Spatial Vibrations Research Articles

Mathematical modeling of articulated passenger train spatial vibrations

The problem of high-speed railway transport development is important for Ukraine. In many countries articulated trains are used for this purpose. As the connections between cars in such a train differ from each other, to investigate its dynamic characteristics not a separate car, but a full train vibrations model is necessary. The article is devoted to the development of the mathematical model for articulated passenger train spatial vibrations. The considered train consists of 7 cars: one motor-car, one transitional car, three articulated cars, one more transitional car and again one motor-car. Differential equations of the train motion along the track of arbitrary shape are set in the form of Lagrange’s equations of the second kind. All the necessary design features of the vehicles are taken into account. Articulated cars have common bogies with adjoining cars and a transfer car and the cars are united by the hinge. The operation of the central hinge between two cars is modeled using springs and dampers acting in the horizontal and vertical directions. Four dampers between two adjacent car-bodies act as dampers for pitching and hunting and are represented in the model by viscous damping. The system of 257 differential equations of the second order is set, which describes the articulated train motion along straight, curved, and transitional track segments with taking into account random track irregularities. On the basis of the obtained mathematical model the algorithm and computational software has been developed to simulate a wide range of cases including all possible combinations of parameters for the train elements and track technical state. The study of the train self-exited vibrations has shown the stable motion in all the range of the considered speeds (40 km/h – 180 km/h). The results obtained at the train motion along the track maintained for the speedy motion have shown that all the dynamic characteristics and ride quality index insure train safe motion and comfortable conditions for the travelling passengers.

ДИНАМІКА ТРАКТОРА ЯК СКЛАДНОЇ МЕХАНІЧНОЇ СИСТЕМИ ТИПУ ПРОСТОРОВОЇ РАМИ

Purpose of the study is to develop a matrix method for studying the dynamics of a tractor as a multi-mass spatial system of rigid bodies with an arbitrary arrangement of elastic suspension of bodies on shock absorbers relative to a fixed support surface and the presence of elastic connections between the bodies, made in the form of beam elements. Research methods. The methodological basis of the work is the generalization and analysis of well-known scientific results regarding the dynamics of two-mass systems in resonance modes and the use of a systematic approach. The analytical method and comparative analysis were used to form a scientific problem, determine the goal and formulate the research objectives. When creating empirical models, the main provisions of the theory of stability of systems, methodology of systems analysis and research of operations were used. The results of the study. A wheeled vehicle is presented as an amortized continuous frame type structure with assemblies and assembly units located on it, as well as a methodology for calculating individual block matrices of stiffness and damping coefficients. In this case, it is assumed that a viscous damper can be connected in parallel to each elastic element. In this construction of the stiffness and damping matrix of the block matrix are formed in the same way. Damping matrices are derived from the corresponding matrices by substituting damping constants instead of stiffness constants. To determine the natural frequencies and vibration modes of an undamped system using a PC, the most effective method of diagonalization by successive rotations. This method provides a complete solution to the problem, allowing all frequencies and shapes to be determined simultaneously, and good convergence. Conclusions. The considered method for analyzing and calculating the dynamics and vibration damping of a tractor as a complex mechanical system is based on a matrix record of the problem of spatial vibrations of a system of rigid bodies with elastic bonds. Matrix equations seem to be especially useful in the study of complex tightly coupled systems with the obligatory use of a PC. The presented work provides a complete methodology for calculating a tractor as a complex mechanical system such as a spatial frame with equipment installed on it.

