are presented, which are in full agreement with the obtained analytical results.

On resonances in a weakly nonlinear microbeam due to an electric actuation

In this paper, the oscillations of an actuated, simply supported microbeam are studied for which it is assumed that the electric load is composed of a small DC polarization voltage and a small, harmonic AC voltage. Bending stiffness and mid-plane stretching are taken into account as well as small viscous or structural damping. No tensile axial force is assumed to be present. By using a multiple time-scales perturbation method, approximations of the solutions of the initial-boundary value problem for the microbeam equation are constructed. This analysis is performed without truncating the infinite series representation in advance as is usually done in the existing literature. It is shown in which cases truncation is allowed for this problem. Moreover, accurate and explicit approximations of the natural frequencies up to order \(\varepsilon ^3\) of the actuated microbeam are also obtained. Intriguing and new modal vibrations are found when the frequency of the harmonic AC voltage is (near) half or twice a natural frequency of the microbeam, i.e., near a superharmonic or a subharmonic resonance.

Adaptive versus Conventional Positive Position Feedback Controller to Suppress a Nonlinear System Vibrations

The nonlinear vibration control of a nonlinear dynamical system modeled as the well-known Duffing oscillators is investigated within this article. The conventional Positive Position Feedback (PPF) controller is proposed to mitigate the considered system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow-flow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability analysis is investigated based on Lyapunov’s first method. The effects of the different control parameters on both the main system and controller are explored. The obtained analytical and numerical results illustrated that the PPF controller can eliminate the main system nonlinear vibrations once the controller natural frequency is tuned to be the same value as the external excitation frequency, otherwise, the controller adds excessive vibrational energy to the main system rather than suppressing it. In addition, the PPF controller can destabilize the main system motion when excited by strong excitation force. Therefore, a modified version of the PPF controller named the Adaptive Positive Position Feedback (APPF) controller is proposed to overcome the main drawbacks of the conventional PPF controller. The idea is to track the external excitation frequency using an adaptive frequency measurement technique to update continuously the PPF controller natural frequency to become the same value of the excitation frequency. Based on this strategy, the system mathematical model is analyzed again by making the controller’s natural frequency equal to the external excitation frequency. The obtained analytical and numerical simulations showed that the adaptive positive position feedback controller can suppress the main system nonlinear vibration close to zero regardless of the excitation force amplitude and excitation frequency.

Stability characteristics of multi-frequency parametrically-excited rotor-oscillator systems

The effect of multi-frequency parametric excitation on the stability of an oscillator is examined in the context of a rotating system where the rotor and the stator have comparable masses and hence the ‘stator’ cannot be considered to be mechanically fixed. The differential equations governing the dynamics of such systems have time-varying terms which are periodic in the rotor frequencies. Floquet theory is employed to numerically develop stability diagrams which distinguish stable from unstable parameter combinations. Reasons for the occurrence of instabilities are explained using the two time scale perturbation method. It was found that the most significant resonance in the single rotor case occurred when the rotor speed equaled the sum of two unforced natural frequencies. In addition, harmonic resonances were found to arise when the rotor speed matched the unforced natural frequencies. Furthermore, the excitation from a second rotor caused instabilities to arise around regimes where multiple combination resonances occurred together. The occurrence of certain experimentally-observed resonances and regimes of low amplitude oscillations were correlated with the model, and the corresponding rotor trajectories were explored. This investigation is useful for identifying operating regimes so as to avoid catastrophic failure in multi-frequency rotating machines such as multi-rotor helicopters and gas turbines with contra-rotating stages.

Response Analysis and Controlling the Nonlinear Vibration of Van Der-Pol Duffing Oscillator Connected to the NIPPF Controller

In this paper, we introduced response analysis and controlling the nonlinear vibration of van der pol duffing oscillator subject to parametric and external excitations by connecting the nonlinear integral positive position feedback (NIPPF). The controller combines the characteristics of the positive position feedback controller (PPF) and integral resonant control (IRC) and these controllers are viewed in a blocked scheme for the closed-loop system (system under control). The analytical solution and all resonance cases are obtained by applying the multiple time scales perturbation method (MSPM). The stability analysis is studied using the frequency response equations at the supposed worst resonance case. For numerical simulation, the system is examined before and after connecting the NIPPF controller using Matlab software program (package ode45). The influences of different parameters are examined numerically for the vibrating system and controller. The approximate analysis is confirmed with numerical simulation outcomes. Finally, making a comparison with previously related published work.

