Showing papers on "Multiple-scale analysis published in 2015"
TL;DR: In this paper, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) subjected to thermal environment including the effect of surface stress.
Abstract: In the present investigation, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) subjected to thermal environment including the effect of surface stress. The material properties of functionally graded (FG) nanobeams vary through the thickness direction on the basis of a simple power law. The geometrically nonlinear beam model, taking into account the surface stress effect, is developed by implementing the Gurtin–Murdoch elasticity theory together with the classical Euler–Bernoulli beam theory and using a variational approach. Hamilton’s principle is utilized to obtain the nonlinear governing partial differential equation and corresponding boundary conditions. After that, the Galerkin technique is employed in order to convert the nonlinear partial differential equation into a set of nonlinear ordinary differential equations. This new set is then solved analytically based on the method of multiple scales which results in the frequency–response curves of FG nanobeams in the presence of surface stress effect. It is revealed that by increasing the beam thickness, the surface stress effect diminishes and the maximum amplitude of the stable response is shifted to the higher excitation frequencies.
105 citations
TL;DR: In this article, the authors investigate theoretically and experimentally the potential of a vibro-impact type nonlinear energy sink (VI-NES) to mitigate vibrations of a linear oscillator (LO) subjected to a harmonic excitation.
Abstract: Recently, it has been demonstrated that a vibro-impact type nonlinear energy sink (VI-NES) can be used efficiently to mitigate vibration of a linear oscillator (LO) under transient loading. The objective of this paper is to investigate theoretically and experimentally the potential of a VI-NES to mitigate vibrations of a LO subjected to a harmonic excitation (nevertheless, the presentation of an optimal VI-NES is beyond the scope of this paper). Due to the small mass ratio between the LO and the flying mass of the NES, the obtained equation of motion are analyzed using the method of multiple scales in the case of 1 : 1 resonance. It is shown that in addition to periodic response, system with VI-NES can exhibit strongly modulated response (SMR). Experimentally, the whole system is embedded on an electrodynamic shaker. The VI-NES is realized with a ball which is free to move in a cavity with a predesigned gap. The mass of the ball is less than 1% of the mass of the LO. The experiment confirms the existence of periodic and SMR response regimes. A good agreement between theoretical and experimental results is observed.
84 citations
TL;DR: In this article, the influence of small scale coefficient on the nonlinear frequency ratio of the first nonlinear normal mode (NNM) for the double layered viscoelastic nanoplates with simply supported boundary condition was investigated.
Abstract: This figure presents the influence of small scale coefficient on the nonlinear frequency ratio of the first nonlinear normal mode (NNM) for the double layered nanoplates with simply supported boundary condition. The figure shows that the frequency ratio increases with the augment of the nonlocal parameter for a given mode amplitude (a1/h). This fact reveals that with the increase of the nonlocal coefficient the nonlinearity for the first NNM is enhanced. abstract The nonlinear flexural vibration properties of double layered viscoelastic nanoplates are investigated based on nonlocal continuum theory. The von Kaman strain-displacement relation is employed to model the geometrical nonlinearity. Based on the classical plate theory, the formulations are derived by the Hamilton's principle in conjunction with Eringen's nonlocal elasticity theory, and are further discretized by the Galerkin's method. The coordinate transformation is adopted to obtain the nonlinear governing equations of motion in the modal coordinate system. On the basis of these equations, the frequency responses of double layered nanoplates with simply supported and clamped boundary conditions are derived by the method of multiple scales. The influences of small scale and other structural parameters (e.g. the aspect ratio of the plate, van der Walls (vdW) interaction and the viscidity of the plate) on the nonlinear vibration characteristics are discussed. From the result, the vdW interaction has obvious effects on the nonlinear frequency corresponding to the second nonlinear normal mode (NNM). The non- existence of the internal resonance is also induced from the vdW forces between the plates. The influ- ence of the elastic matrix is also discussed. The hardening nonlinearity is observed for the primary resonance. Additionally, some interesting phenomena different from the linear vibration are observed.
