Dynamical Systems in Applications

This chapter contributes to the modeling, analysis and control of terrestrial artificial locomotion systems. Inspired by previous models, we set up an unconventional model for a snake-like locomotion systems in form of a chain of visco-elastically interconnected mass points in a plane with passive joints, but – in contrast to literature – active links (time-varying link-length) and rotatable skids to change the movement direction and to avoid obstacles. We investigate this model in a dynamical way and focus on controlling these link lengths to achieve a global movement, steered by the skids. From dynamics, the actuator forces have to adjust the prescribed link length for the locomotion. Since it is impossible to determine the necessary actuator forces a-priori, we apply an adaptive lambda-tracking controller to enable the system to adjust these force outputs on-line on its own. Prescribed motion patterns, i.e. specific gaits, are required to guarantee a controlled movement that differ in the number of resting mass points, the load of actuators and spikes, and the lateral forces of the skids. In contrast to literature, the investigated system of n = 10 mass points exhibit a large variety of possible gaits. To determine the most advantageous gaits, numerical investigations are performed and a weighting function offers a decision of best possible gaits. Using these gaits, a gait transition algorithm, which autonomously changes velocity and number of resting mass points depending on the spike, actuator and lateral skid force load, is presented and tested in numerical simulations.

Methods for the experimental determination of dynamic coefficients are commonly used for the analysis of various types of bearings, including hydrodynamic, aerodynamic and foil bearings. There are currently several algorithms that allow estimating bearing dynamic coefficients. Such algorithms usually use various excitation techniques applied to rotor–bearings systems. So far only a small number of scientific publications show how calculated dynamic coefficients of bearings change as speed rises. In the literature, there are no computation results that demonstrate changes in these coefficients either in a broad range of speeds (that would cover resonant speeds) or at speeds at which a phenomenon of hydrodynamic instability can be observed. This article fills the literature gap in question. For calculation purposes, the impulse response method based on an in-house algorithm (with a linear approximation using the least squares method) was applied. On its basis, the stiffness, damping and mass coefficients of a rotor–bearings system were calculated. It turns out that some of the obtained values of damping coefficients are negative at the resonant speed. Moreover, if the values are calculated at a speed at which the hydrodynamic instability phenomenon is present they are accompanied by considerably higher standard deviations. On the basis of our computation results and the literature review, capabilities and limitations of the method used for the experimental identification of dynamic coefficients of hydrodynamic bearings were discussed.

Rail vibration signals are complex data represent different phenomena occurred by the wheel-rail contact or others excitations in railway infrastructure. Such signals can be considered as source of interesting information or unwanted degradation or annoying effect. Thus proper analysis of rail vibration signals are important task for the engineering research. The paper presents results of the preliminary analysis of rail vibration signal time and frequency structures generated by different mechanisms. The research assumption was to generate vibrations in the railway rail as a result of various dynamic interactions, which arise as a result of exploitation of the railway infrastructure and vehicle traffic, including the passage of the car through the railway crossing. The aim of the research was to compare the amplitude and frequency of vibration signals and to analyze the possibilities of determination of distinguishing features from different sources signals of vibrations as forced mechanisms. In order to analyze the propagation properties of the vibration wave during the experiments, signals were recorded in 3 perpendicular axes, as carriers of information about longitudinal, transverse and vertical waves. As a result of the conducted research, data sets were collected as orthogonal vibration signals generated by rail vehicle passage and the car passage, by switching of the point mechanism and as a result of single impulse of force, as an example of repair work on the tracks. Further research will be conducted towards statistical analysis of the collected data sets and the use of dedicated signal processing methods.

The paper presents 2 DOF dynamic model of the piston-crankshaft system of HONDA NHX 110. The piston-crankshaft system was combined with pressure curve, which was obtained from the experimental research, measured in the cylinder during the execution of the working cycle of ICE engine. In the article the theoretical and analytical dependencies, which described heat generation in the cylinder, have been presented. Based on research results obtained from the experiment and simulation model the value of angular displacement, velocity and acceleration value of flywheel and shaft have been illustrated. Moreover piston displacement, velocity and acceleration has been showed. In the paper, the research results and simulation results at different angles of the ignition advance of ICE engine powered by compressed natural gas—CNG have been analyzed and compared.

