Artykuły (WM)

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Analysis of stability of the human gait.
(Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2007) Mrozowski, Jerzy; Awrejcewicz, Jan; Bamberski, Piotr
Analysis of the gait stability of a man moving along an even surface with a constant velocity is presented. The stability criteria applied to biped robots, namely: the Zero Moment Point (ZMP) and the Ground projection of the Center Of Mass (GCOM) have been employed in the investigations. The analysis has been carried out on the basis of measurement data obtained from the human gait recorded wit h a digital camera.
Asymptotic study of elastic half-plane with embedded punch.
(Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2007) Andrianov, Igor V.; Awrejcewicz, Jan; Kirichek, Yurii A.; Koblik, Steve G.
A contact problem for an elastic half-plane and an embedded rigid punch is studied. The employed mathematical model describes the behavior of a soil with embedded foundation. The analytical solution geverning the stress field behaviour is derived. Singular perturbation and complex analysis techniques are used.
Axially excited spatial double pendulum nonlinear dynamics.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Ludwicki, Michał; Kudra, Grzegorz; Awrejcewicz, Jan
Analysis 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.
Decomposition of the Equations of Motion in the Analysis of Dynamics of a 3-DOF Nonideal System
(2014) Awrejcewicz, Jan; Starosta, Roman; Sypniewska-Kamińska, Grażyna
The dynamic response of a nonlinear system with three degrees of freedom, which is excited by nonideal excitation, is investigated. In the considered system the role of a nonideal source is played by a direct current motor, where the central axis of the rotor is not coincident with the axis of rotation. This translation generates a torque whose magnitude depends on the angular velocity. During the system operation a general coordinate assigned to the nonideal source grows rapidly as a result of rotation. We propose the decomposition of the equations of motion in such a way to extract the solution which is directly related to the rotation of an unbalanced rotor. The remaining part of the solution describes pure oscillation depending on the dynamical behaviour of the whole system. The decomposed equations are solved numerically. The influence of selected system parameters on the rotor vibration is examined. The presented approach can be applied to separate vibration and rotation of motions in many other engineering systems.
Double pendulum colliding with a rough obstacle.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Sypniewska-Kamińska, Grażyna; Starosta, Roman; Awrejcewicz, Jan
The 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.
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, Stanisław
The 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.
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, Jan
The 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.
Dynamics simulations of a rigid body lying on a vibrating table with the use of special approximations of the resulting contact forces.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Szewc, Michał; Wojtunik, Igor; Kudra, Grzegorz; Awrejcewicz, Jan
The work presents simulations and dynamics of a rigid body lying on a vibrating table. An attempt of shaping and control the body dynamical behavior, by the use of manipulation of parameters of the table oscillations, is presented. This work is also an implementation of the specially prepared mathematical models of friction between table and the moving body. Those models are based on the integral model assuming the fully developed sliding on the plane contact area with the foundation of any pressure distribution. In order to simplify the calculations and reduce their computational cost, special approximations of the integral models of friction force and moment are used. They are based on Padé approximants and their generalizations.
Experimental and numerical investigations of a pendulum driven by a low-powered DC motor.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Wasilewski, Grzegorz; Kudra, Grzegorz; Awrejcewicz, Jan; Kaźmierczak, Maciej; Tyborowski, Mateusz; Kaźmierczak, Marek
The work is a continuation of numerical and experimental investigations of a system consisting of a single pendulum with the joint horizontally driven by the use of a crank-slider mechanism and DC motor. The power supplied to the DC motor is relatively small when compared with our earlier investigations, which results in clear return influence of the pendulum dynamics on the DC motor angular velocity and much more rich bifurcation dynamics of the whole system, including regions of chaotic behaviour. In the experiments, the motor is supplied by the use of different but constant in time voltages. A series of experiments allow for accurate estimation of the model parameters and, in the further step, for prediction of the real system behaviour, also for other functions representing input voltage, including the time-varying ones.
Guidance of the resonance energy flow in the mechanism of coupled magnetic pendulums
(Elsevier, 2022) Pilipchuk, Valery N.; Polczyński, Krystian; Bednarek, Maksymilian; Awrejcewicz, Jan
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.
