Rigid Body Dynamics

4 Key excerpts on "Rigid Body Dynamics"

• 

  eBook - ePub

  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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  eBook - ePub

  Classical Mechanics

  • Tom W B Kibble, Frank H Berkshire(Authors)
  • 2004(Publication Date)
  • ICP

    (Publisher)

  Chapter 9

  Rigid Bodies

  The principal characteristic of a solid body is its rigidity. Under normal circumstances, its size and shape vary only slightly under stress, changes in temperature, and the like. Thus it is natural to consider the idealization of a perfectly rigid body, whose size and shape are permanently fixed. Such a body may be characterized by the requirement that the distance between any two points of the body remains fixed. In this chapter, we shall be concerned with the mechanics of rigid bodies.

  9.1Basic Principles

  It will be convenient to simplify the notation of the previous chapter by omitting the particle label i from sums over all particles in the rigid body. Thus, for example, we shall write

  in place of (8.3) and (8.11). The motion of the centre of mass of the body is completely specified by (8.6):

  Our main interest in this chapter will be centred on the rotational motion of the rigid body. For the moment, let us assume, as we originally did in §8.3 , that the internal forces are central. Then, according to (8.14),

  We shall see later that these two equations are sufficient to determine the motion completely.

  One very important application of (9.1) and (9.2) should be noted. For a rigid body at rest, = 0 for every particle, and thus both P and J vanish. Clearly, the body can remain at rest only if the right hand sides of both (9.1) and (9.2) vanish, i.e., if the sum of the forces and the sum of their moments are both zero. In fact, as we shall see, this is not only a necessary, but also a sufficient, condition for equilibrium.

  Under the same assumption of central internal forces, we saw in §8.5 that the internal forces do no work, so that

  This might at first sight appear to be a third independent equation. However, we shall see later that it is actually a consequence of the other two. It is of course particularly useful in the case when the external forces are conservative, since it then leads to the conservation law

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  eBook - ePub

  Applied Engineering Mechanics

  Statics and Dynamics

  • Boothroyd(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  17 Dynamics of Rigid Bodies 17.1  Introduction

  In the previous three chapters, the motion of particles due to the application of external forces was studied. There, each body was assumed to be either infinitesimally small so that its mass and all the external forces acting were concentrated at the same point, or the orientation of the body in space was unimportant so that it was not necessary to take into account the body’s shape or the exact location of the forces. In this chapter, we shall be concerned with a study of the relationships between the forces that act on any rigid body and the corresponding translational and rotational motions of the body.

  Only two equations of motion are needed to determine either the planar motion of a single particle or the planar motion of the center of mass of a group of particles. For the planar motion of a rigid body, an additional equation of motion is needed in order to determine the rotational motion of the body.

  17.2  Planar Motion of a Rigid Body

  Figure 17.1a shows the free-body diagram for a rigid body moving in the xy plane. Since a rigid body is merely a collection of particles with no relative linear motion between them, the motion of the center of mass of the body can be determined from the translational equations of motion in the x and y directions as follows:

  Fig. 17.1 Rigid body with planar motion.

  F x = m a c x = m x ¨ c

  (17.1)

  F y = m a c y = m y ¨ c

  (17.2)

  where m is the total mass of the body; Fx and Fy are the sums of the components of the external forces in the x and y direction, respectively; and acx and acy are the components of the acceleration of the center of mass in these two directions, respectively.

  To develop the rotational equation of motion, we can examine first the motion of a single particle of mass mp located at P (Fig. 17.1b ). This motion consists of the rotation of point P about the center of mass C superimposed on the motion of the center of mass. Thus, the total inertia forces for the particle will be the sum of the individual inertia force components shown in the two diagrams in Fig. 17.1b . Now if Fex and Fix represent the x components of the external and internal forces acting on the particle, respectively, and Fey , Fiy represent the y components, the free-body diagram for the particle will be as shown in Fig. 17.1c . Taking moments about C and equating the moments of the forces in Fig. 17.1c to the sum of the moments of all the individual inertia forces in Fig. 17.1b

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  eBook - ePub

  An Elementary Treatise on Theoretical Mechanics

  • Sir James H. Jeans(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER XI

  MOTION OF RIGID BODIES

  229. The present chapter is devoted to a discussion of the motion of rigid bodies, when the motion is such that the bodies may not be treated as particles.

  It has already been proved in § 66 that the most general motion possible for a rigid body is one compounded of a motion of translation and a motion of rotation. As a preliminary to discussing the general motion of a rigid body under the action of forces of any description, we shall examine in greater detail than has so far been done the properties of a motion of rotation.

  ANGULAR VELOCITY

  230. We have seen (§ 67) that for every motion of a rigid body in which a point P remains fixed, there is an axis of rotation, which is a line passing through P , of which every point remains fixed. If a rigid body is moving continuously we may analyze its motion in the following way. We select a definite particle P of the rigid body, and we refer the motion to a frame of reference having P as origin, and moving so as always to remain parallel to its original position. Relative to this frame, the motion of the body between any two instants is a motion of rotation about P

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