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Pozycja Double pendulum colliding with a rough obstacle.(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Sypniewska-Kamińska, Grażyna; Starosta, Roman; Awrejcewicz, JanThe externally excited and damped vibrations of the double pendulum in the vertical plane are considered. The pendulum can collide with a rough obstacle many times during its motion. The pendulum is modeled as a piecewise smooth system. The differential equations govern the motion of the system in the relatively long time between the collisions. When a contact with the obstacle occurs, the pendulum exhibits a discontinuous behaviour. The velocities of both parts of the pendulum and the reaction forces are changing stepwise. An important element of the solving algorithm is aimed on the continuous tracking of the position of the pendulum in order to detect the collision with the unilateral constraints and to determine the state vector of the pendulum at the impact time instant. A single collision is described by the Euler’s laws of motion in the integral form. The equations are supplemented by the Poisson's hypothesis and Coulomb’s law of friction. The friction law is formulated for the instantaneous values of the reaction forces. The values of their impulses depend on the existence of a slip between the contacting bodies. Furthermore, during the collision the dynamic behaviour may change. Therefore the Coulomb law cannot be generalized for the linear impulses of the forces in a simple way. We have applied the Routh method in order to solve the problem. The method has a simple geometrical interpretation in the impulse space.Pozycja Internal resonances in nonlinear vibrations of a continuous rod with microstructure.(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Andrianov, Igor V.; Awrejcewicz, Jan; Danishevskyy, Vladyslav V.; Markert, BerndNonlinear longitudinal vibrations of a periodically heterogeneous rod are considered. Geometrical nonlinearity is described by the Cauchy–Green strain tensor. Physical nonlinearity is modelled expressing the energy of deformation as a series expansion in powers of the strains. The governing macroscopic dynamical equation is obtained by the higher-order asymptotic homogenization method. An asymptotic solution is developed by the method of multiple time scales. The effects of internal resonances and modes coupling are predicted. The specific objective of the paper is to analyse how the presence of the microstructure influences on the processes of mode interactions. 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 realised.Pozycja Dynamics of two coupled 4-DOF mechanical linear sliding systems with dry friction.(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Kosińska, Angelika; Grzelczyk, Dariusz; Awrejcewicz, JanThe paper introduces a model of two identical coupled 4-DOF mechanical linear sliding systems with dry friction coupled with each other by a linear torsional spring. The appropriate components (bodies) of the coupled systems are riding on two separated driving belts, which are driven at constant velocities, and stick-slip vibrations can be observed. In this case the physical interpretation of the considered model could be two rows of carriages laying on the guideways and coupled by an elastic shaft, which are moving at constant velocity with respect to the guideways as a foundation. From a mathematical point of view the analyzed problem is governed by eight nonlinear ordinary second order differential equations of motion yielded by the second kind Lagrange equations. Numerical analysis is performed in Mathematica software using the qualitative and quantitative theories of differential equations. Some interesting non-linear system dynamics are detected and reported using the phase portraits and the Poincaré maps. Next, power spectra obtained by the FFT technique are reported. The presented results show periodic, quasi-periodic, chaotic and hyperchaotic orbits. Moreover, synchronization effects between the coupled systems are also detected and studied.Pozycja Axially excited spatial double pendulum nonlinear dynamics.(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Ludwicki, Michał; Kudra, Grzegorz; Awrejcewicz, JanAnalysis of a 3D spatial double physical pendulum system, coupled by two universal joints is performed. External excitation of the mechanism is realized by axial periodic rotations of the first joint of the pendulum. System of ODEs is solved numerically and obtained data are analyzed by a standard approach, including time series, phase plots and Poincaré sections. Additionally, FFT (Fast Fourier Transform and the wavelet transformation algorithms have been applied. Various wavelet basic functions have been compared to find the best fit, e.g. Morlet, Mexican Hat and Gabor wavelets. The so far obtained results allowed for detection of a number of non-linear effects, including chaos, quasi-periodic and periodic dynamics, as well the numerous and different bifurcations. Scenarios of transition from regular to chaotic dynamics have been also illustrated and studied.Pozycja Dynamics of articulated vehicles by means of multibody methods.(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Adamiec-Wójcik, Iwona; Awrejcewicz, Jan; Grzegożek, Witold; Wojciech, StanisławThe paper presents modelling of an articulated vehicle by means of joint coordinates, which enable us to describe the motion of the system with a minimal number of generalised coordinates. We consider a model of a semi-trailer formulated using joint coordinates and homogenous transformations. Such an approach enables us to treat the vehicle as a kinematic chain consisting of three single units with an even number of wheels. This means that a single unit vehicle has a structure of an open kinematic chain in a tree form. The contact of wheels with the road is modelled by the Dugoff-Uffelman model. The model is validated by comparing the simulation results with those obtained from experimental measurements. Friction in the fifth wheel is one of the important parameters influencing the motion of a tractor with a semi-trailer. The model presented enables us to analyse different models of friction in the fifth wheel. The influence of those friction models on the results is presented and discussed.