1. |
Bajkowski J.M.^{♦}, Dyniewicz B., Bajkowski J.^{♦}, Bajer C.I., Modelling and identifying a pressurised dilatant sand to be used as a smart damping material,
MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 0888-3270, DOI: 10.1016/j.ymssp.2022.109680, Vol.184, pp.1-14, 2023Abstract: An experimental and modelling study of the properties of a prototype layered beam with a core made of a non-Newtonian sand mixture is presented. The non-typical dilatant sand was covered with an elastic envelope that restricted its movement, which allowed us to pressurise the grains by evacuating the air from within the cover. By applying controlled underpressure, the compressed sand grains become jammed, which resulted in an increased stiffness and damping. This gives the possibility to attenuate vibrations of a cantilever in an adaptive manner. The experiment was performed for free vibrations and prescribed sinusoidal base motion, to demonstrate the possibility of tuning material parameters in a vast range. The experimental amplitude, frequency and damping capacity of the kinetic sand are discussed. An analytical model is proposed to verify how many parameters are necessary to describe the material behaviour. Based on the experimental results, a parameter identification of a custom rheological model is performed and practical simplifications reducing complexity of the problem are elaborated. The performed parameter identification is indispensable for the further development of potential control strategies for effective vibration abatement of dynamic systems using such types of alternative smart materials. Keywords: Granular material, Damping, Dilatant sand, Identification, Control Affiliations:
Bajkowski J.M. | - | Warsaw University of Technology (PL) | Dyniewicz B. | - | IPPT PAN | Bajkowski J. | - | other affiliation | Bajer C.I. | - | IPPT PAN |
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2. |
Dyniewicz B., Bajkowski J.M.^{♦}, Bajer C.I., Effective Viscoplastic-Softening Model Suitable for Brain Impact Modelling,
Materials, ISSN: 1996-1944, DOI: 10.3390/ma15062270, Vol.15, pp.1-13, 2022Abstract: In this paper, we address the numerical aspects and implementation of a nonlinear viscoplastic model of the mechanical behaviour of brain tissue to simulate the dynamic responses related to impact loads which may cause traumatic injury. Among the various viscoelastic models available, we deliberately considered modifying the Norton–Hoff model in order to introduce non-typical viscoplastic softening behaviour that imitates a brain’s response just several milliseconds after a rapid impact. We describe the discretisation and three dimensional implementation of the model, with the aim of obtaining accurate numerical results in a reasonable computational time. Due to the large scale and complexity of the problem, a parallel computation technique, using a space–time finite element method, was used to facilitate the computation boost. It is proven that, after calibrating, the introduced viscoplastic-softening model is better suited for modelling brain tissue behaviour for the specific case of rapid impact loading rather than the commonly used viscoelastic models Keywords: brain biomechanics, brain injury, mechanical properties of brain tissue, viscoplastic
materials, numerical modelling, finite element method (FEM), dynamic response, acceleration,
space–time FEM Affiliations:
Dyniewicz B. | - | IPPT PAN | Bajkowski J.M. | - | Warsaw University of Technology (PL) | Bajer C.I. | - | IPPT PAN |
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3. |
Bajkowski J.M.^{♦}, Dyniewicz B., Bajer C.I., Bajkowski J.^{♦}, An experimental study on granular dissipation for the vibration attenuation of skis,
Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology, ISSN: 1754-338X, DOI: 10.1177/1754337120964015, pp.1-8, 2020Abstract: Due to the continuous deformations and irregularities of the surface of snow, alpine skis exhibit dynamic excitation, leading to drastic vibrations and decreased manoeuvrability. Therefore, attenuating these unwanted vibrations, while ensuring that the ski experience is not compromised, is an important challenge. The possibility of using granular material in a damping device is studied in this paper. A container that was partially filled with loose granules was fixed at the tip of an alpine ski to suppress vibrations by dissipating energy through collisions. The performance was verified experimentally by studying the transient response of a ski mounted in a horizontal cantilever orientation. Moreover, on-snow tests were performed. Different numbers of plastic granules were used as a dissipating material. To identify the nonlinear damping characteristics of the system, a Hilbert transform was used. In the laboratory test, the displacement amplitude decay was up to 16 percentage points higher when a granular dissipator was attached to the ski than without the damper. During field testing, acceleration amplitudes were 9% lower compared to the ski without the dissipator. This solution could possibly be adapted to other boardsports on a wide variety of terrain, including ground, water and snow. Keywords: vibration damping, alpine ski, granular material, Hilbert transform, bending beam Affiliations:
Bajkowski J.M. | - | Warsaw University of Technology (PL) | Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN | Bajkowski J. | - | other affiliation |
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4. |
Bajkowski J.M.^{♦}, Dyniewicz B., Gębik-Wrona M.^{♦}, Bajkowski J.^{♦}, Bajer C.I., Reduction of the vibration amplitudes of a harmonically excited sandwich beam with controllable core,
MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 0888-3270, DOI: 10.1016/j.ymssp.2019.04.024, Vol.129, pp.54-69, 2019Abstract: We consider a theoretical analysis and experimental test of a sandwich beam, with a core layer made of controllable material that can change its properties over time. We show that this dynamically excited beam can be sequentially controlled to obtain higher amplitude attenuation and resistance to the amplitude growth in resonant ranges than when the smart beam parameters are constant over time. Numerical simulations were performed to study the possibility of shifting beam vibration frequency towards ranges distant from resonance. An experimental study on a layered beam consisting of two steel bars with a pneumatically controlled core made of pressurized granular material was considered. A simplified control was performed to detune the beam from the resonance frequencies and reduce the vibrations by 30% in simulations and 10% in experiment. Keywords: smart material, granular structures, structural control, semi-active control, layered beam Affiliations:
Bajkowski J.M. | - | Warsaw University of Technology (PL) | Dyniewicz B. | - | IPPT PAN | Gębik-Wrona M. | - | other affiliation | Bajkowski J. | - | other affiliation | Bajer C.I. | - | IPPT PAN |
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5. |
Wasilewski M.^{♦}, Pisarski D., Bajer C.I., Adaptive optimal control for seismically excited structures,
Automation in Construction, ISSN: 0926-5805, DOI: 10.1016/j.autcon.2019.102885, Vol.106, pp.102885-1-18, 2019Abstract: A novel adaptive control for structures subjected to seismic excitation is presented. The aim of the control is to provide a high stabilizing performance involving a limited computational burden while allowing for frequent update of the control decision to cope with the changes in the excitation characteristics. Consequently, the control is based on a computationally efficient solution to the infinite-horizon linear optimal control problem, which employs the autoregressive model for excitation signals and the alpha-shift method for a performance index. Based on numerical simulations involving an actively controlled 20-story building subjected to different earthquake scenarios, we demonstrate that the adaptive control outperforms the standard LQG and H∞ regulators. Our analysis of the controller's computational complexity has confirmed that the presented method can be successfully implemented in large-scale structures that are equipped with active control devices. Our follow up research will validate the performance of the designed control on a real environment platform and we will design an adaptive controller to mitigate vibration in semi-active structures. Keywords: adaptive control, structural control, autoregressive model, vibration, optimal stabilization Affiliations:
Wasilewski M. | - | other affiliation | Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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6. |
Wasilewski M.^{♦}, Pisarski D., Konowrocki R., Bajer C.I., A new efficient adaptive control of torsional vibrations induced by switched nonlinear disturbances,
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND COMPUTER SCIENCE, ISSN: 1641-876X, DOI: 10.2478/amcs-2019-0021, Vol.29, No.2, pp.285-303, 2019Abstract: Torsional vibrations induced in drilling systems are detrimental to the condition of the machine and to the effectiveness of the engineering process. The cause of vibrations is a nonlinear and unknown friction between a drill string and the environment, containing jumps in its characteristics. Nonlinear behaviour of the friction coefficient results in self-excited vibration and causes undesirable stick-slip oscillations. The aim of this paper is to present a novel adaptive technique of controlling vibrating systems. The scheme is based on the linear quadratic regulator and uses direct measurements of the friction torque to synthesize its linear dynamic approximation. This approach allows generating a control law that takes into account the impact of the friction on the system dynamics and optimally steers the system to the desired trajectory. The controller's performance is examined via numerical simulations of the stabilization of the drilling system. The proposed solution outperforms the comparative LQG regulator in terms of the minimization of the assumed cost functional and the overall stability of the control system under the nonlinear disturbance. Keywords: vibration control, adaptive control, linear-quadratic-regulator, drilling control Affiliations:
Wasilewski M. | - | other affiliation | Pisarski D. | - | IPPT PAN | Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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7. |
Dyniewicz B., Bajer C.I., Kuttler K.L.^{♦}, Shillor M.^{♦}, Vibrations of a Gao beam subjected to a moving mass,
Nonlinear Analysis: Real World Applications, ISSN: 1468-1218, DOI: 10.1016/j.nonrwa.2019.05.007, Vol.50, pp.342-364, 2019Abstract: This paper models, analyzes and simulates the vibrations of a nonlinear Gao beam that is subjected to a moving mass or a massless point-force. Such problems arise naturally in transportation systems such as trains or trams. The dynamics of the system as the mass or the force move on the beam are investigated numerically in the cases when the vibrations are about a buckled state, and in the cases when the mass is positive or vanishes. The simulations are compared to those of the Euler–Bernoulli linear beam and the differences are highlighted. It is seen that the linear beam may be used only when the loads are small, while the Gao beam allows for moderate loads. The simulations are based on a time-marching finite elements algorithm for the model that has been developed and implemented. The results of representative and interesting computer simulations are depicted. The existence of weak solutions of the model is established using a variational formulation of the problem and results about variational set-inclusions. Keywords: dynamic vibrations, buckling of a Gao beam, moving point-load Affiliations:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN | Kuttler K.L. | - | other affiliation | Shillor M. | - | Oakland University (US) |
