On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
R. Wiebe ◽  
P. S. Harvey
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Dynamic Equilibrium ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Double Pendulum ◽  
Governing Equations ◽  
Lumped Mass ◽  
Planar Systems

The Euler–Lagrange equation is frequently used to develop the governing dynamic equilibrium expressions for rigid-body or lumped-mass systems. In many cases, however, the rectangular coordinates are constrained, necessitating either the use of Lagrange multipliers or the introduction of generalized coordinates that are consistent with the kinematic constraints. For such cases, evaluating the derivatives needed to obtain the governing equations can become a very laborious process. Motivated by several relevant problems related to rigid-body structures under seismic motions, this paper focuses on extending the elegant equations of motion developed by Greenwood in the 1970s, for the special case of planar systems of rigid bodies, to include rigid-body rotations and accelerating reference frames. The derived form of the Euler–Lagrange equation is then demonstrated with two examples: the double pendulum and a rocking object on a double rolling isolation system. The work herein uses an approach that is used by many analysts to derive governing equations for planar systems in translating reference frames (in particular, ground motions), but effectively precalculates some of the important stages of the analysis. It is hoped that beyond re-emphasizing the work by Greenwood, the specific form developed herein may help researchers save a significant amount of time, reduce the potential for errors in the formulation of the equations of motion for dynamical systems, and help introduce more researchers to the Euler–Lagrange equation.

The Use of Vector Techniques in Variational Problems

1986 ◽  
Vol 108 (2) ◽  
pp. 141-145 ◽  
Author(s):  
L. J. Everett ◽  
M. McDermott
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Double Pendulum ◽  
Coordinate Transformations ◽  
Elastic Deformations ◽  
Variational Techniques ◽  
Convenient Means ◽  
Moving Coordinate ◽  
Rigid Body Motions

A convenient means for applying vector mathematics to variational problems is presented. The total and relative variations of a vector are defined and results which follow from these definitions are developed and proved. These results are then used to express the variation of a functional using vector techniques rather than the classical scalar or matrix techniques. The simple problems of deriving equations of motion for a rigid body and for a rigid double pendulum are presented as examples of the technique. The key advantages of the method are that (1) it allows the investigator who is familiar and proficient with vector techniques to apply these skills to variational problems and (2) it greatly simplifies the application of variational techniques to problems which include both rigid body motions and elastic deformations. This is accomplished by providing the techniques necessary for computing the variation of a vector defined in a moving coordinate system without using coordinate transformations.

On Analytical Evaluation of Gear Dynamic Factors Based on Rigid Body Dynamics

1985 ◽  
Vol 107 (2) ◽  
pp. 301-311 ◽  
Author(s):  
C. C. Wang
Keyword(s):  
System Dynamics ◽  
Rigid Body ◽  
Time Derivative ◽  
Transmission Error ◽  
Rigid Bodies ◽  
Rigid Body Dynamics ◽  
Dynamic Factor ◽  
Lumped Mass ◽  
Body Dynamics ◽  
Cubic Spline Curve

This paper proposes an initial step to rationalize the dynamic factor calculation and bring it under the control of the laws of mechanics. The theory is straight forward. The concept of mathematical scaling is utilized to simplify the system dynamics’ formulation. The rigid body dynamics accounts for the gear dynamic tooth loads resulting from the prescribed transmission error of each gear step—including the artificial ones. The latter converts a lumped-mass-elastic system into a rigid-bodies-transmission-error system subjected to the solution of the rigid body system dynamics according to Newton’s law. The entire concept of the solution has been implemented into a FORTRAN program approximately 600 statements in length. The results obtained through computer simulation of various test cases demonstrate the potential and effectiveness of the proposed concept. Contrary to the current practice of grossly ignoring the inertial and system effects, this paper has taken all these important factors into account. The transmission-error-induced acceleration is approximated by the second-order time derivative of one of the cubic spline curve-fitting methods. The approach is cost effective and numerically satisfactory. The model can be further improved to reduce the extent of basic assumptions, or to increase the number of conditional constraints without losing economical attractiveness.

scholarly journals Dynamic model of multi-rigid-body systems based on particle dynamics with recursive approach

2005 ◽  
Vol 2005 (4) ◽  
pp. 365-382 ◽  
Author(s):  
Hazem Ali Attia
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Equations Of Motion ◽  
Particle Dynamics ◽  
Rigid Bodies ◽  
Constrained System ◽  
Open Chain ◽  
Chain System ◽  
Closed Chain ◽  
Serial Chains

A dynamic model for multi-rigid-body systems which consists of interconnected rigid bodies based on particle dynamics and a recursive approach is presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. A closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

1995 ◽  
Vol 62 (1) ◽  
pp. 193-199 ◽  
Author(s):  
M. W. D. White ◽  
G. R. Heppler
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Rigid Bodies ◽  
The Body ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
First Moment

The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

A Fully Coupled Flow and 6-DOF Motion Solver in Multiple Reference Frames

2015 ◽  
Author(s):  
Hua Shan ◽  
Sung-Eun Kim ◽  
Bong Rhee
Keyword(s):  
Reference Frames ◽  
Inertial Frame ◽  
Equations Of Motion ◽  
Body Motion ◽  
Governing Equations ◽  
Flow Equations ◽  
Fixed Frame ◽  
Actuator Disk ◽  
Multiple Reference Frames ◽  
Multiple Reference

