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This section addresses the problem of how to characterize forces acting on a rigid body. To begin, we adopt axioms that apply to static forces acting on particles. We hypothesize: 1. A force applied to a particle can be described as a vector.

2. The motion of a particle is determined by the vector sum of the forces applied to the particle.

3. In particular, a particle will remain at rest only if the total force acting on it is zero. Now we define the moment of force about a line, or torque about a line. Let l be some line with direction ˆl, passing through the origin. Suppose a force f acts at x. Then the moment of f about l is defined to be

nl = ˆl · (x × f) (5.1)

We can also define the moment of force or torque about a point O to be

nO = (x − O) × f (5.2)

If the origin is O this reduces to

n= x × f (5.3)

If n gives the moment about the origin, and nl gives the moment about the line l through

the origin, then note that

nl = ˆl · n (5.4)

Now suppose we have a rigid body. Forces acting on the rigid body can be divided into two classes: internal forces are those forces acting between particles of the body; and external forces are those forces acting from outside the body. We define the total force F acting on the body to be the sum of all the external forces, and the total moment (or total

N F x1 x2 f f line of action Figure 5.1

A force may be applied at any point on its line of action without changing its effect on a rigid body.

torque) N to be the sum of all the corresponding moments about the origin:

F=fi (5.5)

N=xi× fi (5.6)

It is a consequence of Newton’s laws that the effect of any system of forces, acting on a single rigid body, is completely determined by the total force F and the total moment N. (This point is considered in greater detail in chapter 8.) Any two systems of forces giving identical F, N, are said to be equivalent. If there is a single force with the same F, N, it is called the resultant.

Line of action. When a force is applied to a particle in three dimensions, that force is completely characterized as a three-dimensional vector. But when a force is applied to a rigid body, we have to consider at which point the force is applied. Consider a force f applied at some point x. The total force F is just the given force: F = f But the total moment N depends on the point of application: N = x × f. However, N is not changed if the force is moved along the line in which it lies—its line of action. In other words, if (x2− x1)  f, then it does not matter whether the force is applied at x1or at x2. Since f can vary freely along a line, it is sometimes called a line vector. Vectors that are fixed at a single point are sometimes called bound vectors, and ordinary vectors are called free vectors when it is necessary to distinguish them.

Resultant of two forces on intersecting lines of action. Suppose two forces f1and f2, acting on L1 and L2 respectively, are applied to some rigid body. If the lines of action intersect, then it is simple to construct the resultant—the single force equivalent to the original system of two forces. We can slide both f1and f2along their respective lines of action to the intersection. Now we have two forces acting at a common point. The resultant force is the vector sum f1+ f2, acting at the intersection.

Rigid Body Statics 95 L1 L2 f1 f2 f1+ f2 Figure 5.2

The resultant of two forces with intersecting lines of action.

Change of reference. Suppose some system of forces yields a total force and moment FQ, NQ, taken with respect to the point Q. What are the force and moment FR, NR, for

a different reference point R? Write the total force and total moment for each choice of reference point: FR=  fi (5.7) FQ=  fi (5.8) NR=  (xi− R) × fi (5.9) NQ =  (xi− Q) × fi (5.10)

From which it follows

FR= FQ (5.11) and NR− NQ=  (Q − R) × fi (5.12) which gives NR= NQ+ (Q − R) × F (5.13)

DEFINITION5.1: A couple is a system of forces whose total force F = fi is zero.

Notice that the moment N of a couple is independent of reference point.

For an arbitrary couple it is trivial to construct an equivalent system of just two forces, which might explain the origin of the name. However, there is no equivalent system of just one force. That is, a couple of non-zero torque does not have a resultant. That means that not every system of forces can be characterized by a resultant.

Figure 5.3

Construction for proof of theorem 5.2.

Equivalent systems of forces

We saw in the previous section that we can characterize a system of forces by its total force and moment, at least as far as its effect on a rigid body. We also saw that a resultant is not a general way of characterizing systems of forces, because some systems, couples for example, do not have resultants. There are other ways of characterizing a system of forces that are more general. In particular, we will define a wrench, which is analogous to the twist used to characterize rigid body motion. First we must tend to some preliminaries.

THEOREM5.1: For any reference point Q, any system of forces is equivalent to a single force through Q, plus a couple.

Proof Let F be the total force, let NQbe the total moment at Q. If we apply a single force F at Q, and construct a couple with moment NQ, then the total force and moment will be F and NQ, as required.