Identification of a non-linear damping coefficient characteristics in the free decay test of a single pendulum with friction.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Awrejcewicz, Jan; Olejnik, Paweł
A pendulum in form of an equal arms angle body being a part of a two degrees-of-freedom mechanical system with friction is identified with respect to the observed in uence of some resistance of its rotational motion in ball bearings. It is damped in a much more complex manner, what could be considered as a non-linear damping. There is supposed between others, that the effective non-linear damping characteristics depends on a few effects such as fluid friction caused by vibrations of the pendulum with two springs in the air, as well as unknown kinds of a frictional resistance existing in ball bearings. The model under investigation finds its real realization on a laboratory rig designed for experimental investigations of viscous and structural frictional effects. A transient response oscillations of the pendulum are described by the explicitly state-dependent free decay. A free decay test of the pendulum with the state dependent non-linear parameters of damping and stiffness has been performed in this paper. It provided interesting observations that led to elaboration of a method of the overall damping coefficient identification. Effects of application of the proposed semi-empirical method of identification of the overall damping and stiffness coefficients have been illustrated and discussed.
Influence of position of an anvil beam on vibrations of collecting electrodes.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Nowak, Andrzej; Nowak, Paweł; Awrejcewicz, Jan
The paper presents experimentally verified computational model of the collecting electrodes in electrostatic precipitator (ESP). Discretisation of the whole system, both the beams and the electrodes, is carried out by means of the modified rigid finite element method (RFEM). The RFEM allows easy to reflect mass and geometrical features of the considered. Moreover, the method is convenient for the introduction of additional concentrated masses which reflects elements such as the joints that fasten the electrodes to the suspension beam, distance-marking bushes, riveted or screw joints, etc... The system of nine electrodes 16 m long and several various positions of an anvil beam in this system are analyzed. Position of the anvil beam in the system has essential influence on tangent and normal accelerations at different points of the plates, and thus on the effectiveness of the dust removal process. The aim of the simulations is to find optimal position of an anvil beam, which guaranties the maximum amplitudes of vibrations and their proper distribution in the plates.
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, Bernd
Nonlinear 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.
Mathematical model for two-dimensional dry friction modified by dither.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Wijata, Adam; Awrejcewicz, Jan; Matej, Jan; Makowski, Michał
A new dynamic two-dimensional friction model is developed which is based on the bristle theory. Actually it is the Reset Integrator Model converted into a two-dimensional space. Usually two-dimensional friction models are indeed onedimensional models which are rotated into the slip velocity direction. However, this often used approach cannot be applied to the bristle model. That is why an idea of a two-dimensional bristle is presented. Bristle’s deformation is described using polar coordinates. The carried out numerical simulation of a planar oscillator has proved that the new model correctly captures the mechanism of smoothing dry friction by dither applied via perpendicular and co-linear way regarding the body velocity. Furthermore, the introduced mathematical model captures two-dimensional stick-slip behaviour. The Cartesian slip velocity components are the only inputs to the model. In addition, our proposed model allows to describe a friction anisotropy using the bristle parameters. The paper contains results of an experimental verification of the new friction model conducted on the special laboratory rig being used to investigate the two dimensional motion in the presence of dither as well as to validate our numerical results.
Mathematical model of a multi-parameter oscillator based on a core-less three-phase linear motor with skewed magnets.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Gajek, Jakub; Kępiński, Radosław; Awrejcewicz, Jan
This paper uses the example of a three-phase core-less linear motor to create a mathematical model of single-dimension multi-parameter oscillator. The studied linear motor consists of: a stator, an U-shaped stationary guide-way with permanent magnets placed askew to the motor’s movement’s direction; and a forcer, a movable set of three rectangular coils subjected to alternating external electrical voltage. The system's parameters are both mechanical (number of magnets and coils, size of magnets, distances between magnets, size of coils) and electromagnetic (auxiliary magnetic field, permeability, coil’s resistance). Lorentz force allows for the transition from electromagnetic parameters to mechanical force and Faraday’s law of induction creates a feedback between the forcer’s speed and coils voltage. An Ampere’s model of permanent magnet is used to determine the simplified function of auxiliary magnetic field distribution throughout the stator. In the model the external voltage applied to each coil serves as the excitation while displacement of the forcer is the output parameter. The solution to the introduced mathematical model of the system is compared with the experimental results showing a good coincidence.