This article presents an experimental evaluation based on a mathematical model and an artificial neural network (ANN) model of an energy storage system. Because of a nonlinear description of charging/discharging dynamics in subsequent cycles and a coupling of the terminal voltage and temperatures of a battery, the recurrent artificial neural network structure (R-ANN) is proposed. Both models, analytical and R-ANN were employed to predict a behavior of the VRLA AGM battery. A training and testing data were gathered at the laboratory stand in different working conditions. As a result, we present the analysis of differences between proposed modeling approaches.

We have introduced and patented a novel type of choppers with rotational shafts which can be operated at much higher speeds than classical choppers with rotational disks or with oscillatory elements. The aim of this preliminary study is to perform a Finite Element Analysis (FEA) using ANSYS of the fundamental issues of choppers with shafts, i.e. their structural stability and deformations—for their maximum possible rotational speed, of 120 krpm. The main steps of the FEA are pointed out. While different possible materials can be explored with regard to the maximum speed that the choppers are operated at, steel is considered in this study. The multi-parametric analysis is approached taking into account geometrical parameters of the device: radius and axial dimension of the shaft, shape, dimensions, and number of slits. The performances of the choppers with regard to their deformations and the highest possible rotational speed can be obtained from the analysis. Due to space limitations, preliminary results of this analysis for a one-slit shaft and the corresponding rules-of-thumb that can be extracted from it are briefly presented.

Real tests of a crash of a vehicle with lighting column are an expensive but necessary part of norm EN-126767:2008. During the design process of lighting columns, it is necessary to provide some modifications to receive sufficient safety class. To not repeat the experiments and to reduce the costs, numerical simulations are carried out, and when finally good results are received, a real crash test of a vehicle with real size streetlight can be performed. However, a truly challenging problem is the validation of the numerical models without experiments and making a good conclusion based on them. To investigate if something else than a real car crash experiments can be used, a small model of a Charpy Impact testing machine was created. The pendulum mass, the location of striking edge and radius of the striking edge can be altered in the model. The location of the mass center of the obstacle with which the crash is to be observed is also important. Experimental results from the testing machine were collected and compared with results from numerical simulations and conclusions were drawn.

In the paper the interaction of wake and the cable with passive damper is analyzed. Vortex induced vibrations (VIV) are caused by vortex shedding, located behind the cable. On the basis of formulation of the fluid forces applied to the cable in the direction of lift, it is described the coupled model where force is resulting from the coupling of cable and wake. The system of dynamical motion equations, corresponding to the coupled model, is supplemented by the equation representing the oscillating lift force acting on the vibrating cable. This equation models the near wake dynamics describing the fluctuating nature of vortex shedding. The numerical simulations are performed for the circular section of the cable. The vertical vibrations, perpendicular to the wind directions, are analyzed. In the first part of the paper the analysis of lift equation is performed for different variation of damping coefficients to fit the curve that fulfilled the fluid equations. The behavior of lift coefficient in time domain and velocity in location domain are presented. Then, for the chosen parameters of oscillating equation coming from the first part of numerical research, the dynamical behavior of the system: cable with and without damper and wake is performed. The displacements and velocities of vibrating cable with and without damper as well as the dynamical response of the wake in time domain are presented. Dimensionless amplitude of motion of the cable and the damper for different damping coefficient of the damper and amplitudes of derivatives of displacement are presented.

A planar 2 DOF model of an unbalanced rigid disc on a massless rigid shaft (rigid Jeffcott rotor) is extended considering nonlinear forces in plain journal bearings. To express the fluid-film forces in the journal bearings, several approximate analytical solutions of the Reynolds equation are used, including widely used approximations for infinitely long and infinitely short journal bearing and a method using correction polynomial functions to extend the area of aspect ratios. The differences in steady-state response of such a rotor are studied. The influence of the approximate solution type, eccentricity ratio and aspect ratio is analysed. The aim is to find out the more effective approach to journal bearing description which could be further used in detailed dynamical analyses of both stable and unstable dynamic behaviour along with nonlinear phenomena like bifurcations and transitions to chaotic motions.

Shaking table testing is the most commonly adopted method to simulate earthquake forces. This approach allows us to analyze the dynamic performance and provides a valuable insight into the dynamics of building structures, which helps to improve their future safety and reliability. The present study aims to conduct a numerical evaluation of dynamic response of a multi-storey steel structure model, which was previously examined during an extensive shaking table investigation. The experimental model was subjected to a number of different earthquake ground motions and a mining tremor. In order to perform this numerical research, the analyzed two-storey steel structure model was considered as a 2-DOF system with lumped parameters, which were determined by conducting free vibration tests. The results obtained demonstrate that not only seismic excitations but also mining tremors may considerably deteriorate structural behaviour by inducting strong structural vibrations. The time-acceleration history plots computed for the multi-storey structure model idealized as a 2-DOF system are consistent with those recorded during the previously conducted shaking table investigation, which confirms high accuracy in assuming lumped parameters to characterize the analyzed two-storey steel structure model.

