Rigid Body Rotation

12 Key excerpts on "Rigid Body Rotation"

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  eBook - PDF

  An Introduction to Mathematics for Engineers

  Mechanics

  • Stephen Lee(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Dynamics of rigid bodies rotating about a fixed axis The human mind has first to construct forms, independently, before we can find them in things. Einstein, 1879–1955 20.1 A rigid body rotating about a fixed axis You might not be able to answer all these questions fully now, but the issues involved should become clearer as you work through this chapter. It is reasonable to treat a large object as a particle when every part of it is moving in the same direction with the same speed, but clearly this is not always the case. The particles in a rotating wheel have different velocities and accelerations and are subject to different forces. The laws of particle dynamics which you have used so far need to be developed so that they can be applied to the rotation of large objects. 20 Q UESTION 20.1 So far you have modelled moving objects as particles. In many circumstances this is reasonable, but how would you model the motion of the sails of a windmill or the other objects illustrated in the pictures above? Do the two children on the roundabout have the same kinetic energy? What is the kinetic energy of a rotating wheel? Definitions You are already familiar with many aspects of rotation such as the angular speed and acceleration of a particle and you have also taken moments to determine the turning effect of a force, but it is as well to be clear about what is meant by some of the terms involved before continuing with the discussion. Rigid bodies Wheels can be modelled as rigid bodies. A rigid body is such that each point within it is always the same distance from any other point. You are not a rigid body but a hard chair is one (molecular vibrations being ignored). The axis of rotation When you lean back on your chair, it might rotate about a point, say A, at the end of one leg. You will have more control, however, if it rotates about the axis formed by the line joining the ends, A and B, of two legs.

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  Newtonian Dynamics

  An Introduction

  • Richard Fitzpatrick(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 8 Rotational Motion

  DOI: 10.1201/9781003198642-8

  8.1 Introduction

  Up to now, we have only analyzed the dynamics of point masses (i.e., objects whose spatial extent is either negligible, or plays no role in their motion). Let us now broaden our approach in order to take extended objects into account. The only type of motion that a point mass object can exhibit is translational motion; that is, motion by which the object moves from one point in space to another. However, an extended object can exhibit another, quite distinct, type of motion by which it remains located (more or less) at the same spatial position, but constantly changes its orientation with respect to other fixed points in space. This new type of motion is called rotation. Let us investigate rotational motion.

  8.2 Rigid Body Rotation

  Consider a rigid body executing pure rotational motion (i.e., rotational motion that has no translational component). It is possible to define an axis of rotation (which, for the sake of simplicity, is assumed to pass through the body); this axis corresponds to the straight-line that is the locus of all points inside the body that remain stationary as the body rotates. A general point located inside the body executes circular motion that is centered on the rotation axis, and orientated in the plane perpendicular to this axis. In the following, we shall tacitly assume that the axis of rotation remains fixed.

  Figure 8.1 shows a typical rigidly rotating body. The axis of rotation is the line AB. A general point, P, lying within the body executes a circular orbit, centered on AB, in the plane perpendicular to AB. Let the line QP be a radius of this orbit that links the axis of rotation to the instantaneous position of P at time t. Obviously, this implies that QP is normal to AB. Suppose that at time

  t + δ t

  point P has moved to P′ , and the radius QP has rotated through an angle

  δ ϕ

  . The instantaneous angular velocity of the body,

  ω ( t )

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  General Physics

  Mechanics and Molecular Physics

  • L D Landau, A. I. Akhiezer, E.M. Lifshitz(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER III

  MOTION OF A RIGID BODY

  Publisher Summary

  This chapter discusses some types of motion of a rigid body. In mechanics, rigid body means that the relative position of the parts of a body remains unchanged during the motion. The body, thus, moves as a whole. The simplest motion of a rigid body is one in which it moves parallel to itself; this is called translation. For example, if a compass is moved smoothly in a horizontal plane, the needle will retain a steady north-south direction and will execute a translational motion. In translational motion of a rigid body, every point in it has the same velocity and describes a path of the same shape, there being merely a displacement between the paths. Another simple type of motion of a rigid body is rotation about an axis. The chapter describes the energy of a rigid body in motion, rotational angular momentum, gyroscope, and inertia forces.

