By Brian H. Chirgwin, Charles Plumpton
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Extra info for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics
Angular velocity is a line vector. So far we have considered the motion of a body when it has angular velocities about 42 A COURSE OF MATHEMATICS each of two intersecting axes simultaneously. We now consider the motion of a body which has two angular velocities, simultaneously, about each of two non-intersecting axes. We must define precisely how a body can have two such angular velocities and we do this by employing three frames of reference : a frame A$1 e2 e, which is fixed in the body; a frame Ox1 x2 x3in which the point A is fixed; and a frame 0 X, X2 X3which we shall take as "fixed in space".
Hence eqn. 44) v = vA + co x (r — rA). This is the fundamental equation for the motion of a rigid body and gives the velocity of an arbitrary point P as the sum of the velocity of x3 'x2 FIG. 10. a "base point" A and that of a rotation co about A. If we use another base point B, about which the angular velocity is (Di , vB = vA + to x (rB rA) v vB + co, x (r — rB). •. r = VA co x (r5— rA) + co x — rB) = VA ± co x (r — rA). w x (r — rB) = co, x (r — rB). Since r is arbitrary we conclude that w = cot .
48) In eqn. 47), XA varies with the time because of the variations in R1. The coordinates of P in 0X1 X2 X3 are X = Ri xA R i Rg • Differentiating this relation gives the resolutes of the velocity of P in the frame 0X1 X2 X3 . = iti XA (h i R2 = R1 xA (R1 R2 =Rl Ri`Y A = it,R;XA RA)g Rik)R2 (x — x A) + R1 112) — XA) (kRj R1ll2R2R1) (X — XA). h. 30)] giving resolutes in the frame 0 X1 X2 X3. 51) where w =w1 co,. 51) shows that the angular velocity of the body is the vector sum of the angular velocity co, of the frame § 1: 10] KINEMATICS IN THREE DIMENSIONS 43 0x1 x2 x3and the angular velocity co, of the body relative to the frame, even when the axes do not intersect.