77 citations
Abstract: Prior work on a disbonded aluminum honeycomb panel showed evidence of a quadratic stiffness nonlinearity, as well as the presence of an unknown cubic nonlinearity. Approximations to higher order nonlinear single degree of freedom (SDOF) models were solved using the method of multiple scales. These approximations were then used to fit displacement data from a sinusoidal excitation test and determine the coefficients of the model as a function of damage size. Confirmation of the quadratic stiffness nonlinearity was achieved through examination of force restoration curves excited at one-half the primary resonance in conjunction with coefficient fitting of the test data to the model. The data were fit against the higher order models to determine whether the cubic nonlinearity could be stiffness or damping related. The coefficient fitting shows that the cubic nonlinearity is a stiffness nonlinearity. This confirmed what was seen in the force restoration curves when the system was excited at one-third the primary resonance. The ability to match the vibratory behavior of the damage to a SDOF model shows that the use of single frequency excitation at lower frequencies can isolate the nonlinear behavior of the damaged area and identify what damage mechanisms may be involved.
57 citations
TL;DR: In this article, the effects of internal resonances on the steady-state responses to external excitations in the nonlinear boundary problem of the partial differential equations were investigated for two elastically connected cantilevers, under harmonic base excitation.
52 citations
TL;DR: The new Nonlinear Integral Positive Position Feedback approach and the Integral Resonant Control method are implemented and analyzed for nonlinear vibration control and it is demonstrated that controller structure has the most salient role in the suppression performance.
44 citations
TL;DR: In this paper, a nonlinear modified positive position feedback (NMPPF) control approach for nonlinear vibration suppression at primary resonance is proposed, which consists of a resonant second-order nonlinear compensator, which is enhanced by a lossy integrating compensator.
Abstract: This paper introduces Nonlinear Modified Positive Position Feedback (NMPPF) control approach for nonlinear vibration suppression at primary resonance. Nonlinearity in the system is due to large deformations caused by high-amplitude disturbances, while this control approach is applicable to all types of nonlinearities in resonant structures. NMPPF controller consists of a resonant second-order nonlinear compensator, which is enhanced by a lossy integrating compensator. The two compensators create a combination of exponential and periodic control inputs, which needs innovative time scaling for using the Method of Multiple Scales to obtain the analytical solution of the closed-loop system. The results of the analytical solution for the closed-loop NMPPF controller are presented and compared with the result of the conventional PPF controller. Effects of the control parameters on the system response are comprehensively studied by parameter variations. The approximate solution is then verified using numerical simulations. According to the results, the NMPPF controller provides a higher level of suppression in the overall frequency domain, as the peak amplitude at the neighborhood frequencies of the primary mode is reduced by 44 %, compared to the PPF method. The tunable control parameters also give more flexibility to create the expected type of system response.
41 citations
TL;DR: In this paper, an analytical model for vibration analysis of a thin orthotropic and general functionally graded rectangular plate containing an internal crack located at the center is presented, where the continuous line crack is parallel to one of the edges of the plate.
40 citations
TL;DR: In this paper, an analytical model is presented for vibration analysis of a thin orthotropic plate containing a partial crack located at the centre of the plate. And the effect of varying elasticity ratio on fundamental frequency of cracked plate is also established.
Abstract: An analytical model is presented for vibration analysis of a thin orthotropic plate containing a partial crack located at the centre of the plate. The continuous line crack is parallel to one of the edges of the plate. The equation of motion of an orthotropic plate is derived using the equilibrium principle based on Classical Plate Theory. The crack terms are formulated using the Line Spring Model. Using Berger's formulation for the in-plane forces and Galerkin's method for solution, the derived equation involving a cubic nonlinear term is converted into the Duffing equation. The effect of nonlinearity is established by deriving the frequency response equation for the cracked plate using the method of multiple scales. The influence of crack length and boundary conditions on the fundamental frequency of square and rectangular plate is demonstrated for three boundary conditions. It is found that the vibration characteristics are affected by the presence of crack. Further, it is deduced that the presence of a crack across the fibres affects the fundamental frequency more as compared to a crack along the fibres. The effect of varying elasticity ratio on fundamental frequency of cracked plate is also established.