  §25. Types of motion of a rigid body

  So far we have considered the motion of bodies which might be regarded as particles under certain conditions. Let us now go on to consider motions in which the finite size of bodies is important. Such bodies will be assumed to be rigid . In mechanics, this term means that the relative position of the parts of a body remains unchanged during the motion. The body thus moves as a whole.

  The simplest motion of a rigid body is one in which it moves parallel to itself; this is called translation . For example, if a compass is moved smoothly in a horizontal plane, the needle will retain a steady north-south direction and will execute a translational motion.

  In translational motion of a rigid body, every point in it has the same velocity and describes a path of the same shape, there being merely a displacement between the paths.

  Another simple type of motion of a rigid body is rotation about an axis. In rotation, the various points in the body describe circles in planes perpendicular to the axis of rotation. If in a time dt the body rotates through an angle dφ , the path ds traversed in that time by any point P of the body is clearly ds = r dφ , where r is the distance of P from the axis of rotation. Dividing by dt , we obtain the velocity of P

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  Understanding Physics

  • Michael M. Mansfield, Colm O'Sullivan(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  7 Rigid body dynamics

  AIMS

  • To show how the principles governing the motion of many‐body systems, discussed in the previous chapter, can be applied to the motion of rigid bodies
  • To develop a formalism for the treatment of the motion of a rigid body about a fixed axis
  • To outline the formal equivalence between the principles which determine the motion of rigid bodies about a fixed axis and those which determine the motion of a point particle, as discussed in earlier chapters
  • To consider the more complicated situation where the axis of rotation is not fixed, as in the case of gyroscopic motion

  7.1 Rigid bodies

  A rigid body is a many‐body system in which the distance between each pair of particles remains fixed, that is, the system keeps its shape despite the action of any external forces. The motion of such a system under the influence of a net external force

  FEXT

  and of a net moment

  MEXT

  is determined by the Equations (6.5) and (6.25) which were derived in the previous chapter, namely

  (7.1)

  and

  (7.2)

  In Equation (7.1) P is the total momentum which is related to

  VC

  , the velocity of the centre of mass of the rigid body, by Equation (6.16) and hence, from Equation (6.17) ,

  (7.3)

  where ℳ is the mass of the body. Note that, in this and in some subsequent sections, the symbol ℳ is used for the mass of a body to avoid any possible confusion with the symbol for moment (M).

  As an example of a rigid body, consider a body of arbitrary shape, as illustrated in Figure 7.1 , which can be thought of as comprising a large number of particles of masses

  m1 , m2 , m3 ,

  … etc. If all of the body is near the Earth's surface so that g is effectively constant over its extent, then each component mass m

  i

  of a body near the Earth's surface experiences a force

  m

  i

  g

  . In the absence of any other external forces, the net external force on the body is given by , where is the mass of the body. The net external moment, relative to the origin O, is given by

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  Elementary Plane Rigid Dynamics

  • H. W. Harkness(Author)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  These elements will all be constrained to rotate about a fixed axis, if such an axis is fixed in the body, and they will all have the same angular acceleration which will equal the angular acceleration of the rigid body as a whole. The Rotation of a Rigid Body around a Fixed Axis We first discuss a rigid body which is free to rotate round a fixed axis and we may ask the following questions. (i) What is the relation between an applied torque and the acceleration produced ? (ii) What is the momentum of the body when it has attained a given angular velocity ? (iii) What is the kinetic energy of the body ? (i) Consider the body represented in Figure 1-2 free to rotate round the fixed axis O normal to the plane of the paper. Let the resultant of the applied forces be F. This produces a torque FIG. 1-2 4 I. RIGID BODY MOTION AROUND A FIXED AXIS FR = L about O. The effect of the forceF is transmitted throughout the body and appears as the elementary forces acting on the particles of the body. These will be of such magnitude as to give each particle the linear acceleration appropriate to its position with respect to the axis so that the angular acceleration of all the particles throughout the body is the same. This latter condition must be fulfilled since the body is rigid. Consider any sample particle dM, a distance r from the axis; dM is the mass of the element measured in slugs and for convenience is so written rather than d(Wjg). The elementary force on dM may be in any direction and its component, dF, mormal to the radius r will be effective in giving dM a rotational motion around O. The radial component, if any, will develop a stress on the axis only. Internal stresses will also be developed; but these and their torques will appear in pairs throughout the body and their sum over the body will be zero.

