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Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system that correlates with two devices, namely a piezoelectric device and an electromagnetic one, to produce two novel models. These devices are connected to a nonlinear damping spring pendulum with two degrees of freedom. The damping spring pendulum is supported by a point moving in a circular orbit. Lagrange's equations of the second kind were utilized to obtain the equations of motion. The asymptotic solutions of these equations were acquired up to the third approximation using the approach of multiple scales. The comparison between the approximate and the numerical solutions reveals high consistency between them. The steady-state solutions were investigated, and their stabilities were checked. The influences of excitation amplitudes, damping coefficients, and the different frequencies on energy-harvesting device outputs are examined and discussed. Finally, the nonlinear stability analysis of the modulation equations is discussed through the stability and instability ranges of the frequency response curves. The work is significant due to its real-life applications, such as a power supply of sensors, charging electronic devices, and medical applications. [ABSTRACT FROM AUTHOR]

This work investigates a novel design of an electromagnetic vibration energy harvester array based on pendulum structures under harmonic base excitation. The design includes combined magnetic-solenoid pendulum harvesters with different mechanical coupling configurations. A comparison is made with standalone magnetic and solenoid pendulum based harvester arrays under similar excitation conditions. Additionally, the positions of harvesters are shuffled and analyzed. The effects on harvested power, power density and frequency bandwidth are evaluated by considering dimensionless coupling ratios, damping ratios, mass ratios and length ratios as parameters. Further, the efficiency of the harvester against these parameters has also been reported. In addition, experimental validation of the numerical formulation is also conducted. The study shows that the proposed magnetic-solenoid harvester array can harvest sustained energy in a wider bandwidth compared to standalone magnetic and solenoid pendulum harvester arrays. [ABSTRACT FROM AUTHOR]

This work focuses on analyzing the dynamical behavior of a mechanical system consisting of a double spatial pendulum in contact with a movable obstacle. The pendulum's ability to move in space is achieved by the use of special Cardan–Hook joints as the links of the pendulum. The mechanical system is equipped with a special movable obstacle, i.e. a rotating circular plate situated below the pendulum, which limits the space of admissible positions. A significant part of this work is devoted to the modeling of the contact between the pendulum and the obstacle. In this regard, a special class of reduced models is derived with the resultant friction force and moment acting on the finite size of the contact area along with a compliant model of impact based on the Hertz stiffness. Finally, the effective model suitable for fast and realistic numerical simulations is obtained. The contact model is tested numerically and the effect of its parameters on the system bifurcation dynamics is investigated. [ABSTRACT FROM AUTHOR]

A parametric pendulum excited by a discrete wave-modulated step function of length is subjected to a mathematical analysis and numerical modeling. We observe an existence of almost periodic solutions of ordinary differential equations with linear boundary value conditions. An exemplary oscillator subject to both an almost periodic step elongation and forcing synchronizes with the forcing, tending to almost periodic motions like stable limit cycles. Conditions for that synchronization as well as trajectories of numerical solutions on time history plots and phase planes are shown to confirm correctness of the analytical derivations and dedicated numerical modeling. [ABSTRACT FROM AUTHOR]

The article deals with the problem of the stability of the equilibrium of a double pendulum under the action of a follower force. The latter is assumed to be imperfect, that is, the direction of its action does not coincide with the axis of the second link of the pendulum. The relationships between the masses of the pendulums, the stiffnesses of the hinges, and the damping coefficients are assumed to be arbitrary. Conditions of stability in the absence of damping and for a weakly damped system are obtained and analyzed. These conditions are formulated for the linearized motion equations in the form of an estimate of the dimensionless parameter corresponding to the value of the critical load. The influence of damping ratio and stiffness ratio on the permissible value of this parameter has been studied. It is shown that near the boundary of the region of flutter instability, the value of the critical load can be increased by decreasing the stiffness of the hinge at the attachment point of the pendulum. [ABSTRACT FROM AUTHOR]

