Direct Dynamic Problem Research Articles

Numerical Analysis of Harmful Environmental Impact of Accidental Explosion at a Hydrogen Filling Station Environmental Impact of an Accident Hydrogen Explosion at a Gas Station

The purpose of this study is to evaluate numerically the consequences of a hydrogen-air mixture explosion at a vehicle hydrogen filling station. The physical process of the explosion, which takes place after the hydrogen gas release into the air due to the accidental destruction of a number of high-pressure storage cylinders, is considered. The blast pressure wave moves in all directions from the epicenter of the accident, gradually losing its intensity and having a negative shock impact on the service personnel of the filling station and the structures of the surrounding buildings. The scale of the accident depends on the number of destroyed cylinders, which determines the size of the hydrogen-air cloud and the power of the explosion. The degree of negative consequences for the environment depends on the maximum overpressure in the blast wave front. Numerical obtaining of spatial pressure distributions in the area of the accident based on a hydrogen explosion mathematical modeling makes it possible to separate zones that are dangerous to human health and building structures strength. The direct problem of gas dynamics of combustion products of a hemispherical cloud of a stoichiometric hydrogen-air mixture in the surface layer of the atmosphere is considered. The mathematical model of an instantaneous hydrogen explosion takes into account the three-dimensional and non-stationary nature of the propagation of the explosion wave, the compressibility of the gas flow, the complex terrain, the shape and initial concentration of the cloud of hydrogen combustion products, and their thermodynamic parameters. The model makes it possible to obtain three-dimensional fields of maximum overpressure, which are the basis for deterministic assessment of the consequences of an accident for human health and the integrity of structures in the area of the explosion. The presented computer technology allows security experts to identify potential dangerous zones by means of mathematical modeling, recommend effective protective measures to mitigate or even eliminate the negative consequences of the blast wave environmental impact.

Information support to solve direct dynamic problem for the previouslydisturbed electromechanicalsystems

Information support to solve direct dynamic problem for the previouslydisturbed electromechanicalsystems

INVERSE AND FORWARD KINEMATICS AND DYNAMICS OF A TWO LINK ROBOT ARM

This paper deals with the kinematic and dynamic analysis of a two-element robot model. We have solved the problem using inverse kinematics. The result is the plotted trajectory of the robot's end point along defined points of the robot's workspace. The angular quantities computed are the rotation angle, angular velocity and angular acceleration in each kinematic pair. Furthermore, an inverse dynamics problem is solved. The moments in the kinematic pairs are computed. Subsequently, using the direct dynamics problem, the correctness of the obtained solution is confirmed. The programs designed for the simulation of dynamical systems - Matlab/Simulink and SimMechanics - were used in the solution. The results are presented in the form of graphs and tables.

Bayesian forces identification in cable networks with small bending stiffness

The regular monitoring of cable forces is essential for ensuring the safety of cable structures both during construction and throughout their lifetime. This paper aims at developing a vibration-based identification procedure of the axial forces, bending stiffness, and, secondarily, the crossing point position of cable networks. A model constituted by two crossing stays having small bending stiffness and negligible sag effects is considered. The in-plane direct dynamic problem is solved both numerically and through a perturbation approach. The obtained results are compared to the outcomes of a finite element model for verification purposes. The theoretical studies are also supported by experimental tests performed on a real cable-stayed bridge (Haccourt bridge), which provide insights into the dynamics of the system showing that models of cables with small bending stiffness are more appropriate than taut string models. The inverse analysis based on non-linear Bayesian regression is developed and the closed-form asymptotic formulations are used to prove that the bending stiffness, the cable forces, and the crossing point position can be separately identified from a set of observed frequencies. The implemented procedure is then applied to the tested bridge as a proof of concept, showing that the proposed in-plane identification strategy provides satisfactory results.

