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We obtain the Schrödinger wave functions of a generalized pendulum under time-dependent gravitation by making use of the Lewis and Riesenfeld invariant method. As an example, we consider a generalized pendulum with constant gravitation and exponentially increasing mass. We also present a canonical approach to the generalized time-dependent pendulum.

Recently, Green’s function approach is used to approximate (initial) boundary value problems for nonlinear dynamical systems. Here, we use this approach for derivation of a simple criterion for approximate controllability of nonlinear dynamical systems. Simple and easy-to-check criteria on system parameters are derived for process controllability. Examples from existing references are considered.

In this work, we analyze the quantum dynamics of a generalized pendulum with a time-varying mass increasing exponentially and constant gravitation. By using Lewis–Riesenfeld invariant approach and Fock states, we solve the time-dependent Schrödinger equation for this system and write its solutions in terms of solutions of the Milne–Pinney equation. We also construct coherent states for the quantized pendulum and use both Fock and coherent states to investigate some important physical proprieties of the quantized pendulum such as eigenvalues of the angular displacement and momentum, their quantum variances as well as the respective uncertainty principle. Finally, we derive the geometric, dynamical and Berry phases for the time-dependent generalized pendulum.

We show that two coupled pendulums that are coupled and can synchronize, are mathematically equivalent to one "horizontal" parametrically driven pendulum. We have fabricated a horizontal pendulum and present data from this horizontal pendulum which we believe to be the first physical realization of such a mechanical "synchronization machine." A description of intermittent synchronization that can occur when two coupled pendulums are in a chaotic state is given in terms of the data from the horizontal pendulum. We discuss the relationship between the modes of the horizontal pendulum and the corresponding synchronization of the two coupled pendulums. Finally, we show that when a horizontal pendulum is driven by any random source, not necessarily chaotic, intermittent synchronization can occur.

We consider oscillatory, nonrotating periodic solutions of the parametrically excited pendulum. We determine the zones in the parameter space in which different types of oscillatory solutions are realized. In particular we determine the zones in which the T-periodic and 2T-periodic harmonic and subharmonic oscillatory solutions are the dominant type of stable solutions. The symmetry properties of these solutions and the associated scenarios of transition to tumbling chaos are also discussed.

We repeat Huygens' experiment using real pendulum clocks in the same way as it was done originally, i.e. we hang two clocks on the same beam and observe the behavior of the pendulums. The clocks in the experiment have been selected in such a way so as to be as identical as possible. It has been observed that when the beam is allowed to move horizontally, the clocks can synchronize both in-phase and anti-phase. We perform computer simulations of the clocks' behavior to answer the question how the nonidentity of the clocks influences the synchronization process. We show that even the clocks with significantly different periods of oscillations can synchronize, but their periods are modified by the beam motion so they are no more accurate.

A pendulum excited by the combination of vertical and horizontal forcing at the pivot point was considered and the period-1 rotational motion was studied. Analytical approximations of period-1 rotations and their stability boundary on the excitation parameters (ω, p)-plane are derived using asymptotic analysis for the pendulum excited elliptically and along a tilted axis. It was assumed that the damping is small and the frequency of the base excitation is relatively high. The accuracy of the approximations was examined for different values of the parameters e and κ controlling the shape of excitation, and it was found that using the second and third order approximations ensures a good correspondence between analytical and numerical results in the majority of cases. Basins of attractions of the coexisting solutions were constructed numerically to evaluate the robustness of the obtained rotational solutions. It was found that the horizontal component of excitation has a larger effect on the shift in position of the saddle node bifurcations for the elliptically excited case than for the pendulum excited along a tilted axis. For the elliptically excited pendulum with pivot rotating in the same direction as the pendulum the stability boundary is shifted downwards providing a larger region of the solution existence. When the pendulum and the pivot rotate in opposite directions, the boundary is shifted upwards significantly limiting the region of the solution existence. In contrast, for the pendulum excited along the tilted axis, the direction of the rotation has a minor effect for low frequency values and the addition of the horizontal component always results in a larger region of the solution existence.

This paper studies a non-linear model for the damped oscillations of a pendulum.

In this paper, we propose a novel concept of a harvester-absorber system. The idea is based on usage of the dynamic absorber for energy harvesting. The device consists of three elements: a main system (damped body), a pendulum (tuned mass absorber) and an electromagnetic harvester. The ultimate goal of this investigation is to achieve simultaneous mitigation of vibrations and energy harvesting. We show the dynamics of the system and ranges of parameters where we observe good damping properties and high energy recovery. Additionally, an optimization of the harvester device parameters has been performed.

Elastically mounted double aerodynamic pendulum is an aeroelastic system with two rotational degrees of freedom. A wing is attached to the second link of the pendulum. It is shown that it is possible to select values of parameters in such a way as to make the trivial equilibrium (where both links of the pendulum are stretched along the flow) unstable. Numerical simulation of behavior of the system in such situations is performed, and arising limit cycles are studied. Experimental investigation of such aerodynamic pendulum is performed in the subsonic wind tunnel of the Institute of Mechanics of Lomonosov Moscow State University. Characteristics of periodic motions are registered for different values of parameters of the system. It is shown that experimental data are in qualitative agreement with results of numerical simulation.