The resonance phenomena occurs when the frequency of the excitation force in an oscillating system is equal to the natural frequency of the system. It will be shown, that in resonance, the inertial forces are compensated with the spring forces and the energy delivered in one cycle of vibrations is equal to the energy dissipated in an oscillator. The mechanical energy will be stored in an oscillator. In this paper the differences between energy accumulation in an oscillator and in the flywheel will be discussed. Finally the possibilities of use of the energy from resonance in machines will be shown.

Gas Metal Arc Welding (GMAW) is a popular method of material joining, widely used for a variety of critical industrial structures. Assuring high quality of joints is than a vital task. Welding is a highly dynamic and non-linear process, thus an application of time-domain or frequency-domain methods is often not suitable for evaluation of welded joints quality. To fully describe the correspondence between the geometry of welding arc, parameters that express the quality of joint, and the welding arc current, being the most important steerable parameter of a GMAW, time-frequency methods (TFM) of signal analysis should be applied. In the paper application of ensemble of STFT and EMD (Empirical Mode Decomposition)-based estimators to evaluate the stability of a GMAW process, that results in the quality of joint. Proposed method of feature extraction was applied on the real data taken during several GMAW realizations with different conditions (changes in welding current, arc voltage, shield gas flow, wire feed speed, etc.). In the active experiment process parameters were acquired. Performed investigations revealed that in comparison to traditional as well as separately used TFM, ensemble of TF estimators gave better performance in a GMAW condition assessment.

In this study we consider a 3-link biomechanical model of a human for simulating a forward fall. Individual segments of the human body are modelled as rigid bodies connected by the rotary elements which correspond to the human joints. The model implemented in Mathematica is constructed based on a planar mechanical system with a non-linear impact law modelling the hand-ground contact. Due to kinematic excitation in the joints corresponding to the hip and the shoulder, the presented fall model is reduced to a single-degree-of-freedom system. Parameters of the model are obtained based on the three-dimensional scanned human body model created in Inventor, while its kinematics (time histories of the angles in hip and shoulder joints) are obtained from the experimental observation with the optoelectronic motion analysis system. Validation of the model is conducted by means of comparing the simulation of impact force with experimental data obtained from the force plate. Finally, the obtained ground reaction forces can be useful in further studies, as a load conditions, for finite element analysis of the numerical model of the human upper extremity.

In this paper we calculated reflection and transmission coefficients of the electromagnetic radiation (light) incident on the cholesteric liquid crystal sandwiched between two isotropic optical media and a pair of polarizers. To model optical phenomena (i.e. propagation and interference of the light waves) in liquid crystal, we applied the 4 × 4 matrix method. As a result of the performed computer simulation, we obtained some interesting reflection/transmission spectra and polar plots for different parameters of the considered system and arbitrary incident monochromatic light. The illustrated and discussed results can be useful for understanding different optical systems, especially liquid crystal displays. Moreover, the applied mathematical approach can be potentially used for modelling of more advances contemporary optical systems, i.e. photonic crystals.

Rotating electrical machines are widely used in engineering and industry applications due to their reliability. The machines under study are subjected to a parametric excitation caused by an inertial thrust and a forcing excitation caused by an unbalanced force of the rotor while the entire system is being supported by nonlinear bearings with nonlinear stiffness characteristics and damping properties. The nonlinear suspension makes the analytical study very difficult, leading to strong nonlinear differential equations, which are hard to be solved through classical methods. Supplementary problems could arise in case of some horizontal rotating machines, when the gravity effect is not negligible for certain stiffness conditions. Also, the misalignment could occur in the electrical machine after some amount of running. In our paper, a new analytical approach, namely the Optimal Auxiliary Functions Method (OAFM) is employed to solve the problem of an electrical machine supported by nonlinear bearings characterized by nonlinear stiffness of Duffing type and the entire system is subjected to a parametric excitation due to the axial thrust and a forcing excitation caused by an unbalanced force of the rotor. This study contains an effective and easy to use procedure which is independent of the presence of small or large parameters in the nonlinear equations. The approximate analytical solution is in very good agreement with the numerical simulation results, which prove the reliability of this procedure.

