171 lines
9 KiB
Markdown
171 lines
9 KiB
Markdown
---
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title: 38-rigid-bodies-in-contact
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---
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must cyclically permute the suffixes 1, 2, 3 to 3, 1, 2. Substitution of (1) in these formulae
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then gives cos 0 = tanh T, = lot + constant, tan =
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It is seen from these formulae that, as t 8, the vector SC asymptotically approaches the
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x2-axis, which itself asymptotically approaches the Z-axis.
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$38. Rigid bodies in contact
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The equations of motion (34.1) and (34.3) show that the conditions of
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equilibrium for a rigid body can be written as the vanishing of the total force
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and total torque on the body:
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F = f = 0 ,
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K ==~rxf=0. =
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(38.1)
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Here the summation is over all the external forces acting on the body, and r
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is the radius vector of the "point of application"; the origin with respect to
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which the torque is defined may be chosen arbitrarily, since if F = 0 the
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value of K does not depend on this choice (see (34.5)).
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If we have a system of rigid bodies in contact, the conditions (38.1) for
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each body separately must hold in equilibrium. The forces considered must
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include those exerted on each body by those with which it is in contact. These
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forces at the points of contact are called reactions. It is obvious that the mutual
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reactions of any two bodies are equal in magnitude and opposite in direction.
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In general, both the magnitudes and the directions of the reactions are
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found by solving simultaneously the equations of equilibrium (38.1) for all the
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bodies. In some cases, however, their directions are given by the conditions
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of the problem. For example, if two bodies can slide freely on each other, the
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reaction between them is normal to the surface.
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If two bodies in contact are in relative motion, dissipative forces of friction
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arise, in addition to the reaction.
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There are two possible types of motion of bodies in contact-sliding and
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rolling. In sliding, the reaction is perpendicular to the surfaces in contact,
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and the friction is tangential. Pure rolling, on the other hand, is characterised
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by the fact that there is no relative motion of the bodies at the point of
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contact; that is, a rolling body is at every instant as it were fixed to the point
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of contact. The reaction may be in any direction, i.e. it need not be normal
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to the surfaces in contact. The friction in rolling appears as an additional
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torque which opposes rolling.
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If the friction in sliding is negligibly small, the surfaces concerned are
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said to be perfectly smooth. If, on the other hand, only pure rolling without
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sliding is possible, and the friction in rolling can be neglected, the surfaces
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are said to be perfectly rough.
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In both these cases the frictional forces do not appear explicitly in the pro-
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blem, which is therefore purely one of mechanics. If, on the other hand, the
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properties of the friction play an essential part in determining the motion,
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then the latter is not a purely mechanical process (cf. $25).
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Contact between two bodies reduces the number of their degrees of freedom
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as compared with the case of free motion. Hitherto, in discussing such
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§38
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Rigid bodies in contact
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123
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problems, we have taken this reduction into account by using co-ordinates
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which correspond directly to the actual number of degrees of freedom. In
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rolling, however, such a choice of co-ordinates may be impossible.
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The condition imposed on the motion of rolling bodies is that the velocities
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of the points in contact should be equal; for example, when a body rolls on a
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fixed surface, the velocity of the point of contact must be zero. In the general
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case, this condition is expressed by the equations of constraint, of the form
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E caide = 0,
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(38.2)
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where the Cai are functions of the co-ordinates only, and the suffix a denumer-
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ates the equations. If the left-hand sides of these equations are not the total
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time derivatives of some functions of the co-ordinates, the equations cannot
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be integrated. In other words, they cannot be reduced to relations between the
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co-ordinates only, which could be used to express the position of the bodies
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in terms of fewer co-ordinates, corresponding to the actual number of degrees
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of freedom. Such constraints are said to be non-holonomic, as opposed to
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holonomic constraints, which impose relations between the co-ordinates only.
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Let us consider, for example, the rolling of a sphere on a plane. As usual,
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we denote by V the translational velocity (the velocity of the centre of the
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sphere), and by Sa the angular velocity of rotation. The velocity of the point
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of contact with the plane is found by putting r = - an in the general formula
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V = +SXR; a is the radius of the sphere and n a unit vector along the
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normal to the plane. The required condition is that there should be no sliding
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at the point of contact, i.e.
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V-aSxxn = 0.