INFLUENCE OF ROLLING STOCK VIBROACOUSTICAL PARAMETERS ON THE CHOICE OF RATIONAL VALUES OF LOCOMOTIVE RUNNING GEAR

Purpose.The success of the traffic on the railways of Ukraine depends on the number and the operational fleet of electric locomotives. Today, the locomotive depot exploit physically and morally outdated locomotives that have low reliability. Modernization of electric locomotives is not economically justified. The aim of this study is to improve the safety of the traction rolling stock by the frequency analysis of dynamical systems, which allows conducting the calculation of the natural (of resonant) frequencies of the design and related forms of vibrations.Methodology.The study was conducted by methods of analytical mechanics and mathematical modeling of operating loads of freight locomotive when driving at different speeds on the straight and curved track sections. The theoretical value of the work is the technique of choice of constructive schemes and rational parameters of perspective electric locomotive taking into account the electric inertia ratios and stiffness coefficients of Lagrange second-order equations.Findings. The problems of theoretical research and the development of a mathematical model of the spatial electric vibrations are solved. The theoretical studies of the effect of inertia ratios and stiffness coefficients on the dynamic values and the parameter values of electric locomotive undercarriages are presented.Originality.The set of developed regulations and obtained results is a practical solution to selecting rational parameters of bogies of the freight mainline locomotive for railways of Ukraine. A concept of choice of constructive scheme and rational parameters of perspective locomotive is formulated. It is developed the method of calculation of spatial electric locomotive oscillations to determine its dynamic performance. The software complex for processing the data of experimental studies of dynamic parameters of electric locomotive and comparing the results of the theoretical calculations with the data of full-scale tests is implemented.Practicalvalue.The theoretical study allows objectively assessing the impact of the partial frequency of electric locomotive elements on both the rational parameters of its underframe, and the performance and physiological condition of the locomotive crew. Thus, the practical value of the whole complex of theoretical research was to develop an algorithm and program, choosing rational parameters and design scheme of electric locomotive running gear. Using a software package for the study of spatial fluctuations of electric locomotive made it possible to evaluate the dynamic performance of the electric locomotive underframe with sufficient accuracy for practical problems.

Modeling of spatial natural oscillations of axisymmetric systems

A mathematical model, methods and algorithm for estimating the natural vibrations of inhomogeneous axisymmetric systems in a three-dimensional statement are presented in the paper. The problem is solved by the semi-analytical finite element method (FEM). The reliability of the developed methods and algorithms was validated by comparing the results obtained with the exact solution of a number of test problems for three-dimensional bodies, as well as by comparing the results with the results of field experiments. Spatial natural vibrations of inhomogeneous spatial axisymmetric systems, i.e. the models of nuclear power plants (NPPs) shielding shell in a three-dimensional statement were investigated. The results were analyzed to detect mechanical effects and revealed that the presence of multiple (close) eigenfrequencies in real spatial axisymmetric systems is not an exception, but a rule.

Evaluation of the high-frequency oscillation parameters of a liquid-propellant rocket engine with an annular combustion chamber

The high-frequency instability (HF instability) of a liquid-propellant rocket engine (LPRE) during static firing tests is often accompanied by a significant increase in dynamic loads on the combustion chamber structure, often leading to the chamber destruction. This dynamic phenomenon can also be extremely dangerous for the dynamic strength of a liquid-propellant rocket engine with an annular combustion chamber. Computation of the parameters of acoustic combustion product oscillations is important in the design and static firing tests of such rocket engines. The main aim of this paper is to develop a numerical approach to determining the parameters of acoustic oscillations of combustion products in annular combustion chambers of liquid-propellant rocket engines taking into account the features of the configuration of the combustion space and the variability of the physical properties of the gaseous medium depending on the axial length of the chamber. A numerical approach is proposed. The approach is based on mathematical modeling of natural oscillations of a “shell structure of an annular chamber – gas” coupled dynamic system by using the finite element method. Based on the developed finite-element model of coupled spatial vibrations of the structure of the annular combustion chamber and the combustion product oscillations, the oscillation parameters of the system under consideration (frequencies, modes, and effective masses) for its dominant acoustic modes, the vibration amplitudes of the combustion chamber casing, and the amplitudes of its vibration accelerations can be determined. The operating parameters of the liquid-propellant rocket engine potentially dangerous for the development of thermoacoustic instability of the working process in the annular combustion chamber can be identified. For the numerical computation of the dynamic gains (in pressure) of the combustion chamber, a source of harmonic pressure excitation is introduced to the finite element model of the dynamic system “shell structure of an annular configuration – gas” (to the elements at the start of the chamber fire space). The developed approach testing and further analysis of the results were carried out for an engine with an annular combustion chamber (with a ratio of the outer and inner diameters of 1.5) using liquid oxygen – methane as a propellant pair. The system shapes and frequencies of longitudinal, tangential and radial modes are determined. It is shown that the frequency of the first acoustic mode in the case of a relatively low stiffness of the combustion chamber casing walls can be reduced by 40 percent in comparison with the frequency determined for a casing with rigid walls.