Development of a new derivation method of a reference dynamics model of a flexible link manipulator is presented in the paper. The model including flexibility can map dynamics and performance of lightweight and fast manipulators correctly and may serve their motion analysis and control design in the presence of kinematic or programmed constraints, which are assumed to be position or first order nonholonomic. The reference dynamics model is derived using the formalism of joint coordinates and homogeneous transformation matrices. This approach allows generating dynamics equations of a manipulator without formulating additional material constraint equations. The constraints present in the reference dynamics model are the programmed ones only. The flexibility of a link is modelled using the rigid finite element method. The main advantage of this method is its ability of application of the rigid-body approach to modeling dynamics of multi-body systems with flexible links. The novelty of the presented method relies on the combination of dynamics modeling of flexible system models with the programmed constraints satisfaction problem for them. The computational algorithm underlying the derivation method presented in the paper is based on Generalized Programmed Motion Equations (GPME) approach. The reference dynamics model derivation is demonstrated for a flexible link manipulator model.

The liquid flow through a various kind of installation or devices is still not fully clarified issue. The liquid flow is assisted by a stream swirling, local pressure drops or changes of flow rates or temperature. CFD methods have been already implemented for few years to analyze phenomena related to the liquid flow. In this work Autodesk CFD Design Study Environment 2018 was used to simulate a new geometry of cavitational tunnel—a laboratory stand for examinations of cavitational resistance of structural materials. Three different geometries of the tunnel were analyzed in this work: with a cavitation initiator made of barricade and counter-barricade systems, with a cavitation initiator having a cylindrical pocket in counter-barricade and with a cavitation initiator having a double wedge shape. The introduced change of geometry allows multiplying the area of local pressure drop (i.e. the area of cavitation phenomenon). Obtained results of will serve in future (after building the new laboratory stand) to verify CFD simulations in a real testing conditions. The new tunnel geometry developed in CFD simulations should shorten evaluation time, what in turn, will give direct economic benefits (i.e. lower exploitation rate of the laboratory stand as well as lower costs of electrical energy).

10GN2MFA steel is used to produce wire and manufacturing of steam generators, pressure compensators, collectors and other equipment for nuclear power plants. In this area, there is no place to do any mistakes during manufacturing or carrying out extensive tests and producing a lot of prototypes. It is the main reason why we used modern software for numerical simulation of welding and heat treatment processes also on the very early stage of development. The aim of this paper is to describe how can welding processes be optimized by means of the numerical simulations mainly with respect to the structural changes, stresses and hardness distribution in the Heat Affected Zone (HAZ). On the real multilayer weld how to arrange whole experiment in order to obtain not only relevant input data but also verification data will be described. Additional aim of this paper is to propose mathematical description of the computational model that is usable for simulation computations of welding and heat treatment of real structure components.

The article presents the problematic aspects of detection of flat places on tram wheels using time-frequency analysis of acoustic signals. A number of pass-by tests were conducted during real life exploitation. The objects of research are light rail vehicles exploited in Poznan. Some of them were characterized by flat places on tram wheels. The research aimed to apply the wrought method for detection of flat places on tram wheels.

Some earthmoving works performed by single bucket excavators require on-line weighing of material transported by excavator’s bucket. Such measurements are critical e.g. for optimal loading of dump trucks. The first commercially available payload weighing systems for single bucket excavators were called static systems. To properly estimate bucket load, the machine equipped with them had to be kept in a standstill condition for a few seconds. Consequently, productivity of the machine was low. Recently dynamic weighing systems have been introduced. These systems enable to precisely weigh the material collected in the bucket while swinging excavator’s house as well as moving its boom, arm or bucket. Thus, they do not influence productivity of the machine. Such systems are commercially available nowadays. Development of reliable dynamic weighing systems requires certain problems to be resolved. Firstly, the influence of acceleration acting on excavator’s bodies and bucket payload has to be taken into account. Values of pressure inside cylinders supporting excavator’s boom, which are usually used as input signals for payload mass computation, significantly differ in dynamic and static conditions. In order to provide satisfactory weighing performance, internal friction of hydraulic cylinders cannot be also omitted. A prototype dynamic weighing system for single bucket excavator developed in The Department of Off-Road Machine and Vehicle Engineering (Wrocław University of Science and Technology) will be presented in the article. While estimating bucket payload the system allows for acceleration acting on excavator bodies as well as hydraulic cylinders friction.