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8. |
Bajer C.I., Dyniewicz B., Shillor M.^{♦}, A Gao beam subjected to a moving inertial point load,
Mathematics and Mechanics of Solids, ISSN: 1081-2865, DOI: 10.1177/1081286517718229, Vol.23, No.3, pp.461-472, 2018Abstract: A model for the dynamics of a Gao elastic or viscoelastic nonlinear beam that is subject to a horizontally moving vertical point-force 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 time-marching 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:
Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN | Shillor M. | - | Oakland University (US) |
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9. |
Szmidt T., Pisarski D., Bajer C.I., Dyniewicz B., Double-beam cantilever structure with embedded intelligent damping block: Dynamics and control,
JOURNAL OF SOUND AND VIBRATION, ISSN: 0022-460X, DOI: 10.1016/j.jsv.2017.04.033, Vol.401, pp.127-138, 2017Abstract: In this paper, a semi-active 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 state-feedback 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, Double-beam structure, Sandwich beam, Magnetorheological elastomer, Semi-active damping, Stabilization Affiliations:
Szmidt T. | - | IPPT PAN | Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN |
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10. |
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: 0022-460X, DOI: 10.1016/j.jsv.2017.01.046, Vol.394, pp.333-347, 2017Abstract: 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 semi-active mode. The passive mode can be applied with materials exhibiting direction-dependent viscosity. Keywords: plate vibration, moving load, intelligent damping layer, semi-active damping Affiliations:
Bajer C.I. | - | IPPT PAN | Pisarski D. | - | IPPT PAN | Szmidt T. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN |
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11. |
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: 0022-460X, DOI: 10.1016/j.jsv.2016.09.027, Vol.386, pp.265-282, 2017Abstract: 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 time-stepping procedure. The transition from one spatial finite element to another must be energetically consistent. In the case of the moving inertial load the classical time-integration 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:
Dyniewicz B. | - | IPPT PAN | Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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12. |
Bajkowski J.M.^{♦}, Dyniewicz B., Bajer C.I., Semi-active damping strategy for beams system with pneumatically controlled granular structure,
MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 0888-3270, DOI: 10.1016/j.ymssp.2015.09.026, Vol.70-71, pp.387-396, 2016Abstract: The paper deals with a control method for semi-active 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:
Bajkowski J.M. | - | Warsaw University of Technology (PL) | Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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13. |
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: 0022-460X, DOI: 10.1016/j.jsv.2016.05.050, Vol.380, pp.37-50, 2016Abstract: 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, semi-active Affiliations:
Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN | Bajkowski J.M. | - | Warsaw University of Technology (PL) |
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14. |
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: 0093-6413, DOI: 10.1016/j.mechrescom.2016.10.005, Vol.78, pp.51-56, 2016Abstract: 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 relay-type control logic. In the experiment, the deformable granular coupler semi-actively 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 semi-active 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:
Bajkowski J.M. | - | Warsaw University of Technology (PL) | Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN | Pisarski D. | - | IPPT PAN |
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15. |
Pisarski D., Szmidt T., Bajer C.I., Dyniewicz B., Bajkowski J.M.^{♦}, Vibration Control of Double-Beam System with Multiple Smart Damping Members,
SHOCK AND VIBRATION, ISSN: 1070-9622, DOI: 10.1155/2016/2438902, Vol.2016, pp.2438902-1-14, 2016Abstract: 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 bang-bang 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:
Pisarski D. | - | IPPT PAN | Szmidt T. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN | Bajkowski J.M. | - | Warsaw University of Technology (PL) |
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16. |
Wasilewski M., Pisarski D., Bajer C.I., Adaptive stabilization of partially damaged vibrating structures,
Machine Dynamics Research, ISSN: 2080-9948, Vol.40, No.1, pp.65-82, 2016Abstract: In this paper, an online adaptive continuous-time 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, Hamilton-Jacobi-Bellman equation Affiliations:
Wasilewski M. | - | IPPT PAN | Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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17. |
Dyniewicz B., Bajkowski J.M.^{♦}, Bajer C.I., Semi-active control of a sandwich beam partially filled with magnetorheological elastomer,
MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 0888-3270, DOI: 10.1016/j.ymssp.2015.01.032, Vol.60-61, pp.695-705, 2015Abstract: The paper deals with the semi-active 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 electro-magnetic 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.