Assessing the accuracy of modeling the tubbing erector manipulator mechanism in solidworks motion program

Purpose. To estimate the calculation error in determining the kinematic and dynamic characteristics of the UT62 tubbing erector manipulator mechanism movement. Methodology. Modern computer analysis methods are used, which are implemented in a number of computing complexes. Due to the task complexity, a mathematical model for the manipulator mechanism fragment, which is a hydraulic cylinder with two degrees of freedom, has been developed to estimate the error of computer modeling. This model is used as a test model. Findings. On the example of the model study, it is shown that when simulating the mechanism movement, errors in the calculations are possible due to incorrect problem formulation, as well as to the inaccurate settings of the modules for converting the initial data. Under these circumstances, it has been concluded that when the input link movement is specified as a vector, then when determining the kinematic and dynamic characteristics of the mechanism, it is necessary to use a cubic spline in the “interpolator” module. Originality. For the first time, the direct problem of the manipulator mechanism dynamics has been solved, which consists in determining the static and dynamic characteristics of the device according to the given motion law of its drives. It is noted that when the motion is specified in the form of smooth analytical functions, then there is a complete coincidence of the calculation results performed by the SOLIDWORKS MOTION program with those obtained by mathematical modeling. In addition, when the mechanism link input movement is modeled as a vector, which is formed from a discontinuous function, then the cubic spline used in the “interpolator” module provides smooth harmonic functions of the movement, acceleration and jerk processes. As a result of modeling the manipulator mechanism parameters, it turned out that it is not expedient to use more than 50 points of discrete time in the research. Thus, the errors in calculating the maximum power values of the manipulator motors do not exceed 20 % for the power hydraulic cylinder and 5 % for the hydraulic motor. Practical value. The proposed algorithm can be used to model the movement of complex mechanisms in machines.

Electromechanical model of exoskeleton with three mobile links

The article considers an electromechanical model of an exoskeleton with three links, for which a system of differential equations of motion is written and inverse and direct problems of dynamics are solved. The angles between the links that specify anthropoid motion are figured out analytically. The electric drive controlling torques are found as a result of solving the inverse dynamics problem. The found torques are approximated with stepwise piecewise-constant functions simulating the impulse control of the exoskeleton motion. The solution of the direct problem of dynamics is carried out and the link rotation angles as functions of time are found. The results of the numerical solution of the system of differential equations are compared with the initial motion of the links. It has been found that the results of simulation with impulse control are in good agreement with the original motion. The total energy expenditures have been calculated. The exoskeleton simulation, taking into account the electric drive impacts also has been carried out. For this purpose, a system of differential equations of motion has been compiled; a numerical solution of the Cauchy problem for the compiled system has been carried out. The significance of the electric drive impact on the dynamics of the mechanism has been confirmed.

Dynamics of cable-driven parallel manipulators with variable length vibrating cables

A parametric nonlinear model of cable-driven parallel manipulators endowed with a three-dimensional end-effector is formulated and discussed in this paper. The proposed model considers the distributed stiffness, inertia, and damping of time-varying length cables and allows to study and characterize the dynamic response of manipulators equipped with a generic number n of cables. The equations of motion of each of the n cables are first derived via a total lagrangian formulation together with the compatibility equations prescribing the connectivity between the cables and the end-effector mass, while the dynamics of the end-effector are described by enforcing the balance of its linear and angular momentum. A discretization procedure, based on admissible trial functions, is used to reduce the nonlinear partial differential equations of motion of the cables to a set of ordinary differential equations. The resulting equations are coupled with those describing the motion of the end-effector and the approximate solution is calculated via numerical time integration. Direct and inverse dynamic problems are then formulated and solved for selected case-study manipulators; finally, the role on the dynamic response of the system of the main mechanical parameters and of the degree of over-actuation is discussed.

ИССЛЕДОВАНИЕ ДИНАМИКИ ДВИЖЕНИЯ ОПОРНОЙ КОНЕЧНОСТИ ШАГАЮЩЕГО РОБОТА БИОНИЧЕСКОГО ТИПА

The study objective is to develop approaches to making a mobile robotic walking-type system that moves based on bionic principles of movements. The developed system is equipped with six limbs and moves typically for some insects, which increases its cross-country capacity in difficult road conditions. The algorithm of the movement arrangement is described and the dynamic analysis of the movement of the executive mechanism links is carried out. A kinematic scheme is made and an algorithm for planning the trajectories of the robot's basis is developed. To implement the modeling, the task of moving the limb of a six-legged robot, presented in the form of a flat two-link mechanism, is considered separately. Based on the given functions, graphs of changes in the basis coordinates and hinge torques, providing the necessary movements of the robot along a given trajectory, are obtained. The direct dynamic problem for one robot leg, which is considered as a system with two degrees of freedom, is solved on the basis of the Lagrange equation. The developed equations are solved in Matlab environment. The torque values of the robot drives necessary for working out a given trajectory are obtained. The novelty of the work is in the development and research of the mechanisms of the walking drive of a mobile robot implementing an algorithm of step-by-step movement, which provides lower energy costs than existing analogues. The results will be useful in making mobile multifunctional robotic systems created on the basis of the insect-type movement, and operating independently in a large number of practical applications.