Due to the long-term problem of electricity and potable water in most developing and undeveloped countries, predominantly rural areas, a novelty of the pendulum water pump, which uses a vertically excited parametric pendulum with variable-length using a sinusoidal excitation as a vibrating machine, is presented. With this, more oscillations can be achieved, reducing human effort further and having high output than the existing pendulum water pump with the conventional pendulum. The pendulum, lever, and piston assembly are modeled by a separate dynamical system and then joined into the many degrees-of-freedom dynamical systems. The present work includes friction while studying the system dynamics and then simulated to verify the system's harmonic response. The study showed the effect of the pendulum length variability on the whole system's performance. The vertically excited parametric pendulum with variable length in the system is established, giving faster and longer oscillations than the pendulum with constant length. Hence, more and richer dynamics are achieved. A quasi-periodicity behavior is noticed in the system even after 50 s of simulation time; this can be compensated when a regular external forcing is applied. Furthermore, the lever and piston oscillations show a transient behavior before it finally reaches a stable behavior. [ABSTRACT FROM AUTHOR]

A weakly nonlinear 2-DOF system, parametrically and externally excited, is studied. An intensive energy transfer between modes of vibrations is discovered. Multiple scales method is used for recognizing resonances occurring in the system. The amplitude response functions for some chosen cases of resonances are obtained and studied. Many of the engineering systems can be investigated in the presented way. [ABSTRACT FROM AUTHOR]

Nonlinear dynamics of a real plane and periodically forced triple pendulum is investigated experimentally and numerically. Mathematical modeling includes details, taking into account some characteristic features (for example, real characteristics of joints built by the use of roller bearings) as well as some imperfections (asymmetry of the forcing) of the real system. Parameters of the model are obtained by a combination of the estimation from experimental data and direct measurements of the system's geometric and physical parameters. A few versions of the model of resistance in the joints are tested in the identification process. Good agreement between both numerical simulation results and experimental measurements have been obtained and presented. Some novel features of our real system chaotic dynamics have also been reported, and a novel approach of the rolling bearings friction modeling is proposed, among other. [ABSTRACT FROM AUTHOR]

The triple physical pendulum with barriers is used to model a piston — connecting rod — crankshaft system of a mono-cylinder combustion engine. Among other results, a peculiar motion of the piston — connecting rod — crankshaft model is reported, i.e. six sliding stages of the piston along the cylinder per one engine cycle are illustrated. In spite of its simplicity, the introduced model can serve as the first step in a more advanced modeling taking into account various technological details. In particular, the proposed model can be considerably useful for the investigations of impacts between the piston and the cylinder aiming at the reduction or total elimination of this source of harmful noise. [ABSTRACT FROM AUTHOR]

In this work, we analyzed theoretically and experimentally the nonlinear dynamics of a magnetic pendulum driven by a coil-magnet interaction. The force between the magnetic elements and the resulting torque on the pendulum are derived using both the magnetic charges interaction model and the experimentally fitted interaction model. This enables the comparison between the two models. The current in the coil is taken first as a sinusoidal current and then as a square current. The comparison of the structure of each interaction model is conducted and it appears that they give qualitatively similar characteristics. The harmonic balance method is used to approximate the frequency responses of the pendulum leading to both symmetric and asymmetric or one-side (intrawell) oscillations. The two-parameters bifurcation diagrams are plotted showing the different dynamical behaviors considering the current amplitude and frequency as the control parameters. Good agreements are found between our theoretical results and experimental ones. [ABSTRACT FROM AUTHOR]

• Magnetic pendulum system is analysed numerically and experimentally. • Model of the system is discretised into states with one- and two-well potential. • Pulsating excitation signal parameters are obtained for one-side oscillations. • The same oscillations are possible for different duty cycles and the same frequency. • Multiperiodic solutions may arise from localized forcing in kick-exited systems. This paper presents a theoretical and numerical analysis of one-side oscillation in a single magnetic pendulum. The system is composed of a physical pendulum with a neodymium magnet fixed to the end of the rod. The pendulum is driven by a pulsating, repulsive magnetic field generated by an electric coil placed underneath. The pendulum pivot is damped by an elastic element. The excitation current signal has a pulsating rectangular waveform with a controlled frequency and duty cycle. In general, magnetic interaction weakens with increasing distance between the magnet and the coil, making excitation in this system not only time-dependent but also position-dependent. The specific type of solution, referred to as "one-side oscillation" is analysed in terms of a frequency and a duty cycle of the current signal. By one-side oscillations, we mean oscillations of the pendulum characterized by the same sign of angular displacement, without passing through the lowest and the highest vertical equilibrium positions. The analysis is based mainly on the assumption that outside some "active zone" the influence of magnetic interaction on the pendulum dynamics is negligible, and the system can be discretized into two states: with and without magnetic force. The results were confirmed by experimental data showing the different periodicity of one-side oscillations. Limitations of the proposed analysis were found in the case of some variants of the analysed solution type. An overview of the system dynamics is presented in the form of bifurcation diagrams obtained numerically and verified by experimental estimates, which show the existence of chaotic behaviour and multiperiodicity for various values of the frequency of the current signal. [ABSTRACT FROM AUTHOR]