High and slender towers may experience excessive vibrations caused by both wind and seismic loads. To avoid excessive vibrations in towers, tuned mass dampers (TMDs) are often used as passive control devices due to their low cost. The TMDs can absorb part of the energy of vibration transmitted from the main structure. These devices need to be finely tuned in order to work as efficient dampers; otherwise, they can adversely amplify structural vibrations. This paper presents the optimal parameters of a pendulum TMD (PTMD) to control the vibrations of slender towers subjected to an external random force. The tower is modeled as a single-degree-of-freedom (SDOF) mass–spring system via an assumed-mode procedure with a pendulum attached. A genetic algorithm (GA) toolbox developed by the authors is used to find the optimal parameters of the PTMD, such as the support flexural stiffness/damping, the mass ratio and the pendulum length. The chosen fitness function searches for a minimization of the maximum frequency peaks. The results are compared with a sensibility map that contains the information of the maximum amplitude as a function of the pendulum length and the mass ratio between the pendulum and the tower. The optimal parameters can be expressed as a power-law function of the supporting flexural stiffness. In addition, a parametric analysis and a time-history verification are performed for several combinations of mass ratio and pendulum length.

Wave Energy is a widespread, reliable renewable energy source. The early study on Wave Energy dates back in the 70’s, with a particular effort in the last and present decade to make Wave Energy Converters (WECs) more profitable and predictable. The PeWEC (Pendulum Wave Energy Converter) is a pendulum-based WEC. The research activities described in the present work aim to develop a pendulum converter for the Mediterranean Sea, where waves are shorter, thus with a higher frequency compared to the ocean waves, a characteristic well agreeing with the PeWEC frequency response. The mechanical equations of the device are developed and coupled with the hydrodynamic Cummins equation. The work deals with the design and experimental tank test of a 1:12 scale prototype. The experimental data recorded during the testing campaign are used to validate the numerical model previously described. The numerical model proved to be in good agreement with the experiments.

Major analysis and synthesis of dynamic models and their controllers imply deeper understanding of closed-loop response. Nevertheless, the numerical solution is useful in practice as long as it reflects a real behavior, that is the solution is as close as possible to reality as long as the model embodies the real complexity of the system and a number of factors are considered for the closed-loop simulation, for instance: state variables, mass distributions, un-modeled disturbances, friction, input and output noise, quantization, actuator saturation, bandwidths, or any signals that excite the dynamical system beyond the modeling hypotheses. However, the complexity of the simulator may increase exponentially if all these factors are included. Conventional techniques of closed-loop models is routinely made by manipulation of ordinary differential equations, which obtains results nearly to the reality, but with lack of validity if each factor is considered isolated, because each factor imposes diverse constraints to other elements of the closed-loop system. In this article, a novel close-loop approach for the modeling, simulation and control of dynamic systems, specifically applied to complex electromechanical systems, is presented based on the so-called Computational Mechatronics scheme (CMk) to obtain results of similar realness but with less complex models in comparison to conventional modeling and simulation procedures. We obtained these tools by using the synergetic mechatronics approach for design and advanced computational CAE tools. The main idea of the CMk is to provide a simulator/modeler that can be executed on PC in the continuous and discreet time under different conditions of operation, with application of parametric uncertainties, kinematics and dynamics constraints, electronics noise, and quantization or filtering.

Establish a rigid flexible feed system device coupling virtual prototype model and calculate the pendulum strength. Using reticulated shell of the pendulum's finite element to establish the ANSYS model. Input finite element model to Adams to put the pendulum flexible through the interface; through the introduction of before and after pendulum model of quality information in contrast to verify the correctness of the import; under the most dangerous conditions and simulation of rigid flexible coupling model that pendulum and the maximum stress; import ADAMS load in ANSYS, recount the maximum stress of the pendulum body, and the relevant conclusions are drawn from the comparison of the two calculation results; use the the rigid flexible coupling model spectrum to gain several dangerous nodes as the basis for the calculation of fatigue.

We overview several analytic methods of predicting the emergence of chaotic motion in nonlinear oscillatory systems. A special attention is given to the second method of Lyapunov, a technique that has been widely used in the analysis of stability of motion in the theory of dynamical systems but received little attention in the context of chaotic systems analysis. We show that the method allows formulating a necessary condition for the appearance of chaos in nonlinear systems. In other terms, it provides an analytic estimate of an area in the space of control parameters where the largest Lyapunov exponent is strictly negative. A complementary area thus comprises the values of controls, where the exponent can take positive values, and hence the motion can become chaotic. Contrary to other commonly used methods based on perturbation analysis, such as e.g., Melnikov criterion, harmonic balance, or averaging, our approach demonstrates superior performance at large values of the parameters of dissipation and nonlinearity. Several classical examples including mathematical pendulum, Duffing oscillator, and a system of two coupled oscillators, are analyzed in detail demonstrating advantages of the proposed method compared to other existing techniques.

This paper presents small pendulum type wave energy generation system, and explains its working principle. Motion model and equation of pendulum based on wave theory are established. Si7mulation by MATLAB and generation performance analysis are carried out.