Rotational Dynamics

3 Key excerpts on "Rotational Dynamics"

• 

  eBook - ePub

  Engineering Science

  For Foundation Degree and Higher National

  • Mike Tooley, Lloyd Dingle(Authors)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  Part II Dynamics In this part of the book we consider the dynamic behaviour of bodies, that is, the motion of bodies under the action of forces. The study of dynamics is normally split into two major fields, kinematics, the study of the motion of bodies without reference to the forces that cause the motion, and kinetics, where the motion of bodies is considered in relationship to the forces producing it. With the exception of the study of mechanisms in Chapter 9, we will not, in this short study of dynamics, differentiate a great deal between these two areas, rather concentrating on the macroscopic effects of forces and resulting motions in dynamic engineering systems and machinery. We start our study of dynamics in Chapter 8 by looking at a number of fundamental concepts concerning linear and angular motion and the forces that create such motion, including a review of Newton’s laws. We then consider momentum and inertia, together with the nature and effects of friction that acts on linear and angular motion machines and systems. Next, mechanical work and energy transfer are considered and their relationship to linear and angular motion systems is explored. Finally in Chapter 8, as a prelude to rotating machinery in Chapter 10, we take a brief look at circular motion and the forces created by such motion. In Chapter 9, we first consider the motion of one or two single and multilink mechanisms. In particular, we look at the nature of the relative velocities and accelerations of these mechanisms, together with the power and efficiencies of particular engineering machines that utilise such mechanisms. Then, very briefly, we consider the geometry and output motions of various types of cams. We will then look at a number of gyroscopic motion parameters, in particular gyroscopic rigidity, precession and reaction torque

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  eBook - ePub

  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  In the absence of external torque, angular momentum is conserved.

  12.5.3   3D Rotational Dynamics

  Now let’s extend the basic principles developed in Section 12.5.2 into three dimensions. First, let’s review the 3D rotational kinematics quantities. The single angle θ is replaced by a rotation tensor of some kind, with a rotation matrix R or a quaternion q being the most common methods of describing orientation in general rigid body simulations. The angular velocity ω and acceleration α become vector quantities and get bolded as ω and α , respectively.

  To extend the dynamics principles into three dimensions, we start with torque. Not surprisingly, torque becomes a vector quantity denoted τ , and the direction of this vector indicates the axis about which the torque is tending to induce rotation. (Later we consider what happens if the object is already rotating about a different axis.) The formula for computing the torque for an applied force f and lever arm I is actually simpler in 3D than the corresponding 2D formula!

  Torque in Three Dimensions

  τ = 1 × f .

  (12.29)

  Compare Equation (12.29) to τ = Fl sin ϕ (Equation (12.24)), and notice that the cross product has the magnitude and sin ϕ terms built in.

  Angular momentum likewise becomes a vector L , with a similar formula for its relation to the linear quantity:

  L = r × P .

  Orbital angular momentum of a particle in three dimensions with radial vector r

  Compare this to Equation (12.28).

  A reader who is paying attention might note that Equation (12.28) is only one of two equations we gave for angular momentum in the plane—the one we deemed to be more appropriate for orbital angular velocity of a particle—and wonder about the other formula, Equation (12.27), which was more appropriate for spin angular velocity. That formula was L = Jω , and to get its three-dimensional equivalent, we must understand how to extend J , the moment of inertia, into three dimensions. Luckily, the link between the two momentum equations is an excellent way to get this understanding. Let’s start by expanding L = r × P , with the goal of ending up with something that looks like L = Jω

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  eBook - ePub

  Classical Mechanics

  From Newton to Einstein: A Modern Introduction

  • Martin W. McCall(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Nevertheless, providing we keep these limitations in mind, the rigid body concept is very useful. We must not get carried away, though. You might think that if all the bodies respond to each other at the same time, then effectively the dynamical problem reduces to that of a single particle. This is not so. A force applied to the centre of mass of a rigid body does indeed cause the body as a whole to accelerate. However, an external force acting through any other point exerts a torque about the centre of mass that causes the body to spin (see Figure 8.1). The spinning motion maintains the spatial separations, as required for a rigid body, but has no analogue for single-particle systems. Nevertheless, the problem has been effectively reduced to describing the centre-of-mass motion and the twisting effect induced by external forces and torques. With respect to the potential complexity of a many-particle system, the rigid body constraint is very severe. Our aim in this chapter is to analyse the residual dynamics consistent with this constraint. Figure 8.1 A free rigid body translating and rotating in response to an external force. 8.2 Torque and Angular Momentum for Systems of Particles For a single particle we had (Equation (7.8)) G = d L / dt, and our first task is to generalise this to a system of particles. Figure 8.2 shows an arbitrary system, with as yet no rigid body constraint being applied. Forces acting on a particular particle are of two kinds. The external forces acting may be summed to give F ext 1, the total external force acting on particle 1, for example. Otherwise, there are the forces due to other particles in the system, where F 12 denotes the force on particle 1 due to particle 2. The total torque on particle 1 is given by (8.1) Similarly for mass 2: (8.2) Now calculate the total torque as G total = G 1 + G 2 + · · ·

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