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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]

A damped parametric pendulum with friction is identified twice by means of its precise and imprecise mathematical model. A laboratory test stand designed for experimental investigations of nonlinear effects determined by a viscous resistance and the stick-slip phenomenon serves as the model mechanical system. An influence of accurateness of mathematical modeling on the time variability of the nonlinear damping coefficient of the oscillator is proved. A free decay response of a precisely and imprecisely modeled physical pendulum is dependent on two different time-varying coefficients of damping. The coefficients of the analyzed parametric oscillator are identified with the use of a new semi-empirical method based on a coupled oscillators approach, utilizing the fractional order derivative of the discrete measurement series treated as an input to the numerical model. Results of application of the proposed method of identification of the nonlinear coefficients of the damped parametric oscillator have been illustrated and extensively discussed. [ABSTRACT FROM AUTHOR]

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement. [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]

Abstract: The paper is dedicated to mathematical analysis of the idealized model of a string moving at a variable speed. Resonance case is studied and it is shown that an infinite system of equations cannot be reduced. Some exact solutions are obtained. [Copyright &y& Elsevier]

Dynamical regular response of an oscillator with two serially connected springs with nonlinear characteristics of cubic type and governed by a set of differential-algebraic equations (DAEs) is studied. The classical approach of the multiple scales method (MSM) in time domain has been employed and appropriately modified to solve the governing DAEs of two systems, i.e. with one- and two degrees-of-freedom. The approximate analytical solutions have been verified by numerical simulations. [ABSTRACT FROM AUTHOR]

Friction-induced stick–slip vibration has been identified as the main reason for various types of disc brake noises like creep groan, squeak and creak. In this work, we investigate the suppression of stick–slip vibrations in disc brakes due to the application of a small-amplitude normal harmonic force on the brake pad. The discontinuity induced bifurcations that lead to such suppression are studied in detail. The frequency range of the normal force in which suppression of stick–slip vibration occur is identified. A 3 DoF discontinuous model of the disc brake is considered. The discontinuous oscillator is modelled as a Filippov system with smooth vector fields separated by a discontinuity surface. It is shown that the normal harmonic force entrains the non-smooth limit cycle even for very small forcing amplitudes. Different frequency intervals exhibiting harmonic, super-harmonic, and sub-harmonic entrainments are identified. It is demonstrated that the initiation and termination of these entrainment intervals are associated with Neimark–Sacker bifurcations. It is also shown that discontinuity induced grazing-sliding mechanism brings about these bifurcations. Switching-sliding bifurcations happening in the entrainment region give rise to a sub-interval which is associated with suppression of stick–slip vibrations. We also report interesting discontinuity induced bifurcation scenarios in higher order entrainment regimes like grazing-sliding involving different orbit loops at lower and upper limits of the interval and splitting–merging of the sliding surface. • Stick–slip vibration suppression in disc brakes by using normal force is studied. • Various entrainment regions are detected and studied by Filippov's method. • Discontinuity induced bifurcations causing these entrainments are analysed. • Entrainment sub-intervals with suppressed stick–slip vibration are identified. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the dynamic properties of an existing experimental stand of 1DOF mechanical parametric oscillator, with a focus on approximate analytical solutions of the observed isolated branches of periodic orbits. The experimental stand involves a cart moving along a rolling guide, with the stiffness consisting of two components: a time-varying linear element created by a rotating rod with a rectangular cross-section and a nonlinear hardening stiffness caused by magnetic springs. It was demonstrated that a rolling bearing's nonlinear resistance to motion consists of viscous damping and a second component analytically compared to dry friction. The study utilises multiple scales and harmonic balancing methods to provide analytical solutions. It is then successfully validated using numerical simulations and experimental data. The study investigates how dry friction influences oscillator response and applies the modified Mathieu–Duffing equation to represent the system's dynamics. Different branches of periodic orbits are researched to determine their function in energy harvesting and mechanical system improvement. This research demonstrates the distinctions across analytical, numerical, and experimental methodologies, providing a comprehensive understanding of investigating intricate nonlinear systems. • Analytical derivation of the periodic solutions of 1DOF mechanical parametric oscillator with dry friction. • The existence of isolated branches of periodic orbits has been observed. • Very good qualitative agreement was achieved between analytical, numerical and experimental solutions. • Numerically computed basins of attraction allows for a better understanding of the investigated system dynamics. [ABSTRACT FROM AUTHOR]

