Periodic Parametric Excitation Research Articles

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the ‘transmitted’ Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.

Parametric Excitation of a Single Railway Wheelset

This paper presents the results of analytical, numerical and experimental investigations of a single railway wheelset. Periodic parametric excitation is added to one of the simplest linear mechanical models. This extended model describes, for example, the geometric deviations often experienced on roller rigs. Above a certain critical speed, the stationary running of the wheelset loses its stability. To verify the analytical and numerical results for the critical speed, experiments were carried out on a simple roller rig having a large ratio of the radii of the roller and the railway wheel.

Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems

In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.

Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.

THE DYNAMICS OF A NON-LINEAR SYSTEM SIMULATING LIQUID SLOSHING IMPACT IN MOVING STRUCTURES

In this paper the non-linear modal interaction is examined between liquid hydrodynamic impact and an elastic support structure. The liquid impact is modelled based on a phenomenological concept by introducing a power non-linearity with a higher exponent. A special saw-tooth time transformation (STTT) technique is used analytically to describe the in-phase and out-of-phase strongly non-linear periodic regimes. Based on explicit forms of analytical solutions, all basic characteristics of the non-linear free and forced response regimes, such as the time history, the amplitude–frequency dependence and the non-linear parametric resonance curves, are estimated. The response behaviour reveals that a high frequency out-of-phase non-linear mode takes place with a relatively small tank amplitude, and is more stable than the in-phase oscillation mode under small perturbations. The in-phase mode has relatively large tank amplitudes and does not preserve its symmetry under periodic parametric excitation.

Global dynamics of parametrically excited nonlinear reversible systems with nonsemisimple 1:1 resonance

In this paper, we analytically investigate the global dynamics associated with the nonlinear reversible systems that exhibit Hopf bifurcation in the presence of one-to-one nonsemisimple internal resonance. The effect of periodic parametric excitations is examined on such systems near the principal subharmonic resonance in presence of dissipation. The nonlinear and nonautonomous system is simplified considerably by reducing it to the corresponding four-dimensional normal form. The normal form associated with the reversible systems is obtained as a special case from the general normal form equations obtained in [N. Sri Namachchivaya, M.M. Doyle, W.F. Langford and N. Evans, Normal form for generalized hopf bifurcation with non-semisimple 1:1 resonance, Z. Angew. Math. Phys. (ZAMP) 45 (1994) 312–335]. Under small perturbations arising from parametric excitations and nonreversible dissipation, two mechanisms are identified in such systems that may lead to chaotic dynamics. Explicit restrictions on the system parameters are obtained for both of these mechanisms which lead to this complex behavior. Finally, the results are demonstrated through a two-degree-of-freedom model of a thin rectangular beam vibrating under the action of a pulsating follower force.

Microparticle driven by parametric and random forces: Theory and experiment.

The confined motion of a charged microparticle within the Paul Trap (also known as the electrodynamic levitator trap) in an atmosphere near the standard temperature T and pressure ${\mathit{P}}_{\mathrm{atm}}$ is studied both theoretically and experimentally. The suggested theoretical model is based on the Mathieu differential equation with damping term and stochastic source. This equation describes the damped microparticle motion subjected to the combined periodic parametric and random external excitations. To solve the equation in an experimentally investigated regime of extremely strong damping and periodic excitations, the singular perturbation theory (WKB theory) is applied. In order to compare experimental data obtained in the long-time imaging limit with an analytical solution obtained for the autocorrelation function, the last is averaged by employing the Bogoliubov general averaging principle. This comparison is performed in terms of the standard deviation of the microparticle confined stochastic motion. It results almost in the perfect agreement between the analytical result and the data obtained experimentally in an entire region of the investigated experimental parameters. The only theoretical restrictions imposed on the model parameters are 1/\ensuremath{\alpha}\ensuremath{\ll}1 and 4\ensuremath{\beta}/${\mathrm{\ensuremath{\alpha}}}^{2}$\ensuremath{\ll}1 (where \ensuremath{\alpha} and \ensuremath{\beta} are the dimensionless drag and drive parameters). It is discovered both experimentally and theoretically that there is a minimum equal to [8kT/(m${\mathrm{\ensuremath{\omega}}}^{2}$)${]}^{1/2}$ in the standard deviation of the microparticle confined stochastic motion (m is the microparticle mass and \ensuremath{\omega} is the drive force frequency). The presence of this minimum, which takes place at \ensuremath{\beta}\ensuremath{\approxeq}1.518\ensuremath{\alpha}, reduces the thermal noise effects, providing unique opportunities for the spectroscopic studies. Comparison with the numerical simulation schemes developed in papers [Arnold, Folan, and Korn, J. Appl. Phys. 74, 4291 (1993); Blatt et al., Z. Phys. D 4, 121 (1986); Zerbe, Jung, and Hanggi, Phys. Rev. E 49, 3626 (1994)] is discussed.

