be in as before the deviation of angular velocity to be produced rotation with the angular velocity a about an axis O may be considered as compounded of a rotation with the same angular velocity about an axis drawn through G parallel to O and a translation with the velocity o.. OG, OG being the perpendicular distance between the two axes. Hence the required deviation B may be regarded as compounded of a
deviation of translation dv./=OG. da, to produce which there would be required, according to equation (82), a force applied at G perpendicular to the plane OGgy
in the interval dt. A
P -é OG .E (34)
and a deviation da of rotation about an axis drawn through G parallel to O, to produce which there would be required a couple of the moment M given by equation (83). According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose GC=M/P=R2/OG. (85)
Thus is determined the position of the centre of percussion C, corresponding to the axis of rotation O. It is obvious from this equation that, for an axis of rotation parallel to O traversing C, the centre of percussion is at the point where the perpendicular OG meets O.
§ I2S.* To jind the moment of inertia of a bo? about an axis through its centre of gravity experimentally.-Suspen the body from any conveniently selected axis O (fig. 48) and hang near it a small plumb bob. Adjust the length of the plumb-line until it and the body oscillate together in unison. The length of the plumb-line, measured from its point of suspension to the centre of the bob, is for all practical purposes equal to the length OC, C being therefore the centre of percussion corresponding to the selected axis O. From equation (85
~ R2 = CG XOG = (OC-OG)OG.
The position of G can be found experimentally; hence OG is known, and the quantity R2 can be calculated, from which and the ascertained weight W of the body the moment of inertia about an axis through G, namely, W/gXR', can be computed.
§ 126.* T o and the force competent to produce the instantaneous acceleration of any link of a mechanism.-In many practical problems it is HCCBSSEHR/ to know the magnitude and position of the forces acting to pro uce the accelerations of the several links of a mechanism. For a given link, this force is the resultant of all the accelerating forces distributed through the substance of the material of the link required to produce the requisite acceleration of each particle. and the determination of this force depends upon the principles of the two preceding sections. The investigation of the distribution of the forces through the material and the stress consequently produced belongs to the subject of the STRENGTH or MATERIALS (q. v.). Let BK (fig. 134) be any link moving in any manner in a plane, and B let G be its centre of gravity.
Then its motion may be analysed
into (1) a translation of
G its centre of gravity; and (2) a
rotation about an axis through
its centre of gravity perpen-
dicular to its plane of motion.
K Let a be the acceleration of
the centre of gravity and let A
be the angular acceleration
1, about the axis through the
centre of gravity; then the
) force required to produce the
g, g translation of the centre of
/ gravity is F=Wa/g. and the
/" couple required to produce the
k -/ angular acceleration about the
Fm' !34 centre of gravity is M-IA/g,
W and l being respectively the
weight and the moment of inertia of the link about the axis through the centre of gravity. The couple M may be produced by shifting the force F parallel to itself through a distance x, such that Fx=M. When the link forms part of a mechanism the respective accelerations of two points in the link can be determined by means of the velocity and acceleration diagrams described in § 82, it being understood that the motion of one link in the mechanism is prescribed, for instance, in the steam-engine's mechanism that th e crank shall revolve uniformly. Let the acceleration of the two points B and K therefore be su posed known. The problem is now to find the acceleration a and Take an pole O (55 49), and set out Ob equal to the acceleration of B and Ok equal to the acceleration of K. join bk and take the point g so that KG: GB=kg: b. Og is then the acceleration of the centre of gravity and the force F can therefore be immediately calculated. To find the angular acceleration A, draw kt, bt respectively parallel to and at right angles to the link KB. Then tb represents the angular acceleration of the point B relatively to the point K and hence tb/KB is the value of A, the angular acceleration of the link. Its moment of inertia about G can be found experimentally by the method explained in § 125, and then the value of the couple M can be computed. The value of x is found immediately from the quotient M/F. Hence the magnitude F and the position of F relatively to the centre of gravity of the link, necessary to give rise to the couple M, are known, and this force is therefore the resultant force required.
§ 127.* Alternative construction for finding the position of F relatively to the centre of gravity of the link.-Let B and K be any two points in the link which for greater K genera lit are taken in fig. 135, so that the centre ofygravity G is not in the line joining them. First find the value of R experimentally. Then produce the given Y
directions of acceleration of B and K to meet in O; draw a circle through the three points B, K and O; produce the line joining O and G to cut the circle in Y; and
ta e a point Z on the line OY so that
¢G><GZ=R2. Then Z is a point in the
line of action of the force F. This useful theorem is due to G. T. Bennett, of
Emmanuel College, Cambridge. A proof
of it and three corollaries are given In appendix 4 of the second edition of Dalby's Balancing of Engines (London, 1906). It is to be noticed that only the directions of the accelerations of two points are required to find the point Z.
For an example of the application of the principles of the two preceding sections to a practical problem see Valveand Valve Gear Mechanisms, by W. E. Dalby (London, 1906), where the inertia stresses brought upon the several links of a ]oy valve gear, belonging to an express passenger engine of the Lancashire & Yorkshire railway, are investigated for an engine-speed of 68 m. an hour. § 128.* The Connecting Rod Problem.-A particular problem of practical importance is the determination o the force producing the motion of the connecting rod of a steam-engine mechanism of the usual type. The methods of the two preceding sections may be used when the acceleration of two points in the rod are known, In this problem it is usually assumed that the crank pin K (fig. 136) ia
Gite L” K '
ll'/ o I if
moves with uniform velocity, so that if o. is its angular velocit and r its radius, the acceleration is azr in a direction along the cranl; arm from the crank pin to the centre of theshaft. Thus the accelera tion of one point K is known completely. The acceleration of a second point, usually taken at the centre of the cross head pin, can be found by the principles of § 82, but several special geometrical Constructions have been devised for this purpose, notably the construction of Klein# discovered also independently by Kirsch.” But probably the most convenient is the construction due to G. T. Bennett” which is as follows: Let OK be the crank and KB the connecting rod. On the connecting rod take a point L such that KLXKB=KO2. Then, the crank standing at any angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the acceleration of the point B to the scale on which KO represents the acceleration of the point K. The proof of this construction is given in The Balancing of Engines. I
The finding of F may be continued thus: join AK, then AK is the acceleration image of the rod, OKA being the acceleration diagram. Through G, the centre of gravity of the rod, draw Gg parallel to the line of stroke, thus dividing the image at g in the proportion that the connecting rod is divided by G. Hence Og represents the acceleration of the centre of gravity and, the weight of the connecting 1 ]. F. Klein, “ New Constructions of the Force of Inertia of Connecting Rods and Couplers and Constructions of the Pressures on their Pins, " Journ. Franklin Inst., vol. 132 (Sept. and Oct., 1891); 2 Prof. Kirsch, “' Uber die gra hische Bestimrnun der Kolbenbeschleunigung, " Zeitsch. Verein zikutsche Ingen. (18 5, p. 1320. 3 Dalby, The Balancing of Engines (London, 19063, app. 1.