Prof. Czesław Bajer, Ph.D., Dr. Habil., Eng. 

Habilitation thesis
1998  Numeryczne modelowanie czasoprzestrzenne dynamicznych zagadnień kontaktowych 
Professor
20101223  Title of professor 
Supervision of doctoral theses
1.  20141027  Bajkowski Jacek (PW)  Vibrations of sandwich beams controlled by smart materials  1075  
2.  20120426  Pisarski Dominik  SemiActive Control System for Trajectory Optimization of a Moving Load on an Elastic Continuum  656  
3.  20090326  Dyniewicz Bartłomiej  Dynamiczne właściwości układu hybrydowego poddanego ruchomym źródłom zaburzeń  628  
4.  20070410  Konowrocki Robert  Oddziaływanie koła z drogą z uwzględnieniem poślizgów bocznych  605 
Recent publications
1.  Bajer C.I., Pisarski D., Szmidt T., Dyniewicz B., Intelligent damping layer under a plate subjected to a pair of masses moving in opposite directions, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2017.01.046, Vol.394, pp.333347, 2017 Abstract: Reducing displacements of a plate vibrating under a pair of masses traveling in opposite directions can be improved by adding a smart subsoil instead of a classical damping layer. We propose a material that acts according to the instantaneous state of the plate, i.e., its displacements and velocity. Such an intelligent damping layer reduces vertical displacements even by 40%–60%, depending on the type of load and the assumed objective function. Existing materials enable the application of the proposed layer in a semiactive mode. The passive mode can be applied with materials exhibiting directiondependent viscosity. Keywords:Plate vibration, Moving load, Intelligent damping layer, Semiactive damping Affiliations:
 
2.  Szmidt T., Pisarski D., Bajer C.I., Dyniewicz B., Doublebeam cantilever structure with embedded intelligent damping block: Dynamics and control, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2017.04.033, Vol.401, pp.127138, 2017 Abstract: In this paper, a semiactive method to control the vibrations of twin beams connected at their tips by a smart damping element is investigated. The damping element can be made of a magnetorheological elastomer or a smart material of another type, for instance, vacuum packed particles. What is crucial is the ability to modify the storage and loss moduli of the damping block by means of devices attached directly to the vibrating structure. First, a simple dynamical model of the system is proposed. The continuous model is discretized using the Galerkin procedure. Then, a practical statefeedback control law is developed. The control strategy aims at achieving the best instantaneous energy dissipation of the system. Numerical simulations confirm its effectiveness in reducing free vibrations. The proposed control strategy appears to be robust in the sense that its application does not require any knowledge of the initial conditions imposed on the structure, and its performance is better than passive solutions, especially for the system induced in the first mode. Keywords:Vibration control, Doublebeam structure, Sandwich beam, Magnetorheological elastomer, Semiactive damping, Stabilization Affiliations:
 
3.  Dyniewicz B., Pisarski D., Bajer C.I., Vibrations of a Mindlin plate subjected to a pair of inertial loads moving in opposite directions, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2016.09.027, Vol.386, pp.265282, 2017 Abstract: A Mindlin plate subjected to a pair of inertial loads traveling at a constant high speed in opposite directions along arbitrary trajectory, straight or curved, is presented. The masses represent vehicles passing a bridge or track plates. A numerical solution is obtained using the space–time finite element method, since it allows a clear and simple derivation of the characteristic matrices of the timestepping procedure. The transition from one spatial finite element to another must be energetically consistent. In the case of the moving inertial load the classical timeintegration schemes are methodologically difficult, since we consider the Dirac delta term with a moving argument. The proposed numerical approach provides the correct definition of force equilibrium in the time interval. The given approach closes the problem of the numerical analysis of vibration of a structure subjected to inertial loads moving arbitrarily with acceleration. The results obtained for a massless and an inertial load traveling over a Mindlin plate at various speeds are compared with benchmark results obtained for a Kirchhoff plate. The pair of inertial forces traveling in opposite directions causes displacements and stresses more than twice as large as their corresponding quantities observed for the passage of a single mass. Keywords:Mindlin plate, Mass moving at varying speed, Arbitrary trajectory, Inertial load, Space–time finite element method Affiliations:
 
4.  Bajer C.I., Dyniewicz B., Shillor M.^{♦}, A Gao beam subjected to a moving inertial point load, Mathematics and Mechanics of Solids, ISSN: 10812865, DOI: 10.1177/1081286517718229, pp.112, 2017 Abstract: A model for the dynamics of a Gao elastic or viscoelastic nonlinear beam that is subject to a horizontally moving vertical pointforce is modeled and computationally studied. In particular, the behavior and vibrations of the beam as the mass is moving on it is investigated. Such problems arise naturally in transportation systems with rails. A timemarching finite element numerical algorithm for the problem is developed and implemented. Results of representative simulations are depicted and compared to the behavior of a linear Euler beam with a moving mass. Keywords:Gao beam, moving inertial point load, Finite Element simulations, beam vibration Affiliations:
 
5.  Bajkowski J.M.^{♦}, Dyniewicz B., Bajer C.I., Semiactive damping strategy for beams system with pneumatically controlled granular structure, MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 08883270, DOI: 10.1016/j.ymssp.2015.09.026, Vol.7071, pp.387396, 2016 Abstract: The paper deals with a control method for semiactive damping of a double beam system with a smart granular structure placed in a thin silicone envelope. The damping properties of the system are controlled pneumatically, by subjecting the granular material to underpressure at particular moments. A mathematical model of the layered beam with a granular damping structure is represented by the two degrees of freedom, modified Kelvin–Voigt model of two masses, a spring with controllable stiffness, and a viscous damper with a variable damping coefficient. The optimal control problem is posed, using the concept of switching of the parameters to increase the efficiency of suppressing the displacement׳s amplitude. The resulting control strategy was verified experimentally for free vibrations of a cantilever system. The research proved that the appropriate, periodic switching of the properties of the considered structure enables reducing the vibration more effectively than if the material is treated passively. Keywords:Granular materials, Smart materials, Adaptive control, Vibration damping Affiliations:
 
