Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.     

Locating order-chaos-order transition in elastic pendulum

An undamped elastic pendulum being a nonintegrable Hamiltonian system always has some chaotic trajectories observable on choosing appropriate initial conditions. This is true even if the pendulum is in libration with small amplitude; in this situation, the pendulum may be seen as a nearly integrable system. Since the measure of the set of the local chaotic trajectories in the phase space may be very small, the trajectories are hard to locate. However, the emergence of widespread chaos when the elastic pendulum is at autoparametric resonance is well-documented. The transition from the local and the widespread chaos is typically established through the Chirikov overlap criterion that approximates the phase portrait around a resonance using a one degree-of-freedom pendulum Hamiltonian. We argue in this paper that the aforementioned transition in the elastic pendulum is due to interaction between two resonances of same kind and their coexistence can be analytically located using perturbation methods, like the method of averaging, whereas the technique of the pendulum Hamiltonian is inapplicable. Furthermore, in the course of validating the result numerically, we also showcase the order-chaos-order transition in the elastic pendulum using the fast Lyapunov indicator.

     

Chaos control in a pendulum system with excitations and phase shift

Melnikov methods are used for suppressing homoclinic and heteroclinic chaos of a pendulum system with a phase shift and excitations. This method is based on the addition of adjustable amplitude and phase-difference of parametric excitation. Theoretically, we give the criteria of suppression of homoclinic and heteroclinic chaos, respectively. Numerical simulations are given to illustrate the effect of the chaos control in this system. Moreover, we calculate the maximum Lyapunov exponents (LEs) in parameter plane, and study how to vary the maximum LE when the parametric frequency varies.     

Nonlinear oscillations of an elastic two-degrees-of-freedom pendulum

Nonlinear oscillations of the vertical plane swinging spring pendulum in the resonance case are studied (frequencies ratio regarding horizontal and vertical directions is equal to 1:2). Square and cubic terms of the Hamiltonian are taken into account. Novel normal form method, i.e., the so called invariant normalization is applied to solve the stated problem. Full system of integrals exhibits equations of the normal form, and solution for the pendulum coordinates is expressed via elementary functions. Frequencies of modes of oscillations are proportional to the first power of amplitude, and not to the second power as it is exhibited by one dimensional Duffing oscillator. Amplitudes of the modes are changed periodically, and energy from one mode is transited to energy of the second one, whereas the period of oscillations depends on the initial conditions. It is illustrated that asymptotic solution with small amplitudes approximates well numerical solution of the governing equations. In addition, an example of a periodic stable solution with constant amplitudes of the oscillation modes is given. Stability of this solution is proved.     

Crack growth in planar elastic fiber materials

Particularly attention is here given to crack growth in opening mode in fiber networks. Low- and high-density cellulose fiber materials are used in synchrotron X-ray microtomography tensile experiments to illustrate phenomena arising during crack growth. To capture the observed fundamental mechanisms, significantly different from classical continua, a mechanical model based on a strong nonlocal theory is applied in which an intrinsic length reflects a characteristic length of the microstructure. Nonlocal stress and strain tensor fields are estimated by analytical solutions on closed form to a modified inhomogeneous Helmholtz equation using LEFM crack-tip fields as source terms. Justified by experimental observations, physical requirements of finite stresses and strains at infinity and at the tip are applied to remove singularities. The near-tip nonlocal hoop stress field is used to estimate crack growth directions and sizes of fracture process zones. Experimental observations are shown to be qualitatively well in accordance with numerical predictions, which justifies the adopted approach.     

Compact waves on planar elastic rods

Planar Kirchhoff elastic rods with non-linear constitutive relations are shown to admit traveling wave solutions with compact support. The existence of planar compact waves is a general property of all non-linearly elastic intrinsically straight rods, while intrinsically curved rods do not exhibit this type of behavior.     

Rotary motion of the parametric and planar pendulum under stochastic wave excitation

In this paper a rotary motion of a pendulum subjected to a parametric and planar excitation of its pivot mimicking random nature of sea waves has been studied. The vertical motion of the sea surface has been modelled and simulated as a stochastic process, based on the Shinozuka approach and using the spectral representation of the sea state proposed by Pierson–Moskowitz model. It has been investigated how the number of wave frequency components used in the simulation can be reduced without the loss of accuracy and how the model relates to the real data. The generated stochastic wave has been used as an excitation to the pendulum system in numerical and experimental studies. For the first time, the rotary response of a pendulum under stochastic wave excitation has been studied. The rotational number has been used for statistical analysis of the results in the numerical and experimental studies. It has been demonstrated how the forcing arrangement affects the probability of rotation of the parametric pendulum.     

Estimation of parameters of various damping models in planar motion of a pendulum

Meccanica - In this work, planar free vibrations of a single physical pendulum are investigated both experimentally and numerically. The laboratory experiments are performed with pendula of...     

Second order analysis of elastic frames

Summary A second order theory for the analysis of elastic frames is presented. Its derivation, by successive simplifications from the theory of finite deformations, allows for the range of the theory to be clearly established.The physical meaning of the variables which are introduced will be self-evident.A simple way of evaluating the critical load is also presented.