• Novel magnet – coil interaction model development for fast analytical calculations. • Magnetic dipole moment determination experimental technique. • Experimental validation on precisely constructed stand. • Dry friction neglection due to aerostatic support application. • Magnetic and electromagnetic non-linearity sources. The paper investigates a unique configuration of magneto-electro-mechanical system in terms of numerical simulation and experimental analysis. The harmonically excited mass of the oscillator with magnets being suspended by the magnetic spring interacts with the electric coil, inducing current in its' circuit. Existing and often empirical models found in the literature do not fully describe the interaction between the permanent magnet and the multi-layer and multi-turn electric coil. This work fills this gap by proposing and validating a novel, easy to use model of magnet-coil interaction for either analytical or numerical calculations. An additional point of interest lies in the developed experimental technique utilized to determine the magnetic dipole moment of the magnet. The behaviour of the system and the parameters influence have been extensively described and illustrated in various frequency characteristics and phase portraits. The identification of the key parameters has been carried out based on experimental studies led on a precisely constructed experimental stand including aerostatic supports, which allow for exact, axial guidance and eliminate dry friction phenomena in the system. A good agreement between the numerical simulation and experimental results has been achieved and illustrated, especially in terms of the values and shapes of the generated electromotive force and electrical power, which cannot be predicted on the basis of empirical models. [ABSTRACT FROM AUTHOR]

Non-stationary effects in the friction-induced dynamics of a two-degree-of-freedom brake model are examined in this paper. The belt–spring–block model is designed to take into account variations of the normal load during the braking process. It is shown that due to the adiabatically slowing down velocity of the belt, the system response experiences specific qualitative transitions that can be viewed as simple mechanical indicators for the onset of squeal phenomenon. In particular, the creep-slip leading to a significant widening of the spectrum of the dynamics is observed at the final phase of the process. [ABSTRACT FROM AUTHOR]

A new method of consideration and reduced analysis of long-time series is suggested. The method is based on the scaling (fractal) properties of the beta-distribution function that allows finding the stable parameters and reducing the long-time series containing 5×10 5 –10 6 data points to analysis of 10–20 stable parameters. The new procedure of clusterization with the usage of the generalized Pearson correlation function allows to take into account the influence of external factor and combine/separate different parameters into a cluster with respect to the influence of the qualitative external parameter considered. The method is rather flexible and can be applied to wide set of large data. As an example the triple pendulum data are considered. This method opens new possibilities in creation of the reduced database (specific fingerprints) of different long-time series for their comparison and subsequent analysis. [ABSTRACT FROM AUTHOR]

• Experimental and mathematical models of a one-degree-of-freedom impact oscillator are presented. • A new discontinuous impact model with finite collision duration is developed. • The impact model is validated during numerical and experimental bifurcation analysis of the investigated system. In the work it is proposed a new discontinuous impact model based on the restitution coefficient and the finite duration of the collision, which are functions of the velocity before the collision. The model was validated during modelling, experimental and simulation studies of the bifurcation dynamics of an oscillator of one-degree-of-freedom with a one-sided limiter. The simulation results using the new discontinuous impact model were compared with numerical solutions obtained with the use of the continuous five-parameter Hunt-Crossley model assuming the obstacle compliance. It was shown that the new discontinuous model produces results that are as realistic and consistent with the experimental data as the compliant impact model. Moreover, the significance of the dependence of the restitution coefficient on the speed before the collision and the contact time during the collision in obtaining a good agreement of the simulation results with the experimental data were shown. [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]

Chaotic and regular behavior of the system consisting of a double physical pendulum with two repulsive permanent magnets is studied. We are focused on mathematical modelling, numerical simulations, experimental measurements and in particular on a novel magnetic interaction modelling. System parameters are identified by matching the output signals from experiments and numerical solutions to the developed mathematical model governed by a strongly non-linear set of two second order ODEs including the friction and the magnetic interaction torques. Considered system shows chaotic and periodic dynamics. Few chaotic zones have been detected numerically and confirmed experimentally. Scenarios of transition from regular to chaotic motion and vice versa, as well as the bifurcation diagrams are illustrated and discussed. Good agreement between the numerical simulation and experimental measurement is achieved. [ABSTRACT FROM AUTHOR]