6.  Pisarski D., Bajer C.I., Dyniewicz B., Bajkowski J.M.^{♦}, Vibration control in smart coupled beams subjected to pulse excitations, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2016.05.050, Vol.380, pp.3750, 2016 Abstract: In this paper, a control method to stabilize the vibration of adjacent structures is presented. The control is realized by changes of the stiffness parameters of the structure׳s couplers. A pulse excitation applied to the coupled adjacent beams is imposed as the kinematic excitation. For such a representation, the designed control law provides the best rate of energy dissipation. By means of a stability analysis, the performance in different structural settings is studied. The efficiency of the proposed strategy is examined via numerical simulations. In terms of the assumed energy metric, the controlled structure outperforms its passively damped equivalent by over 50 percent. The functionality of the proposed control strategy should attract the attention of practising engineers who seek solutions to upgrade existing damping systems. Keywords:vibration, damping, smart materials, control, semiactive Affiliations:
 
7.  Bajkowski J.M.^{♦}, Bajer C.I., Dyniewicz B., Pisarski D., Vibration control of adjacent beams with pneumatic granular coupler: an experimental study, Mechanics Research Communications, ISSN: 00936413, DOI: 10.1016/j.mechrescom.2016.10.005, Vol.78, pp.5156, 2016 Abstract: A novel type of pneumatic device filled with granular material is proposed in the implementation of a switched control strategy to stabilize the vibration of slender structures. The analytically obtained control law for the airtight, elastic, granular coupler is implemented in a test structure with a relaytype control logic. In the experiment, the deformable granular coupler semiactively damps an initially deflected pair of adjacent, aluminum beams. Two cases of initial excitation are considered, showing an improvement of up to 33 percent in vibration abatement efficiency compared to the passive case. Although this semiactive device is conceptually simple, its ease of operation and low cost should attract the attention of engineers who seek solutions that can be used to build new structures and upgrade existing ones. Affiliations:
 
8.  Pisarski D., Szmidt T., Bajer C.I., Dyniewicz B., Bajkowski J.M.^{♦}, Vibration Control of DoubleBeam System with Multiple Smart Damping Members, SHOCK AND VIBRATION, ISSN: 10709622, DOI: 10.1155/2016/2438902, Vol.2016, pp.2438902114, 2016 Abstract: A control method to stabilize vibration of a double cantilever system with a set of smart damping blocks is designed and numerically evaluated. The externally controlled magnetorheological sheared elastomer damping block is considered, but other smart materials can be used as well. The robust bangbang control law for stabilization the bilinear system is elaborated. The key feature of the closed loop controller is the efficiency for different types of initial excitement. By employing the finite element model, the performance of the controller is validated for strong wind blow load and concentrated impact excitement of the particular point of one of the beams. For each of the excitations, the closed loop control outperforms the optimal passive damping case by over 27% for the considered energy metric. Affiliations:
 
9.  Wasilewski M., Pisarski D., Bajer C.I., Adaptive stabilization of partially damaged vibrating structures, Machine Dynamics Research, ISSN: 20809948, Vol.40, No.1, pp.6582, 2016 Abstract: In this paper, an online adaptive continuoustime control algorithm will be studied in the vibration control problem. The examined algorithm is a Reinforcement Learning based scheme able to adapt to the changing system’s dynamics and providing control converging to the optimal control. Firstly, a brief description of the algorithm is provided. Then, the algorithm is studied by the numeric simulation. The controlled model is a simple conjugate oscillator with a sudden change of its rigidity. The effectiveness of the adaptation of the algorithm is compared to the simulation results of controlling the same object by the traditional Linear Quadratic Regulator. Because of the lack of constraints for a system size or its linearity, this algorithm is suitable for optimal stabilization of more complex vibrating structures. Keywords:Vibration control, Adaptive control, Optimal control, Policy iterations, HamiltonJacobiBellman equation Affiliations:
 
10.  Dyniewicz B., Konowrocki R., Bajer C.I., Intelligent adaptive control of the vehiclespan/track system, MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 08883270, DOI: 10.1016/j.ymssp.2014.12.007, Vol.5859, pp.114, 2015 Abstract: This paper presents the strategy of semiactive damping of vibrations of a beam span subjected to a moving load. Intermediate supports as controlled dampers significantly decrease transverse displacements in comparison with a system with permanently active dampers. The gain can reach 40% in the case of high speed loads. In a real structure with a load moving at 3 m/s, considered in this paper, the improvement is about 10%. The control is determined by a minimization procedure. Numerical simulations are confirmed experimentally on a stand with a length of 4 m. Controlled dampers can be replaced with an intelligent material. The potential applications are in transport or robotics. Keywords:Moving inertial load, Intelligent adaptive control, Semiactive damping, Beam vibration Affiliations:
 
11.  Dyniewicz B., Bajkowski J.M.^{♦}, Bajer C.I., Semiactive control of a sandwich beam partially filled with magnetorheological elastomer, MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 08883270, DOI: 10.1016/j.ymssp.2015.01.032, Vol.6061, pp.695705, 2015 Abstract: The paper deals with the semiactive control of vibrations of structural elements. Elastomer composites with ferromagnetic particles that act as magnetorheological fluids are used. The damping coefficient and the shear modulus of the elastomer increases when it is exposed to an electromagnetic field. The control of this process in time allows us to reduce vibrations more effectively than if the elastomer is permanently exposed to a magnetic field. Semiactive control, Beam vibration, Magnetorheological elastomer, Sandwich beam, Damping Affiliations:
 
12.  Bajkowski J.M.^{♦}, Dyniewicz B., Bajer C.I., Damping properties of a beam with vacuumpacked granular damper, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2014.12.036, Vol.341, pp.7485, 2015 Abstract: An experimental study of the properties of a layered beam partially treated with a damping element based on a granular material is presented. The beam structure comprises two aluminium face strips connected at the tip by a hermetic, elastic envelope, filled with bulk granules. Changing the underpressure value inside the airtight envelope allows variation of the mechanical properties of such a complex system, like stiffness or damping coefficients. Four types of granules, different in size, shape, and material, were examined to find the most promising one. A detailed discussion of the experimental amplitude, frequency, and damping capacity of the cantilever is given. The Zener, Kelvin–Voigt, and classic Maxwell models were employed for modelling and parameter identification. The range of applicability and limitations of the proposed solution has been given, as well as the benefits from the application. Keywords:Granular materials, smart materials, vibrations Affiliations:
 
