Chaotic dynamics of a whirling pendulum

A physical system is considered consisting of a rigid frame which is free to rotate about a vertical axis and to which is attached a planar simple pendulum. This system has “one and a half” degrees of freedom due to the fact that the frame and pendulum may freely rotate about the vertical axis, i.e., conservation of angular momentum holds for the “ideal”, or unperturbed, system. Using a Hamiltonian formulation we reduce the unperturbed equations of motion to a conservative planar system in which the constant angular momentum plays the role of a parameter. This system is shown to possess one or two sets of homoclinic motions depending on the level of the angular momentum. When this system is perturbed by external excitations and dissipative forces these homoclinic motions can break into homoclinic tangles providing the conditions for chaotic motions of the horseshoe type to exist. The criteria for this to occur can be formulated using a variation of Melnikov's method developed for slowly varying oscillators [1, 2]. For the present problem, the angular momentum becomes a slowly varying parameter upon addition of the disturbances. These ideas are used to rigorously prove the existence of chaotic motions for this system and to compute, to first order, global bifurcation parameter conditions. Since two types of homoclinic motions can occur, two different chaotic modes of motion can result and physical interpretations of these motions are given. In addition, a limiting case is considered in which the system becomes a single degree of freedom oscillator with parametric excitation.     

Nonlinear vibrations of clamped-free circular cylindrical shells

Only experimental studies are available on large-amplitude vibrations of clamped-free shells. In the present study, large-amplitude nonlinear vibrations of clamped-free circular cylindrical shell are numerically investigated for the first time. Shells with perfect and imperfect shape are studied. The Sanders-Koiter nonlinear shell theory is used to calculate the elastic strain energy. Shell displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series, i.e. harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. All boundary conditions are satisfied. The system is discretized by using natural modes of the shell and Lagrange equations by an energy approach, retaining damping through Rayleigh's dissipation function. Different expansions involving from 18 to 52 generalized coordinates are used to study the convergence of the solution. The nonlinear equations of motion are numerically studied by using arclength continuation method and bifurcation analysis. Numerical responses to harmonic radial excitation in the spectral neighborhood of the lowest natural frequency are compared with experimental results available in literature. The effect of geometric imperfections and excitation amplitude are numerically investigated and fully explained.     

Instabilities in the main parametric resonance area of a mechanical system with a pendulum

Vibrations of an autoparametric system, composed of a nonlinear mechanical oscillator with an attached damped pendulum, around the principal resonance region, are investigated in this paper. Approximate analytical solutions of the model are determined on the basis of the Harmonic Balance Method (HBM). Correctness of the analytical results is verified by numerical simulations and experimental tests performed on an especially prepared experimental rig. The influence of all essential parameters such as damping, excitation amplitude and frequency, nonlinear stiffness of the spring, on the localisation of the instability region and the system dynamics is presented in the work. Regions of regular system oscillations, chaotic motions, and full rotation of the pendulum are confirmed experimentally.     

参数激励与晶体摆动场辐射的稳定性

寻找新光源, 特别是短波长相干光源备受关注. 本文讨论了晶体摆动场辐射作为短波长激光的可能性和必须满足的基本条件; 指出了至今尚未获得可利用的短波长激光可能不只是技术原因, 而且还有物理原因. 利用参数激励方法对这个问题进行了分析. 在经典力学框架内和偶极近似下, 引入正弦平方势, 把粒子运动方程化为具有阻尼项和参数激励项的摆方程. 利用Melnikov方法讨论了系统的稳定性, 并对系统的临界条件进行了分析. 结果表明: 系统的稳定性与其参数有关, 只需适当调节这些参数, 系统的稳定性就可以原则上得到保证. 关键词： 晶体摆动场辐射 沟道辐射 参数激励 稳定性     

复摆运动状态的研究

从复摆的运动方程出发,利用计算机对其各种运动情况进行模拟,研究复摆从周期运动转化到混沌运动的过程,深化复摆实验内容,拓宽复摆实验的研究空间．     

Dynamics of a horizontal pendulum driven by high-frequency rocking

This paper investigates the dynamics of a horizontal pendulum subjected to high frequency rocking. The method of direct partition of motion is applied to the governing equation to separate the fast and slow dynamics. It is shown that two stable equilibria may coexist for certain parameter values. It is also shown that the high frequency excitation can stabilize an unstable equilibrium for a horizontal pendulum driven by a rocking motion. The aforementioned theoretical results show good agreement with numerical investigations. A series of experimental tests were also performed to corroborate the bifurcation threshold where forcing parameters can cause a change in stability.     