This article deals with designing internal artificial lighting as part of the working environment, which is subject to certain rules, derived from the nature of lighting. Good lighting exerts an impact on visual comfort, which contributes to overall psychological well-being, and indirectly also to the quality and productivity of performance, to reliability and to visual performance. Currently the development of computer graphics software products exist to enable a comprehensive design and calculation of the parameters of lighting systems, which would reflect light effects that arise in artificial and day lighting. For the purposes of this paper, as to the possibilities utilisation simulations of light—technical parameters are presented simulations of the lighting design of mechanical engineering workshop created in the software DIALux.

Optimal design of multibody systems (MBS) is of primary importance to engineers and researchers working in various fields, e.g.: in robotics or in machine design. The goal of this paper is a development and implementation of systematic methods for finding design sensitivities of multibody system dynamics with respect to design parameters in the process of optimization of such systems. The optimal design process may be formulated as finding a set of unknown parameters such that the objective function is minimized under the assumption that design variables may be subjected to a variety of differential and/or algebraic constraints. The solutions of such complex optimal problems are inevitably connected with evaluation of a gradient of the objective function. Herein, a multibody system is described by redundant set of absolute coordinates. The equations of motion for MBS are formulated as a system of differential-algebraic equations (DAEs) that has to be discretized and solved numerically forward in time. The design sensitivity analysis is addressed by using the adjoint method that requires determination and numerical solution of adjoint equations backwards in time. Optimal design of sample planar multibody systems are presented in the paper. The properties of the adjoint method are also investigated in terms of efficiency, accuracy, and problem size.

In the paper authors present a concept and a modelling method of the static and dynamic loads of human body’s parts: bones and muscles during movement. Currently the problem of human body modelling is very important for many domains of our “better life” programs e.g.: an automotive—to find the best solution for human protection during accidents, sport—to find the most efficient and least tiring movements, to the health protection or extend the active life of the elderly. The human body is not a rigid multi-body system, but elastic, flexible and varying according to time. It consists of semi-stiff bones, elastic muscles and tendons. Other parts like stomach or liver are hanging on elastic wires and move relatively to each other during the whole acceleration process etc. All those elements can be broken or fatigued under some load. So the model of human body is not linear and shouldn’t be modelled by linear equation sets. Authors present the concept of human body modelling based on three types element chain. One type are “bones” that are coupled in joints and conduct loads to the support surface; the elastic tendons that keep the joints and propel bones rotation in joints, and muscles that generate forces for stabilization system or for propelling the bones. In paper the simplified model of human body built in MATLAB/Simulink software is presented. Some results of simulation e.g. load of a knee during squat or landing after jump are compared with real test results.

The paper describes a model of a light module handling system (LMHS) developed for IMR (Inspection, Maintenance and Repair) services typically performed for the seabed-located oil and gas production facilities. In order to describe dynamic performance and loads during the operation, the system is characterized by means of a multi-body model consisting both rigid and flexible links. Using the joint coordinates and homogeneous transformations the dynamics can be described by a set of differential equations of the second order and some constraint equations. The system forms several tree-like structures of bodies. The interaction between them takes place on guiding elements and lifting ropes. An important features of the handling system are flexible guide beams and prongs. The flexibility provided by those elements helps to limit some impact loads during the module docking phase. This functionality is modelled by the rigid finite elements. Lifted objects (subsea modules) are described by a set of special elements defining the hydrodynamic interaction. The work will also show some simulation results reflecting a typical operation.

This paper aims to propose a new spectral element with additional mass. Methodologies for structural health monitoring are used to include additional auxiliary mass in the structure to change of natural frequencies. Therefore, the additional auxiliary mass can enhance the effects of discontinuities in the structural dynamics response, which could improve the identification and location of the discontinuities. The proposed approach deals with the wave propagation in structures regarding the spectral analysis method. The change in the natural frequencies due the mass is examined by comparing the differences between the dynamic responses of the beam with and without additional auxiliary mass. Similar analyses also performed with the Galerkin assumed modes technique to validate the new spectral element. The proposed technique is validated with numerical simulation and then compared to experimental data.

Air temperature has a significant influence on the dynamic properties of pipes supplied with pulsating flows. In many applications (power plants, pipelines, intake and exhaust systems for internal combustion engines), air temperature has an effect on resonance. Depending on the air temperature and its influence on transient flow parameters (pressure, temperature, density, speed of sound), there may be significant changes in the dynamic properties of the test pipe, such as resonant frequencies and the damping coefficient. In this study, experiments were conducted in an air temperature range of between 288 and 343 K, with a very a short air temperature step of around 5 K. Each measurement series was performed in triplicate. The results were processed in the Matlab environment using Fast Fourier Transforms. The empirical coefficients were visualized as 3D maps, including the influence of air temperature on pulsation dynamics in pipes. Finally, the experimental results were compared with the author’s 1D model (based on the method of characteristics). The results are significant both for the theoretical understanding of flows in pipelines with pulsating flows and for practical applications in industry..

