Lagrange Equations Of Second Kind Research Articles

Исследование по динамике рамных конструкций

Determining the natural and forced oscillations of frame structures simulated by the rods with distributed masses (an infinite number of degrees of freedom) is quite difficult. Therefore, in the article, the frame model is endued with a finite number of degrees of freedom: the mass is placed in a certain number of nodes that elastically interact with rods that have no mass. The rods work only for bending. Longitudinal displacements are not taken into account, since the frequency of longitudinal oscillations is two orders of magnitude higher than the frequency of bending ones. Such a model leads to the construction of expressions of the kinetic and potential energy, which then allows using the Lagrange equations of second kind to obtain a system of differential oscillation equations of the structure. The problem of free oscillations of the G-shaped frame was solved in the article using Green's functions, matrices of stiffness, masses, malleability, etc. The obtained approximate results were compared with little-known exact results and demonstrated good convergence, especially with an increase in the number of degrees of freedom (the number of concentrated masses simulating the distributed mass of the rods of the G-shaped frame).

Control of wheeled platforms straight motions taking into account jerk restrictions under speeding-up from the state of rest

The generalized mathematical model of wheeled platforms straight motions on the ideal horizontal plane under speeding-up from the state of rest mode is proposed, and the controls satisfying the restrictions of motion jerks are find. The pure mechanical and electromechanical wheeled platforms are considered, as well as the computer simulations of the researched processes are made. The jerks restrictions are reduced to limiting the value of the wheeled platform acceleration time derivative. The proposed approaches are based on the holonomic systems mechanics and on the electromechanical analogies allowing to consider the different kinds of the wheeled platforms taking into account the electric on-board systems like the drive electric motors and the control systems by using the Lagrange equations of second kind. The examples of the proposed approaches using to define the controls satisfying the jerks restrictions under speeding-up from the state of rest are considered for the pure mechanical and electromechanical wheeled platforms. It is obtained the inequality allowing to chose the instantly supplied driving mechanical couple which will provide the admissible jerks of the motion of the wheeled platform under speeding-up from the state of rest. It is shown that the rolling friction and the viscous damping are the principal causes of the wheeled platforms jerks under speeding-up from the state of rest. It is obtained the inequality defining the voltage instantly supplied on the drive electric motors which will provide the admissible jerks of the motion of the electromechanical wheeled platform during speeding-up from the state of rest, and it is shown that the proposed general approaches are suitable for considering the different kinds of wheeled platforms. The computer simulations of the processes of speeding-up from the state of rest for the electromechanical wheeled platform are considered to show results correctness and to illustrate satisfying the restrictions of the motion jerks. The obtained results of the computer simulations are in the full agreement with the well-known fundamental property inherent for the wheeled platforms. The results for the jerks show that the maximum value of the jerk is really at the initial time as was suggested before, and it is shown that the jerks values at the initial time obtained by using the computer simulations are in full agreement with the theoretically defined correspondent exact values. The big jerks of the considered electromechanical wheeled platform are due to the voltage instantly supplying on the drive electric motors at the initial time, and it is understandable that limiting of such instantly supplied voltage value cannot provide any wished small jerks. The smooth time depending for the voltages supplying on the drive electric motors are required to provide any wished small jerks of the electromechanical wheeled platforms.

Метод исследования динамики управляемой механической бесступенчатой передачи с упругим элементом и механизмом свободного хода

Mechanical continuously variable transmissions have high potential for torque conversion, are capable of providing significant technical and economic advantages when used in shafts of technological and other machines, this making investigations in this area to be pertinent. This paper represents a mechanical and mathematical technique, which enables to do research of dynamic processes, that occur in functioning of mechanical continuously variable transmission with a resilient member and free-wheel mechanism, making it possible to transmit torque to a driven shaft only in the same direction. The objective of this paper is to work out a mathematical model of the mechanical system, which determines dynamic force of transmission elements. Computational model of transmission is shown, in obedience to which there was built a mathematical model based on Lagrange equation of second kind with undetermined multipliers. A set of assumptions was made on working out the dynamic system mathematical model. In particular, possibility of excitation of resonant modes in a resilient member was not considered. Moreover, nonlinearity of free-wheel mechanisms was not considered as well as hysteresis losses of energy in on-off use cycle and significant capacity costs in disconnected machinery. In future it is necessary to carry out experimental investigations in order to make transmission dynamics more exact. By the results of simulation modeling regularities were obtained which determine dynamic moments on transmission elements. The results of the investigation may be used for making an output characteristic of transmission, its adaptation with the motor and forecasting of operational features of shafts of different technological equipment.

Study on Transient Contact-Impact Characteristics and Driving Capability of Piezoelectric Stack Actuator.

The transient contact-impact mechanism and driving capability of the piezoelectric stack actuator is analyzed using both experimental and theoretical methods. An experimental setup and its corresponding measurement approaches for the transient responses are designed. The launch range of the object resulting from the first contact-impact is measured through laser doppler vibrometer and the motion process is captured by high-speed camera. Experimental results illustrate that the launch range increases firstly and decreases subsequently as the frequency of the sine driving voltage increases. Meanwhile, considering the local viscoelastic contact deformation, a theoretical methodology including the mechanics model for the driving process is proposed. Based on the Lagrange equations of second kind, the governing equation of the driving system is derived. Transient responses are calculated using the fourth-order Runge–Kutta integration method. Contact forces and Poisson’s coefficient of restitution are calculated by the proposed theoretical method. The results of launch range show that the theoretical solutions have a good agreement with the experimental data. The peak value of contact force increases firstly and decreases subsequently with the increase of voltage frequency. In addition, the coefficient of restitutions is roughly 0.9 when f is greater than 3.5 kHz.