THEOREM5.2: Every system of forces is equivalent to a system of just two forces. Proof It was remarked earlier that for any couple there is an equivalent system of two forces, and that couples can be moved rigidly without affecting their force or torque. So, take the construction of figure 5.3, using only two forces in the construction of the couple. We now have three forces: two to construct the couple, and one passing through Q. Move the couple so that one of its two forces passes through Q. Then replace the two forces at Q with their vector sum. Thus the three forces have been reduced to two.

THEOREM5.3: A system consisting of a single non-zero force plus a couple in the same plane, i.e. a torque vector perpendicular to the force, has a resultant.

Rigid Body Statics 97

F

− F N/F

Figure 5.4

Construction for proof of theorem 5.3.

Proof Let F be the force, acting at P. Let N be the moment of the couple. Construct an equivalent couple as in figure 5.4 and translate it so that−F is applied at P. This will cancel the original F, leaving one resultant force.

THEOREM5.4 POINSOT’S THEOREM: Every system of forces is equivalent to a single force, plus a couple with moment parallel to the force.

Proof Let F and N be the force and moment, respectively, of a given system of forces. Decompose the moment into two components: N_parallel to F, and N_⊥perpendicular to F. By theorem 5.3 we can replace F and N⊥by a single force F parallel to F. Now we construct a couple with moment N_to obtain the desired result: a force and a couple with moment parallel to the force.

Poinsot’s theorem is analogous to Chasles’s theorem (theorem 2.7). And, like Chasles’s theorem, it can be phrased in terms of screws. First we define a wrench.

DEFINITION5.2: A wrench is a screw plus a scalar magnitude, giving a force along the screw axis plus a moment about the screw axis. The force magnitude is the wrench magnitude, and the moment is the twist magnitude times the pitch. Thus the pitch is the ratio of moment to force.

Using the language of screws, Poinsot’s theorem is succinctly stated: every system of rigid body forces reduces to a wrench along some screw.

We can also extend screw coordinates to include wrenches. Let f be the magnitude of the force acting along a line l, and let n be the magnitude of the moment about l. The magnitude of the twist is the magnitude of the force f . Starting from the definition of screw coordinates based on Pl¨ucker coordinates, we can write the screw coordinates of the wrench

w= f q (5.14)

where(q, q0) are the normalized Pl¨ucker coordinates of the wrench axis l, and p is the pitch, which is defined to be

p= n/f (5.16)

Let r be some point on the wrench axis, so we obtain

q0= r × q (5.17)

Then by substituting equations 5.16 and 5.17 into equations 5.14 and 5.15 we can write

w= f (5.18)

w0= r × f + n (5.19)

By comparing with equation 5.13 we can write

w= f (5.20)

w0= n0 (5.21)

where n0is just the moment of force at the origin. Thus we find that screw coordinates of a wrench are actually a familiar representation(f, n0). This yields a vector space, so that we can scale wrenches or add wrenches, just as with differential twists. For a wrench in the x -y plane, the fz, n0x, and n0y terms are all zero, so planar wrenches can be written

( fx, fy, 0, 0, 0, nOz), or more simply as ( fx, fy, n).

The reciprocal product of a differential twist and a wrench is meaningful and useful. Using screw coordinates:

(ω, v0) ∗ (f, n0) = f · v0+ n0· ω (5.22) which is the power produced by the wrench(f, n0) and differential twist (ω, v0). Thus we can immediately observe that a differential twist is reciprocal to a wrench if and only if no power would be produced. In section 3.3, we developed a first order analysis of kinematic constraint using the reciprocal product between a velocity twist and a constraint screw. In section 5.3, we will use reciprocal product between a velocity twist and a wrench instead, but the result is the same.

Undoubtedly, the reader has observed that some conventions for wrench coordinates seem to be reversed from the conventions taken for twist coordinates. In particular, pitch p= n/f has the angular component over the linear component, and the screw coordinates (f, n0) have the linear component before the angular component. Both of these are the reverse of the conventions for twist coordinates. This is not a peculiarity of our conventions; it reflects a deeper fact that is fundamental to the dual relationship of motion and force. For example, in comparing Chasles’s and Poinsot’s theorems, we find that an axis of rotation is

Rigid Body Statics 99

analogous to a line of force. Let us summarize some points of comparison between motion and force:

Motion Force

A zero-pitch twist is a pure rotation. A zero-pitch wrench is a pure force. For a pure translation, the direction

of the axis is determined, but the location is not.

For a pure moment, the direction of the axis is determined, but the loca- tion is not.

A differential translation is equiva- lent to a rotation about an axis at in- finity.

A moment of force is equivalent to a force along a line at infinity. In the plane, any motion can be

described as a rotation about some point, possibly at infinity.

In the plane, any system of forces reduces to a single force, possibly at infinity.