---

Sommario Viene presentata una teoria del secondo ordine per l'analisi delle strutture intelaiate elastiche. La sua derivazione per successive approssimazioni dalla teoria delle deformazioni finite permette di definire chiaramente i limiti di applicabilità.Il significato fisico delle grandezze introdotte risulta immediato. Viene anche presentato un semplice metodo per la valutazione del carico critico.

     

Evaluation of the second order elastic moduli

The expressions of the apparent linear elastic moduli and their first and second derivatives, with respect to hydrostatic pressure, are obtained according to the second order elasticity theory. As a particular case when the material is hyperelastic, formulae of the first derivatives of the linear elastic moduli reduce to those obtained by Seeger and Buck.     

Higher order model for soft and hard elastic interfaces

The present paper deals with the derivation of a higher order theory of interface models. In particular, it is studied the problem of two bodies joined by an adhesive interphase for which “soft” and “hard” linear elastic constitutive laws are considered. For the adhesive, interface models are determined by using two different methods. The first method is based on the matched asymptotic expansion technique, which adopts the strong formulation of classical continuum mechanics equations (compatibility, constitutive and equilibrium equations). The second method adopts a suitable variational (weak) formulation, based on the minimization of the potential energy. First and higher order interface models are derived for soft and hard adhesives. In particular, it is shown that the two approaches, strong and weak formulations, lead to the same asymptotic equations governing the limit behavior of the adhesive as its thickness vanishes. The governing equations derived at zero order are then put in comparison with the ones accounting for the first order of the asymptotic expansion, thus remarking the influence of the higher order terms and of the higher order derivatives on the interface response. Moreover, it is shown how the elastic properties of the adhesive enter the higher order terms. The effects taken into account by the latter ones could play an important role in the nonlinear response of the interface, herein not investigated. Finally, two simple applications are developed in order to illustrate the differences among the interface theories at the different orders.     

Non-linear dynamics of an elastic cable under planar excitation

The phenomena of the finite forced dynamics of a suspended cable associated with the quadratic and cubic non-linearities in the equations of motion are studied. A high-order perturbation analysis for the primary resonance is accomplished and numerical results are presented for the frequency-response equation and the region of instability of the steady-state solutions. Multivaluedness of the response curves is shown to occur with different characteristics depending on the cable and forcing parameters. The dependence of the response on the initial conditions is examined by means of the trajectories of the unsteady-state motions.     

Oscillation of an elastic bar rigidly linked to a kinematically excited pendulum

The oscillation of a mechanical system consisting of an elastic bar rigidly linked at the middle to a kinematically excited pendulum is studied. A system of integro-differential equations with appropriate boundary and initial conditions for the deflections of the bar axis and the rotation angle of the pendulum is derived using the Hamilton-Ostrogradsky principle. Given kinematic excitation conditions, the rotation angle is found as a solution to an inhomogeneous Hill equation in the form of a double power series in the amplitude of kinematic excitation. It is shown that the oscillation of the bar is the linear superposition of three oscillations __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 107–115, October 2006.     

Non-uniqueness of time-domain reflection from 3D planar elastic layers

Recently, some explicit results were obtained regarding non-uniqueness for the traction to displacement maps of bounded elastic bodies in 2D and 3D, under the assumption of an internal kinematic constraint. The approach utilized is that of transformation optics. As the approach could, under the usual continuum assumptions, handle all frequencies without resorting to active materials, it could potentially be directly applied also to time domain problems. In the present paper we cover the extension to the time domain of these recent results in the case of reflection from a composite slab of rather general anisotropy, and derive the required material properties of different slabs with identical reflection properties. In particular we describe how homogeneous and inhomogeneous slabs of very different thicknesses may be indistinguishable with respect to elastic wave reflection properties. It should be noted that the approach retains both the minor and major symmetries of the stiffness tensor, and does not require an anisotropic mass density tensor to be used.     

Wave propagation in elastic laminates using a second order microstructure theory

A generalization of the simpler microstructure theory developed earlier for elastic laminates by Sun, Achenbach and Herrmann is used to analyze steady state plane wave propagation. This new version incorporates higher-order thickness variations in the displacement functions and includes restrictions on both displacement and stress at the laminate interfaces.To assess the potential of a second-order microstructure theory for accurate modeling of mechanical processes in laminates, dispersion results and especially mode shape data for both displacements and stresses are obtained and compared to corresponding solutions obtained by the theory of elasticity. The comparisons indicate that while dispersion results may be nearly identical, extremely significant differences may be observed in the mode shapes.     

Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory

Several results are presented concerning symmetry properties of the tensor of third order elastic moduli. It is proven that a set of conditions upon the components of the modulus tensor are both necessary and sufficient for a given direction to be normal to a plane of material symmetry. This leads to a systematic procedure by which the underlying symmetry of a material can be calculated from the 56 third order moduli. One implication of the symmetry conditions is that the nonlinearity parameter governing the evolution of acceleration waves and nonlinear wave phenomena is identically zero for all transverse waves associated with a plane of material symmetry.