37 citations
TL;DR: In this article, the static and dynamic characteristics of a doubly clamped micro-beam-based resonator driven by two electrodes were investigated, which is essentially nonlinear due to its cubic stiffness and electrostatic force.
Abstract: This paper investigates the static and dynamic characteristics of a doubly clamped micro-beam-based resonator driven by two electrodes. The governing equation of motion is introduced here, which is essentially nonlinear due to its cubic stiffness and electrostatic force. In order to have a deep insight into the system, static bifurcation analysis of the Hamiltonian system is first carried out to obtain the bifurcation sets and phase portraits. Static and dynamic pull-in phenomena are distinguished from the viewpoint of energy. What follows the method of multiple scales is applied to determine the response and stability of the system for small vibration amplitude and AC voltage. Two important working conditions, where the origin of the system is a stable center or an unstable saddle point, are considered, respectively, for nonlinear dynamic analysis. Results show that the resonator can exhibit hardening-type or softening-type behavior in the neighborhood of different equilibrium positions. Besides, an attractive linear-like state may also exist under certain system parameters if the resonator vibrates around its stable origin. Whereafter, the corresponding parameter relationships are deduced and then numerically verified. Moreover, the variation of the equivalent natural frequency is analyzed as well. It is found that the later working condition may increase the equivalent natural frequency of the resonator. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.
36 citations
TL;DR: In this paper, a coupled longitudinal-transverse dynamic model due to geometrical nonlinearity is established by Hamilton's principle and then is discretized by Galerkin method.
TL;DR: In this article, the non-linear vibrations of fixed-fixed tensioned pipe with vanishing flexural stiffness and conveying fluid with constant velocity are considered and the fractional calculus approach is introduced in the constitutive relationship of viscoelastic material.
Abstract: In this study, the non-linear vibrations of fixed–fixed tensioned pipe with vanishing flexural stiffness and conveying fluid with constant velocity are considered. The fractional calculus approach is introduced in the constitutive relationship of viscoelastic material. The pipe is on fixed support and the immovable end conditions result in the extension of the pipe during vibration and hence are introduced further nonlinear terms to the equation of motion. Analytical solutions are obtained by using the method of multiple scales. Nonlinear frequencies versus the amplitude of deflection are calculated. For frequencies close to one times the natural frequency, stability of steady-state solutions is analysed.
TL;DR: In this article, linear vibrations of an axially moving Euler-Bernoulli beam under non-ideal support conditions have been investigated; the main difference from the other studies is that the nonideal clamped support allow minimal rotations and non-IDEAL simple support carry moment in minimal orders.
Abstract: In this study, linear vibrations of an axially moving beam under non-ideal support conditions have been investigated. The main difference of this study from the other studies; the non-ideal clamped support allow minimal rotations and non-ideal simple support carry moment in minimal orders. Axially moving Euler-Bernoulli beam has simple and clamped support conditions that are discussed as combination of ideal and non-ideal boundary with weighting factor (k). Equations of the motion and boundary conditions have been obtained using Hamilton\'s Principle. Method of Multiple Scales, a perturbation technique, has been employed for solving the linear equations of motion. Linear equations of motion are solved and effects of different parameters on natural frequencies are investigated.
TL;DR: In this paper, the dynamic properties of a simply supported at four edges truss core sandwich plate with tetrahedral core are investigated by using the Hamilton's principle, the von Karman type equation for geometric nonlinearity and a Zig-Zag theory.
TL;DR: In this paper, the stability of a rotor system, which is comprised of a simply supported nonlinear spinning shaft with multi-rigid disk, near to the major critical speeds is investigated.
