Energy and Work

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Rigid Bodies ◽  
Second Law ◽  
Spherically Symmetric ◽  
Centre Of Mass ◽  
Unbalanced Force ◽  
Vector Calculus ◽  
Salient Points ◽  
Coordinate Origin ◽  
Bare Minimum ◽  
Kinetic And Potential Energies

This chapter discusses the work–energy theorem, which is developed from Newton’s second law, and defines the kinetic and potential energies of the system. While there is some vector calculus involved, it has been kept to the bare minimum and the reader should not require in-depth knowledge to understand the salient points. If there is a net force on the particle, it accelerates in the direction of the unbalanced force. The force is a central force if it depends only on the distance between the point on which the force acts and the coordinate origin. Using Stokes’s theorem, potential energies are thoroughly discussed. The chapter also discusses spherically symmetric potentials, isotropic force, force on systems of particles, centre of mass coordinates and rigid bodies.

Formulation of the Equations of Motion of RRPR Robot Manipulator

2000 ◽  
Author(s):  
Hazem Ali Attia ◽  
Tarek M. A. El-Mistikawy ◽  
Adel A. Megahed
Keyword(s):  
Dynamic Analysis ◽  
Efficient Solution ◽  
Robot Manipulator ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Second Law ◽  
Equivalent System ◽  
Constraint Forces ◽  
Dynamic Formulation ◽  
System Of Particles

Abstract In this paper the dynamic analysis of RRPR robot manipulator is presented. The equations of motion are formulated using a two-step transformation. Initially, a dynamically equivalent system of particles that replaces the rigid bodies is constructed and then Newton’s second law is applied to derive their equations of motion. The equations of motion are then transformed to the relative joint variables. Use of both Cartesian and joint variables produces an efficient set of equations without loss of generality. For open chains, this process automatically eliminates all of the non-working constraint forces and leads to an efficient solution and integration of the equations of motion. The results of the simulation indicate the simplicity and generality of the dynamic formulation.

II. The angular motion of a freely falling rocket

1964 ◽  
Vol 279 (1379) ◽  
pp. 523-533
Keyword(s):  
Angular Velocity ◽  
Unit Length ◽  
Free Fall ◽  
Axisymmetric Body ◽  
Rigid Bodies ◽  
Angular Motion ◽  
Time Interval ◽  
Angular Momentum Vector ◽  
Centre Of Mass ◽  
Aerodynamic Forces

The motion of a rocket with its propellant exhausted and above the heights where aerodynamic forces can be used to control its motion, can be considered as that of a rigid body in free flight, subjected to small perturbations by weak aerodynamic forces. This permits the separate consideration of the motion of the centre of mass of the rocket along an approximately ‘free fall’ trajectory and the rotation of the rocket about its centre of mass. The rotational motion of free rigid bodies is well known and may be readily visualized by means of Poinsot’s construction (Corben & Stehle 1960). This analysis may be applied to the motion of a rocket with an accuracy which depends on the smallness of the residual aerodynamic forces and the time interval over which the ‘free fall’ approximation is applied. The Skylark rocket vehicle is a long axisymmetric body of approximately uniform mass per unit length. The momental ellipsoid of such a body is a long ellipsoid of revolution with its major axis along the spin axis of the rocket. In this case, the angular motion will consist only of roll and regular precession. In the early stages of the flight the rocket is given some spin motion by aero­dynamic forces on the fins. The angle between the geometrical axis of the rocket and the angular momentum vector is small and can change only slowly because of the aerodynamic forces which are important during the initial stages of the flight. The rate of precession of the rocket axis is much smaller than the rate of spin. In these circumstances, the angular motion will be as shown in figure 11 and can be regarded as roll about the vehicle axis OV with angular velocity ω and precession of this axis about an invariant direction OC with angular velocity Ω. The semi-angle, COV = ρ , of the precession cone is given by cos p = I L / I T ω / Ω , where I L and I T are the moments of inertia about longitudinal and transverse axes passing through the centre of mass.

Central force fields and Kepler's laws

2018 ◽  
Vol 102 (554) ◽  
pp. 270-279
Author(s):  
Luis Blanco ◽  
María García-García ◽  
Joaquín Gutíerrez ◽  
Andrea Rios
Keyword(s):  
Undergraduate Students ◽  
Force Fields ◽  
Central Force ◽  
Second Law ◽  
Double Integral ◽  
Universal Gravitation ◽  
Conservation Of Energy ◽  
Vector Calculus ◽  
Kepler's Laws ◽  
Kepler’S Laws

In this survey we give very simple proofs of Kepler's Laws and other facts about central force fields using only Newton's second law, Newton's law of universal gravitation, basic notions of vector calculus, and an elementary double integral.Hopefully, this article will help undergraduate students of mathematics and engineering who wish to understand these fundamental scientific discoveries.In many textbooks (see, for instance, [1, 2, 3, 4, 5]), Kepler's Laws are obtained using conservation of energy and angular momentum, differential equations, mobile reference systems, or notions not so well-defined such as differentials or ‘infinitesimal elements’. Some of the arguments appear to be rather involved if one is not accustomed to them, whereas the proof of Kepler's Laws may actually be obtained from quite simple facts.

scholarly journals The Stability of Solutions of Sitnikov Restricted Problem of three Bodies When the Primaries are Triaxial Rigid Bodies

2015 ◽  
Vol 19 (2) ◽  
pp. 76-78
Author(s):  
R.R. Thapa
Keyword(s):  
Restricted Problem ◽  
Principal Axis ◽  
Rigid Bodies ◽  
The Body ◽  
Stability Of Solutions ◽  
Centre Of Mass ◽  
Axis Of Symmetry ◽  
The Stability ◽  
Axes Of Symmetry ◽  
Infinitesimal Mass

The paper deals with the stability of the solutions of Sitnikov's restricted problem of three bodies if the primaries are triaxial rigid bodies. The infinitesimal mass is moving in space and is being influenced by motion of two primaries (m1>m2). They move in circular orbits without rotation around their centre of mass. Both primaries are considered as axis symmetric bodies with one of the axes as axis of symmetry whose equatorial plane coincides with motion of the plane. The synodic system of co-ordinates initially coincides with inertial system of co-ordinates. It is also supposed that initially the principal axis of the body m1 is parallel to synodic axis and are of the axes of symmetry is perpendicular to plane of motion.Journal of Institute of Science and Technology, 2014, 19(2): 76-78

A Variational-Vector Calculus Approach to Machine Dynamics

1986 ◽  
Vol 108 (1) ◽  
pp. 25-30 ◽  
Author(s):  
E. J. Haug ◽  
M. K. McCullough
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Linear Structure ◽  
Rigid Bodies ◽  
Machine Dynamics ◽  
System A ◽  
Vector Calculus ◽  
The Matrix ◽  
Generalized Force ◽  
System Equations

A variational-vector calculus approach is presented to define virtual displacements and rotations and position, velocity, and acceleration of individual components of a multibody mechanical system. A two-body subsystem with both Cartesian and relative coordinates is used to illustrate a systematic method of exploiting the linear structure of both vector and differential calculus, in conjunction with a variational formulation of the equations of motion of rigid bodies, to derive the matrix structure of governing multibody system equations of motion. A pattern for construction of the system mass matrix and generalized force terms is developed and applied to derivation of the equations of motion of a vehicle system. The development demonstrates an approach to multibody machine dynamics that closely parallels methods used in finite-element structural analysis.

Student understanding of the application of Newton's second law to rotating rigid bodies

2013 ◽  
Vol 81 (6) ◽  
pp. 458-470 ◽  
Author(s):  
Hunter G. Close ◽  
Luanna S. Gomez ◽  
Paula R. L. Heron
Keyword(s):  
Student Understanding ◽  
Rigid Bodies ◽  
Second Law ◽  
Newton's Second Law
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