• Mechanism composed of two coupled magnetic pendulums is constructed. • Nonlinear features of the system are analyzed analytically and numerically. • Energy exchange between pendulums using magnetic field is monitored. • Open-loop and feedback control strategies for the energy flow are presented. This paper presents a methodology of controlling the resonance energy exchange in mechanical system consisting of two weakly coupled magnetic pendulums interacting with the magnetic field generated by coils placed underneath. It is shown that properly guided magnetic fields can effectively change mechanical potentials in a way that the energy flow between the oscillators takes the desired direction. Studies were considered by using a specific set of descriptive functions characterizing the total excitation level, its distribution between the pendulums, and the phase shift. The developed control strategies are based on the observation that, in the case of antiphase oscillation, the energy is moving from the pendulum subjected to the repelling magnetic field, to the oscillator under the attracting field. In contrast, during the inphase oscillations, the energy flow is reversed. Therefore, closed-loop controller requires only the information about phase shift, which is easily estimated from dynamic state signals through the coherency index. Advantage of suggested control strategy is that the temporal rate of inputs is dictated by the speed of beating, which is relatively slow compared to the carrying oscillations. [Display omitted] [ABSTRACT FROM AUTHOR]

• Experimental and mathematical models of a pendulum driven by DC motors are developed. • Good agreement between numerical simulations and experimental data is obtained. • Bifurcation dynamics is investigated by experimental and numerical methods. • Passive control of chaos with magnetic rheological rotational damper is proposed. In the present work, we deal with a dynamical analysis and passive control of chaos with magnetic rheological (MR) rotational damper in a pendulum driven by a DC motor via slider mechanism. A mathematical model for electromechanical system composed of a pendulum driven horizontally by through a DC motor and a slider-crank mechanism is presented and the parameters are estimated based on experimental data. Numerical and experimental results demonstrate that for certain values of the motor input voltage they can lead the system to chaotic behavior. For dynamic analysis, bifurcation diagrams, Poincaré sections, phase diagrams and 0–1 test are considered. In order to suppress the chaotic behavior, it is proposed to include MR rotational damper, as a passive control. In the case of the passive rotational MR damper, the influence of the introduction of the MR damper in a pendulum is performed considering the bifurcation diagrams. The numerical results show that the introduction of a passive rotational MR damper suppresses the chaotic behavior of the system. Additionally it is shown that it is possible to keep the pendulum oscillating with periodic behavior using the rotational MR damper with energizing discontinuity. [ABSTRACT FROM AUTHOR]

• Coupled pendulums excited magnetically are studied experimentally and numerically. • The modified mathematical formula of an experimental magnetic torque is proposed. • Two-harmonic analytical approximations for different periodic regimes are derived. • Discontinuous piecewise linear maps generated by chaotic attractors are presented. The nonlinear forced vibrations of two pendulums coupled by an elastic element are considered. One of the pendulums is subjected to the magnetic stimulus produced by a pair of a periodically powered coil and a permanent magnet mounted at the end of the pendulum. We extend our previous experimental and numerical results presented in Polczyński et al. (2019) and proposed a new rational approximation of a magnetic torque to enable the application of analytical methods. Based on this approximation, the numerical and analytical studies of equations of motion are carried out. In particular, the resonant phenomena and periodic regimes are studied in detail. The procedure to approximate the subharmonic and almost harmonic regimes is developed using the balance harmonic method. The successive maxima map has been employed to study more complicated regimes. This technique allows one to reveal changes in profile's shapes for various driving frequencies and shows the existence of strongly chaotic trajectories. [ABSTRACT FROM AUTHOR]