Mathematical model of the cabin suspension system for grain and forage harvesters taking into account the dynamic properties of the supporting system

The paper is devoted to the study of the vibrational state of the operator's workplace of a transport-technological machine with a supporting system that undergoes torsional and bending elastic deformations during operation. Based on the previously carried experimental studies, the relevance of the work was substantiated and the goal was set to develop a computational model of spatial vibrations of the cabin on a non-existent system in the form of elastic cantilever beams. An original mathematical model of cabin vibrations on an elastic foundation is presented. It takes into account the nonlinear elastic and dissipative properties of the supporting system structure. A variant of the implementation of the developed mathematical model using the example of the mathematical modeling environment Mathcad is proposed. As an example of calculating the dynamic properties of a bearing system, a case of using the MSC Adams software package with an integrated finite element calculation module Flex is shown. It is presented that taking into account the dynamic properties of the carrier system in the model makes it possible to calculate the vibration load of the cabin with high accuracy and reproduce the resonance phenomena caused by the natural modes of vibration of the carrier system and disturbances from technological sources. The results of the verification of the computational model based on the analysis of the convergence of the values of the full corrected vibration acceleration in the center of mass of the grain harvester cabin, as well as the vibration spectra in linear directions obtained by experimental and computational methods, are presented. Directions for the development of measures and technical solutions for improving the vibration load of operators of transport and technological machines who have active sources of power disturbances on board, as well as a supporting system that experiences elastic deformations during operation are proposed.

Influence of the micro profile of the treated surface on the strength and performance characteristics of machine parts

The influence of spatial vibrations of the cutting tool on the roughness of the machined surface is considered. The influence of the parameters of roughness and the height of micro-roughness on the strength and performance properties of the executive surfaces of parts is also considered. The parameters of the cutting process that affect the intensity and frequency of vibrations of the cutting tool tip are considered.

SIMULATION OF A MULTI-FREQUENCY STOCKBRIDGE VIBRATION DAMPER OSCILATIONS WITH ENERGY SCATTERING HYSTERESIS

Spatial vibrations of a system containing a cable and a mass (solid body of arbitrary spatial configuration) are modeled. The problem is solved in a geometrically linear formulation, taking into account the hysteresis of energy scattering that is based on the kinematic equation. Identification of its parameters is carried out on the basis of experimental data on hysteresis loops of the limit cycle.

To the problem of train running safety

The paper is presenting the procedure used for establishing the cause of the wagon derailment. To do this, the computer simulations and the computational software, developed in the Dnipro National University of Railway Transport (DIIT) were used. The level of longitudinal forces and the wagons dynamic performance have been evaluated using the mathematical models of longitudinal oscillations of a train and the spatial vibrations of wagons, in particular of tank wagons. As a result of modeling we obtained oscillograms of longitudinal forces in each inter-wagon connection, the dependence of the largest longitudinal forces on travel time and distance traveled, the distribution of the maximum longitudinal forces along the train length, the speed dependence on travel time and track coordinates. We also obtained the dynamic performance of wagons: the vertical dynamics coefficients of the axle-box and central suspension, the horizontal dynamics coefficients and the derailment stability coefficient. The influence of the movable load in the tank wagons and the characteristics of rail irregularities on the stability coefficient against wheel climbing onto the rail is also considered. The presented methodology was used to determine the cause of the tank wagon derailment in a non-homogenous freight train consisting of 50 wagon tanks on an existing track section of the Lithuanian railways. When simulating the train movement, it was assumed that the train was equipped with elastic-friction absorbing devices and air distributors, turned on to the average operation mode. As a result of numerical experiments, an assumption was made about the cause of the train derailment.