First the analytical solution for the vibrations of a sandwich beam filled with an elastomer is given. Then the control problem is defined and applied to the analytical formula. The numerical solution of the minimization problem results in a periodic, perfectly rectangular control function if free vibrations are considered. Such a temporarily acting magnetic field is more efficient than a constantly acting one. The surplus reaches 20–50% or more, depending on the filling ratio of the elastomer. The resulting control was verified experimentally in the vibrations of a cantilever sandwich beam.
The proposed semi-active control can be directly applied to engineering vibrating structural elements, for example helicopter rotors, aircraft wings, pads under machines, and vehicles. Keywords: Semi-active control, Beam vibration, Magnetorheological elastomer, Sandwich beam, Damping Affiliations:
Dyniewicz B. | - | IPPT PAN | Bajkowski J.M. | - | Warsaw University of Technology (PL) | Bajer C.I. | - | IPPT PAN |
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18. |
Dyniewicz B., Konowrocki R., Bajer C.I., Intelligent adaptive control of the vehicle-span/track system,
MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 0888-3270, DOI: 10.1016/j.ymssp.2014.12.007, Vol.58-59, pp.1-14, 2015Abstract: This paper presents the strategy of semi-active 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, Semi-active damping, Beam vibration Affiliations:
Dyniewicz B. | - | IPPT PAN | Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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19. |
Bajkowski J.M.^{♦}, Dyniewicz B., Bajer C.I., Damping properties of a beam with vacuum-packed granular damper,
JOURNAL OF SOUND AND VIBRATION, ISSN: 0022-460X, DOI: 10.1016/j.jsv.2014.12.036, Vol.341, pp.74-85, 2015Abstract: 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:
Bajkowski J.M. | - | Warsaw University of Technology (PL) | Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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20. |
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: 0042-3114, DOI: 10.1080/00423114.2014.982659, Vol.53, No.1, pp.51-67, 2015Abstract: 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 real-life 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN | Matej J. | - | other affiliation |
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21. |
Dyniewicz B., Pręgowska A., Bajer C.I., Adaptive control of a rotating system,
MECHANICAL SYSTEMS AND SIGNAL PROCESSING, ISSN: 0888-3270, DOI: 10.1016/j.ymssp.2013.09.006, Vol.43, pp.90-102, 2014Abstract: In the present paper, an adaptive control of structural vibrations is presented. Based on earlier research, we claim that the periodical switching on of magneto-rheological 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, Semi-active control, Vibration control, Shaft vibrations, Torsional vibrations, Magneto-rheological dampers Affiliations:
Dyniewicz B. | - | IPPT PAN | Pręgowska A. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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22. |
Dyniewicz B., Bajer C.I., New Consistent Numerical Modelling of a Travelling Accelerating Concentrated Mass,
World Journal of Mechanics, Vol.2, No.6, pp.281-287, 2012Abstract: 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, non-iterative 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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23. |
Pisarski D., Bajer C.I., Smart suspension system for linear guideways,
JOURNAL OF INTELLIGENT AND ROBOTIC SYSTEMS, ISSN: 0921-0296, DOI: 10.1007/s10846-010-9450-7, Vol.62, pp.451-466, 2011Abstract: This paper presents a new method for the semi-active 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 semi-active 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, bang-bang 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: Semi-active control, Smart suspension system, Vibration control, Linear guideway, Moving load Affiliations:
Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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24. |
Pisarski D., Bajer C.I., Semi-active control of 1D continuum vibrations under a travelling load,
JOURNAL OF SOUND AND VIBRATION, ISSN: 0022-460X, DOI: 10.1016/j.jsv.2009.09.006, Vol.329, pp.140-149, 2010Abstract: The paper presents a method for computing the response of a 1D elastic continuum supported by a set of semi-active 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 time-marching scheme allows us to continue a solution in successive layers with initial conditions taken from the end of previous stages. The semi-active 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:
Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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25. |
Dyniewicz B., Bajer C.I., New feature of the solution of a Timoshenko beam carrying the moving mass particle,
ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.62, No.5, pp.1-15, 2010Abstract: 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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26. |
Dyniewicz B., Bajer C.I., Numerical methods for vibration analysis of Timoshenko beam subjected to inertial moving load,
VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 0860-6897, Vol.24, pp.87-92, 2010Abstract: 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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27. |
Konowrocki R., Bajer C.I., Vibrations of railroad due to the passage of the underground train,
ARCHIVES OF TRANSPORT, ISSN: 0866-9546, Vol.22, No.1, pp.61-82, 2010Abstract: 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 train-track interaction, ground borne vibrations, curved and straight track, rolling Affiliations:
Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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28. |
Pisarski D., Bajer C.I., On the semi-active control of carrying structures under moving a load,
VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 0860-6897, Vol.24, pp.325-330, 2010Abstract: 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: Semi-active control, structural control, optimization, moving load Affiliations:
Pisarski D. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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29. |
Dyniewicz B., Bajer C.I., Symulacja komputerowa ruchomych obciążeń inercyjnych,
DROGI I MOSTY, ISSN: 1643-1618, Vol.1, pp.5-30, 2010Abstract: W pracy przedstawiono algorytmy numeryczne metody elementów skończonych, 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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30. |
Bajer C.I., Dyniewicz B., Virtual functions of the space–time finite element method in moving mass problems,
COMPUTERS AND STRUCTURES, ISSN: 0045-7949, DOI: 10.1016/j.compstruc.2009.01.007, Vol.87, pp.444-455, 2009Abstract: 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 Newmark-type 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 semi-analytical results. Almost perfect coincidence proves the efficiency of the approach. Keywords: Space–time finite element method, Vibrations, Virtual function, Moving mass Affiliations:
Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN |
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31. |
Bajer C.I., Dyniewicz B., Numerical modelling of structure vibrations under inertial moving load,
ARCHIVE OF APPLIED MECHANICS, ISSN: 0939-1533, DOI: 10.1007/s00419-008-0284-8, Vol.79, pp.499-508, 2009Abstract: Inertial loading of structures by mass travelling with near-critical 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 space-time 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 over-critical 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:
Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN |
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32. |
Dyniewicz B., Bajer C.I., Paradox of the particle's trajectory moving on a string,
ARCHIVE OF APPLIED MECHANICS, ISSN: 0939-1533, DOI: 10.1007/s00419-008-0222-9, Vol.79, No.3, pp.213-223, 2009Abstract: 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 second-order 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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33. |
Konowrocki R., Bajer C.I., Friction rolling with lateral slip in rail vehicles,
JOURNAL OF THEORETICAL AND APPLIED MECHANICS, ISSN: 1429-2955, Vol.47, No.2, pp.275-293, 2009Abstract: The paper deals with the dynamic phenomena accompanying the wheel rolling over a road (rail, track), with lateral slip effects. They occur in rol-ling 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 de-tected 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 degree-of-freedom the-oretical system. In the case of steel/polyester and polyamide/polyester friction pair, qualitatively similar results were obtained. Keywords: rolling, friction, lateral slip, wheel-rail interaction Affiliations:
Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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34. |
Pisarski D., Bajer C.I., Aktywne tłumienie drgań struny i belki pod obciążeniem ruchomym,
DROGI I MOSTY, ISSN: 1643-1618, Vol.8, No.4, pp.71-87, 2009 | |
35. |
Bajer C.I., Dyniewicz B., Space-time approach to numerical analysis of a string with a moving mass,
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 0029-5981, DOI: 10.1002/nme.2372, Vol.78, No.10, pp.1528-1543, 2008Abstract: Inertial loading of strings, beams and plates by mass travelling with near-critical 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:
Bajer C.I. | - | IPPT PAN | Dyniewicz B. | - | IPPT PAN |
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36. |
Bajer C.I., Dyniewicz B., Moving inertial load and numerical modeling,
VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 0860-6897, Vol.23, pp.65-70, 2008 | |