13.  Dyniewicz B., Bajer C.I., Matej J.^{♦}, Mass splitting of train wheels in the numerical analysis of high speed train–track interactions, Vehicle System Dynamics, ISSN: 00423114, DOI: 10.1080/00423114.2014.982659, Vol.53, No.1, pp.5167, 2015 Abstract: We demonstrate that the dynamic simulation of a vehicle moving on a track requires the correct mass distribution in the wheel–rail system. A wheel travelling on a rail should be modelled as a pair of masses coupled as a double mass oscillator. One of the masses is attached to the rail and carries the moving inertial load, while the second one is treated classically, being connected to the rail only through an elastic spring. This model is called the ‘mass splitting model’. The classical approach overestimates the accelerations by a factor of 10. The presented method produces displacements and velocities which agree well with the results of a precise finite element method and with measurements. Some reallife problems of a vehicle moving on a track at high speed are solved numerically by own computer program and the results are compared with measurements and with the solutions obtained using other codes. Keywords:moving mass, numerical mass modelling, wheel–rail interaction Affiliations:
 
14.  Dyniewicz B., Pręgowska A., Bajer C.I., Adaptive control of a rotating system, MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 08883270, DOI: 10.1016/j.ymssp.2013.09.006, Vol.43, pp.90102, 2014 Abstract: In the present paper, an adaptive control of structural vibrations is presented. Based on earlier research, we claim that the periodical switching on of magnetorheological controlled dampers results in the reduction of the amplitudes of vibrations more than does their permanent actuation. This statement, when applied to a moving load problem, was mathematically proved in earlier papers. In the present paper we determine the efficiency of such a control applied to a rotating shaft. The earlier mathematical analysis allows us to propose a control strategy. A finite element simulation together with the solution of the control problem shows that the dampers should act only during a short period of the highest displacements of the structure. The same conclusion is found in experimental tests. Although high frequency control with MR dampers is less efficient than in the theoretical investigations, we have found an amplitude reduction in the range of 10–20%. Keywords:Adaptive control, Semiactive control, Vibration control, Shaft vibrations, Torsional vibrations, Magnetorheological dampers Affiliations:
 
15.  Dyniewicz B., Bajer C.I., New Consistent Numerical Modelling of a Travelling Accelerating Concentrated Mass, World Journal of Mechanics, Vol.2, No.6, pp.281287, 2012 Abstract: This paper deals with vibrations of structures subjected to moving inertial loads. In literature structures are usually subjected to massless forces. In numerical applications, however, the direct influence of the inertia of a moving object is usually neglected since the characteristic matrices, although simple, can not be easily derived. The paper presents a direct, noniterative treatment of the motion of a mass along the finite element edge. The general characteristic matrices of finite elements that carry an inertial particle are given an d can be applied directly to almost all types of structures. Numerical tests and a comparison with examples from the literature and especially with analytical results, prove the efficiency and accuracy of our analysis. Keywords:Vibrations, Moving Mass, Moving Inertial Load, Time Integration Affiliations:
 
16.  Pisarski D., Bajer C.I., Smart suspension system for linear guideways, JOURNAL OF INTELLIGENT AND ROBOTIC SYSTEMS, ISSN: 09210296, DOI: 10.1007/s1084601094507, Vol.62, pp.451466, 2011 Abstract: This paper presents a new method for the semiactive control of the span system of linear guideways subjected to a travelling load. Two elastic beams are coupled by a set of controlled dampers. The relative velocity of the spans provides an opportunity for efficient control via semiactive suspension. The magnitude of the moving force is assumed to be constant by neglecting inertial forces. The response of the system is solved in modal space. The full analytical solution is based on the power series method and can be given over an arbitrary time interval. The control strategy is formulated by using bilinear optimal control theory. As a result, bangbang controls are taken into account. The final solution is obtained as a numerical mean value. Several examples demonstrate the efficiency of the proposed method. The controlled system outperforms passive solutions over a wide range. Due to the simplicity of its design, the presented solution should be interesting to engineers. Keywords:Semiactive control, Smart suspension system, Vibration control, Linear guideway, Moving load Affiliations:
 
17.  Pisarski D., Bajer C.I., Semiactive control of 1D continuum vibrations under a travelling load, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2009.09.006, Vol.329, pp.140149, 2010 Abstract: The paper presents a method for computing the response of a 1D elastic continuum supported by a set of semiactive viscous dampers and induced by a load travelling over it. The magnitude of the moving force has been assumed to be constant by neglect of the inertia forces. Full analytical solution is based on the power series method and is given in an arbitrary time interval. The timemarching scheme allows us to continue a solution in successive layers with initial conditions taken from the end of previous stages. The semiactive open loop control strategy is proposed. Shapes of damping functions are defined as a form of piecewise constant function. The control strategy is suboptimal and it outperforms the passive case. Numerical results are presented for the cases of a string and a Bernoulli–Euler beam. Affiliations:
 
18.  Dyniewicz B., Bajer C.I., New feature of the solution of a Timoshenko beam carrying the moving mass particle, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.62, No.5, pp.115, 2010 Abstract: The paper deals with the problem of vibrations of a Timoshenko beam loaded by a travelling mass particle. Such problems occur in a vehicle/track interaction or a power collector in railways. Increasing speed involves wave phenomena with significant increase of amplitudes. The travelling speed approaches critical values. The moving point mass attached to a structure in some cases ca n exceed the mass of the structure, i.e. a string or a beam, locally engaged in vibrations. In the literature, the travelling inertial load is often replaced by massless forces or oscillators. Classical solution of the motion equation may involve discussion concerning the contribution of the Dirac delta term, multiplied by the acceleration of the beam in a moving point in the differential equation. Although the solution scheme i s classical and successfully applied to numerous problems, in the paper the Lagrange equation of the second kind applied to the problem allows us to obtain the final solution with new features, not reported in the literature. In the case of a string or the Timoshenko beam, the inertial particle trajectory exhibits discontinuity and this phenomenon can be demonstrated or proved mathematically in a particular case. In practice, large jumps of the travelling inertial load is observed. Keywords:moving mass, travelling inertial load, Timoshenko beam, Lagrange equation Affiliations:
 
