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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]

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]

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]

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]

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]

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]