# Page:EB1911 - Volume 17.djvu/977

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MECHANICS

of the three links in the above example). They may be called (in a generalized sense) the co-ordinates of the lamina. The lamina when perfectly free to move in its own plane is said to have three degrees of freedom. By a theorem due to M. Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J. For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since AB, BC, are two different positions of the same line in the lamina they are equal, and it is evident that the rotation could have been effected by a rotation about ], B C the centre of the circle ABC, through an angle AJB. As a special case A the three points A, B, C may be in a straight line; I is then at infinity and the displacement is equivalent to a pure translation, since every point of the lamina is now displaced parallel to AB through a space equal to AB. Next, consider any continuous motion of the lamina. The latter may be brought from any one of its positions to a neighbouring one by a rotation about the proper centre. The limiting position ] of this centre, when the two positions are taken infinitely close to one another, is called the instantaneous centre. If P, P' be consecutive positions of the same point, and 50 the corresponding angle of rotation, then ultimately PP' is at right angles to IP and equal to IP 60. The instantaneous centre will have a certain locus in space, and a certain locus in the lamina. These two loci are called pole-curves or cent rodes, and are sometimes distinguished as the space-centrode and the body-centrode, respectively. In the continuous motion in question the latter curve rolls without slipping on the former (M. Chasles). Consider in fact any series of successive positions Fic. xo. (3) 1, 2, 3. . of the lamina (ng. L9 JS. Ju J lu Ju JOE . -FIG. rx. will be assumed in succession 11); and let Jn, Jw, Ju . . . be the positions in space of the centres of the rotations by which the lamina can be brought from the first position to the second, from the second to the third, and so on. Further, in the position 1, let Lg, ]'u, ]'34 . . be the points of the lamina which have become the successive centres of rotation. The given series of positions if we imagine the lamina to general complicated, but in one case, that of the “crossed parallelogram ” (fig- 13), they assume simple forms. We then ave AB=DC and AD=BC, and C from the symmetries of the figure it is plain that A] + ]B=C] + ]D=AD. Hence the locus of {~relative to AB, and the locus re ative to CD are e ual ellipses of which A, B and %, D are respectively the foci. It may be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but admits of infinitesimal displacement, whenever the directions of the three links are Concurrent (or parallel). The matter may of course be treated analytically, but we shall only require the formula for infinitely small displacements. If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are X, p, and if the axes are simultaneously turned through an angle e, the coordinates of a point of the lamina, rel ative to the original axes, are changed from x, y to A-|-x cos e-y sin e, p.+x sin e-I-y cos e, or) + x-ye, ;.i+ xe+ y, ultimately. Hence the component displacements are ultimately D

A B Flo. 13. 5x=>-ye,6y=;t+xe (1) If we equate these to zero we get the co-ordinates of the instantaneous centre. § 4. Plane Statics.-The statics of a rigid body rests on the following two assumptions 1(i) A force may be supposed to be applied indifferently at any point in its line of action. In other words, a force is of the nature of a “ bound ” or “ localized ” vector; it is regarded as resident in a certain line, but has no special reference to any particular point of the line. (ii) Two forces in intersecting lines may be replaced by a force which is their geometric sum, acting through the intersection. The theory of parallel forces is included as a limiting case. For if O, A, B be any three points, and m, n any scalar quantities, we have in vectors m.5A+n.(?3=(m+n)&, (1) provided A m CA + n . C; = o. Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided m= P/OA, n= Q/OB; (2) also R=P OC/OA + Q OC/OB, 1'0ta£€ f1fSl about JIZ until Veg comes into coincidence with ]¢3, then about In until ]'34 comes into coincidence with J34, and so on. This is equivalent to imagining the polygon 'In ]'¢3 ]'34 ., supposed fixed in the lamina, to roll on the polygon J12 Ira lm ., which is supposed fixed in space. By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated. The particular case where both cent rodes are circles is specially important in mechanism. The theory may be illustrated by the case of “ three-bar motion.” formed of jointed links. If, AB being held fixed, the Let ABCD 'be any quadrilateral J quadrilateral be slightly de/°'. ormed, it is obvious that the D.; ' instantaneous centre ] will ~"~-.» ' be at the intersection of the 1 . u ~¢~ straight lines AD, BC, since the displacements of the points D, C are necessarily at right angles to AD, BC, respectively. Hence these displacements are proportional to IID, JC, and therefore to DD, CC', where C'D' is an line drawn parallel to CS, meeting BC, AD in C', D', respectively. The determination of the cent rodes in three-bar motion is in S A B Fig. 12. and P.AC:Q.CB=OA:OB. (4) These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action. If we now imagine the point O to recede to ininity, the forces P, Q and the resultant R are parallel, and we have R=P+Q, P.AC=Q.CB. (5) When P, Q have opposite signs the point C divides AB externally on the side of the greater force. The investigation fails when P+Q=O, since it leads to 9 an infinitely small resultant acting in an infinitely distant line. A combination of two equal, parallel, but oppositely directed forces cannot in fact be replaced by 0 A anything simpler, and must therefore be recognized as an FIG, 14, independent entity in statics. It was called by L. Poinsot, who first systematically investigated its properties, a couple. We now restrict ourselves for the present to the systems of forces in one plane. By successive applications of (ii) any