Control of vibrations in fluid transmitting pipes has emerged as a major engineering challenge. For higher fluid flow rate, support vibration, and excitation load, the vibrations in pipes that convey fluid can be complex and, at times, catastrophic. Recent works have demonstrated that Nonlinear Energy Sink (NES) can be effectively employed for passive vibration control of fluid-conveying pipes. The present work explores the possibility of using NES with nonlinear damping to overcome the limitations of conventional NES-based vibration control in fluid-transmitting pipes. Geometric nonlinear damping (velocity-displacement dependent damping), whose physical realization is possible by a suitable geometric configuration of the linear viscous damper, is considered in this study. Euler–Bernoulli beam theory models the fluid conveying pipe under an external harmonic load. The reduced order model obtained through the Galerkin procedure is investigated analytically using the complex averaging method. The occurrence of strongly modulated and weakly modulated responses are demonstrated, and their effect on vibration mitigation is shown. The frequency range exhibiting strongly modulated response in the system is significantly increased with the addition of nonlinear damping to NES. This assists in mitigating pipe vibrations by initiating targeted energy transfer from pipe to NES. The Saddle–Node and Hopf bifurcation boundaries in the model are identified using the slow-flow dynamics obtained by complex averaging. The effect of nonlinear damping, external excitation, the location of NES, and the flow rate on the bifurcation boundaries are investigated in detail. It is shown that the external excitation needed to trigger Saddle–Node bifurcation is largely reduced for NES having nonlinear damping. Furthermore, a study of Hopf bifurcation in the frequency domain shows that using NES with nonlinear damping significantly decreases the frequency range in which high-amplitude isolated solutions occur. The method of multiple scales is used to derive the analytical expression for the slow invariant manifold of the pipe-NES system. The vibration suppression mechanism and the occurrence of strongly modulated responses are explained with the help of flow on slow invariant manifolds. A comparison of the topology of the slow invariant manifolds shows that using nonlinear damping with NES substantially reduces the vibration amplitude of the pipe structure. The percentage of energy transferred to the NES is quantified to show the effectiveness of the proposed absorber over normal NES for different flow speeds. [ABSTRACT FROM AUTHOR]

In this work, we have mathematically modeled a 1DOF parametric oscillator with stiffness-hardening characteristics and dry friction and investigated both experimentally and numerically the bifurcation dynamics. The system consists of a cart moving along a linear rolling guide widely used in the industry. It has a stiffness comprised of two components: a linear time-variable part generated by a rotating rod of a rectangular cross-section and a nonlinear hardening stiffness caused by magnetic springs. In the case of nonlinear resistance of motion in a rolling bearing, regardless of the true nature of this phenomenon, it was modeled as the sum of viscous damping and the second component mathematically equivalent to dry friction. The trivial solution was observed to be stable in the whole range of parametric excitation frequencies. But there is a frequency range where the system is bistable, a periodic attractor coexists with the stable equilibrium position, and the branch of the periodic orbit is isolated, i.e., not connected with the equilibrium position, and forms an Isola. This distinguishes the analyzed system from the commonly investigated parametric oscillators, including the classical Mathieu equation and its various versions. Our work perfectly agreed between the numerical simulations and the experimental data. Moreover, the mathematical model for different friction values is investigated, showing the transition between the system without dry friction and the actual rig. The stability of the equilibrium position is tested, and the bifurcation dynamics of periodic orbits are presented using numerical continuation methods, obtaining complete agreement between the results obtained using different ways. • 1DOF parametric oscillator with dry friction is investigated experimentally. • Bistability of experimental oscillator is observed. • Bifurcation dynamics of the oscillator is investigated. • Isolated branches of periodic solutions are shown. • Effect of dry friction on the properties of the parametric oscillator is revealed. [ABSTRACT FROM AUTHOR]

The work presents the possible applications and effectiveness of certain class of models of the resultant friction force and rolling resistance. The friction models are based on the integral model constructed under assumption of fully developed sliding on the plane contact area of general shape and any pressure distribution. Then the integral model of friction force and moment are approximated based on Padé approximants and their generalizations. These models are expected to be computationally effective in numerical simulations of rigid bodies with frictional contacts, such like billiard balls, Thompson top, the wobble stone and many others. In the present work two different examples of application of the developed contact models are presented and tested: a) a billiard ball rolling and sliding on the plane horizontal table; b) a full ellipsoid of revolution in contact with plane and horizontal base. [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]

• Experimental and mathematical models of 2DOF oscillator with magnetic springs are presented. • Good agreement between numerical simulations and experimental data is obtained. • Regular and chaotic dynamics is investigated by numerical methods. In this paper, oscillations in the physical model consisting of two carts mounted on a guide are considered. The movement of the carts is restricted by the repelling permanent magnets. The interaction between carts is provided by a linear spring. The harmonic external loading is applied to one of the carts. According to the experimental studies, the period doubling cascade leading to the creation of a chaotic attractor occurs with an increase in the forcing frequency in the system. Chaotic attractor crises, quasi-periodic regimes, and finite periodic doubling cascade are revealed as well. To describe system's dynamics, the nonlinear mathematical model is developed. Based on the experimental data, the procedure of model validation is introduced. The numerical and qualitative analysis of the mathematical model allows one to identify bifurcation types, reveal unstable modes, and discover phenomena existing beyond the main period doubling cascade. [ABSTRACT FROM AUTHOR]