19.  Konowrocki R., Bajer C.I., Vibrations of railroad due to the passage of the underground train, ARCHIVES OF TRANSPORT, ISSN: 08669546, Vol.22, No.1, pp.6182, 2010 Abstract: In the paper we present results of vibration measurements in the train and on the base of the railroad in tunnels of Warsaw Underground. Measurements were performed at straight and curved sections of the track. The paper is focused on the influence of the lateral slip in rail/wheel contact zone on the generation of vibrations and a noise. Vibrations were analyzed in terms of accelerations, velocities or displacements as a function of time and frequency. Results ware compared with the experiment of rolling of the wheel with lateral sleep. In both cases we observed double periodic oscillations. Keywords:dynamic traintrack interaction, ground borne vibrations, curved and straight track, rolling Affiliations:
 
20.  Dyniewicz B., Bajer C.I., Symulacja komputerowa ruchomych obciążeń inercyjnych, DROGI I MOSTY, ISSN: 16431618, Vol.1, pp.530, 2010 Abstract: W pracy przedstawiono algorytmy numeryczne metody elementów czńczonych, dotyczące analizy drgań konstrukcji pod ruchomym obciążeniem bezwładnościowym. Niektóre problemy dynamiki konstrukcji trudno jest rozwiązać metodą elementów skończonych, stosowaną do zmiennych przestrzennych i metodą Newmarka, stosowaną do zmiennej czasu. Osobliwe cechy analitycznych rozwiązań równań różniczkowych, opisujących drgania wywołane ruchomym punktem masowym, muszą znaleźć swoje odzwierciedlenie również w ich rozwiązaniach numerycznych. Duże gradienty przebiegu rozwiązań, skoki wartości lub nieciągłości rozwiązań trudno jest uzyskać numerycznymi metodami dyskretnymi. Metody te same wymagają przybliżeń i wnoszą błędy, których oszacowanie jest trudne. W pracy omawiamy rozwiązania numeryczne, pozwalające uzyskać wyniki dokładne w pełnym zakresie prędkości przejazdu obciążenia bezwładnościowego. Affiliations:
 
21.  Pisarski D., Bajer C.I., On the semiactive control of carrying structures under moving a load, VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 08606897, Vol.24, pp.325330, 2010 Abstract: In this paper we address a group of recent research focused on the semi active control problems in carrying structures systems subjected to a travelling load. The magnitude of the moving force is assumed to be constant by neglecting inertial forces. The response of the system is solved in modal space. The optimal control problem is stated and it is solved by using of Pontryagin Maximum Principle. Switching control method is verified by numerical examples. The controlled system widely outperforms passive solutions. Due to its simplicity in practical design, the presented solution should be interesting to engineers. Keywords:Semiactive control, structural control, optimization, moving load Affiliations:
 
22.  Dyniewicz B., Bajer C.I., Numerical methods for vibration analysis of Timoshenko beam subjected to inertial moving load, VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 08606897, Vol.24, pp.8792, 2010 Abstract: The paper deals with the problem of modelling of the moving mass particle in numerical computation by using the finite element method in one dimensional wave problems in which both the displacement and angle of the pure bending are described by linear shape functions. The analysis is based on the Timoshenko beam theory. We consider the simply supported beam, in a range of small deflections with zero initial conditions. Keywords:numerical method, moving mass, moving inertial load, vibrations Affiliations:
 
23.  Bajer C.I., Dyniewicz B., Virtual functions of the space–time finite element method in moving mass problems, COMPUTERS AND STRUCTURES, ISSN: 00457949, DOI: 10.1016/j.compstruc.2009.01.007, Vol.87, pp.444455, 2009 Abstract: Classical time integration schemes fail in vibration analysis of complex problems with moving concentrated parameters. Moving mass problems and moving support problems belong to this group. Commercial systems of dynamic simulations do not support such an analysis. Moreover, the classical finite element method with the Newmarktype time integration method does not allow us to obtain convergent results at all. The reason lies in the impossibility of full mathematical consideration of the time integration stage and the analysis of inertial terms of a travelling mass. Both of them, unfortunately, are decoupled. In this paper we propose an efficient and exact numerical approach to the problem by using the space–time finite element method. We derive characteristic matrices of the discrete element of the string and the Bernoulli–Euler beam that carry the concentrated mass. We present four types of virtual functions in time and we apply two of them to the practical analysis. Displacements in time obtained numerically are compared with semianalytical results. Almost perfect coincidence proves the efficiency of the approach. Keywords:Space–time finite element method, Vibrations, Virtual function, Moving mass Affiliations:
 
24.  Dyniewicz B., Bajer C.I., Paradox of the particle's trajectory moving on a string, ARCHIVE OF APPLIED MECHANICS, ISSN: 09391533, DOI: 10.1007/s0041900802229, Vol.79, No.3, pp.213223, 2009 Abstract: This paper deals with the paradoxical properties of the solution of string vibration under a moving mass. The solutions published to date are not simple enough and cannot be applied to investigations in the entire range of mass speeds, including the overcritical range. We propose a formulation of the problem that allows us to reduce the problem to a secondorder matrix differential equation. Its solution is characteristic of all features of the critical, subcritical, and overcritical motion. Results exhibit discontinuity of the mass trajectory at the end support point, which has not been previously reported in the literature. The closed solution in the case of a massless string is analyzed and the discontinuity is proved. Numerical results obtained for an inertial string demonstrate similar features. Small vibrations are analyzed, which is why the effect discussed in the paper is of purely mathematical interest. However, the phenomenon results in complexity in discrete solutions. Keywords:Moving mass, Vibrations of string, Inertial load Affiliations:
 