Energy consumption is a significant issue for public transport providers, because of the relatively high mass of the vehicles and the huge amounts of energy used to overcome inertial forces. In city buses, the kinetic braking energy is usually transformed into waste heat. In this paper, we propose a system to enable the recovery of braking energy in city buses, using a Kinetic Energy Recuperation System (KERS). The main assumptions of the system were elaborated in a Matlab/Simulink model of city bus dynamics, based on the real driving cycles of buses in Lodz (Poland). The following components were modelled: the flywheel supported in an active magnetic bearing, two reverse electric motors, the high ratio gears and the control system. The main objective of the control system was to balance operation of the braking and drive systems with optimal usage of the flywheel system. The charging and discharging phases of the KERS were analysed, as well as the energy stored. Fuel consumption savings were calculated via a comparison with normal city bus driving cycles without KERS. This model could be a very useful tool for research into city bus dynamics with KERS systems, providing a wide range of operational parameters for city bus powertrains.

In the paper a sensor-less and self-sensing control scheme for a bilateral teleoperation system with force-feedback based on a prediction of an input of a non-linear inverse model by prediction blocks was presented. As a part of the paper a method of a time constant estimation of the prediction block was also proposed. The prediction method of an input of an inverse model was designed to minimize the effect of the transport delay and the phase shift of sensors, actuators and mechanical objects. The solution is an alternative to complex non-linear models like NARX or artificial neural networks, which requires complex stability analysis, and control systems with high computing powers. The effectiveness of the method has been verified on the hydraulic manipulator’s test stand.

This paper uses the approach of Vibrational Mechanics (VM) performing the Method of Direct Separation of Motions (MDSM) for the analysis of a non-ideal rotor mechanism with a limited power source. We employ a modification of the method to study the governing equations of a non-ideally excited electromechanical pendulum system consisting of three masses (block, rotating, pendulum) and a DC motor. The mechanism has three degrees of freedom and we derive the main equations of the slow component of motion from the initial governing equations to allow for a derivation of analytical solutions in the stability domain. The paper focuses on a purely analytical approach.

The wave problem of propagation of shock perturbation in a semi-infinite elastic rod interacting with the medium is investigated using the model of elastic-plastic resistance with decreasing relation between shear stress and jump of displacement on the lateral surface. An exact solution of the initial-boundary problem is obtained using the Laplace transforms. A wave pattern of perturbation including the prefront zone of rest, the area of motion and the domain of stationary residual stresses has been built. The three-dimensional diagrams for nonstationary fields of displacement, velocity and stresses have been constructed too.

The achievements of the authors in the analytical modeling of hysteretic energy dissipation in the systems of shells with deformable filler at the expense of dry friction are presented. Four design variants are considered: solid shell with filler, cut shell with filler and the same systems with internal coaxial shell. Based on the one-dimensional models of shells and filler the non-conservative quasi-static problems for the dampers under nonmonotonic loading are formulated and solved. The distribution of the stresses and displacements in contact system has been studied for the processes of active loading, unloading and repeated loading. The loop of structural damping (the force-displacement diagram) is constructed too. The last obtained result describes the effect of maximum energy absorption by a shell damper. Importance of tribology settings of contact system, for which the dissipated energy of the external load reaches the maximum, is revealed.

The article deals with the numerical solution of modal analysis of a simple model. It is a system of rigid bodies resiliently mounted and bound. The solution was done in the Ansys simulation program. The article describes how to build the program. Further, some of the results of the actual frequencies and shapes of the symmetrically loaded system are shown. The results served to refine the mathematical model that solves the vertical oscillation of the symmetrically or asymmetrically loaded model with different kinematic excitation. The numerical solution of vehicle model vibration was done in MSC Adams. The results of the vertical vibration measurement of the vehicle model are also given in the article. After adjusting the boundary conditions of the numerical solution, good agreement between experimental and numerical solution (more than 90%) was achieved.