Abstract: In this study the stability of a rotor system, which is comprised of a simply supported nonlinear spinning shaft with multi-rigid disk, near to the major critical speeds is investigated. The nonlinearity is due to the stretching and large amplitude. The influence of rotary inertia and gyroscopic effects are included, however, shear deformation is ignored. To analyze the nonlinear equations of motion, the method of multiple scales is applied to the ordinary differential equations of motion. The influences of different parameter such as number of disks, disk mass moment of inertia, rotational speed, external damping, and position of disks on the forward and backward linear frequencies, steady state response, stability and bifurcations of the rotor system are investigated. It is seen that in the higher rotational speeds, the backward frequency is increasing with an increase of number of disks, and in the lower rotational speeds, the backward frequency is decreasing with an increase of number of disks. By increasing number of disks, bifurcations occur in the lower speeds therefore, the instability occurrence for large number of disk is at speeds lower than that regarding to the low number of disks. By an increase of disk mass moment of inertia, the amplitude and the hardening effect decrease.
TL;DR: Nonlinear vibrations of a cylindrical shell embedded in a fractional derivative viscoelastic medium and subjected to the different conditions of the internal resonance are investigated and the energy exchange could occur between two or three subsystems at a time.
TL;DR: In this paper, the nonlinear response and nonlinear power distribution of parametrically excited space cable-beam structures under the effects of simultaneous internal and external resonances were analyzed.
TL;DR: In this article, the Hamiltonian and corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed and the approximate analytical solutions of the coupled differential equations involving the noncommuting field operators are solved up to the second order in the an-harmonic coupling.
Abstract: The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The approximate analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second order in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second order variances and hence the squeezing in pure and in mixed modes. The higher order quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher order quadrature and higher ordered amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.
TL;DR: In this paper, the nonlinear response of a beam on elastic foundation subjected to the harmonic excitation is investigated, and the effect of soil-structure interaction on the primary resonance of the beam is analyzed.
Abstract: The nonlinear response of a beam on elastic foundation subjected to the harmonic excitation is investigated, and the effect of soil–structure interaction on the primary resonance of the beam is analyzed. Considering the inextensional condition, the nonlinear equation of motion of a beam on elastic foundation is proposed via the Hamilton principle. Then, the method of multiple scales is used to obtain the frequency–response equation and second-order approximate solution of the dynamic response of the beam on elastic foundation. Finally, numerical results are presented to investigate the effects of the second-order moment, foundation models, Winkler parameter and excitation amplitude on the primary resonance of the beam on elastic foundation.
TL;DR: This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators using geometrically exact mechanical formulations and obtains a nonlinear model that governs the transverse and longitudinal dynamics of multilayer microbeams, and takes into account rotary inertia effects.
Abstract: This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.
TL;DR: In this paper, the classical Bernoulli−Euler model is employed for analysis of the plane wave propagation in an infinitely long elastic layer with periodically varying thickness, and the Floquet theory is applied to derive asymptotic formulas defining location and broadness of frequency stop bands for several corrugation shapes with different levels of discontinuity.
TL;DR: In this article, the principal resonance response of a stochastically driven elastic impact (EI) system with time-delayed cubic velocity feedback is investigated, and a design criterion is proposed to suppress the jump phenomenon, which is induced by the saddle-node bifurcation.
Abstract: In this paper, the principal resonance response of a stochastically driven elastic impact (EI) system with time-delayed cubic velocity feedback is investigated. Firstly, based on the method of multiple scales, the steady-state response and its dynamic stability are analyzed in deterministic and stochastic cases, respectively. It is shown that for the case of the multi-valued response with the frequency island phenomenon, only the smallest amplitude of the steady-state response is stable under a certain time delay, which is different from the case of the traditional frequency response. Then, a design criterion is proposed to suppress the jump phenomenon, which is induced by the saddle-node bifurcation. The effects of the feedback parameters on the steady-state responses, as well as the size, shape, and location of stability regions are studied. Results show that the system responses and the stability boundaries are highly dependent on these parameters. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.
01 Jan 2015
TL;DR: In this article, a nonlinear transverse vibration of a beam moving with a harmonically fluctuating velocity and subjected to parametric excitation at a frequency close to twice the natural frequency in presence of internal resonance is analyzed using the method of multiple scales.