Experiments on periodic and chaotic motions of a parametrically forced pendulum

An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic route to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.     

Frequency response of sloshing in an annular cylindrical tank subjected to pitching excitation

Frequency responses of stable planar and rotary motions in a partially filled annular cylindrical tank, subjected to a pitching excitation at a frequency in the neighborhood of the lowest resonant frequency, are investigated. The nonlinearity of the liquid surface oscillation and the nonlinear coupling between the dominant modes and other modes (e.g., an axisymmetric mode) are considered in the response analysis of the sloshing motion. The basic equations of the liquid motion are derived by using the variational principle and the nonlinear equations of motion of the liquid surface displacement are formulated. The characteristics of the liquid motion in an annular cylindrical tank are discussed. The equations governing the amplitude of the stable planar and rotary liquid motions are derived and the stability of each motion is analyzed. An experiment was carried out using a model tank. It is shown that the nonlinear characteristic of the liquid motion in an annular cylindrical tank is more complicated than that in a circular cylindrical tank. Furthermore, it is shown that the nonlinear analysis is important for estimating the sloshing responses.     

Free induction decay with radiation damping II. Possible limitation of NMR thermometers at very low temperatures

Solution of Bloch equations with a term including the radiation damping for the pulse excitation of a two-level spin system shows, that the shape of the envelope of the free induction decay can be, under certain conditions, temperature dependent. This result is quantitatively exact for spin systems obeying these equations. Qualitatively, we show that this fact can give a limitation to the widespread use of NMR thermometers at very low temperatures (in the region of and below 1 [mK]).     

Magneto-elastic dynamics and bifurcation of rotating annular plate

In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton's principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincare′ maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions,and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos.     

Occurrence of stable periodic modes in a pendulum with cubic damping

Dynamical systems with nonlinear damping show interesting behavior in the periodic and chaotic phases. The Froude pendulum with cubical and linear damping is a paradigm for such a system. In this work the driven Froude pendulum is studied by the harmonic balancing method; the resulting nonlinear response curves are studied further for resonance and stability of symmetric oscillations with relatively low damping. The stability analysis is carried out by transforming the system of equations to the linear Mathieu equation.     

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh's dissipation function. Four different non-linear thin shell theories, namely Donnell's, Sanders-Koiter, Flügge-Lur’e-Byrne and Novozhilov's theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnell's non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.     

Extended Kalman filter for modal identification of structures equipped with a pendulum tuned mass damper

Pendulum tuned mass dampers (PTMDs) have been employed in several full-scale applications to attenuate excessive structural motions, which are mostly due to wind. Conducting periodic condition assessments of the devices to ascertain their health is necessary to ensure their continued optimal performance, which involves identifying the modal parameters of the underlying (bare) structure to which they are tuned to. Such an identification is also necessary for the design of control systems such as adaptive tuned mass dampers. Existing methods of arresting the motion of the damper to estimate the modal properties are expensive, time-consuming, and not suitable for online tuning. Instead, in this paper, parameter estimation using the Extended Kalman Filter (EKF) is proposed to undertake this task. The central task accomplished here is to estimate the dynamic characteristics of the bare structure (structure without the PTMD) from response measurements of the coupled main structure and PTMD system. The proposed methodology relies on ambient acceleration measurements of TMD-attenuated responses to estimate the bare structural modal frequencies, damping, and mode shapes, which can then be used either for condition assessment or for control. The application of EKF to modal parameter estimation is not new. However, a methodology to address the problem in wind engineering arising from stochastic disturbances present in both the measurement and state equations, and unknown process and noise covariances arising due to ambient excitations, is presented for the first time. Extensively studied for synthetic data, these two challenges have limited thus far the application of Kalman filtering to practical wind engineering parameter estimation problems using experimentally obtained measurements. In this paper, a detailed methodology is presented to address these challenges by using a modified form of the standard EKF equations, together with an algorithm to estimate the unknown disturbance and measurement noise covariances. Numerical simulations and an experimental study are both presented. Results demonstrate that the method proposed provides reliable estimates for the modal parameters required to perform condition assessment and control tasks for pendulum tuned mass dampers.     