Parametrically driven microparticle in the presence of a stationary zero-mean stochastic source: Model for thermal equilibrium in the Paul trap.

An analytical approach is developed to consider confined motion of a charged microparticle within the Paul trap (an electrodynamic levitator trap) in an atmosphere near the standard temperature and pressure. The suggested approach is based on a second-order linear stochastic differential equation which describes dampled microparticle motion subjected to the combined periodic parametric and random external excitations. To solve this equation a new ansatz is developed. This ansatz is a generalization of the Bogoliubov-Krylov decomposition technique, which is usually used to reduce the order of a differential equation. The solution is obtained in the long time imaging limit by applying the Bogoliubov general averaging principle. In spite of the second-order form of the initial stochastic differential equation, the microparticle motion can be understood as a one-dimensional Markov process. Comparison in the long time imaging limit of the calculated data obtained from the analytically derived expression for the standard deviation of confined microparticle stochastic motion with the experimentally obtained data demonstrates asymptotic agreement for regions where the dimensionless parameter \ensuremath{\kappa} is much less than 1 (\ensuremath{\kappa}\ensuremath{\le}0.005). Simple extremum analysis of the expression obtained for the standard deviation reveals that for the particular case of a large drag parameter \ensuremath{\alpha} (\ensuremath{\alpha}\ensuremath{\gg}8 \ensuremath{\surd}12 ) there is a minimum in the standard deviation which is only \ensuremath{\alpha} dependent.

Chaotic behavior of a parametrically excited nonlinear mechanical system

The chaotic dynamics of a single-degree-of-freedom nonlinear mechanical system under periodic parametric excitation is investigated. Besides the well known type-I and type-III intermittent transitions to chaos we give numerical evidence that the system can follow an alternative route to chaos via intermittency from an equilibrium state to a chaotic one, which was not found in the previous simulations of the dynamics of the system.

Experimental approximate solutions of nonlinear discrete oscillation with parametric excitation

AbstractIn the analysis of a discrete nonlinear system, one of the important problems is to derive a steady solution for the nonlinear difference equation characterizing the system. However, the study for the solution method of the difference system has not been sufficient, and examples of the methods being applied to the parametric oscillation are few. This paper presents a method which derives the steady solution for the quasi‐linear difference equation containing a periodic parametric excitation in its weakly nonlinear function with small parameter e. In this method the harmonic balance and the averaging methods used in the solution of the continuous nonlinear differential equation are applied to the discrete system. Since the nonlinear term in the equation is bounded by ε the discrete waveform in the steady state is almost sinusoidal. This paper presents numerical examples for the solutions by the two methods, assuming the solution as the discrete oscillation corresponding to the sinusoid of the continuous system. As a result, the same solution is obtained by either method. Comparing the result with the output waveform obtained directly by the numerical calculation for the fundamental equation, the validity of the approximate solution by the proposed method was verified.

Periodically Perturbed Hopf Bifurcation

In this paper the influence of small periodic perturbations on systems exhibiting Hopf bifurcation is studied in detail. In two-dimensional systems that ordinarily exhibit Hopf bifurcation, the addition of small periodic parametric excitation gives rise to interesting “secondary phenomena.” Explicit results for various primary and secondary bifurcations, along with their stabilities, are obtained. In this work, the ideas related to method of averaging, Poincare–Birkhoff normal forms, and center manifold theorem are used appropriately at different stages. It is found that various results obtained using these techniques agree in their common regions of validity.