Eigen and temporal characteristics of multi-frequency parametrically-excited rotor-oscillator systems

Traditional investigation on the dynamics of rotating systems assumes the axis of the rotor to be fixed. This assumption is valid as long as the stator is significantly heavier than the rotor, or is otherwise constrained to have negligible motion. However, in certain engineering applications where the rotor and the stator have comparable inertia and compliances, the stator’s motion is not only significant but also coupled with that of the rotors’. This paper examines the complicated temporal and eigen behaviors of such a generic multi-rotor system in which the (non-static) ‘stator’ and the rotors have comparable inertia and compliances. The multi-frequency parametrically-excited equations governing the dynamics of a representative dual-rotor case were numerically-integrated to obtain the responses. The effects of system parameters on the responses were analyzed using the two time-scale perturbation method. It was found that the frequency spectra comprised side-bands centered around the unforced natural frequencies. The genesis of the complicated frequency spectra was traced to the cyclically time-varying eigen-structure. Further, the underlying eigen-structure was noted to be periodic at a frequency related to that of the individual rotor speeds. Frequency components of the obtained responses were validated against measurements performed on an analogous experimental system. These results suggest that even a seemingly simple multi-frequency parametrically excited rotating system that considers the coupling between the rotor and the stator exhibits complicated eigen as well as temporal characteristics.

On applicability of truncation method for damped axially moving string

In this paper, the detailed study of the transversal vibrations of a damped axially moving string is considered. Two end pulleys of the string are taken to be fixed and the initial conditions are assumed to be of general displacement field and the general velocity field. The axial speed of the string is considered to be sinusoidal, time-dependent and small compared to wave-velocity. A two timescales perturbation method with a combination of Fourier-sine series which fits the boundary conditions is employed in order to formulate the valid and uniform asymptotic approximations of the exact solutions for the equation. It is found that there are infinitely many values of frequency parameter Ω which cause the internal resonances in system. The fundamental resonant frequency, the non-resonant frequency and the detuning cases have been discussed and analyzed in detail. It has been found explicitly that the total mechanical energy of the infinite dimensional system decreases for two cases of the damping parameter, that is, for δ=2 and for δ&gt;2. By truncation method it has been shown that the mode-amplitude response for first few modes is stable. So, Galerkin’s truncation method may be possible for these two cases of the parameter δ. But for case δ&lt;2 the total mechanical energy of belt system is increasing exponentially. Therefore, it is evident that the Galerkin’s truncation method cannot be applied in order to obtain valid approximations on long timescales, that is, on timescales of O1/ε.

Lateral vibration control and stabilization of the quasiperiodic oscillations for rotor-active magnetic bearings system

This work aims to improve the vibrational behaviors of the rotor-active magnetic bearings (AMB) system via proposing a new control methodology for such lateral vibrations. Accordingly, the rotor-AMB system model is modified by integrating the conventional proportional–derivative (PD) controller along with the positive position feedback (PPF) controller to the considered system. The control currents in the rotor-AMB system are designed to become a combination of two control signals: One of them is generated via the PD controller, and the other one comes from the PPF controller. Based on the suggested control method, the whole dynamical system model is derived and then analyzed by the multiple time-scale perturbation method. Then, the bifurcation diagrams of both the spinning speed and rotor eccentricity response curves are investigated before and after control. The obtained analytical results showed that the integrating of the PPF controller to the considered system could mitigate the system lateral vibration close to zero. Moreover, the system quasiperiodic motions have been stabilized after control and the sensitivity to the initial conditions has eliminated. Finally, numerical validations for the obtained analytical results are performed that illustrated an excellent agreement with the analytical ones.

On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate’s method will be employed to establish the approximate analytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response. Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order ε -1

Three-to-one internal resonance of inclined marine riser

In this paper, we investigate the three-to-one internal resonance of an inclined marine riser. The model accounts for the flexural rigidity of the riser, variable axial load, nonlinear geometry, and initial static deflection due to self-weight. The applied tension with the static deflection are tuned such that the ratio between the fifth and first natural frequencies is three. Then, the multiple time scales (MTS) perturbation method is applied to study the internal resonance interactions when the structure is harmonically excited. The system equation is solved using a multi-mode Galerkin model and its results are compared to the perturbation results showing good agreement. Moreover, the frequency response curves of the fifth and first modes amplitudes exhibit Hopf and saddle node bifurcations. In addition, the interaction of the fifth mode with the first mode during internal resonance results into new emerging solutions and states. These phenomena are well observed in the force response curves and confirmed by the time history of the response of the structure, which can lead to complex dynamics that hinder the life of the riser by fatigue failure.