Creation of an electromechanical model of an exoskeleton link in the form of Lagrange-Maxwell equations for agricultural mechanization

The electromechanical model of exoskeleton link has been developed for agricultural mechanization. First, a differential equation of link motion was compiled without taking into account electric drives in the form of the Lagrange equation of the second kind. For it, the inverse and direct problems of dynamics are solved. Then, this equation takes into account the influence of the rotating rotor and the gear ratio of the gearbox on the dynamics of the link. For this model, the direct problem of dynamics is solved. A significant influence on the results of the numerical solution for taking into account the rotating rotor of the electric drive has been established. Then the system of differential equations, describing dynamics of the controlled motion of the model, in the form of Lagrange-Maxwell equations has been composed. The local system of coordinates has been applied in the model, since the electric drive changes the angles between the links. Direct and inverse dynamics problems have been solved. The comparative analysis of the models with electric drive and without it has been made. It has been established that taking into account the electrical drive system allows achieving good results in modeling accuracy.

A Nonlinear PDE for Families of Orbits on a Given Surface

We study a nonlinear PDE which descibes monoparametric families of orbits on a certain surface produced by two-dimensional potentials. We face the following version of the direct problem of Newtonian Dynamics: Given a surface Sand a two-dimensional potentialV = V(u,v), determine all the isoenergetic families of orbitsf(u,v) = c (c = const.), that is, families of orbits which are traced by a test particle with the same preassigned value of the total energy {mathcal {E}} = {mathcal {E}}_{0}. We are interested especially in those orbits which are described by energy {mathcal {E}}_{0} = 0. Thus, using Merten’s equation (ZAMM 61:252–253, 1981), we establish a new, nonlinear PDE for the “slope function” gamma = {frac{{f_{v}}}{{f_{u}}}} which represents well the corresponding family of orbits f(u,v) = c on the given surface S. We find two necessary and sufficient differential conditions, one for the potential V = V (u, v) and another one for the slope function gamma , so that the above PDE has solution. Furthermore, we determine the general solution of the above PDE. Not only real but also complex potentials can produce these families of orbits on the given surface S. Several examples are offered.

Free oscillations of a dissipative oscillator with double quadratic nonlinearity

Approximate formulas for calculating the amplitudes of free damped oscillations of the oscillator with a quadratic nonlinear elastic characteristic under the action of a resistance force that is proportional to the square of the velocity are derived by the energy balance method. Two variants of this method have been implemented. In the first, an approximate differential equation of the envelope graph of oscillations is composed and its analytical solution is constructed. As a result, for the calculation of amplitudes, iterative relations were obtained using the Lambert W function. The argument of this special function is positive for the hard power characteristic and negative for the soft one. Asymptotic approximations of the Lambert W function, which simplify the practical implementation of analytical solutions, are proposed, and the possibility of using known tables of this special function is indicated. In presenting the second variant of the energy balance method, the recurrent relation between the amplitudes of oscillations related to the analytical solution of the cubic equation is derived. Unlike the first option, it does not require iterations. It has been made a comparison of numerical results, which these methods of calculating the amplitudes lead to, and numerical computer integration of the differential equation of oscillations of the oscillator. Satisfactory consistency of the results obtained in different ways confirmed the suitability of the derived approximate formulas for engineering calculations. The main advantage of this method is that it does not involve the construction and use of an exact solution of the double nonlinear differential equation of motion of the oscillator. In addition to the direct problem, the inverse problem of determining the coefficient of quadratic resistance of the medium based on the results of measuring the amplitudes of free oscillations on the oscillogram is also analytically solved here. The obtained solution of the direct dynamics problem was used to check the accuracy of the coefficient identification.

To continue the optical-mechanical analogy

On the basis of the V-function method, an optical-mechanical analogy is carried out; and further it receives a new continuation. In particular, the equality of the phase velocity of the wave and the velocity of the particle is shown. It is noted that according to its physical meaning the wave characterizes the properties of the action that manifests itself in the motion of the particle. The V-function method is based on the concept of a process-state and it consists of a local variational principle, a new formulation of the direct and inverse problem of dynamics. According to the V-function method, the trajectory motion of an object is associated with its wave motion. In this case it is assumed that there is an explicit connection between the trajectory and wave equations of the object. The application of the V-function method for the case of rectilinear motion of an object is shown.