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  Available until 16 Feb |Learn more

  Classical Mechanics

  A Computational Approach with Examples Using Mathematica and Python

  • Christopher W. Kulp, Vasilis Pagonis(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 11

  Rigid Body Motion

  We begin this chapter by reviewing the rotational motion of a single particle around an arbitrary axis, and the concepts of the moment of inertia and the center of mass. In particular we focus on how the center of mass simplifies the description of the translational and rotational motion of a system of particles. After the review, we explore generalized definitions of the moment of inertia, including products of inertia and the inertia tensor, and we demonstrate how to calculate these quantities for a variety of solids. We will see that the moment of inertia tensor of a solid depends on the choice of the coordinate system. Furthermore, we will discuss the parallel axis theorem for rigid bodies and show how it can be used to calculate the moment of inertia tensor. This is followed by a discussion of eigenvalues and eigenvectors of matrices, and how they can be used to describe the principal axes of a rigid body. The chapter will conclude with a discussion of the Euler equations and how they can be used to describe the precessional motion of spinning tops and gyroscopes.

  11.1Rotational motion of particles around a fixed axis

  In this section, we will review rotational motion concepts that you likely learned in your introductory physics course. Here we will focus on the rotational motion of a particle that is confined to move in a circle of radius r in the xy-plane with an angular velocity

  ω =

  ω z

  k ^

  , as illustrated in Figure 11.1

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  Arthur Haas: Introduction to Theoretical Physics. Volume 1

  • Arthur Haas, T. Verschoyle(Authors)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  THE MOTION OF RIGID BODIES 87 two particles, the direction of these two opposite accessory forces must be that of the connecting line ; i.e., they must be so-called central forces. Hence, in virtue of the earlier considerations of §§ 19 and 20, we can treat a rigid body as a system of free particles, upon which there act, besides the external forces actually exerted, imaginary internal accessory forces which are central forces, and which obey Newton's Third Law of Motion. It follows that we can apply to a rigid body not only the Principle of the Conservation of the Centre of Mass, but also that of the Conservation of the Total Angular Momentum. If we denote the total momentum of the rigid body by G, its total angular momentum referred to a given point by U, and the acting forces by K, the following relations must therefore be satisfied for every rigid body :— (5) and (6) where r is the radius-vector drawn from the reference point to the point of application of the force in question. The Law of the Conservation of Energy is unimportant for the mechanics of rigid bodies, inasmuch as the internal potential of a rigid body must be invariable, owing to the constancy of the distances within it. Hence, the Law of the Conservation of Energy, in its application to rigid bodies, expresses no more than in its application to a single particle. The kinetic energy of a rigid body, upon which no external forces act, is thus an invariable quantity. § 24. Translation and Rotation of a Rigid Body. Let 0 ' be a point rigidly connected to a solid body, and let us suppose it to be the origin of a coordinate system which is also rigidly connected with the body. The velocity of the point 0' will then represent the velocity of translation of the coordinate system, which we shall call v/. Let the coordinate system rotate with an angular velocity w'.

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  Game Physics Engine Development

  How to Build a Robust Commercial-Grade Physics Engine for your Game

  • Ian Millington(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Part III Rigid-Body Physics 9 The Mathematics of Rotations hus far we have covered a lot of ground on our way to building the first two T physics engines. We have built a sophisticated system capable of simulating par-ticles, either individually or connected into aggregates. To arrive at our goal of build-ing a complete physics engine, we are missing two things: A robust, general-purpose collision detection system (currently we’re using quite an ad hoc system of hard constraints and special case code). The ability of objects to rotate as well as move around linearly. Collision detection is often a self-contained problem solved by a piece of code more or less independent of the physics engine. It will be the subject of Part IV of the book. The second impacts the code we’ve already written: it is the difference between a complete rigid-body physics system and the mass aggregate systems we’ve seen so far. To add rotations, we’ll need to go backward in the capability of our engine. We’ll need to remove a good deal of functionality and rebuild it based on full rotating rigid bodies. Such treatments comprise this part of the book and Part V, most of the rest of the book. This chapter focuses on the properties of rotating bodies, and considers the math-ematical structures needed to represent and manipulate them. As in Chapter 2, we’ll focus on understanding and implementing the mathematics here before building it into our physics engine in Chapter 10. Copyright © 2010, Elsevier Inc. All rights reserved. DOI: 10.1016/B978-0-12-381976-5.00009-7 157 158 Chapter 9 The Mathematics of Rotations 9.1 Rotating Objects in 2D Before we look at rotations in three dimensions, it is worth understanding them in two. I will not implement any code from this section, but thinking about the 2D case is a good first step toward understanding three dimensions. In two dimensions, we can represent any object’s configuration in space by its 2D position and an angle that shows how it is oriented.