Abstract: The present work deals with nonlinear transverse vibration of a beam moving with a harmonically fluctuating velocity and subjected to parametric excitation at a frequency close to twice the natural frequency in presence of internal resonance. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the beam. The analysis is carried out using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first- order ordinary differential equations governing the modulation of amplitude and phase of the first two modes are analyzed numerically to obtain steady state and dynamic bevaviour along with stability as well as bifurcation of the travelling system. The system exhibits trivial, single mode and two mode solutions with pitchfork, saddle-node and Hopf bifurcations. The sensitivity of the system towards the variation in frequency and amplitude of fluctuating velocity component, variation in damping, flexural stiffness and initial points are studied.
TL;DR: Asymptotic homogenization via the method of multiple scales is considered for problems in which the microstructure comprises inclusions of one material embedded in a matrix formed from another as discussed by the authors.
Abstract: Asymptotic homogenisation via the method of multiple scales is considered for problems in which the microstructure comprises inclusions of one material embedded in a matrix formed from another. In particular, problems are considered in which the interface conditions include a global balance law in the form of an integral constraint; this may be zero net charge on the inclusion, for example. It is shown that for such problems care must be taken in determining the precise location of the interface; a naive approach leads to an incorrect homogenised model. The method is applied to the problems of perfectly dielectric inclusions in an insulator, and acoustic wave propagation through a bubbly fluid in which the gas density is taken to be negligible.
01 Jan 2015
TL;DR: In this article, a cylindrical shell embedded in a fractional derivative viscoelastic medium and subjected to the different conditions of the internal resonance of the order of ε, where ε is a small value, is investigated.
Abstract: Non-linear vibrations of a cylindrical shell embedded in a fractional derivative viscoelastic medium and subjected to the different conditions of the internal resonance of the order of \(\varepsilon \), where \(\varepsilon \) is a small value, are investigated. The displacement functions are determined in terms of eigenfunctions of linear vibrations. The procedure resulting in decoupling linear parts of equations is proposed with the further utilization of the method of multiple scales for solving nonlinear governing equations of motion, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales. The influence of viscosity on the energy exchange mechanism is analyzed. It is shown that each mode is characterized by its damping coefficient connected with the natural frequency by the exponential relationship with a negative fractional exponent. Comparison of the results obtained in this paper for the nonlinear shallow cylindrical shell in the cases of the internal resonance of the order of \(\varepsilon \) with those for a nonlinear plate, the motion of which is described also by three coupled nonlinear equations in terms of three displacements, reveals the fact that the shell equations could produce much more diversified variety of internal resonances, including combinational resonances of the additive and difference types, than the plate equations.
TL;DR: In this article, the dynamics and chaos control of the two-degree-of-freedom nonlinear electromechanical system, in which the magnetic field is modeled as being time-varying (periodic in fact), is investigated.
Abstract: The dynamics and chaos control of the two-degree-of-freedom nonlinear electromechanical system, in which the magnetic field is modeled as being time-varying (periodic in fact), will be investigated. The system is modeled by a coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the mechanical and electrical components of the considered model. The steady-state response and stability of the solutions for various parameters are studied numerically, using the frequency response function and the time-series solution. Effects of system parameters including external forces and time-varying magnetic field on the solutions of nonlinear equations are investigated numerically. The amplitudes have maximum peaks at the simultaneous primary resonance case $$({\varOmega } = \omega _{1}=\omega _{2}=\omega =1.0)$$
and hence is considered as the worst resonance case of the system behavior. It can be seen that the best control law is the negative linear velocity feedback. Comparison between numerical solution and perturbation solution is obtained.
14 Dec 2015
TL;DR: In order to get an approximate analytical solution of coupled van der Pol's (CVDP) oscillator system, a well-known perturbation technique, Method of Multiple Scales (MMS), is used.