Demanding higher travel velocities of rigid body articulated wheeled vehicles and maintaining high safety while moving on stiff ground requires finding more accurate methods of adjusting steering angle between rigid frames and using special solutions allowing the driver keeping desired path. In this article, identification of factors affecting directional stability is discussed. A new mathematical algorithm for estimating vehicles directional stability is proposed and tested. Computer simulations of methods counteracting snaking behavior indicates that the speed limit of 50–60 km/h for articulated rigid body vehicles can be exceeded by using new solutions.

The paper addresses dynamic interaction in the pantograph–catenary system present in a rail vehicle. The contact force, which is measured between pantograph and catenary, may significantly fluctuate during ride due to nonlinear properties of the entire system. This may cause unexpected drops of the current flow efficiency and further power decreases. Following the relevant significance of the addressed issue, the authors performed an analysis of the influence of suspension properties of the critical pantograph’s passive components on the improvement of electric current collection on a train. The analysis was performed based on a co-simulation model for the pantograph–catenary interaction elaborated by the authors. The Finite Element catenary model assumes nonlinear droppers, large displacements and contact with the pantograph slider. To overcome limits of widely used lumped parameters model of the pantograph, the relatively more realistic Multibody model was considered. Reported in other works, the use of Multibody models, in which all properties of pantograph keep physical sense, provide wide range of design improvements for better current collection. By using the adapted model, the ability of implementation additional dampers in a pantograph structure for improvement of contact quality was investigated. Utilized pantograph model takes into account friction forces, suspension springs and aerodynamic effects. Presented results proof the ability to effectively improve current collection merely by adjusting pantograph’s passive components.

The model of water hammer in viscoelastic pipelines was considered. Additional term describing the retarded deformation of the pipe wall was added to continuity equation. System of partial differential equations describing this type of flow was analyzed using the method of characteristics and finite difference method. To determine the unsteady wall shear stress, a new effective method of solution which corresponds to Zielke (laminar flow) and Vardy-Brown (turbulent flow) models were used. The convolution integral of local pressure history and derivative from the material creep function is found similarly to the efficient Zielke convolution solution presented by Schohl. The research was carried out with the assumption of a quasi-steady and unsteady character of resistance. The comparison of numerical simulation and experimental results was presented.

The vibration of a vehicle treated as an object can be varied in two basic ways. Parametric or structural modifications can be made. Applying parametric modifications is not always possible. Examples are the suspension of motor vehicles. Changing the elasticity factor is limited by the allowable deflection arrow for different loads. Structural modifications are used for vibration isolation or vibration elimination. Vibro-isolation tasks are somewhat contradictory and practically impossible to achieve in passive systems. The alternative is to use in the suspension system elements with adjustable characteristics. On the vibrations object of the specified mass acts on the control signal. The control signal has a force dimension. This force is produced by a vibration isolator for which vibration parameters are input signals. The control force is the weighted sum of the forces of elasticity and damping. These components of the control force are proportional to relative displacement and relative velocity, respectively. In the developed concept of autonomous vibration damper, a control algorithm is applied which deactivating damper if the damping force influences the increase acceleration of the object’s vibration.

The article presents the validation of a numerical model of the electric multiple unit (EMU) driver’s cab. The subject of the study was the cab of the driver of the Impuls I rail vehicle of Newag S.A. The numerical model was developed in the LS-Dyna environment based on the documentation received from the manufacturer. The driver’s cab was modelled as shell elements, the additional parts required for the crash test were modelled as solid elements. Experimental research was carried out on the order of Newag S.A. on the experimental track of the Railway Institute in Węglewo near Zmigrod according to PN-EN 15227. The collision was recorded by 3 cameras used for fast changing phenomena. Additionally, acceleration sensors were placed at specific locations of the construction. The article presents results from experimental research and their comparison with the results of numerical simulation.

The purpose of the study is elaboration of approach for determination of functioning of chosen muscles that are essential for gait performance (Tibialis Anterior, Rectus Femoris, Gastrocnemius Medialis, Biceps Femoris). The scope of the study involves the analysis of the symmetric planar motion performing in the sagittal plane of the body by applying planar multibody model and electromyography signals (EMG) registered over normal gait performance. The analysis is performed by applying two types of multibody model: six degree of freedom system and seven degree of freedom system. Inverse dynamics task was used to calculated joint moments influenced ankle joints, knee joints and hip joints. Applied model also described single support phase and double support phase by taking into consideration the model of interaction between the ground and the contact foot. The activity states of considered muscles are determined on the base of their average activations and sequences in time.