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  Introduction to Physics

  Mechanics, Hydrodynamics Thermodynamics

  • P. Frauenfelder, P. Huber(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  A plane perpendicular to this axis, and intersecting it at point g , is called a plane of rotation. Such a plane is shown in Fig. 4-5. All external forces are assumed parallel to this plane. Particle i, of mass m i9 is acted on by an external FIG. 4-5. Rotation of a rigid body about a fixed axis through the point β. The axis of rotation is perpendicular to the plane of the figure. DYNAMICS OF RIGID BODIES 167 force F eA9 which we can resolve into a radial component F €ti cos φ ι and a tangential component F eJ sin (p t . The radial force components are balanced by reaction forces acting on the axis. The tangential components result in a torque M q about the axis, where r t lies in the plane of rotation and is the moment arm of the tangential component. The force component j F ^ s i n ^ accelerates the /th particle, and hence the entire rigid body, with an angular acceleration ώ. To calculate this acceleration, we apply the second law to the /th particle in the form developed in Section 42-1: / ^ ¿ s i n ^ i + Gt = nifti, where G t is the tangential component of the resultant of all internal forces exerted on the /th particle by the other particles of the body. Since we are concerned with rotational motion, it is convenient to use, instead of the forces, their moments about the axis, and instead of the acceleration a i9 the angular acceleration ώ. We have r{F e¡i sin y, = M eJ , G t r t = M iti , a t = r,tí>. Hence, from the second law, we have for the /th particle and for the entire rigid body, Σ M e, i + Σ M i, i = Σ Ή,Γίώ. But £ M eJ equals the resultant moment of the external forces about the axis, and Υ ^Μ ΙΛ = 0 since this sum represents internal forces only. Further-i more, the quantity ώ can be taken outside the summation sign, since the angular acceleration is the same for all particles of a rigid body. The remain-ing term, £ w¿r 2 , is called the moment of inertia of the body, I q , with respect to the fixed axis (q).

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  Rigid Body Mechanics

  Mathematics, Physics and Applications

  • William B. Heard(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  3.1 Vectorial Mechanics 3.1.1 Translational and Rotational Motion The equation of translational motion of a rigid body follows from Newton’s second law for particle motion m p ¨ r p = F p For a rigid body, this is used in an integral form as B dm ¨ r = B dm f where f is the force per unit mass acting on an infinitesimal element of mass. 1 Now ˙ r is expressed in terms the velocity of the reference point r 0 and the an-gular velocity ω (2.17), ˙ r = ˙ r 0 + ω × ( r − r 0 ) ¨ r = ¨ r 0 + ˙ ω × ( r − r 0 ) + ω × ( ˙ r − ˙ r 0 ) = ¨ r 0 + ˙ ω × ( r − r 0 ) + ω × [ ω × ( r − r 0 )] and upon integration we obtain M { ¨ r 0 + ˙ ω × ( ¯ r − r 0 ) + ω × [ ω × ( ¯ r − r 0 )] } = F (3.1) where F = B dm f When r 0 = ¯ r this reduces to M ¨ ¯ r = F (3.2) and the body behaves as if it were a point mass concentrated at the center of mass. 1 ) The use of f dm or dF to derive the equations of motion for rigid bodies is attributed to Euler [21]. 62 3 Dynamics The vector form of the equation of rotational motion of a rigid body is de-rived from the moment, relative to r 0 , of Newton’s second law for particle motion m ( r p − r 0 ) × ¨ r p = ( r p − r 0 ) × F p which can be rewritten as m d dt ( r p − r 0 ) × d dt ( r p − r 0 ) + m ( r p − r 0 ) × ¨ r 0 = ( r p − r 0 ) × F p and this yields the equation for the evolution of the angular momentum of a particle d dt h p + m ( r p − r 0 ) × ¨ r 0 = ( r p − r 0 ) × F p This is used in an integral form to obtain the rate of change of angular mo-mentum for a rigid body, B dm d dt ( r − r 0 ) × d dt ( r − r 0 ) + m ( r − r 0 ) × ¨ r 0 = B dm ( r − r 0 ) × f or d dt h + m ( r − r 0 ) × ¨ r 0 = τ (3.3) where τ = B dm ( r − r 0 ) × f is the total torque about r 0 . Again f is the force per unit mass acting on an infinitesimal element of mass dm . As in the derivation of Eq. (3.2) the time derivative can be taken outside the integral.

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  Guide to Mechanics

  • Philip Dyke, Roger Whitworth(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  Further work can be carried out when these properties are applied to the many problems of general motion, however the principles involved will not be explored here as they belong securely in more advanced texts (see Phil Dyke's, Work Out Mechanics , Macmillan ± now Palgrave, 1995). Here, we shall be content to consider a simple case of a rigid body travelling in a two-dimensional plane, where the position vector of the centre of mass is in Cartesian form so that: r ˆ x y and the velocity and acceleration of the centre of mass are respectively: _ r ˆ _ x _ y and  r ˆ  x  y The angle of rotation about an axis perpendicular to the plane of Ox and Oy is and the corresponding angular velocity and angular acceleration are _ and  . Newton's laws relating force couples and acceleration applied to the planar motion of a rigid body of mass m whose moment of inertia about an axis through its centre of mass is I give: Translation (linear momentum): impulse ˆ change in linear momentum Rotation (angular momentum): moment of impulse ˆ change in angular momentum Recall the practical exploration of the motion of the rule at the beginning of this chapter. The following example explores the dynamics of the same pro-blem. An Introduction to Rigid Body Dynamics 281 Example 10.19 A thin uniform rod AB of mass m and length 2 a rests in equilibrium on a smooth horizontal plane. An impulse J acts in the horizontal plane on the rod at A perpendicular to AB . Find (a) the result-ing motion of the rod and (b) the linear velocity of the ends A and B immediately after the impact (see Figure 10.30). Solution (a) Immediately after the impulse has been applied: (i) i the centre of mass of the rod moves with velocity: v ˆ u v (ii) the angular speed of the centre of mass of the rod is ! .

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  General Physics Mechanics Thermodynamics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2022(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  An example is shown in the figure where the only fixed point in the absolute system is clearly the centre of mass of the cylinder in the gimbals and all rotations ω x , ω y and ω z , respectively about axes x, y and z, are τ G,1 G m m τ G,2 R R θ r 2 r 1 F F 2 1 ω G ω m m r 2 r 1 ω z ω y ω x x y z G Chapter 8 Dynamics of a Rigid Body 191 independent. When the body in the centre is at rest, any rotation about any axis can be eas- ily obtained; when on the contrary the cylinder rotates about axis y with angular velocity  ω = ω y  u y , the existing rotation exerts a resistance against the change of direction of the rotation axis. Since the rotation axis is a central inertia axis, the direction of the rotation axis is coincident with the direction of the angular momentum, in fact  L G = I yy  ω . But an- gular momentum is conserved if no external torque is present since  τ G E ( ) = d  L G dt = 0 ⇒  L G = I yy  ω = const ⇒  ω = const , thus the angular velocity is constant both in magnitude and direction, causing the latter to be fixed. These devices are mechanical stabilisers and their stabilising effect increases with the magnitude of the angular velocity. 8.20 Precession A much more complex, but very important, situation occurs when the equilibrium of a rotating gyroscope is perturbed by the application of an external torque. Suppose that a force  F E ( ) , perpendicular to the rotation axis, is applied at distance R from the centre of mass G of a gyroscope constrained in G and in rotational equilibrium about an axis y with angular velocity  ω . Such a force exerts a torque  τ G E ( ) =  r ×  F E ( ) = d  L G dt perpendicular to the rotation axis.

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