Higher-order differential variational principle and differential equations of motion for mechanical systems in event space

2014 ◽  
Vol 23 (10) ◽  
pp. 104501
Author(s):  
Xiang-Wu Zhang ◽  
Yuan-Yuan Li ◽  
Xiao-Xia Zhao ◽  
Wen-Feng Luo
Keyword(s):  
Variational Principle ◽  
Differential Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Higher Order ◽  
Event Space

New Formulations on Motion Equations in Analytical Dynamics

2016 ◽  
Vol 823 ◽  
pp. 49-54 ◽  
Author(s):  
Iuliu Negrean ◽  
Kalman Kacso ◽  
Claudiu Schonstein ◽  
Adina Duca ◽  
Florina Rusu ◽  
...  
Keyword(s):  
Differential Equations ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Higher Order ◽  
Analytical Dynamics ◽  
Lagrange Principle ◽  
Motion Equations ◽  
Differential Motion

Using the main author's researches on the energies of acceleration and higher order equations of motion, this paper is devoted to new formulations in analytical dynamics of mechanical multibody systems (MBS). Integral parts of these systems are the mechanical robot structures, serial, parallel or mobile on which an application will be presented in order to highlight the importance of the differential motion equations in dynamics behavior. When the components of multibody mechanical systems or in its entirety presents rapid movements or is in transitory motion, are developed higher order variations in respect to time of linear and angular accelerations. According to research of the main author, they are integrated into higher order energies and these in differential equations of motion in higher order, which will lead to variations in time of generalized forces which dominate these types of mechanical systems. The establishing of these differential equations of motion, it is based on a generalization of a principle of analytical differential mechanics, known as the D`Alembert – Lagrange Principle.

scholarly journals A Higher-Order Thermomechanical Vibration Analysis of Temperature-Dependent FGM Beams with Porosities

2016 ◽  
Vol 2016 ◽  
pp. 1-20 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Ali Jafari
Keyword(s):  
Differential Equations ◽  
Porous Material ◽  
Power Law ◽  
Material Properties ◽  
Solution Method ◽  
Equations Of Motion ◽  
Higher Order ◽  
Functionally Graded ◽  
Thickness Direction ◽  
Temperature Dependent

In the present paper, thermomechanical vibration characteristics of functionally graded (FG) Reddy beams made of porous material subjected to various thermal loadings are investigated by utilizing a Navier solution method for the first time. Four types of thermal loadings, namely, uniform, linear, nonlinear, and sinusoidal temperature rises, through the thickness direction are considered. Thermomechanical material properties of FG beam are assumed to be temperature-dependent and supposed to vary through thickness direction of the constituents according to power-law distribution (P-FGM) which is modified to approximate the porous material properties with even and uneven distributions of porosities phases. The governing differential equations of motion are derived based on higher order shear deformation beam theory. Hamilton’s principle is applied to obtain the governing differential equations of motion which are solved by employing an analytical technique called the Navier type solution method. Influences of several important parameters such as power-law exponents, porosity distributions, porosity volume fractions, thermal effects, and slenderness ratios on natural frequencies of the temperature-dependent FG beams with porosities are investigated and discussed in detail. It is concluded that these effects play significant role in the thermodynamic behavior of porous FG beams.

scholarly journals Lagrangian for Circuits with Higher-Order Elements

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1059 ◽  
Author(s):  
Zdenek Biolek ◽  
Dalibor Biolek ◽  
Viera Biolkova
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Higher Order ◽  
The State ◽  
Even Order ◽  
Independent Variable ◽  
Necessary And Sufficient ◽  
Derivatives Of ◽  
State Functions

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.

Differential equations of motion of mechanical systems with variable parameters

1974 ◽  
Vol 10 (6) ◽  
pp. 671-674
Author(s):  
V. A. Lazaryan ◽  
L. A. Manashkin ◽  
A. V. Yurchenko
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variable Parameters

Transient Analysis of Forced Vibrations of Complex Structural-Mechanical Systems

1962 ◽  
Vol 66 (619) ◽  
pp. 457-460 ◽  
Author(s):  
S. P. Chan ◽  
H. L. Cox ◽  
W. A. Benfield
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Finite Differences ◽  
Transient Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Forced Vibrations ◽  
Degree Of Freedom ◽  
Dynamic Forces ◽  
Complex Structural

This paper presents a numerical method, derived directly from the basic differential equations of motion and expressed in the form of recurrence-matrix of finite differences, that can be generally applied to all multi-degree-of-freedom structures subjected to dynamic forces or forced displacements on any masses at any instants of time. The movements of the system may be described by any form of generalised co-ordinates.

Application of Euler Parameters to the Dynamic Analysis of Three-Dimensional Constrained Mechanical Systems

1982 ◽  
Vol 104 (4) ◽  
pp. 785-791 ◽  
Author(s):  
P. E. Nikravesh ◽  
I. S. Chung
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Dependent Variables

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are the Euler parameters. Euler parameters, in contrast to Euler angles or any other set of three rotational generalized coordinates, have no critical singular cases. The maximal set of generalized coordinates facilitates the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matrix relating variations in dependent and indpendent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.

scholarly journals INVESTIGATION OF THE DYNAMICS OF NONLINEAR MECHANICAL SYSTEMS WITH LONG POWER LINES THROUGH DIGITAL MODELING

2019 ◽  
Vol 16 (33) ◽  
pp. 668-680
Author(s):  
L. A. KONDRATENKO ◽  
L. I. MIRONOVA ◽  
V. G. DMITRIEV ◽  
O. V. EGOROVA ◽  
A. O. SHEMIAKOV
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Hydraulic Drive ◽  
Operating Conditions ◽  
Dynamic Features ◽  
Good Convergence ◽  
Operating Modes

Many of the mechanisms used in industry contain input and output links connected by long lines of force. Increasing the efficiency and service life of mechanical systems with long lines is of great importance for the country's economy. For a more rational use of these devices, it is important to maintain these operating modes with maximum accuracy, usually including the required speed of the actuator and the voltage in the lines. Such parameters can spontaneously change depending on the operating conditions of the system. In the presence of various influences, similar tasks to determine the marked regimes and parameters indicating the need for their change can be solved only with the help of the corresponding theory and research methods. The article presents the problems and the method of studying two-tier mechanical systems with an infinite number of degrees of freedom on the basis of the equations of momentum and moment of momentum in differential form. Transformations with the use of well-known wave equations are proposed, which made it possible to explicitly take into account the oscillations of the speeds of motion and stresses in the force lines of mechanical systems when describing dynamic processes. The solution of systems of partial differential equations is given using the Laplace transform, which made it possible to obtain general equations of motion and, after some simplifications, proceed to ordinary differential equations that take into account the dynamic features of systems with distributed parameters. The modernized Runge-Kutta method obtained solutions and carried out numerical simulation of transient processes in the hydraulic drive, the results of which have good convergence with full-scale experiments.

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