-aX a dt where the last equality follows from the explicit form of A'. The second term consists of the components in the body frame of the vector da'/d/, where the time derivatives are evaluated in the reference frame. If we denote this vector by (da'/dt)b, we have
If the components of a along the body axes, au,av,aw, are constant, then da/d/=0 and tiXa (16-19)
This expression gives the derivatives of a in the reference coordinate system, but with the vector components resolved along the body coordinate axes. Because it is a vector equation, a and a can be resolved into components along any set of coordinate axes, including the reference axes; therefore, the prime and the subscript b will be omitted in future applications where the distinction is clear from the context. An alternative, geometric derivation can be given which derives Eq. (16-19) directly in the reference coordinate system. Figure 16-1 shows the vector a at times t and f+Af. The motion of a is in a cone with <o as the axis, with fixed cone angle t). In the time between / and /+A/, the rotation angle is «Ar, and the magnitude of Aa, the change in a, is
A a =» 2 (a sini})sin£uAf where tj is the angle between a and a. Then
In the limit Ar-^0, the direction of Aa is tangent to the circle, perpendicular to the plane containing a and Thus frwxa which is Eq. (16-19) in the reference coordinate system.
16.13 Angular Momentum, Kinetic Energy, and Moment of Inertia Tensor
The fundamental quantity in rotational mechanics is the angular momentum, L, as discussed in Section 15.1. For a collection of n point masses, the angular momentum is given by n
/= i where m,, r,, and v( are the mass, position, and velocity, respectively, of the /th point mass. Newton's laws of motion, which are valid only in an inertia! coordinate system, will be used to derive an equation of motion for L, so it is important to assume for the present that r( and v, are the position and velocity in an inertial reference frame. It is convenient to write r, as the sum of two terms r, = R+ft (16-21)
where R is the position of a fixed reference point, O', in the rigid body, and p, is the position of the /th mass relative to O', as shown in Fig. 16-2. Differentiating Eq. (16-21) with respect to time gives v,=V+^ (16-22)
where V is the velocity of O' in the inertial frame. Substituting Eqs. (16-21) and
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