The paper presents the results of theoretical considerations, experimental studies and control of electro-hydraulic servo control of manipulator. The structure of the device together with an analysis of its kinematic structure is described. The structure of dynamic manipulator model is derived from each axis dynamic model. The issue that has been discussed, focused on providing robust of electro-hydraulic servo controller to change the dynamic properties. The results of simulation and experimental tests data are presented and analyzed.

The horizontal coordinate’s maximization problem as well as related the brachistochrone problem are considered. The particle is moving in the vertical plane under influence of gravity, viscous drag that proportional to n-th degree of the velocity. The reaction of the basement is considered as a control. The optimal control problem is reduced to the boundary value problem for the system of two nonlinear equations. It was established that the reaction force of the basement could change its sign no more than one time, moreover, it changes only from the negative value to the positive value. The qualitative features of the optimal control allows to elaborate the results obtained in other studies.

In calculations of first natural frequency of transversal vibrations of clamped-free bars with variable cross section, one can use an energetic method. The aim of this paper is an analysis of vibrations of solid and hollow slender posts having the shape of solid of revolution with the generatrix described by an exponential curve. The authors used the Rayleigh’s method with the assumption that the shape of the post axis deflected during vibration is the same as a shape of the axis of a beam deflected by a uniform continuous static load. It was also assumed that the bars are made of an elastic and continuously distributed material what required numerous integrations of complicated expressions, therefore the calculations were carried out using the MATHEMATICA environment. Apart of the elastic energy of the bar, the authors considered also the potential energy connected to changes of location of the gravity center of elementary material slices during the post axis deflection. It was considered both the kinetic energy resulting from the replacement of the material slices perpendicular to the post axis as well as the energy connected to their rotation. The obtained results are very close to those obtained in FEM.

Our research aims the study of balancing on a rolling balance board with respect to dynamic properties such as stability and stabilizability. The goal is to identify the parameter regions where human subjects are able to keep themselves stable in the upright position for at least 60 s. The radius of the balance board and the height of the foot platform are adjusted for each individual test, which is a time demanding process. We give a preliminary design of a substituting four-bar mechanism in order to speed up the balance board experiments and to extend the limits of the parameter study. The mechanism is tunable quickly in order to imitate the motion of the balance board with different radii and platform heights; whilst the agreement of the kinematic behaviour is almost perfect for tilt angles within the region of $$\pm 30 ^{\circ }$$ . The dynamic behaviour of the mechanism and the balance board are compared based on theoretically derived stability diagrams associated with the underlying mechanical models. The balancing process is modelled by a proportional-derivative delayed feedback controller in order to account with the reaction time delay of the subject. We show that the stable parameter regions of the balance board and the mechanism are in good agreement, therefore the mechanism can be used as a substituting device for balance board.

This paper presents a numerical and experimental modal analysis of laminated glass beams, i.e. a multilayer composite structure made of glass panes bonded to an interlayer foil. These polymer foils provide shear coupling of glass layers, damping of vibrations, and play a key role in post-breakage performance. In this contribution, three-layer beams with ethylene-vinyl acetate interlayer are investigated. Using a finite element discretization and the Newton method, we solve numerically a complex eigenvalue problem which is nonlinear due to the frequency/temperature-sensitive viscoelastic behavior of the interlayer. In our experimental investigations, a roving hammer test was carried out to identify the mode shapes, natural frequencies, and modal damping. The validation shows that there is a good agreement between the numerical predictions and experimental data in natural frequencies. However, the errors in loss factors can be high, because these values are very sensitive to the material properties of polymer, frequency, temperature, and boundary conditions. These effects are discussed in the concluding part of our study.

The paper presents special methods of modeling and simulation based sensitivity analysis of torsional vibrations in the motorcycle steering system. The vibrations generated in the motorcycle steering system in the presence of freeplay and friction phenomena have a strong non-linear nature because of stick-slip processes. Due to the threshold character of these nonlinearities and the variability of the model structure, simulation-type investigations of such vibrations are difficult and still require extensive research. For solution these difficult problems, special methods of modeling and special methods of simulation analysis have been applied. The luz(…) and tar(…) projections with their original mathematical apparatus give new facilities for modeling and analysis strong non-linear vibrations. Among other, they can be used for synthesis substitutive formulas expressing time lag phenomena in such systems, they are very useful also when the model of the system is reduced parametrically. Application of Lissajou portraits and Poincare maps seems to be attractive methods not only for visualization of the non-linear vibrations, but also effective methods for analysis these spectacular signals what has been done in a simulation software.