Calculation of Controls, Not Generating Singularities for Four Control Moment Gyrodynes

This paper demonstrates the technology for calculating spacecraft’s (SC’s) programmed laws of orientation that does not generate special, singular, states of the control moment gyro systems and consists of two collinear gyrodyne pairs. The calculation of the programmed orientation laws that do not contain singular states for two pairs of gyrodynes is possible due to the use of the kinematic configuration of the system of the control moment gyrodynes in the new technology for calculating the SC’s controls, which allows, if required, adjusting the position of the virtual flywheel axes for new attitude control problems. Today, the laws of programmed attitude control are calculated as a solution to the direct problem of dynamics. These laws contain singular states of the gyro system in which the SC remains uncontrollable. The controllability is restored by withdrawing the gyro system from these states using an additional gyro system and additional control. At the same time, the attitude control process of the SC is interrupted for the duration of the output. These material and temporary losses require searching for attitude control laws that do not contain special states of the gyro system. The calculation of such laws is demonstrated in solving the problem of orienting a space telescope with two collinear gyrodyne pairs.

Nonparametric Method for Aircraft State Prediction

This paper is a part of a series of articles on nonparametric methods in dynamic systems theory. Here the authors propose a new method for direct dynamic problem solution based only on some past measurements of system input-output signals. The described nonparametric state prediction method needs no any priori information on system model parameters and does not require identification, training or statistical calculations. The methodological examples of nonparametric aircraft state prediction are presented.

Comparison of near-surface layers spectral characteristics under the seismic stations «Trosnyk», «Uzhgorod», «Mezhgirya» calculated using finite element method and obtained experimentally

The purpose of this work was to calculate the theoretical transfer characteristics of the sedimentary layers under the seismic stations «Trosnyk», «Uzhgorod», «Mizhgirya» of the Carpathian seismic network using the Finite element method (FEM). These results are used for comparison with the dynamic parameters of near-surface layers received experimentally; for checking the conformity of investigated environment resonance properties obtained by these methods for the possibility of their application in regional seismological studies. The research was carried out by solving the direct dynamic seismic problem using the finite element method. This method of mathematical modeling allows making calculations for models that are complex in their structure. When solving the direct dynamic seismic problem using FEM, the opportunity to take into account various exchange effects inside the model does not forfeit. Also, we can calculate models with the complex geometric environment structure and various inclusions. Using the developed technique of wave field simulation by finite element method, oscillations for models of the seismic section under the «Trosnyk», «Uzhgorod» and «Mizhgirya» stations were investigated. According to the simulation results, theoretical spectral ratios for these stations were calculated. For the Seismic Station «Mizhgirya», we observe the coincidence of two interference maxima at a frequency of 4 and 10 Hz for both charts: experimental and theoretical. For the Seismic Station «Uzhgorod», the graphs obtained by the experimental and modeling method showed identical behavior throughout the frequency range. These charts provide a stable source of noise within the city. For the Seismic Station «Trosnyk» there is a discrepancy in the behavior of experimental and model charts above 6 Hz. Such results are due to the high level of groundwater under the seismic station, which was not taken into account in the creation of the model for calculating using the finite element method. The spectral characteristics of near-surface layers under the «Trosnyk», «Uzhgorod» and «Mizhgіrya» seismic stations were calculated by the finite element method for the first time. The results of mathematical modeling were compared with experimental measurements. The knowledge about the dynamic parameters of the upper sediment layers under seismic stations, which are the largest filter of oscillation frequencies in the spectral range, will allow more accurate interpretation of registered by these stations occurrences. The calculated spectral ratios give a possibility to estimate the degree of environmental influence on stations’ seismic signals recording, which will be the largest at frequencies corresponding to the received resonance maxima and minima. These results should be taken into account when evaluating the parameters of possible seismic effects in the territory.