25.  Bajer C.I., Dyniewicz B., Numerical modelling of structure vibrations under inertial moving load, ARCHIVE OF APPLIED MECHANICS, ISSN: 09391533, DOI: 10.1007/s0041900802848, Vol.79, pp.499508, 2009 Abstract: Inertial loading of structures by mass travelling with nearcritical velocity has been intensively debated. In the literature a moving mass is replaced by an equivalent force or an oscillator that is in permanent contact with the structure. A direct mass matrix modification method frequently implemented in the finite element approach gives reasonable results only in the range of relatively low velocities and for low mass value if compared with the mass of a structure. However, existing solutions are incorrect and are not implemented in commercial computer codes. In this paper we present the space–time finite element approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the spacetime finite element domain. Resulting characteristic matrices include inertia, Coriolis and centrifugal forces. Simple modification of matrices in the discrete equations of motion allows us to gain accuracy in a wide range of velocity, up to the overcritical speed. Numerical examples of string and beam vibrations prove the simplicity and efficiency of the method. Keywords:moving mass, inertial load, space–time finite element method, vibrations Affiliations:
 
26.  Konowrocki R., Bajer C.I., Friction rolling with lateral slip in rail vehicles, JOURNAL OF THEORETICAL AND APPLIED MECHANICS, ISSN: 14292955, Vol.47, No.2, pp.275293, 2009 Abstract: The paper deals with the dynamic phenomena accompanying the wheel rolling over a road (rail, track), with lateral slip effects. They occur in rolling of a wheel and wheelset on a straight track in the case of lateral load and especially on curves. Different curvature radii and rotary oscillations of wheelsets result in skew rolling and, in turn, in lateral slip oscillation in the contact zone between the wheel and rail. It significantly increases noise and wear in real structures. Double periodicity of motion was detected in an underground train. Hitherto, this phenomenon has not been reported in the literature. Experimental investigation was performed on a test stand for various parameters: the angle of skew rolling, velocity and contact pressure. Results were related to a two degreeoffreedom theoretical system. In the case of steel/polyester and polyamide/polyester friction pair, qualitatively similar results were obtained. Keywords:rolling, friction, lateral slip, wheelrail interaction Affiliations:
 
27.  Pisarski D., Bajer C.I., Aktywne tłumienie drgań struny i belki pod obciążeniem ruchomym, DROGI I MOSTY, ISSN: 16431618, Vol.8, No.4, pp.7187, 2009  
28.  Bajer C.I., Dyniewicz B., Spacetime approach to numerical analysis of a string with a moving mass, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 00295981, DOI: 10.1002/nme.2372, Vol.78, No.10, pp.15281543, 2008 Abstract: Inertial loading of strings, beams and plates by mass travelling with nearcritical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms Affiliations:
 
29.  Dyniewicz B., Bajer C.I., Discontinuous trajectory of the mass particle moving on a string or a beam, Machine Dynamics Research, ISSN: 20809948, Vol.32, No.3, pp.6679, 2008  
30.  Dyniewicz B., Bajer C.I., Stringbeam under moving inertial load, VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 08606897, Vol.23, pp.115120, 2008 Abstract: The paper deals with the original analyticalnumerical approach to the Bernoulli—Euler beam with additional tensile effect under a moving inertial load. The authors applied the 2nd kind Lagrange equation to derive a motion differential equation of the problem. The moving mass can travel through the stringbeam with a whole range constant speed, also overcritical. The analytical solution requires a numerical calculation in the last stage and is called a semi—analytical one. Keywords:moving mass, inertial load, string, beam Affiliations:
 
31.  Konowrocki R., Bajer C.I., Vibrations due to the passage of a railway vehicle on straight and curved tracks, VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 08606897, Vol.23, pp.199204, 2008 Abstract: The paper presents the results of vibration measurements on line of railway during passages of a train at a constant speed. The measurements have been performed on a railway track at straight and curve sections as well as and inside the train on the floor. The experimental results exhibited higher amplitudes of vibrations on the curve of the track than on its straight segments. The lateral slip in rail/wheel contact zone is considered as a possible reason of such a phenomenon. Keywords:railway vibrations, dynamic traintrack interaction, ground borne vibrations, moving load Affiliations:
 
32.  Bajer C.I., Dyniewicz B., Moving inertial load and numerical modeling, VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 08606897, Vol.23, pp.6570, 2008  
33.  Konowrocki R., Bajer C.I., Oddziaływanie między nawierzchnią drogi i kołem w obecności poślizgu bocznego, ZESZYTY NAUKOWE INSTYTUTU POJAZDÓW / POLITECHNIKA WARSZAWSKA, ISSN: 1642347X, Vol.1, No.64, pp.7582, 2007 Abstract: Praca dotyczy analizy dynamicznych zjawisk towarzyszących toczeniu koła po drodze (torze), przy jednoczesnym oddziaływaniu czynników wywołujących poślizg w kierunku prostopadłym do kierunku toczenia. Przypadki takie występują przy toczeniu się koła kolejowego po szynie na odcinkach prostych w przypadku oddziaływania wiatru oraz na łukach. Występowanie poślizgów bocznych przy jeździe na łukach spowodowane jest różnicą promieni krzywizny obu szyn i oscylacjami obrotowymi zestawu kołowego w płaszczyźnie toru. Na skutek ukośnego ustawienia płaszczyzn kół względem chwilowego kierunku toczenia dochodzi do bocznych poślizgów w strefie kontaktu koła z szyną. Oscylacje te mają niekorzystny wpływ zarówno na zużycie kół i szyn, jak i na otoczenie, ze względu na emisję hałasu. Otrzymane wyniki eksperymentalne pozwoliły na opracowanie modelu matematycznego zjawiska. Zaobserwowano dwuokresowy charakter drgań wywołany siłą boczną. Zbadano wpływ prędkości toczenia, siły nacisku oraz kąta między płaszczyzną toczącego koła a drogą na dwuokresowe drgania w kierunku osiowym koła. Keywords:nawierzchnia, transport szynowy, poślizg boczny, kontakt kołoszyna Affiliations:
 
34.  Tokaj P.^{♦}, Bajer C.I., Dynamika toru z podkładami stalowymi klinowymi, DROGI I MOSTY, ISSN: 16431618, Vol.5, No.2, pp.535, 2006  
35.  Konowrocki R., Bajer C.I., Investigation of the friction phenomenon in the wheelroad interaction, VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 08606897, Vol.15, pp.173178, 2006 Abstract: Rolling contact has been investigated intensively during last decade [1, 2]. However, generation of corrugations, the fundamental problem of rolling in railway transportation, is still not completely explained. In the paper we consider dynamic phenomenon, which occur in rolling of the wheel over the rail, with lateral slip. Such cases are involved by wind blows (both on curved and straight tracks), deformation of wheels, wheelsets and rails, different linear velocity of wheels on curves and rotatory oscillations of wheelsets. Experimental investigations, Affiliations:
 
36.  Tokaj P.^{♦}, Bajer C.I., Dynamika toru typu „Y”  narzędzia komputerowe w analizie dynamicznej toru, ZESZYTY NAUKOWE INSTYTUTU POJAZDÓW / POLITECHNIKA WARSZAWSKA, ISSN: 1642347X, Vol.2, pp.151160, 2006  
37.  Bajer C.I., Bogacz R., Propagation of perturbances generated in classic track, and track with Ytype sleepers, ARCHIVE OF APPLIED MECHANICS, ISSN: 09391533, DOI: 10.1007/s0041900504038, Vol.74, pp.754761, 2005 Abstract: Railway track with classic and Yshaped sleepers or slab track is composed of two rails that are assumed to be infinitely long and joined with sleepers by viscoelastic pads. Numerous assumptions are used in railwaytrack modelling, leading to different simplifications. The twodimensional periodic model of track consists of two parallel infinite Timoshenko beams (rails) coupled with equally spaced sleepers on a viscoelastic foundation. Track dynamics, Vibrations, Moving load, Ytype track Affiliations:
 
38.  Bajer C.I., Numeryczne modelowanie czasoprzestrzenne dynamicznych zagadnień kontaktowych (Praca habilitacyjna), Prace IPPT  IFTR Reports, ISSN: 22993657, No.5, pp.1189, 1997 
List of recent monographs
1. 372  Bajer C.I., Dyniewicz B., Pisarski D., Bajkowski J.M.^{♦}, Vibration control with smart materials, IPPT PAN, pp.1240, 2015 
2. 286  Bajer C.I., Dyniewicz B., Numerical analysis of vibrations of structures under moving inertial load, Springer, pp.1294, 2012 
3. 98  Bajer C.I., Metoda elementów czasoprzestrzennych w obliczeniach dynamiki konstrukcji, Biblioteka Mechaniki Stosowanej, IPPT PAN (Warszawa), pp.1292, 2009 
List of chapters in recent monographs
1. 299  Bajer C.I., Myśliński A.^{♦}, Żochowski A.^{♦}, Dyniewicz B., Interactive computer environment for solving optimal problemsIDOS, rozdział: Optimal control problems described by PDEs., Springer, pp.101123, 2012 
2. 300  Bajer C.I., Myśliński A.^{♦}, Żochowski A.^{♦}, Dyniewicz B., Pisarski D., Interactive computer environment for solving optimal problemsIDOS, rozdział: Solving optimal control problems described by PDEs, Ed. R. Pytlak, pp.347394, 2012 
3. 486  Pisarski D., Bajer C.I., Theoretical Foundations of Civil Engineering: XVII PolishUkraïnianLithuanian Transactions, rozdział: Active Suspension Control of 1D Continuum under a Travelling Load, Oficyna Wydawnicza Politechniki Warszawskiej, red. nauk.: Wacław Szcześniak, Artur Zbiciak, pp.273278, 2009 
4. 485  Dyniewicz B., Bajer C.I., Theoretical Foundations of Civil Engineering: XIV PolishUkraïnianLithuanian Transactions, rozdział: Moving loads – analytical and numerical approaches, Oficyna Wydawnicza Politechniki Warszawskiej, pp.3748, 2006 
5. 236  Konowrocki R., Bogacz R., Bajer C.I., Symulacja w Badaniach i Rozwoju, rozdział: Badanie oddziaływania koła z drogą z uwzględnieniem poślizgów bocznych, PTSK, PW, IPPT, Warszawa, pp.124131, 2006 
Conference papers
1.  Dyniewicz B., Bajkowski J.M.^{♦}, Bajer C.I., Smart control in vibrations of structures, 6WCSCM, 6th World Conference on Structural Control and Monitoring, 20140715/0717, Barcelona (ES), pp.22272241, 2014 Abstract: A semiactive control of structural vibrations is presented as an efficient method of damping. We consider structures subjected to a load applied in stationary points or to a moving load. We claim that the periodically switched magnetorheological actuators, controlled dampers, or elastic envelopes filled with granular materials subjected to controlled underpressure, give a more efficient vibration reduction than a permanently activated ones. In our work we show the efficiency of such a control strategy applied to a beams under moving inertial load, cantilevers and rotating shaft. The mathematical analysis allowed as to propose the particular control strategy. The finite element simulation, together with the solution for the control problem, proved that the damping devices should act only for a short period of each cycle of vibration. The control function depends on the type of the structure, excitation and the type of vibrations. The efficiency of the concept was proved in the experimental tests. The considered structures exhibit the reduction of amplitudes at the range 1040% in the periodically controlled case, in comparison to the constant damping. Keywords:Mechanical vibrations, semiactive control, smart materials Affiliations:
 
2.  Dyniewicz B., Bajer C.I., General numerical description of a mass moving along a structure, XXV Symposium Vibrations in Physical Systems, 20120515/0519, PoznańBędlewo (PL), Vol.25, pp.135140, 2012 Abstract: The paper deals with vibrations of structures under a moving inertial load. The space time finite element approach has been used for a general description of the moving mass particle. Problems occur when we perform computer simulations. In the case of wave problem numerical description of the moving inertial loads requires great mathematical care. Otherwise we get a wrong solution. There is no commercial computing packages that would enable us direct simulation of moving loads, both gravitational and inertial. Keywords:spacetime finite element method, moving mass, vibrations Affiliations:
 
3.  Dyniewicz B., Bajer C.I., Nonhertzian contact model in wheel/rail or vehicle/track system, XXV Symposium Vibrations in Physical Systems, 20120515/0519, PoznańBędlewo (PL), Vol.25, pp.141146, 2012 Abstract: In the paper the springmass system describing the moving lo ad, determined with the Hertz theory, was replaced with the springmass system with an inertial part being in contact with the beam, rail, or a track. Computational problems can b e reduced significantly. Results are qualitatively and quantitatively improved, especially at higher range of the speed, related to critical values. Keywords:wheel/rail contact, traintrack system, moving loads Affiliations:
 
4.  Dyniewicz B., Bajer C.I., General numerical description of a mass moving along a structure, XXV Symposium Vibrations in Physical Systems, 20120515/0519, PoznańBędlewo (PL), pp.135140, 2012 Abstract: The paper deals with vibrations of structures under a moving inertial load. The space time finite element approach has been used for a general description of the moving mass particle. Problems occur when we perform computer simulations. In the case of wave problem numerical description of the moving inertial loads requires great mathematical care. Otherwise we get a wrong solution. There is no commercial computing packages that would enable us direct simulation of moving loads, both gravitational and inertial. Keywords:spacetime finite element method, moving mass, vibrations Affiliations:
 
5.  Dyniewicz B., Bajer C.I., The influence of inertia in moving load problems, Theoretical foundations of civil engineering : 17 PolishUkraïnianLithuanian transactions, 200909/09, Warszawa (PL), pp.7380, 2009  
6.  Pisarski D., Bajer C.I., Active Suspension Control of 1D Continuum under a Travelling Load, Theoretical foundations of civil engineering : 17 PolishUkraïnianLithuanian transactions, 200909/09, Warszawa (PL), Vol.17, pp.273278, 2009  
7.  Dyniewicz B., Bajer C.I., General approach to problems with moving mass, Theoretical foundations of civil engineering : 16 PolishUkraïnianLithuanian transactions, 200806/06, Warszawa (PL), pp.99106, 2008  
8.  Konowrocki R., Bajer C.I., Bogacz R., Badania eksperymentalne współpracy koła z drogą z uwzględnieniem poślizgów bocznych, XIV Warsztaty Polskiego Towarzystwa Symulacji Komputerowej, 20070926/0929, Zakopane (PL), Vol.14, pp.158171, 2008 Abstract: Praca opisuje zagadnienia dynamicznego oddziaływania podziemnego pojazdu szynowego na tor. Oddziaływanie to wywołuje drgania, które za pośrednictwem konstrukcji toru, tunelu oraz gruntu propagowane są na pobliskie konstrukcję i budynki. Ze względu na zagęszczenie ruchu ulicznego podziemna komunikacja jest obecnie jedyną alternatywą płynnego przemieszczania się osób w wielkich aglomeracjach miejskich. Sprzyja to rozwojowi i modernizacji podziemnej kolei miejskiej, w której jako jednym z celów stawianych projektantom jest minimalizacja powstania drgań propagujących się na otaczające środowisko W nauce również można zaobserwować wzrost zainteresowania zjawiskami powstawania i propagacji drgań w gruncie, wywołanych przejazdem pojazdów kolei podziemnej. Głównym pytaniem nurtującym badaczy jest kwestia kontroli układu pojazd/tor i zmniejszenie powstających szkodliwych wibracji i hałasu. Ze względu na wielorakie rozwiązana konstrukcyjne pojazdów i torowisk kolei podziemnej trudno jest zaproponować jednoznaczne rozwiązanie. Keywords:metro, oddziaływania podziemnego pojazdu szynowego, wibroizolacja, grunt Affiliations:
 
9.  Konowrocki R., Bajer C.I., Drgania boczne w układzie koło szyna wywołane przejazdem kolei podziemnej na łuku i na prostej, XVIII Konferencja Naukowa „Pojazdy Szynowe”, 20080917/0919, Ustroń (PL), Vol.Tom I, pp.352368, 2008 Abstract: Zagadnienia dynamicznego oddziaływania pojazdu metra na tor są analizowane w pracy. Oddziaływanie to wywołuje drgania, które za pośrednictwem konstrukcji toru, tunelu oraz gruntu propagowane są na pobliskie konstrukcję i budynki. Ze względu na zagęszczenie ruchu ulicznego podziemna komunikacja jest obecnie jedyną alternatywą płynnego przemieszczania się osób w wielkich aglomeracjach miejskich. Sprzyja to rozwojowi i modernizacji podziemnej kolei miejskiej, w której jako jednym z celów stawianych projektantom jest minimalizacja powstania drgań propagujących się na otaczające środowisko W nauce również można zaobserwować wzrost zainteresowania zjawiskami powstawania i propagacji drgań w gruncie, wywołanych przejazdem pojazdów kolei podziemnej. Ze względu na wielorakie rozwiązana konstrukcyjne pojazdów i torowisk kolei podziemnej trudno jest zaproponować jednoznaczne rozwiązanie. Głównym pytaniem nurtującym badaczy jest kwestia kontroli układu pojazd/tor i zmniejszenie powstających szkodliwych wibracji i hałasu. Keywords:Metro Warszawskie, tunel metra, kolej podziemna, badania eksperymentalne Affiliations:
 