37. |
Konowrocki R., Bajer C.I., Vibrations due to the passage of a railway vehicle on straight and curved tracks,
VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 0860-6897, Vol.23, pp.199-204, 2008Abstract: 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 train-track interaction, ground borne vibrations, moving load Affiliations:
Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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38. |
Dyniewicz B., Bajer C.I., String-beam under moving inertial load,
VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 0860-6897, Vol.23, pp.115-120, 2008Abstract: The paper deals with the original analytical-numerical 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 string-beam 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:
Dyniewicz B. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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39. |
Dyniewicz B., Bajer C.I., Discontinuous trajectory of the mass particle moving on a string or a beam,
Machine Dynamics Research, ISSN: 2080-9948, Vol.32, No.3, pp.66-79, 2008 | |
40. |
Konowrocki R., Bajer C.I., Oddziaływanie między nawierzchnią drogi i kołem w obecności poślizgu bocznego,
ZESZYTY NAUKOWE INSTYTUTU POJAZDÓW / POLITECHNIKA WARSZAWSKA, ISSN: 1642-347X, Vol.1, No.64, pp.75-82, 2007Abstract: 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 koło-szyna Affiliations:
Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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41. |
Tokaj P.^{♦}, Bajer C.I., Dynamika toru typu „Y” - narzędzia komputerowe w analizie dynamicznej toru,
ZESZYTY NAUKOWE INSTYTUTU POJAZDÓW / POLITECHNIKA WARSZAWSKA, ISSN: 1642-347X, Vol.2, pp.151-160, 2006 | |
42. |
Konowrocki R., Bajer C.I., Investigation of the friction phenomenon in the wheel-road interaction,
VIBRATIONS IN PHYSICAL SYSTEMS, ISSN: 0860-6897, Vol.15, pp.173-178, 2006Abstract: 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.
Skew wheel plane related to the direction of rolling results in lateral slip in rail/wheel contact zone. The rail/wheel system oscillates and generates noise. In the same time the wear increases considerable and negatively influences the safety of transportation. Corrugations are visible results of this oscillatory wear (Fig. 1). However, the phenomenon can be observed also in rolling of a tire over the road and the contact of rolls with guide. In all cases friction is the main reason of the described phenomenon. The friction law influences it qualitatively and qualitatively. Keywords: Experimental investigations, Affiliations:
Konowrocki R. | - | IPPT PAN | Bajer C.I. | - | IPPT PAN |
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43. |
Tokaj P.^{♦}, Bajer C.I., Dynamika toru z podkładami stalowymi klinowymi,
DROGI I MOSTY, ISSN: 1643-1618, Vol.5, No.2, pp.5-35, 2006 | |
44. |
Bajer C.I., Bogacz R., Propagation of perturbances generated in classic track, and track with Y-type sleepers,
ARCHIVE OF APPLIED MECHANICS, ISSN: 0939-1533, DOI: 10.1007/s00419-005-0403-8, Vol.74, pp.754-761, 2005Abstract: Railway track with classic and Y-shaped 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 railway-track modelling, leading to different simplifications. The two-dimensional periodic model of track consists of two parallel infinite Timoshenko beams (rails) coupled with equally spaced sleepers on a viscoelastic foundation. Nowadays the interest of engineers is focused on slab track and track with Y-shaped sleepers. The fundamental qualitative difference between the track with classic and Y-shaped sleepers is related to local longitudinal symmetric or antisymmetric features of the railway track. The sleeper spacing influences the periodicity of the foundation elasticity coefficient, mass density (rotational inertia) and the effective shear rigidity. Track with classic concrete sleepers is affected much more by rotational inertia and shear deflections than track with Y-shaped sleepers. The increase of the elastic-wave velocity in track with Y-shaped sleepers and more uniform load distribution will be proved by analysis and simulation. The analytical and numerical analyses allows us to evaluate the track properties in a range of moderate and high train speeds. However, the correct approach is not simple, since the structure of the track interacts with the wheels, wheelsets and vehicles, which constitute complex inertial loads. We note that the growth of amplitude in selected velocity ranges depends strongly on the track type. Keywords: Track dynamics, Vibrations, Moving load, Y-type track Affiliations:
Bajer C.I. | - | IPPT PAN | Bogacz R. | - | IPPT PAN |
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45. |
Bajer C.I., Numeryczne modelowanie czasoprzestrzenne dynamicznych zagadnień kontaktowych (Praca habilitacyjna),
Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.5, pp.1-189, 1997 | |