Application of Spectral Methods to Inverse Dynamic Problem of Seismicity of a Stratified Medium

Under study is the wave propagation process in the half-space y3 = 0 with the Cartesian coordinates y1, y2, and y3 which is filled with an elastic medium. The parameters of the medium are discontinuous and depend only on the coordinate y3. The wave process is induced by an external perturbation source that generates a plane wave moving from the domain y3 > h > 0. It is proved that the direct dynamic problem is uniquely solvable in the corresponding function space, and a special presentation is found for the solution. The problem of determination of the acoustic impedance of the medium from the wave field measurements on the surface is investigated by the spectral methods of the theory of differential equations.

Development of the Mathematical Model of Movement of the Output Robot Rotating on the Basis of Positioning Possibility Analysis

Рассмотрено построение математической модели движения выходного звена промышленного робота с шестью степенями свободы при исследовании точности его позиционирования. Математическая модель робота включает в себя решение обратной задачи кинематики, решение прямой задачи динамики, а также задачи о положении рабочей точки робота. Существенное влияние на характеристики систем управления сложными мехатронными объектами, такими как манипуляционные роботы, оказывают их кинематические и динамические параметры, для использования которых при реализации СУ необходимо дать их подробное математическое описание. Для решения данной задачи используются два направления. Первое заключается в создании точной кинематической модели манипулятора, позволяющей однозначно определить его пространственную конфигурацию, что, в свою очередь, даст возможность описывать законы перемещения рабочего органа. Вторым направлением является описание динамических характеристик и связей, существующих в манипуляторе, что позволит описать его поведение при перемещении по заданной траектории. Особенностью модели является учет отклонений траектории выходного звена от заданного положения. Разработанная математическая модель позволит получить более полную картину процесса позиционирования выходного звена мехатронного комплекса с учетом инерционных сил и тем самым повысить его точность при отработке пространственной траектории произвольной формы.

Dynamic Analysis of Composite Rods Under Thermal and Force Loads

We propose a general approach to the solution of direct dynamic problems of the numerical analysis of composite rods subjected to thermal and force loads for a large variety of their materials and loading conditions. We take into account the shear effects and interactions between the elements of structures and supporting media. The proposed inhomogeneous rods are regarded as elements of rod systems and characterized by higher parameters of strength and stiffness and lower costs of their production as compared with homogeneous elements. In the equations of motion and physical relations, we introduce four functional characteristics of stiffness and viscosity and three mass functional characteristics equivalently reflecting the dynamic deformation of composite rods with the help of a one-dimensional model. With the help of trigonometric Fourier series, the dynamic loads, displacements, and temperature are represented as products of functions of the coordinate and time. The solution of the homogeneous problem is obtained via the matrizant of the system of equations of the first order. Partial solutions for loads of various types and generalized temperature loads are obtained on the basis of approximation of given quantities and the required displacements by trigonometric series.

Improved general solution for the dynamic modeling of Gough–Stewart platform based on principle of virtual work

This paper presents an improved general dynamic formulation, inverse and direct dynamics, of 6-UPS Gough–Stewart parallel robot based on the virtual work method. The new formulation offers a reduction in the computational time and improves accuracy of the dynamics equations. This method allows elimination of constraint forces/moments at the passive joints from equations of motion. Since, the dynamic formulations are derived in joint space, the concept of direct link Jacobian matrices are employed to obtain all rigid bodies’ twists. The direct link Jacobian matrices convert the twist of the rigid bodies to actuated joints velocities. Moreover, more accurate formulation is obtained by considering the angular velocity and acceleration vectors of the robot’s legs. In the process of solving the direct dynamics problem, a modified hybrid strategy is employed to obtain the near-exact solution for the direct kinematics problem (DKP). The modified hybrid strategy combines the artificial neural network and the third-order Newton–Raphson method. This strategy satisfies both goals to find the nearest exact solution and reduces execution time for the DKP. Next, two numerical examples are presented and the results are verified using a commercial dynamics modeling software. Finally, for comparison, Euler–Lagrange formulation is also obtained. Results indicates that the proposed dynamics formulation offers a significant improvement in both accuracy and execution time.

Optimal Design of Elastic Rod Motions Based on a Projection Approach

Dynamics modelling and control design for elastic rods with distributed parameters are under study. The constitutive laws are specified by an integral equality according with the method of integro-differential relations. A numerical algorithm is developed to solve direct and inverse dynamic problems in linear elasticity based on the Ritz method and finite element technique. The control problem is to obtain the rod motion from some initial state to terminal one at a fixed time with the minimal mean energy. The exact solution is found with a specific triangular space-time mesh and a piecewise polynomial displacement input.