Calculations of vibration isolation systems by means of SCAS KiDyM in a nonlinear setting

The problem of vibration isolation of a device installed on four three-degree elastic shock absorbers with nonlinear characteristics, as an example of vibration isolation of a discrete system of arbitrary structure with several nonlinear elastic shock absorbers is considered in given paper. An analytical description of mechanical models is applied According to this description a special computer algebra system KiDyM (SCAS KiDyM) constructs differential equations of motion, linearizes inertial and elastic terms by means of elimination of the first small terms, carrying out harmonic linearization of the second one. After getting linear multi-mass mathematical model, the KiDyM software package performs calculations of free vibrations and presents the results in the form of skeletal curves. It allows to determine their sensitivity to the values of elastic and inertial parameters, It makes possible to carry out a cycle of calculations with varying model parameters to achieve optimal characteristics of the vibration protection system by shear and deformation of skeletal curves. An example of the problem analysis solution to the above-mentioned vibration protection system of a device performing spatial vibrations is given. Thus, the system contains 12 nonlinearities. The method for determining the experimental elastic characteristics for specimens of two types of shock absorbers is described. The possibilities of narrowing the spectrum of free vibrations by taking into account the nonlinearity of shock absorbers and the problem of skeletal curves construction of systems with several nonlinearities with generalized coordinates as the main coordinates of the oscillatory system.

Spatial natural vibrations of viscoelastic axisymmetric structures

Spatial natural vibrations of viscoelastic axisymmetric structures

Spatial natural vibrations of viscoelastic axisymmetric structures

Spatial natural vibrations of viscoelastic axisymmetric structures

Dynamic calculation of boxed design of buildings

A method for dynamic calculation of a box-like structure, consisting of interconnected longitudinal and transverse plate and beam elements is developed. The problem is posed of spatial vibrations of the box-like structure of a building under dynamic impact determined by its base motion according to a sinusoidal law. It is assumed that the external load-bearing walls of the building, perpendicular to the direction of the seismic effect, work on transverse bending only. The interior panels, located in the direction of external impact, are subjected to tension-compression and shear in their planes. Equations of vibrations of the points of panels, box beams, boundary and initial conditions of the problem are given. In the areas of butt joints of panels, full contact conditions are set to ensure the equality of displacements and stresses. Within the framework of the finite difference method, a methodology was developed for dynamic calculation of box-like structures. Numerical results of stresses over time in the hazardous areas of the box were obtained. The laws of changes in the maximum stress values in characteristic sections of the panels are graphically presented as a function of time.

GONDOLA CARS DYNAMICS FROM THE ACTION OF LONGITUDINAL FORCES

Purpose. The aim of the work is to study the influence of longitudinal quasistatic tensile and compressive forces in gondola cars arising at stationary and transient modes of train movement on their main dynamic indicators and interaction indicators of rolling stock with a rail track, taking into account the possibility of speed increasing. The relevance of this study is related with the need to control the longitudinal forces arising during stationary and transient modes of train movement, with increasing speeds, masses and lengths of trains, especially freight ones, increasing the locomotives` power. Methodology. The main method for studying the dynamic loading of a gondola car on typical three-element bogies is mathematical and computer modeling of the interaction of rolling stock and track structure based on the model of spatial vibrations of freight cars` couplings. In a theoretical study, the influence of quasistatic longitudinal tensile and compressive forces is considered depending on the change in speed and the force value on the tension of 1 MN; 0.5 MN; 0 and before compression of 0.5 MN; 1 MN. Findings. As a result of theoretical studies and after modeling, taking into account the processes of oscillation of gondola cars under the action of quasistatic longitudinal forces limited by norms to ± 1MN (100 tf), the dependencies of the main parameters normalized by technical documentation are obtained taking into account the value of the movement speed. Originality. The influence of longitudinal compressive and tensile forces on the dynamic loading of a freight car is studied in order to solve the problem of forecasting the dynamics of rolling stock, taking into account the value of the speed along curved track sections. Practical value. Application of the results obtained can increase the stability of freight rolling stock and the strength of the railway track, which in turn will remove some existing restrictions on permissible speeds and increase the technical speed of trains. The obtained dependencies of the main normalized indices on the longitudinal quasistatic force will make it possible to predict the development of deviations and prevent their transformation into the dangerous ones for train movement.