Nonlinear Forced Vibration of Curved Microbeam Resting on Nonlinear Foundation Using the Modified Strain Gradient Theory

The nonlinear forced vibrations of a curved micro-beam resting on a nonlinear foundation are examined. The equations of motion are derived using Hamilton’s principle and the modified strain gradient theory which is capable to examine the size effects in microstructures. The nonlinear partial differential equations of motion are reduced to a time-dependent ordinary differential equation containing quadratic and cubic nonlinear terms. The frequency response of the curved microbeam at the primary resonance is determined using the multiple time scales perturbation method. From the application point of view, the frequency response curves may be useful to select the optimum values of design parameters. The effects of geometry parameters and foundation moduli on the vibration behavior of the curved microbeam are illustrated.

Primary and Secondary Resonance of Micro-resonators Based on Couple Stress Theory

In this work, the influence of size effect on the primary and the secondary resonance of micro-resonators is investigated. The micro-resonator, which is considered in this study, is a clamped–clamped beam excited by AC and DC electrostatic voltage. The DC voltage is applied to prevent doubling of excitation frequency and tune the range of system’s working frequency. We computed the beam’s motion equation using Euler–Bernoulli theory for wide beams considering couple stress theory and geometrical nonlinearity. For micro-resonators with high quality factor and small amplitude of harmonic excitation, we captured frequency response curves using shooting method, then we used multiple time scale method for its validation. Due to the limitation of the multiple time scale perturbation method for the computation of frequency responses in large amplitude harmonic excitation, the shooting method has been used, regardless of how big the harmonic excitation amplitude or how small the micro-resonator’s quality factor is. Another contribution of this paper is the analysis of the influence of size effect on the system’s hardening and softening as well as its effect on reduction of pull-in band. This work presents that considering the size effect will result in reduction or elimination of pull-in band and also the nonlinear behavior of the system will be inclined to become linear in primary resonance.

Controlling quarter car suspension system by proportional derivative and positive position feedback controllers with time delay

The active car suspension system is presented here to suppress the vibration of the car by applying proportional derivative (PD) and positive position feedback (PPF) controllers with time delay. The control signal output of the controller is applied electrically to the magneto-rheological (MR) damper or Electrical-rheological (ER) damper which is attached parallel to the passive components to improve the suppression of the vibration. The electrical control signal is produced by electronic circuits or programmable logic controller and the two-position sensor feedback signal which are connected to the controller. The approximate solutions of PD and PPF suspension systems are obtained by applying multiple time scales perturbation method. The effects of parameters variation of both the system and the controllers are investigated to achieve the best performance. Simulation results show good performance of the designed controllers.

Analytical modeling of nonlinear vibrations in a 2-DOF airfoil device based on an unsteady flow model

The objective of this paper is to provide an analytical framework based on Wagner’s unsteady flow model, to study the nonlinear dynamic of pitch and plunge DOFs in airfoils. To this end, the nonlinear equations of motion of an airfoil section model with pitch and plunge DOFs are derived. The 2-DOF model of the system is converted to a 3-DOF model by adding augmented states related to the dynamic of the correction coefficient of the Wagner’s flow model. The resulting 3-DOF system is then normalized and re-expressed around the equilibrium state. As the mass and stiffness matrices of the resulting system are not symmetric, special techniques are used to diagonalize them and facilitate the later nonlinear analysis. Multiple timescales perturbation method is then used to provide analytical solution for the nonlinear self-excited 3-DOF model. The analytical and numerical results are compared at three different situations related to the subcritical, critical and supercritical values of the wind speed. It is observed that the qualitative and quantitative results of the numerical and analytical models are in excellent agreement with each other. To study the effect of nonlinearity on the system’s response, a linear eigenvalue analysis is carried out using a root locus plot of the homogenous linearized system. By comparing the analytical results of the linear and perturbed system with nonlinear numerical simulations, it is shown that the linear model provides a rough over-estimation for the instability speed and the use of a nonlinear model is inevitable for achieving sufficiently accurate findings. The approach and results presented in this paper can be used to provide better prediction for the dynamic behavior of flexible blades with pitch and plunge DOFs.

On Parametric Stability of a Nonconstant Axially Moving String Near Resonances

The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales perturbation method is used to compute the first-order approximation of the solutions of the equations of motion that governs the transverse vibrations of an axially moving string. It turns out that the system can give rise to resonances when the velocity fluctuation frequency is equal (or close) to an odd multiple of the natural frequency of the system. The stability conditions are investigated analytically in terms of the displacement-response and the energy of the system near the resonances. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations.