Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Nonlinear behavior of a rotor-AMB system under multi-parametric excitations

A rotor-active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic nonlinearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor-AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Runge-Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom

In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system. Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions. It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves. It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions. An erratum to this article is available at .     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

Active vibration control of a dynamical system via negative linear velocity feedback

In this paper, a negative velocity feedback is added to a dynamical system which is represented by second-order nonlinear differential equations having quadratic coupling, quadratic, and cubic nonlinearities. The system describes the vibration of the system subjected to multi-parametric excitation forces. The method of multiple scale perturbation technique is applied to obtain the response equation near the simultaneous internal and super-harmonic resonance case of this system. The stability to the system is investigated applying frequency response equations. The numerical solution and the effects of some parameters on the vibrating system are investigated and reported. The simulation results are achieved using MATLAB 7.0 program. A comparison is made with the available published work.     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

Nonlinear vibrations of a shell-shaped workpiece during high-speed milling process

This paper is focused on nonlinear dynamics of a shell-shaped workpiece during high speed milling. The shell-shaped workpiece is modeled as a double-curved cantilevered shell subjected to a cutting force with time delay effects. Equations of motion are derived by using the Hamilton principle based on the classical shell theory and von Karman strain-displacement relation. The resulting nonlinear partial differential equations are reduced to a two-degree-of-freedom nonlinear system by applying the Galerkin approach. The averaging method is used to obtain four-dimensional averaged equations for the case of foundational parametric resonance and 1:2 internal resonance. Using a numerical method, the dynamics of the cantilevered shell-shaped workpiece is studied under time-delay effects, parametric excitation, and forcing excitation. It is found that time-delay parameters have great impact on chaotic motion. With increasing amplitude of forcing and parametric excitations, the shell-shaped workpiece exhibits different dynamic behavior.     

Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities

Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.     

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.     

A refined asymptotic perturbation method for nonlinear dynamical systems

In this paper, a refined asymptotic perturbation method for general nonlinear dynamical systems is proposed for the first time. This method can be considered as an alternative means for the traditional multiple scales method. Moreover, it is easier to be understood and used to carry out higher-order perturbation analysis. In addition, three examples including the Duffing equation, a system with quadratic and cubic nonlinearities to a subharmonic excitation, as well as the coupled van der Pol oscillator with parametrical excitations are investigated to illustrate the validity and usefulness of the proposed technique. The analytical and numerical results show good agreement.     

Dynamic instability of inclined cables under combined wind flow and support motion

In this paper an inclined nearly taut stay, belonging to a cable-stayed bridge, is considered. It is subject to a prescribed motion at one end, caused by traveling vehicles, and embedded in a wind flow blowing simultaneously with rain. The cable is modeled as a non-planar, nonlinear, one-dimensional continuum, possessing torsional and flexural stiffness. The lower end of the cable is assumed to undergo a vertical sinusoidal motion of given amplitude and frequency. The wind flow is assumed uniform in space and constant in time, acting on the cable along which flows a rain rivulet. The imposed motion is responsible for both external and parametric excitations, while the wind flow produces aeroelastic instability. The relevant equations of motion are discretized via the Galerkin method, by taking one in-plane and one out-of-plane symmetric modes as trial functions. The two resulting second-order, non-homogeneous, time-periodic, ordinary differential equations are coupled and contain quadratic and cubic nonlinearities, both in the displacements and velocities. They are tackled by the Multiple Scale perturbation method, which leads to first-order amplitude-phase modulation equations, governing the slow dynamics of the cable. The wind speed, the amplitude of the support motion and the internal and external frequency detunings are set as control parameters. Numerical path-following techniques provide bifurcation diagrams as functions of the control parameters, able to highlight the interactions between in-plane and out-of-plane motions, as well as the effects of the simultaneous presence of the three sources of excitation.     

Modeling of planar nonshallow prestressed beams towards asymptotic solutions

Employing the geometrically exact approach, the governing equations of nonlinear planar motions around nonshallow prestressed equilibrium states of slender beams are derived. Internal kinematic constraints and approximations are introduced considering unshearable extensible and inextensible beams. The obtained approximate models, incorporating quadratic and cubic nonlinearities, are amenable to a perturbation treatment in view of asymptotic solutions. The different perturbation schemes for the two mechanical beam models are discussed.     

Vibration control by nonlocal feedback and jerk dynamics

Jerk dynamics is used for a new method for the suppression of self-excited vibrations in nonlinear oscillators. Two cases are considered, the van der Pol equation and nonlinear oscillator with quadratic and cubic nonlinearities. A nonlocal control force is introduced in such a way to obtain a third order nonlinear differential equation (jerk dynamics). Using the asymptotic perturbation method, two slow flow equations on the amplitude and phase of the response are obtained, and subsequently the performance of the control strategy is investigated. The feedback gains are connected with the stability and response of the system under control. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the self-excitations. Numerical simulation confirms the validity of the new method.     

Nonlinear modeling and control of flexible-link manipulators subjected to parametric excitation

This paper presents nonlinear dynamic modeling and control of flexible-link manipulators subjected to parametric excitation. The equations of motion are obtained using the Lagrangian-assumed modes method. Singular perturbation methodology is developed for the nonlinear time varying equations of motion to obtain a reduced-order set of equations. Control strategies, computed torque control and a composite control, based on the singular perturbation formulation developed, are utilized to reduce mechanical vibrations of the flexible-link and enable better tip positioning. Under the composite control technique, the effect of the value of perturbation parameter on the control signal is investigated. Numerical simulations supported by real-time experiments show that the singular-perturbation control methodology developed for the nonlinear time-varying system offers better system response over the computed torque control as the manipulator is commanded to follow a certain trajectory.     

Quasi-Periodic Oscillations,Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations

An analysis is given of the dynamic of a one-degree-of-freedom oscillator with quadratic and cubic nonlinearities subjected to parametric and external excitations having incommensurate frequencies. A new method is given for constructing an asymptotic expansion of the quasi-periodic solutions. The generalized averaging method is first applied to reduce the original quasi-periodically driven system to a periodically driven one. This method can be viewed as an adaptation to quasi-periodic systems of the technique developed by Bogolioubov and Mitropolsky for periodically driven ones. To approximate the periodic solutions of the reduced periodically driven system, corresponding to the quasi-periodic solution of the original one, multiple-scale perturbation is applied in a second step. These periodic solutions are obtained by determining the steady-state response of the resulting autonomous amplitude-phase differential system. To study the onset of the chaotic dynamic of the original system, the Melnikov method is applied to the reduced periodically driven one. We also investigate the possibility of achieving a suitable system for the control of chaos by introducing a third harmonic parametric component into the cubic term of the system.     

Vibration and stability of an aircraft tail under simultaneous primary-combined and internal resonance

This paper adds a negative velocity feedback to the dynamical system of twin-tail aircraft to suppress the vibration. The system is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to both multi-harmonic and multi-tuned excitations. The method of multiple time scale perturbation is adopted to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the proposed analytic solution near the simultaneous primary, combined and internal resonance is studied and its conditions are determined. The effect of different parameters on the steady state response of the vibrating system is studied and discussed by using frequency response equations. Some different resonance cases are investigated numerically     

Nonlinear dynamic analysis for coupled vehicle-bridge system with harmonic excitation

In this paper, a multi-degree-of-freedom lumped parameter coupled vehicle-bridge dynamic model is proposed considering the nonlinearities of suspension and tire stiffness/damping and the nonlinear foundation of bridge. In terms of modelling, the continuous expressions of the kinetic energy, potential energy and the dissipation function are constructed. The dynamic equations of the coupled vehicle-bridge system (CVBS) are derived and discretized using Galerkin’s scheme, which yield a set of second-order nonlinear ordinary differential equations with coupled terms. The numerical simulations are conducted by using the Newmark-β integration method to perform a parametric study of the effects on excitation amplitude, suspension stiffness and position relation. The bifurcation diagram, 3-D frequency spectrum and largest Lyapunov exponent are demonstrated in order to better understand the vibration properties and interaction between the vehicle and bridge with the key system parameters. It can be found that the nonlinear dynamic characteristics such as parametric resonance, jump phenomena, periodic, quasi-periodic and chaotic motions are strongly attributed to the interaction between vehicle and bridge. Significantly, under the combined internal and external excitations, the vibration amplitudes of the CVBS have a certain degree of dependence on the external excitation. Suspension stiffness could lead to complex dynamics such as the higher-order bifurcations increase and the chaotic regions broaden. The increasing of distance could effectively control the nonlinear vibration of CVBS. The application of the proposed nonlinear coupled vehicle-bridge model would bring higher computational accuracy and make it possible to design the vehicle and bridge simultaneously.     

Global bifurcations and chaos in a harmonically excited and undamped circular plate

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.