Discontinuous and impulsive excitation

In this paper, we study the solution of differential equation with Dirac function and Heaviside function, arising from discontinuous and impulsive excitation. Firstly, according to the theory of differential equation, we suggest, then we derive the equation of x1(t) and x2(t) by terms of property of distribution, and by solving x1(t) and x2(t) we obtain x(t); finally, we make a thorough investigation about periodic impulsive parametric excitation.

On the study of random oscillations in non-autonomous mechanical systems using the fokker-planck-kolmogorov equations

A method of integrating the Fokker-Plank-Kolmogorov equations (FPKE) used in the theory of random oscillations /1–4/ is proposed. The Duffing equation is first studied as an example. The method is then used, together with the method of averaging, to study random oscillations of non-autonomous mechanical systems with one degree of freedom when the eigenfrequency varies in a random manner. The Van-der-Pol equation is considered for the case of a randomly varying eigenfrequency and periodic parametric excitation. When the function sought is replaced, the FPKE transform into another equation whose trivial solutions have the corresponding particular solutions of the FPKE. The condition of integrability of the FPKE is obtained as the direct consequence of the change in question.

A global analysis of a non-linear system under parametric excitation

A strongly non-linear mechanical system consisting of a rigid damped bar subjected to a periodic parametric excitation is treated here in an exact manner. The emphasis is on the global behavior of this system which is carried out by using the point mapping and the cell-to-cell mapping methods. Even though the mechanical system is a simple one, yet it has a very complex global behavior. It is shown here that the newly developed theory of cell-to-cell mappings offers a tremendous advantage in obtaining the global domains of attraction of strongly non-linear dynamical systems.

Steady-state response of a non-linear system under impulsive periodic parametric excitation

The non-linear problem of an elastically restrained and damped bar subjected to a periodic impact load is studied in this paper. The possible steady-state response of such a system is examined in a systematical manner. The approach used is to formulate the problem first in terms of non-linear difference equations and then to investigate the periodic solutions of these equations and their stability character. In this manner the problem is thus linked to the idea of diffeomorphisms in the theory of differentiable dynamics. It is believed that the application of this approach given in this paper is a novel and effective one. For the particular problem investigated, many important non-linear features, such as limit cycle response and bifurcation phenomenon, can be studied in a rather complete fashion. The exact treatment given in this paper and the results thus obtained could provide a useful guide to studies of other much more complex problems of similar nature.

Response of a non-linearly damped oscillator to combined periodic parametric and random external excitation

An analytical solution, using the Fokker-Planck-Kolmogorov equation, is obtained for the problem of response of a non-linearly damped oscillator to combined periodic parametric and random external excitation. The solution yields first-order probability densities of amplitude and phase. These expressions are employed to distinguish between oscillations excited by external and parametric periodic forces in the presence of additional broadband random external excitation. Through decoupling of fast and slow motions an approximate expression is obtained for expected value of time to phase “switch”.

Existence de résonances paramétriques dans des systèmes de transformation d'intérêt biologique

The study of systems exhibiting a band-pass function is completed for systems whose parameters are time-dependent. In the case of periodic parametric excitations, it is demonstrated that some systems can get into “resonance” for a particular frequency. By studying this problem, a new and probably fruitful approach of some rythmic behaviors can be made.

Applications of the Theory of Impulsive Parametric Excitation and New Treatments of General Parametric Excitation Problems

In this paper the stability theory of impulsive parametric excitation developed in [1] is first applied to three mechanical systems. Explicit and exact stability conditions are easily found and some typical stability charts are presented. Also presented in the paper is the use of this theory and a parallel theory involving step functions as approximate methods for treating periodic parametric excitations of more general nature. Exploratory studies along this line have led us to believe that these approximate methods have promise to be very powerful and practical tools for dealing with the stability of general high-order periodic systems.

Impulsive Parametric Excitation: Theory

Given in this paper is the development of a theory for dynamical systems subjected to periodic impulsive parametric excitations. By periodic impulsive parametric excitation we mean those excitations representable by periodic coefficients which consist of sequences of Dirac delta functions. It turns out that for this class of periodic systems the stability analysis can be carried out in a remarkably simple and general manner without approximation. In the paper, after giving the general theory, many special cases are examined. In many instances simple and closed-form analytic stability criteria can be easily established.