## Generalized Active Forces

Given a system 5 consisting of N particles Pi, . . . , PN, suppose that n generalized speeds are defined as in Eq. (1.21.1). Let v/' denote the rth partial velocity of Pi in A (see Sec. 1.21), and let R, be the resultant of all contact and body forces acting on P,. Then F,, . . . , F„, called generalized active forces for 5 in A, are defined as n

Some forces that contribute to R, (/' = 1, . . . , N) make no contributions to Fr(r = 1, . . . , n). (Indeed, this is the principal motivation for introducing generalized forces.) For example, the total contribution to Fr of all contact forces exerted on particles of S across smooth surfaces of rigid bodies vanishes; and, if B is a rigid body belonging to S, the total contribution to Fr of all contact and gravitational forces exerted by all particles of B on each other is equal to zero.

If a set of contact and/or body forces acting on a rigid body B belonging to S is equivalent to a couple of torque T together with a force R applied at a point Q of B, then {Fr)B, the contribution of this set of forces to Fr, is given by

(Fr)a = &)r • T + vr • R (r = 1, . . . , n) (2)

where tar and vr are, respectively, the rth partial angular velocity of B in A and the rth partial velocity of Q in A.

Derivations Let C be a contact force exerted on a particle P of 5 by a smooth rigid body B. Then, if n is a unit vector normal to the surface of B at P,

where C is some scalar. Next, consider A\p, the velocity of P in A. This can be expressed as

where A\B is the velocity in A of that point B of B that is in contact with P, V is the velocity of P in B, and B\p must be perpendicular to n if P is neither to lose contact with, nor to penetrate, B. Moreover,

because otherwise there can exist values of ux, ... , un such that Bvp is not perpendicular to n. Now suppose that B is a part of S. Then

Consequently,

and the contribution to Fr of the forces exerted on each other by P and B is [see Eq. (1)]

A\p ■ (Cn) + AvB ■ (-Cn) = C(Av/ - A\B) • n = 0 (8)

Alternatively, suppose that B is not a part of S. Then «,,...,«„ always can be chosen in such a way that "v8 is independent of u{, . . . , u„, which means that

and that the contribution to Fr of the contact force exerted by B on P is [see Eq. (1)]

In both cases, therefore, the contact forces exerted across a smooth surface of a rigid body contribute nothing to Fr (r = I n).

In Fig, 4.1.1, P, and Pj designate particles of a rigid body B belonging to S, R,; is the resultant of all contact and gravitational forces exerted on P, by Pj, and Rj, is the resultant of all contact and gravitational forces exerted on Pj by P,. To show that the total contribution of R,j and Ry, to Ft (r = 1,...,«) is equal to zero, it is helpful to note (see Prob. 1.31) that the partial velocities \rFi and vrp> of P, and Pj in A are related to the partial angular velocity oir of B in A by v/y = v/' + wr X Py (11)

where p¡j is the position vector from P, to Pj.

The Law of Action and Reaction asserts that Rl; and R„ have equal magnitudes and opposite directions, and that the lines of action of R;, and R;, coincide. Furthermore, the line of action of Ry must pass through P,, and that of Ryi through Pj. Consequently,

The total contribution of R,; and R,v to Fr (r = 1, . . . , n) is thus Jsee Eq. (1)1 v/< - R j + y/i ■ R;r (=(v/' - v/>) ■ Rrj

To establish the validity of Eq. (2), we introduce forces K|, , . . , Ky acting on particles P\ PN< of a rigid body B (see Fig. 4.1.2), and let pt, ... , p^

be the position vectors from a point Q fixed in B to Pi, . . , , Py, respectively. Figure 4.1.1 Figure 4.1.2

Then, by definition of equivalence, the set of forces Ki, . . . , K*,. is equivalent to a couple of torque T and a force R applied at Q if and only if

where p, is the position vector from Q to P,. Also by definition, the contribution of Ki, . . . , KN' to the generalized active force Fr is n'

Now, referring once again to Prob. 1.31, one can write v/< = vre + <or x P,. (18)

(.Frh = E (Vre + iorx Pl) • K, = (Or ■ 2 P, X Ki + vrc • S K, (19)

(17,18) i=1 i=1 i=l and, using Eqs. (15) and (16), one arrives at Eq. (2).

Example Figure 4.1.3 shows a uniform rod B of mass m and length L.B is free to move in a plane fixed in a reference frame N. P is a particle of mass m, fixed in N, and qu q2, and q3 are generalized coordinates characterizing the configuration of B in N. _

Suppose that the resultant force exerted by P on B is approximated with P as defined in Eq. (2.3.6) and that the moment about B* of all forces exerted by P on B is approximated with M as defined in Eq. (2.6.3). Then the set of forces exerted on B by P is equivalent to a couple of torque T together with a force R applied at B*, with

where a, and a2 are unit vectors directed as shown in Fig. 4.1.3, and I is the central inertia dyadic of B. Hence, T can be expressed also as

GmmL2

If Mi, u2, u3 are defined as

0 0