Velocity and Acceleration in an Arbitrarily. Moving Frame

Since in many applications, we want to express the position, inertial velocity, and in­ertial acceleration of a particle in components parallel to the axes of moving frames, we need general theorems that allow for arbitrary motion of the origin, and arbitrary angular velocity of the frame. These theorems are presented below.

Let F^Oxyz) be any moving frame with origin at О and with angular velocity to relative to F,. Let r = r0 + r’ be the position vector of a point P of FM (see Fig. A.6). Let the velocity and acceleration of P relative to F, be v and a. Then in F,

v, = r, a/ = І/

We want expressions for the velocity and acceleration of P in terms of the compo­nents of r’ in FM. Expanding the first of (A.6,1)

V/ = r0/ + r;

= v0, + r;

where v0 is the velocity of О relative to FThe velocity components in FM are given by

= W/V/ = LM/(v0/ + ry) — + L Mli,

From the rule for transforming derivatives (A.4,22)

Ljvf/Г/ — iM + ojmvm (A.6,3)

whence

Vm = VoM + Гм+ "лАм (A.6,4)

The first term of (A.6,4) is the velocity of О relative to F„ the second is the velocity of P as measured by an observer fixed in FM, and the last is the “transport velocity,” that is the velocity relative to F, of the point of FM that is momentarily coincident with P. The total velocity of P relative to F, is the sum of these three components. Following traditional practice in flight dynamics, we denote

(A.6,5)

(When necessary, subscripts are added to the components to identify particular mov­ing frames.)

The scalar expansion of (A.6,4) is then

vx = v0r + x + qz – ry

vv = v0r + у + rx – pz (A.6,6)

vz = u0. + z + РУ – qx

These expressions then give the components, parallel to the moving coordinate axes, of the velocity of P relative to the inertial frame.

On differentiating v7 and using (A.6,4) we find the components of inertial accel­eration parallel to the Fm axes to be

ам Lw, v, vM + toMM

= v0m + FM + + "мГдг + + <aMi’M + (Ьмымг’м

= a0„ + r’M + ojMrM + 2tbMr’M + ojMwMr’M (A.6,7)

where a0vf = v0m + 6>mv0m = LjWvl); is the acceleration of О relative to F,.

The total inertial acceleration of P is seen to be composed of the following parts:

aoM; *’м’- A* f ^ 2a>MrM: m’-

the acceleration of the origin of the moving frame the acceleration of P as measured by an observer fixed in the mov­ing frame

the “tangential” acceleration owing to rotational acceleration of the frame FM

the Coriolis acceleration the centripetal acceleration

Three of the five terms vanish when the frame FM has no rotation, and only r’M re­mains if it is inertial. Note that the Coriolis acceleration is perpendicular to coM and
t’M, and the centripetal acceleration is directed along the perpendicular from P to to. The scalar expansion of (A.6,7) gives the required inertial acceleration components of P as

ax = a0x + x + 2qz ~ 2ry – x(q2 + r ) + y( pq – r) + z(pr + q)

ay = a0y + >’ + 2rx – 2pz + x(pq + r) – y(p2 + t2) + z(qr – p)

az = a0z + z + 2py – 2qx + x(pr – q) + y(qr + p) – zip2 + q2)