Structural Vibration Control of the Spatial Suspended Mass Pendulum

Due to limitations of the simplified model of the suspended mass pendulum (SMP), a spatial model is proposed in the study. The kinetic equations of the host structure with a spatial SMP are established. The numerical analysis is conducted to analyze spatial vibrations of the SMP and their influence on the structural vibration. Comparing the simulation results, it is found that spatial vibrations of the SMP can be simulated better through the spatial model.

Self-adjoint time operator of a quantum field

We study the properties of a quantum field with time as a dynamical variable. Temporal vibrations are introduced to restore the symmetry between time and space in a matter field. The system with vibrations of matter in time and space obeys the Klein–Gordon equation and Schrödinger equation. The energy observed is quantized under the constraint that a particle’s mass is on shell. This real scalar field has the same properties of a zero-spin bosonic field. Furthermore, the internal time of this system can be represented by a self-adjoint operator without contradicting the Pauli’s theorem. Neutrino can be an interesting candidate for investigating the effects of these temporal and spatial vibrations because of its extremely light weight.

Prediction and reduction of spatial transverse vibration of hoisting catenaries induced by drum winding in super-deep mine hoists

The paper focuses on predicting and reducing the spatial transverse vibrations of hoisting catenaries, which may cause disordered rope arrangement, induced by drum winding in super-deep mine hoists. Firstly, the governing equations of spatial transverse vibrations of a hoisting catenary and the rope tension have been derived and experimentally validated; subsequently, the functions of the spatial transverse excitation displacements at the drum end and the hoisting velocity have been precisely modelled and calculated, numerical simulation indicated that large amplitude spatial vibrations will be excited by drum winding and that the quasi-static rope tension can be employed to predict the spatial transverse vibrations of the catenary. Eventually, a vibration isolated system is proposed to reduce the spatial transverse vibrations of a catenary, numerical simulation validated the feasibility. The paper will provide good technical supports for the vibration suppression of hoisting catenaries in super-deep mine hoists.

Stereovision Tracking System for Monitoring Loader Crane Tip Position

This paper presents the idea and experimental tests of a non-stationary stereovision system used for supervision of a loading crane. The system based on two independent gimbals, equipped with cameras, allows simultaneously to measure and track a group of individual spatial points. The four steps calibration procedure is required for each gimbal axis, camera optics, gimbals positions, and loading crane position. Modern camera systems have been widely applied in several fields of the industry allowing for accurate measurements and real-time quality control. The systems based on stereo camera models and appropriate image analysis algorithms can measure a group of spatial points, complex surfaces, small displacements, or vibrations. There are also several vision systems such as laser tracker, camera mounted on fully controlled rotating head (called gimbal), or TOF camera, which can be used to follow and measure real-time object displacements. Unfortunately, they all have significant individual limitations explained in this paper. Accuracy tests of 3D measurements including active tracking of length pattern and crane tip have been performed within a selected working area. Based on reconstructed 3D data points, different types of geometric errors have been presented. The results revealed effects of accuracy degradation on the workspace boundaries and for higher observation angles, as well as 3D data bias suggesting mechanical issues. Several improvements in calibration and gimbal construction might result in far enough accuracy for normal operations of the crane.

Mathematical model of spatial vibrations of a four-axle gondola car on bogies with elastic coupling of wheelsets and side frames

Mathematical model of spatial vibrations of a four-axle gondola car on bogies with elastic coupling of wheelsets and side frames

Investigation of the forced spatial vibrations of a wood shaper, caused by rotor’s unbalance of its electric motor and by cutting forces

Mechanic-mathematical matrix modeling of the forced spatial vibrations of a wood shaper is performed in this study .The wood shaper is modeled as a mechanical system of three rigid bodies, which are connected by elastic and damping elements with each other and with the motionless floor. This mechanical system has 18 degrees of freedom. Formulas and algorithms are developed for computer calculating, analysis and synthesis of designing and investigating of this machine. This study renders an account the geometric, kinematic, mass, inertia, elastic and damping properties of the machine. A system of differential equations is derived. Analytical solutions are presented. The study presnts results of the numerical investigations of the forced spatial vibrations by using parameters of a particular machine. They allow to select parameters that reduce harmful vibrations for people and constructions.