Mathematical model of pennate muscle.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Wojnicz, Wiktoria; Zagrodny, Bartłomiej; Ludwicki, Michał; Awrejcewicz, Jan; Wittbrodt, Edmund
The purpose of this study is to create a new mathematical model of pennate striated skeletal muscle. This new model describes behaviour of isolated flat pennate muscle in two dimensions (2D) by taking into account that rheological properties of muscle fibres depend on their planar arrangement. A new mathematical model is implemented in two types: 1) numerical model of unipennate muscle (unipennate model); 2) numerical model of bipennate muscle (bipennate model). Applying similar boundary conditions and similar load, proposed numerical models had been tested. Obtained results were compared with results of numerical researches by applying a Hill-Zajac muscle model (this is a Hill type muscle model, in which the angle of pennation is taken into consideration) and a fusiform muscle model (a muscle is treated as a structure composed of serially linked different mechanical properties parts).
Mechanical systems with two nonlinear springs connected in series.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Starosta, Roman; Sypniewska-Kamińska, Grażyna; Awrejcewicz, Jan
The aim of the paper is analysis of dynamical regular response of the nonlinear oscillator with two serially connected springs of cubic type nonlinearity. Behaviour of such systems is described by a set of differential-algebraic equations (DAEs). Two examples of systems are solved with the help of the asymptotic multiple scales method in time domain. The classical approach has been appropriately modified to solve the governing DAEs. The analytical approximated solution has been verified by numerical simulations.
Modelling and numerical simulations of a pendulum elastically suspended and driven by frictional contact with rotating disk.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Kudra, Grzegorz; Awrejcewicz, Jan
The work concerns modelling and numerical simulations of a special kind of physical pendulum frictionally driven. The pendulum’s joint is suspended elastically in the plane of the motion resulting in the full plane motion of the pendulum and in tree degrees of freedom of the analysed mechanical system. The pendulum is driven by frictional contact with a disk with a constant angular velocity. Examples of self-excited oscillations and bifurcation dynamics of the pendulum are presented. Majority of the work focuses on efficient approximate modelling of the resultant friction force and moment occurring on the contact surface.
Occurrence of strick-slip phenomenon.
(Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2007) Awrejcewicz, Jan; Olejnik, Paweł
A self-excited system wit h unilateral friction contact modelled by a two degrees-of-freedom mechanical system, where the normaI farce varies during displacement of a block has been studied. The constructed real laboratory rig approximates the investigated system, and it includes feedback reinforcement of the friction force acting on the vibrated block. Some methods of data acquisition and data handling procedures are proposed for experimentally observed results and data collection. A novel static friction force model for both positive and negative velocities of the base is proposed.
On the clutch dynamics with the approximations of resulting contact forces.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Wojtunik, Igor; Szewc, Michał; Kudra, Grzegorz; Awrejcewicz, Jan
The aim of the work is to present effectiveness of numerical simulations of clutch with frictional contacts, elaborated by some of the current articles in the last few years. The friction models assumes full developed sliding and the classical Coulomb friction law on each element of the contact with general shape and any pressure distribution. Then special modification of the integral model of friction force and moment are proposed. The approximants based on Padé approximants and their generalizations. In the work the clutch dynamics with contacts forces model is presented. The system was simplified to friction disk on rotating master disk. Two different configurations are investigated: coaxial and non-coaxial arrangement of the two disks. For the appropriate rotations speed, the system of non-coaxial arrangement can exhibit the ability to self-centering. The models based on generalizations of Padé approximants are compared with the simulation results obtained by the use of approximants based on Taylor’s expansion and models with exact integral expressions for friction force and torque components.