Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part II—System Equations

1990 ◽  
Vol 112 (2) ◽  
pp. 215-224 ◽  
Author(s):  
S. Nagarajan ◽  
D. A. Turcic
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Elastic Response ◽  
Mutual Dependence ◽  
Generalized Coordinates ◽  
Lagrange’S Equation ◽  
Elastic Mechanism

The first step in the derivation of the equations of motion for general elastic mechanism systems was described in Part I of this work. The equations were derived at the elemental level using Lagrange’s equation and the generalized coordinates were both the rigid body degrees of freedom, and the elastic degrees of freedom of element ‘e’. Each rigid body degree of freedom gave rise to a scalar equation of motion, and the elastic degrees of freedom of element e gave rise to a vector equation of motion. Since both the rigid body degrees of freedom and elastic degrees of freedom are considered as generalized coordinates, the equations derived take into account the mutual dependence between the rigid body and elastic motions. This is important for mechanisms that are built using lightweight and flexible members and which operate at high speeds. A schematic diagram of how the equations of motion are obtained in this work is shown in Fig. 1 in Part I. The transformation step in the figure refers to the rotational transformation of the nodal elastic displacements (which were measured in the element coordinate system), so that they are measured in terms of the reference coordinate system. This transformation is necessary in order to ensure compatibility of the displacement, velocity and acceleration of the degrees of freedom that are common to two or more links during the assembly of the equations of motion. This final set of equations after assembly are obtained in closed form, and, given external torques and forces, can be solved for the rigid body and elastic response simultaneously taking into account the mutual dependence between the two responses.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

scholarly journals New Aspects Concerning Dynamics of Rigid Solid Body in Plane-Parallel Motion Subjected to Real Constraints. Part One

2019 ◽  
Vol 24 (2) ◽  
pp. 175-180
Author(s):  
Vladimir Dragoş Tătaru ◽  
Mircea Bogdan Tătaru
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Analysis ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Original Method ◽  
Horizontal Force ◽  
Constraint Forces ◽  
Inclined Plane ◽  
Parallel Motion

Abstract The present paper approaches in an original manner the dynamic analysis of a wheel which climbs on an inclined plane under the action of a horizontal force. The wheel rolls and slides in the same time. The two movements, rolling and sliding are considered to be independent of each other. Therefore we are dealing with a solid rigid body with two degrees of freedom. The difficulty of approaching the problem lies in the fact that in the differential equations describing the motion of the solid rigid body are also present the constraint forces and these are unknown. For this reason they must be eliminated from the differential equations of motion. The paper presents as well an original method of the constraint forces elimination.

scholarly journals Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Awadhesh Kumar Poddar ◽  
Divyanshi Sharma
Keyword(s):  
Perturbation Theory ◽  
Rigid Body ◽  
Coordinate System ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
General Perturbation ◽  
Triaxial Rigid Body ◽  
Three Dimensional Coordinate

AbstractIn this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.

Rigid Body dynamics and decoupled control architecture for two strongly interacting manipulators

Robotica ◽  
1991 ◽  
Vol 9 (4) ◽  
pp. 421-430 ◽  
Author(s):  
M.A. Unseren
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Position Control ◽  
Equations Of Motion ◽  
Joint Space ◽  
Rigid Body Dynamics ◽  
Contact Forces ◽  
Control Architecture ◽  
Reduced Order ◽  
Closed Chain

SUMMARYA rigid body dynamical model and control architecture are developed for the closed chain motion of two structurally dissimilar manipulators holding a rigid object in a three-dimensional workspace. The model is first developed in the joint space and then transformed to obtain reduced order equations of motion and a separate set of equations describing the behavior of the generalized contact forces. The problem of solving the joint space and reduced order models for the unknown variables is discussed. A new control architecture consisting of the sum of the outputs of a primary and secondary controller is suggested which, according to the model, decouples the force and position-controlled degrees of freedom during motion of the system. The proposed composite controller enables the designer to develop independent, non-interacting control laws for the force and position control of the complex closed chain system.

scholarly journals A new pendulum motion with a suspended point near infinity

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Degrees Of Freedom ◽  
Angular Displacement ◽  
Equations Of Motion ◽  
Circular Path ◽  
Large Parameter ◽  
Pendulum Motion ◽  
Third Order ◽  
The Third ◽  
Lagrange’S Equation ◽  
Pendulum Model