Abstract: Nature being so nonlinear provides substantial reasoning to study the nonlinear dynamics inherited by the systems Almost all of the systems that exist are nonlinear and analysis of such type of systems is necessary to understand a certain phenomenon This paper presents a qualitative analysis of a coupled van der Pol's oscillator with the study of limit cycles In order to get an approximate analytical solution of coupled van der Pol's (CVDP) oscillator system, a well-known perturbation technique, Method of Multiple Scales (MMS), is used Stability and periodic construction of the van der Pol's system is examined by means of Floquet Multipliers and Shooting Method Eigenvalues of Monodromy Matrix, a by-product of shooting method, are analysed to distinguish between hyperbolic and non-hyperbolic solutions of the system
TL;DR: Nonlinear acoustic wave propagation is considered in an infinite orthotropic thin circular cylindrical waveguide, where at some frequencies the amplitude modulation is governed by the Nonlinear Schrödinger Equation (NLSE), and at other specific frequencies, interactions occur between the primary wave and its higher harmonics.
Abstract: Nonlinear acoustic wave propagation is considered in an infinite orthotropic thin circular cylindrical waveguide. The modes are non-planar having small but finite amplitude. The fluid is assumed to be ideal and inviscid with no mean flow. The cylindrical waveguide is modeled using the Donnell's nonlinear theory for thin cylindrical shells. The approximate solutions for the acoustic velocity potential are found using the method of multiple scales (MMS) in space and time. The calculations are presented up to the third order of the small parameter. It is found that at some frequencies the amplitude modulation is governed by the Nonlinear Schrodinger Equation (NLSE). The first objective is to study the nonlinear term in the NLSE, as the sign of the nonlinear term determines the stability of the amplitude modulation. On the other hand, at other specific frequencies, interactions occur between the primary wave and its higher harmonics. Here, the objective is to identify the frequencies of the higher harmonic interactions. Lastly, the linear terms in the NLSE obtained using the MMS calculations are validated. All three objectives are met using an asymptotic analysis of the dispersion equation.
Journal Article•
TL;DR: In this article, the primary resonance of a simply supported rotating composite shaft with geometrical nonlineary is studied, where a variational-asymptotical method applied to anisotropic thin-walled closed-cross-sectional beams is used to describe the displacement and strain fields of the composite shafts.
Abstract: The primary resonance of a simply supported rotating composite shafts with geometrical nonlineary is studied. The composite shaft is modeled as a thin-walled Euler-Bernoulli beam. A variational-asymptotical method (VAM) applied to anisotropic thin-walled closed-cross-sectional beams is used to describe the displacement and strain fields of the composite shafts. The geometrical nonlineary is considered in the relationships of strain and displacement of the shaft. The nonlinear extensional-bending-torsional equations of motion for the composite shaft are derived by using the Hamilton principle. In order to emphatically study nonlinear transverse bending vibration, the effects of extensional and torsional deformations are ignored. By means of the method of multiple scales the approximation solution of primary resonance of transverse bending vibration is obtained. The Galerkin method is employed to reduce the governing equations to the ordinary differential equations. By using fourth-order Runge-Kutta method the time histories, phase diagrams and power spectrums are plotted. The study shows the effect of the external damping, ply angle, eccentricity, ratios of length over radius, ratios of radius over thickness and rotating speed on nonlinear dynamic behavior of the shaft. Specifically, the numerical simulation results show that the shaft exhibits the complex dynamic behavior including periodic, quasi-periodic and chaotic motion.
TL;DR: In this paper, a proportional-derivative controller is proposed to reduce the horizontal vibration of a magnetically levitated system having quadratic and cubic nonlinearities to primary and parametric excitations.
Abstract: In this paper, a proportional-derivative controller is proposed to reduce the horizontal vibration of a magnetically levitated system having quadratic and cubic nonlinearities to primary and parametric excitations. A second order approximate solution is sought using the method of multiple scales perturbation technique to clarify the nonlinear behavior for both amplitude and phase of the system. The effect of feedback signal gain is studied to indicate the optimum values for best performance. Validation curves are included to compare the approximate solution and the numerical simulation. A comparison with previously published work is included.