In many computational fluid dynamics (CFD) applications involving a single rotating part, such as the flow through an open water propeller rotating at a constant rpm, it is convenient to formulate the governing equations in a non-inertial rotating frame. For flow problems consisting of both stationary and rotating parts, e.g. the stator and the rotor of a turbine, or the hull and propeller of a ship, the multiple reference frames (MRF) approach has been widely used. In most existing MRF models, the computation domain is divided into stationary and rotating zones. In the stationary zone, the flow equations are formulated in the inertial frame, while in the rotating zone, the equations are solved in the non-inertial rotating frame. Also, the flow is assumed to be steady in both zones and the flow solution in the rotating zone can be interpreted as the phase-locked time average result. Compared with other approaches, such as the actuator disk (body-force) model, the MRF approach is superior because it accounts for the actual geometry of the rotating part, e.g. propeller blades. A more complicated situation occurs when the flow solver is coupled to the six degrees of freedom (6-DOF) equations of rigid-body motion in predicting the maneuver of a self-propelled surface or underwater vehicle. In many applications, the propeller is replaced by the actuator disk model. The current work attempts to extend the MRF approach to the 6-DOF maneuvering problems. The governing equations for unsteady incompressible flow in a non-inertial frame have been extended to the flow equations in multiple reference frames: a hull-fixed frame that undergoes translation and rotation predicted by the 6-DOF equations of motion and a propeller-fixed frame in relative rotation with respect to the hull. Because of the large disparity between time scales in the 6-DOF rigid body motion of the hull and the relative rotational motion of the propeller, the phase-locked solution in the propeller MRF zone is considered a reasonable approximation for the actual flow around the propeller. The flow equations are coupled to the 6-DOF equations of motion using an iterative coupling algorithm. The coupled solver has been developed as part of NavyFOAM. The theoretical framework and the numerical implementation of the coupled solver are outlined in this paper. Some numerical test results are also presented.

Idealisation and Formulation in Structural Dynamics Using Modal Analysis

2011 ◽  
Vol 418-420 ◽  
pp. 1022-1025
Author(s):  
Muhammad Danish ◽  
Vinay Kumar Pingali ◽  
Somnath Chattopadhyaya ◽  
N.K. Singh ◽  
A.K. Ray
Keyword(s):  
Reference Frame ◽  
Structural Dynamics ◽  
Reference Frames ◽  
Dynamic Equilibrium ◽  
Equations Of Motion ◽  
Complex Structure ◽  
Complicated Structure ◽  
Multiple Reference Frames ◽  
Lumped Masses ◽  
Multiple Reference

The crux feature of this paper is the equations of motion in a structural dynamics with respect to single reference frame that can be easily derived, and the results are well defined and converged. However, problem occurs, when the analysis of any complex, complicated structure is considered and its equation of motion is extracted with respect to single reference frame. The results are indecipherable, ambiguous and less converged. Thus, for such a complex structure, the results obtain with respect to multiple reference frames. In present study, an approximated model with a set of lumped masses, properly interconnected, along with discrete spring and damper elements are in consideration for continuous vibrating system. This results in dynamic equilibrium, which in turn results in formulation and idealization. As, rightly said by scientist Steve Lacy- “To me, there is spirit in a reed. It is a living thing, a weed, really and it does not contain spirit of sort. It’s really an ancient vibration”

scholarly journals Collisions of Constrained Rigid Body Systems with Friction

1998 ◽  
Vol 5 (3) ◽  
pp. 141-151 ◽  
Author(s):  
Haijun Shen ◽  
Miles A. Townsend
Keyword(s):  
Rigid Body ◽  
Tangential Velocity ◽  
Rigid Bodies ◽  
Physical Analysis ◽  
Double Pendulum ◽  
Physical Context ◽  
Collision Problem ◽  
Articulated Systems ◽  
Conventional Methods ◽  
Velocity Changes

A new approach is developed for the general collision problem of two rigid body systems with constraints (e.g., articulated systems, such as massy linkages) in which the relative tangential velocity at the point of contact and the associated friction force can change direction during the collision. This is beyond the framework of conventional methods, which can give significant and very obvious errors for this problem, and both extends and consolidates recent work. A new parameterization and theory characterize if, when and how the relative tangential velocity changes direction during contact. Elastic and dissipative phenomena and different values for static and kinetic friction coefficients are included. The method is based on the explicitly physical analysis of events at the point of contact. Using this method, Example 1 resolves (and corrects) a paradox (in the literature) of the collision of a double pendulum with the ground. The method fundamentally subsumes other recent models and the collision of rigid bodies; it yields the same results as conventional methods when they would apply (Example 2). The new method reformulates and extends recent approaches in a completely physical context.

Geometric Algorithms for Robot Dynamics: A Tutorial Review

2018 ◽  
Vol 70 (1) ◽  
Author(s):  
Frank C. Park ◽  
Beobkyoon Kim ◽  
Cheongjae Jang ◽  
Jisoo Hong
Keyword(s):  
Rigid Body ◽  
Inverse Dynamics ◽  
Group Structure ◽  
Equations Of Motion ◽  
Variable Stiffness ◽  
Rigid Bodies ◽  
Robot Dynamics ◽  
Motion Optimization ◽  
Geometric Formulation ◽  
Variable Stiffness Actuators

We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed-loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.

scholarly journals CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN FOUR-DIMENSIONAL SPACE

2017 ◽  
Vol 23 (1) ◽  
pp. 41-58
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Pendulum Motion ◽  
Nonconservative Force ◽  
Similar Force

In this article, we systemize some results on the study of the equations of motion of dynamically symmetric fixed four-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free four-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. We also show the nontrivial topological and mechanical analogies.

Sign in / Sign up

Export Citation Format

Share Document