Possible Resonator Configurations for the Spherical Gravitational Wave Antenna

We solve algebraically the equations of motion for a spherical antenna coupled to an arbitrary number of small resonators, free to move radially, and investigate the conditions under which damping forces can be neglected in the system. We show that in order that the antenna's modes be decoupled a preferred distribution of the resonators on its surface should be used. We find that either 5, 6, 10 or 16 resonators can be used as long as they are conveniently positioned on the antenna's surface. We calculate and analyse the frequency shift and the signal-to-noise ratio of the coupled system for the various distributions studied.     

Time-dependent variance and covariance of responses of structures to non-stationary random excitations

In this paper techniques for the analysis of non-stationary random responses of linear structures, discretized by the finite element method so that they can be analyzed as multi-degree of freedom systems, subjected to non-stationary random excitation are developed. The non-stationary random excitation is represented as a product of (a) an exponentially decaying function and a white noise process, and (b) a modulating function in the form of an exponential envelope and a white noise process. Closed form expressions for the time-dependent variance and covariance of response of structures are presented. Application of these expressions is made for the analysis of non-stationary random responses of a physical model of a class of mast antenna structures subjected to base excitation. It is concluded that (a) the coupling terms do have a definite influence on the response; the magnitude of the influence is proportional to the amount of damping in the structure and proximity of the modes excited; (b) the non-stationary random excitations considered are general in that the modulating functions are not necessarily identical, and therefore the influence of various modulating functions of the excitations applied to different locations of the structure on responses can be examined quantivatively; and (c) for a given damping parameter the magnitudes of the modulating function parameters cannot be chosen arbitrarily though the shapes of normalized modulating functions can be selected to best fit the excitation realizations.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Characteristics of an active feedback system for the control of plate vibrations

In the interests of improving airborne insulation of panels and of controlling room reverberation, a technique is studied for establishing control of the transverse vibrations of a thin plate by the application of active energy feedback. A localized point control force is derived from the sensed motion of some point on the plate surface. The superposition principle is applied in the form of a mobility analysis which shows that open loop gain conditions cannot result in a specific motion, including that of complete damping, of any arbitrary point on the plate surface but can be effective for particular points and for control of resonant modal motions under conditions of light damping. With velocity sensing, the characteristics for stable operation under the convenient condition of constant gain depend on maintenance of like symmetry, in the sense of an identity of velocity magnitude and sign, in the relative motion of sensing and control-force points. Bandwidth limitations are avoidable only by closure of the loop between these points. Two varieties of control force generator are involved: namely, where the generator mass is rigidly mounted and again where a spring mounting on the plate provides a self-supporting role.This is the first of two companion papers on active control of plate vibrations. Systems in which an array of multiple control units is used will be described in the second paper.     

Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt

This paper investigates the multi-pulse global bifurcations and chaotic dynamics for the nonlinear, non-planar oscillations of the parametrically excited viscoelastic moving belt using an extended Melnikov method in the resonant case. Using the Kelvin-type viscoelastic constitutive law and Hamilton's principle, the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:1 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics. The paper demonstrates how to employ the extended Melnikov method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear non-planar oscillations of the viscoelastic moving belt, the Shilnikov-type multi-pulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that the chaos for the Smale horseshoe sense in motion exists.     

Transient vibration phenomena in deep mine hoisting cables. Part 2: Numerical simulation of the dynamic response

A simulation model is presented which investigates the dynamic response of a deep mine hoisting cable system during a winding cycle. The response, namely the lateral motions of the catenary cable and the longitudinal motion of the vertical rope with conveyance is observed on the fast time scale, and the slow time scale is introduced to monitor the variation of slowly varying parameters of the system. The cable equivalent proportional damping parameters, and periodic excitation functions resulting from the cross-over cable motion on the winder drum are identified. Subsequently, the model is solved numerically using parameters of a double-drum multi-rope system. Since the system eigenvalues are widely spread and the problem is of stiff nature, the numerical simulation is conducted using a stiff solver. The results of the simulation demonstrate various transient non-linear resonance phenomena arising in the system during the wind. The nominal ascending cycle simulation results reveal adverse dynamic behaviour of the catenary largely due to the autoparametric interactions between the in- and out-of-plane modes. Principal parametric resonances of the lateral modes also occur, and conditions for autoparametric interactions between the lateral and longitudinal modes arise. Additionally, a transition through a number of primary longitudinal resonances takes place during the wind. The adverse dynamic motions in the system promote large oscillations in the cable tension which must be considered significant with respect to fatigue of the cable. It is noted that a small change in the winding velocity may cause large changes in the dynamic response due to the resonance region shifts. Consequently, the resonance modal interactions can be avoided, to a large extent, if the winding velocity is increased to an appropriate level.