A rigorous proof for the equivalence of the projective Newton–Euler equations and the Lagrange equations of second kind for spatial rigid multibody systems

It is well known that the projective Newton–Euler equation and the Lagrange equation of second kind lead to the same result when deriving the dynamical equations of motion for holonomic rigid multibody systems. It can be shown that both approaches follow from the principles of d’Alembert or Jourdain. However, as to the author’s knowledge, no direct rigorous proof for the equivalence of these approaches is given in literature so far when it comes to spatial systems of rigid bodies. In this paper, we present a novel proof that directly addresses the projective Newton–Euler equation and the Lagrange equation of second kind without the detour via variational principles. The proof is mainly based on vector and matrix manipulations and elementary concepts of differential geometry. Although the mathematical framework is thereby kept simple, the argumentation is considerably more complex compared to the case of planar systems of rigid bodies or spatial systems of particles. An illustrative example is presented.

Geometrical modeling of the shape of a multilink rod structure in weightlessness under the influence of pulses on the end points of its links

We have examined a geometrical model of the new technique for unfolding a multilink rod structure under conditions of weightlessness. Displacement of elements of the links occurs due to the action of pulses from pyrotechnic jet engines to the end points of links in a structure. A description of the dynamics of the obtained inertial unfolding of a rod structure is performed using the Lagrange equation of second kind, built using the kinetic energy of an oscillatory system only. The relevance of the chosen subject is indicated by the need to choose and explore a possible engine of the process of unfolding a rod structure of the pendulum type. It is proposed to use pulse pyrotechnic jet engines installed at the end points of links in a rod structure. They are lighter and cheaper as compared, for example, with electric motors or spring devices. This is economically feasible when the process of unfolding a structure in orbit is scheduled to run only once. We have analyzed manifestations of possible errors in the magnitudes of pulses on the geometrical shape of the arrangement of links in a rod structure, acquired as a result of its unfolding. It is shown at the graphical level that the error may vary within one percent of the estimated value of the magnitude of a pulse. To determine the moment of fixing the elements of a multilink structure in the preset unfolded state, it is proposed to use a «stop-code». It is a series of numbers, which, by using functions of the generalized coordinates of the Lagrange equation of second kind, define the current values of angles between the elements of a rod structure. Results are intended for geometrical modeling of the unfolding of large-size structures under conditions of weightlessness, for example, power frames for solar mirrors, or cosmic antennae, as well as other large-scale orbital facilities.

Work cycle optimization problem of manipulator with revolute joints

In this work the problem of motion modeling and work cycle optimization of manipulator with revolute joints has been considered. The motion equations of the manipulator elements under any spatial work cycle conditions have been formulated. The formulation has been completed by using the classic vector mechanics and Lagrange equations of second kind. The equations of motion of the system have been obtained using commercial software. The chosen motion model for each considered actuator is point-to-point motion model with quasi-trapezoid velocity profile. Additionally, the problem of optimization of a particular work cycle has been presented. The optimization objective has been chosen as minimization of loads (torques) in actuators. The objective function has been formulated using performance indexes and the design variables are rated velocity value and initial time value of work cycle in each considered actuator. The formulated optimization problem has been solved using constrained Multi-Objective Particle Swarm Optimization algorithm. A numerical computation has been completed using specially performed software and results of the computation have been attached to the paperwork.

Computer-based derivation and transformations of spatial motion equations of a large space structure in the course of its assembly

A mechanic system of the form of a large space structure being assembled in orbit is considered. The problem of on-line computer-based derivation of the current mathematical model of the spatial motion of a mechanic system with its structure changing when extra bodies or constraints are either added or removed is studied. The problem is solved using Lagrange equations of second kind. In the problem statement, it is required that the computing system linearize the obtained nonlinear Lagrange mathematical model, transform it by reducing it to the principal (normal) coordinates and then pass to more constructive, in terms of synthesizing control laws, modally physical representation. The mathematical technique for solving the listed problems is given. A number of necessary propositions are proved. The examples are given to illustrate that the proposed methods are constructive and can be implemented sufficiently simply in Maple.

On a generalized form of lagrange equations of second kind: PMM Vol. 37, N°1, 1973, pp. 156–159

We have obtained a new form for the equations of motion of both holonomic as well as nonholonomic systems. In a special case these equations reduce to the Mangeron-Deleanu equations [1 – 3].

On the possible types of critical cases for lagrange equations of second kind: PMM vol. 36, n≗3, 1972, pp. 390–395

We consider the types of critical cases arising in the general equations of a holonomic scleronomous system in independent coordinates. We examine the system's first-approximation matrix and we study the elementary divisors corresponding to this matrix. We prove a theorem on the stability of the trivial solution in one specific critical case when we use a function which is sign-definite in a part of the variables. After Liapunov's original work 1,2 the critical cases in the general problem of stability of motion were considered in [3]. The algebraic unsolvability of stability problems in sufficiently complex critical cases was pointed out in [4].