AbstractIn this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufficiently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates $$\varphi$$ φ and $$\xi$$ ξ are obtained using Lagrange’s equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter $$\varepsilon$$ ε will be defined instead of a small one in previous studies. The influences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations.

scholarly journals Classical n-body system in geometrical and volume variables: I. Three-body case

2021 ◽  
pp. 2150140
Author(s):  
A. M. Escobar-Ruiz ◽  
R. Linares ◽  
Alexander V. Turbiner ◽  
Willard Miller
Keyword(s):  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Symmetric Polynomial ◽  
Body System ◽  
Generalized Coordinates ◽  
New Variables ◽  
Three Body ◽  
Mass Dependent

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

scholarly journals The Motion of a Rigid Body in the Presence of a Gyrostatic : الحركات المغزلية السريعة لجسم متماسك في مجال قوة نيوتوني في حالة وجود العزم

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Ghadir Ahmed Sahli
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
First Integral ◽  
Geometric Interpretation ◽  
The Body ◽  
Parameter Method ◽  
Principal Axes ◽  
Linear Autonomous System ◽  
Newtonian Force Field

In this study، the rotational motion of a rigid body about a fixed point in the Newtonian force field with a gyrostatic momentum  about the z-axis is considered. The equations of motion and their first integrals are obtained and reduced to a quasi-linear autonomous system with two degrees of freedom with one first integral. Poincare's small parameter method is applied to investigate the analytical peri­odic solutions of the equations of motion of the body with one point fixed، rapidly spinning about one of the principal axes of the ellipsoid of inertia. A geometric interpretation of motion is given by using Euler's angles to describe the orientation of the body at any instant of time.

Nature of the Requirements for Reference Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 49-62
Author(s):  
C. A. Lundquist
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Equations Of Motion ◽  
Inertial Reference Frame ◽  
Earth System ◽  
Coordinate Systems ◽  
Dynamical Equations ◽  
The Earth ◽  
Measurement Uncertainties ◽  
Rotational Motions

AbstractThe current need for more precisely defined reference coordinate systems arises for geodynamics because the Earth can certainly not be treated as a rigid body when measurement uncertainties reach the few centimeter scale or its angular equivalent. At least two coordinate systems seem to be required. The first is a system defined in space relative to appropriate astronomical objects. This system should approximate an inertial reference frame, or be accurately related to such a reference, because only such a coordinate system is suitable for ultimately expressing the dynamical equations of motion for the Earth. The second required coordinate system must be associated with the nonrigid Earth in some well defined way so that the rotational motions of the whole Earth are meaningfully represented by the transformation parameters relating the Earth system to the space-inertial system. The Earth system should be defined so that the dynamical equations for relative motions of the various internal mechanical components of the Earth and accurate measurements of these motions are conveniently expressed in this system.

Augmented Perpetual Manifolds and Perpetual Mechanical Systems-Part I: Definitions, Theorem and Corollary for Triggering Perpetual Manifolds, Application in Reduced Order Modeling and Particle-Wave Motion of Flexible Mechanical Systems

2021 ◽  
Author(s):  
Fotios Georgiades
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Mechanical Engineering ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Reduced Order ◽  
Rigid Body Motions ◽  
Definition Of ◽  
External Forces ◽  
Perpetual Points

Abstract Perpetual points in mechanical systems defined recently. Herein are used to seek specific types of solutions of N-degrees of freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, is proven, that the perpetual points are forming the perpetual manifolds and they are associated with rigid body motions, and these systems are called perpetual. The definition of perpetual manifolds herein is extended to the augmented perpetual manifolds. A theorem, defining the conditions of the external forces applied in an N-degrees of freedom system lead to a solution in the exact augmented perpetual manifold of rigid body motions, is proven. In this case, the motion by only one differential equation is described, therefore forms reduced-order modelling of the original equations of motion. Further on, a corollary is proven, that in the augmented perpetual manifolds for external harmonic force the system moves in dual mode as wave-particle. The developed theory is certified in three examples and the analytical solutions are in excellent agreement with the numerical simulations. The outcome of this research is significant in several sciences, in mathematics, in physics and in mechanical engineering. In mathematics, this theory is significant for deriving particular solutions of nonlinear systems of differential equations. In physics/mechanics, the existence of wave-particle motion of flexible mechanical systems is of substantial value. Finally in mechanical engineering, the theory in all mechanical structures can be applied, e.g. cars, aeroplanes, spaceships, boats etc. targeting only the rigid body motions.

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