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(38.3)
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This cannot be integrated: although the velocity V is the total time derivative
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of the radius vector of the centre of the sphere, the angular velocity is not in
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general the total time derivative of any co-ordinate. The constraint (38.3) is
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therefore non-holonomic.t
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Since the equations of non-holonomic constraints cannot be used to reduce
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the number of co-ordinates, when such constraints are present it is necessary
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to use co-ordinates which are not all independent. To derive the correspond-
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ing Lagrange's equations, we return to the principle of least action.
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The existence of the constraints (38.2) places certain restrictions on the
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possible values of the variations of the co-ordinates: multiplying equations
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(38.2) by St, we find that the variations dqi are not independent, but are
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related by
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(38.4)
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t It may be noted that the similar constraint in the rolling of a cylinder is holonomic. In
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that case the axis of rotation has a fixed direction in space, and hence la = do/dt is the total
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derivative of the angle of rotation of the cylinder about its axis. The condition (38.3) can
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therefore be integrated, and gives a relation between the angle and the co-ordinate of the
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centre of mass.
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124
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Motion of a Rigid Body
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§38
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This must be taken into account in varying the action. According to
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Lagrange's method of finding conditional extrema, we must add to the inte-
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grand in the variation of the action
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=
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the left-hand sides of equations (38.4) multiplied by undetermined coeffici-
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ents da (functions of the co-ordinates), and then equate the integral to zero.
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In SO doing the variations dqi are regarded as entirely independent, and the
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result is
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(38.5)
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These equations, together with the constraint equations (38.2), form a com-
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plete set of equations for the unknowns qi and da.
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The reaction forces do not appear in this treatment, and the contact of
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the bodies is fully allowed for by means of the constraint equations. There
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is, however, another method of deriving the equations of motion for bodies in
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contact, in which the reactions are introduced explicitly. The essential feature
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of this method, which is sometimes called d'Alembert's principle, is to write
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for each of the bodies in contact the equations.
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dP/dt==f,
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(38.6)
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wherein the forces f acting on each body include the reactions. The latter
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are initially unknown and are determined, together with the motion of the
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body, by solving the equations. This method is equally applicable for both
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holonomic and non-holonomic constraints.
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PROBLEMS
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PROBLEM 1. Using d'Alembert's principle, find the equations of motion of a homogeneous
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sphere rolling on a plane under an external force F and torque K.
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SOLUTION. The constraint equation is (38.3). Denoting the reaction force at the point of
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contact between the sphere and the plane by R, we have equations (38.6) in the form
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u dV/dt = F+R,
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(1)
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dSu/dt = K-an xR,
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(2)
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where we have used the facts that P = V and, for a spherical top, M = ISE. Differentiating
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the constraint equation (38.3) with respect to time, we have V = aS2xn. Substituting in
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equation (1) and eliminating S by means of (2), we obtain (I/au)(F+R) = Kxn-aR+
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+an(n . R), which relates R, F and K. Writing this equation in components and substitut-
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ing I = zua2 (§32, Problem 2(b)), we have
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R2 = -F2,
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where the plane is taken as the xy-plane. Finally, substituting these expressions in (1), we
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§38
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Rigid bodies in contact
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125
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obtain the equations of motion involving only the given external force and torque:
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dVx dt 7u 5 Ky
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dt
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The components Ox, Q2 y of the angular velocity are given in terms of Vx, Vy by the constraint
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equation (38.3); for S2 we have the equation 2 dQ2/dt = K2, the z-component of equa-
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tion (2).
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PROBLEM 2. A uniform rod BD of weight P and length l rests against a wall as shown in
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Fig. 52 and its lower end B is held by a string AB. Find the reaction of the wall and the ten-
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sion in the string.
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Rc
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h
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P
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RB
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T
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A
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B
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FIG. 52
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SOLUTION. The weight of the rod can be represented by a force P vertically downwards,
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applied at its midpoint. The reactions RB and Rc are respectively vertically upwards and
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perpendicular to the rod; the tension T in the string is directed from B to A. The solution
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of the equations of equilibrium gives Rc = (Pl/4h) sin 2a, RB = P-Rcsin x, T = Rc cos a.
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PROBLEM 3. A rod of weight P has one end A on a vertical plane and the other end B on
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a horizontal plane (Fig. 53), and is held in position by two horizontal strings AD and BC,
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RB
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TA
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A
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RA
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C
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FIG. 53
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126
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Motion of a Rigid Body
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