10.  Konowrocki R., Bajer C.I., Oddziaływanie koła z drogą z uwzględnieniem wpływu sił bocznych, XVII Konferencja Naukowa „Pojazdy Szynowe”, 20060913/0915, Kazimierz Dolny (PL), pp.397403, 2006 Abstract: Analizy dynamicznych zjawisk towarzyszących toczeniu koła po drodze (torze), przy jednoczesnym oddziaływaniu czynników wywołujących poślizg w kierunku prostopadłym do kierunku toczenia są analizowane w artykule. Przypadki takie występują przy toczeniu się koła kolejowego po szynie na odcinkach prostych w przypadku oddziaływania wiatru oraz na łukach. Występowanie poślizgów bocznych przy jeździe na łukach spowodowane jest różnicą promieni krzywizny obu szyn i oscylacjami obrotowymi zestawu kołowego w płaszczyźnie toru. Na skutek ukośnego ustawienia płaszczyzn kół względem chwilowego kierunku toczenia dochodzi do bocznych poślizgów w strefie kontaktu koła z szyną. Oscylacje te mają niekorzystny wpływ zarówno na zużycie kół i szyn, jak i na otoczenie, ze względu na emisję hałasu. Otrzymane wyniki eksperymentalne pozwoliły na opracowanie modelu matematycznego zjawiska. Zaobserwowano dwuokresowy charakter drgań wywołany siłą boczną. Keywords:dwuokresowy charakter drgań, kontakt kołoszyna,numeryczne badania Affiliations:
 
11.  Tokaj P.^{♦}, Bajer C.I., Dynamika toru typu „Y”  narzędzia komputerowe w analizie dynamicznej toru, XVII Konferencja Naukowa „Pojazdy Szynowe”, 20060913/0915, Kazimierz Dolny (PL), pp.537544, 2006 
Conference abstracts
1.  Wasilewski M., Pisarski D., Konowrocki R., Bajer C.I., New efficient adaptive control of torsional vibrations induced by sudden nonlinear disturbances, CMM2017, 22nd International Conference on Computer Methods in Mechanics, 20170913/0916, Lublin (PL), pp.12, 2017 Abstract: The aim of this paper is to present a novel adaptive technique of control of the vibrating drilling systems. The algorithm constitutes an adaptive linear quadratic regulator that uses direct measurements of the disturbance to synthesize its linear dynamic approximation. This approach allows generating control law that includes the impact of the friction on the system dynamics and optimally steers the system to the desired trajectory. The effectiveness of the algorithm is validated via comprehensive numerical simulations of the control of the simplified drilling model. The results are compared to these obtained with the use of the Linear Quadratic Gaussian regulator. Keywords:vibration control, drillstring, adaptive control, autoregressive model Affiliations:
 
2.  Pisarski D., Bajer C.I., Dyniewicz B., Szmidt T., Distributed control of smart cantilever structure, ETAMM, Emerging Trends in Applied Mathematics and Mechanics, 20160530/0603, Perpignan (FR), pp.1, 2016 Abstract: The work presents a novel distributed vibration control method of twin cantilever beams system coupled by a set of magnetorheological elastomers. The control is realized by a change of magnetic field influencing elastomers’ mechanical properties. The objective is to stabilize the system with the optimal rates of decrease of the energy. The control is based on an easy for practical implementation distributed statefeedback structure. It employs a set of communication channels to exchange the state information between the neighboring controllers. The performance of the designed method is validated by means of the numerical experiments. In terms of the assumed metrics, the proposed distributed method significantly outperforms the passive case and is competitive to standard centralized control. Affiliations:
 
3.  Bajkowski J.M.^{♦}, Dyniewicz B., Bajer C.I., Smart materials in semiactive controlled structures, ETAMM, Emerging Trends in Applied Mathematics and Mechanics, 20160530/0603, Perpignan (FR), pp.84, 2016  
4.  Dyniewicz B., Bajer C.I., Bajkowski J.M.^{♦}, Vibration abatement using layered structures with smart core, XXIV FrancuskoPolskie Seminarium Mechaniki, 20161017/1018, Warszawa (PL), pp.2021, 2016  
5.  Szmidt T., Bajer C.I., Finite displacement dynamic model of twin beams with controllable damper, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.813814, 2015 Abstract: Lateral vibrations of two parallel cantilever beams joined at their free ends by a viscoelastic member are studied experimentally and theoretically. A dynamic model of the structure is proposed. It fits the experimental data well and allows to estimate the shear modulus and damping coefficient of the member. The model can be useful for the development of semiactive control strategies for doublebeam systems with controllable damping members. Keywords:sandwich beam, doublebeam system, shear deformation, magnetorheological elastomer, semiactive damping Affiliations:
 
6.  Pisarski D., Bajer C.I., Dyniewicz B., SemiActive Stabilization of Smart Structures Subjected to Impact Exctitation, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.805806, 2015 Abstract: In the work, a novel control method to stabilize vibrations of high structures is presented. The control is realized by changes of the stiffness parameters of the structural couplers. A seismic pulse excitation applied to the structure is submitted as a kinematic excitation. For such a representation the designed control law provides the best rate of the energy dissipation. Performance in different structural settings is studied by means of the stability analysis. Then, the efficiency of the proposed strategy is examined via numerical simulations. In terms of the assumed energy metric, the controlled structure outperforms its passively damped equivalent by over 50 percent. Keywords:Structural control, semiactive control, smart materials, smart buildings, stabilization Affiliations:
 
7.  Pisarski D., Bajer C.I., Moving load passage optimization via semiactive control system, CMM 2011, 19th International Conference on Computer Methods in Mechanics, 20110509/0512, Warszawa (PL), pp.18812, 2011 Abstract: In this paper the optimal control system for straight passage of moving load is considered. The magnitude of the moving force is assumed to be constant by neglecting inertial forces. The response of the system is solved in modal space. The idea of semiactive control manner for 1D continuum subjected to travelling load was first based on the numerical investigations. Then, the switching control strategy was formulated. The methods of computing the optimal switching times was developed by means of adjoint state. We present the efficient way of calculating such a switching times. The effect of predeflected guideway is also considered. Several examples demonstrate the efficiency of the proposed techniques. The controlled system widely outperforms passive solutions. Keywords:optimization, beams, sandwich structures, vibrations, industrial problems Affiliations:
 
8.  Dyniewicz B., Bajer C.I., Inertial load moving on a string  discontinuous solution, Theoretical foundations of civil engineering : 15 PolishUkraïnianLithuanian transactions, 200705/05, Warszawa (PL), pp.141150, 2007 