DYNAMICS OF MACHINERY. When an engine has been proportioned to sufficient capacity and strong enough in all its parts to transmit or develop a given amount of power there still remains the problem of dynamic de sign which deals. with the engine in operation. The requirements in dynamic design are free dom from vibration, constant speed for a given load, and automatic adjustment of the energy developed to the work to be done. Vibrations are produced by the forces which arise from the accelerations of moving masses. An engine or motor in which these forces are counteracted is said to be balanced. The speed for a given load is controlled by means of a flywheel; a governor adjusts the supplied energy to the load.
Rotating Balance.—A particle of weight W
revolving with an angular velocity w radians
per second in a circular path of radius r exerts
a force called the kinetic reaction,
where
W, on the constraint which deflects it
$rom a rectilinear path. If the kinetic reactions
rotating shaft are not in equilibrium,
which is usually the case for very high speeds,
the shaft will bend and therefore whirl and
wear its bearings out of shape.. An elementary
account of whirling, which lack of space
bids here, is given in Stodola's 'Steam
bines.' The problem has been investigated
perimentally by Dunkerly, 'On the Whirling
and Vibrations of Shafts'
When a number of masses all move in one plane perpendicular to the axis of rotation as FIG. 1.
in Fig. 1 the criterion for a balanced system of kinetic reactions is found as follows : Assume the centroid C of all the masses to lie at ro from the axis. Any infinitesimal element of mass situated at (r,,, ni ) exerts a force ("rims along r,. The reaction normal to OC in the plane of rotation is (min sin & mei sin & . . .).
Since the centroid lies on OC the parenthesis is zero, hence there is no force normal to OC. The force along OC is Ems cos 0 so whence the resultant kinetic reaction of a sys tem of masses rotating in one plane normal to the axis is the same as if the total mass were concentrated at the centroid. Therefore if the resultant is to vanish, i.e., if the system is to be in running balance, the centroid must lie on the axis of rotation; from a viewpoint more convenient for the solution of problems, the system is balanced when the kinetic reactions are in equilibrium. u may be taken as unity. But when the centroid is on the axis the system will remain in any position into which it is turned. This is called standing balance. Thus rotating parts such as armatures, turbine runners and flywheels that are constructed to be in standing balance will also be in running balance at any speed provided the centroids of all the parts are coplanar. This condition is usually not fulfilled;
furthermore although perfect balance is inde pendent of .,, unbalanced forces increase as so that deviations from equilibrium so small as to be unnoticed when the system is at rest may yet have an appreciable effect at great speeds. For example, if a system of 1 lb. weight is mounted eccentrically 0.0001 inch, the turning moment due to gravity is only 0.00009 ft.-lb., while at a speed of 1,500 r.p.m. the un balanced radial force is almost 0.7 lb., at 2,500 r.p.m. about 1.9 lb.
Fig. 2 shows the general case of rotating balance where the masses are not coplanar; i is the projection on a plane through the shaft and one arm r, and ii is an end view of i. If the kinetic reactions on the shaft are denoted by C, C,, C,, ... along r, ... in ii, the conditions for equilibrium are found by ing along and perpendicular to • in in and by taking moments about a perpendicular to both r and the shaft, and also about • itself. There results C — C, cos — Co cos IP = 0 (1) CI sin el + C, Sin . . = 0 ( 2)cos 0' + Cal cos . . . . 0 (3) CIA sin & sin & = 0 (4) Four unknowns can be determined from these four equations; but not more than two of them can be lengths I because only two equa tions contain I. The most convenient un knowns for practical work are two pairs of reaction components, one pair at each of two arbitrarily selected points. The reactions themselves may then be found and the proper masses and radii selected accordingly. In short, any system of revolving masses can be balanced by two masses properly placed; the positions of their radii along the shaft are arbitrary but their directions are not. The foregoing solution may be carried out graphi cally. Suppose that in Fig. 2 the magnitudes and positions of M,, Mx, M. are known and it is required to find C = mrw' and C, = in magnitude and direction. Equations (3) and (4) show that if the moment of each force about an axis perpendicular, at any point, to the plane through the shaft and the force be regarded as a vector having the direction of the force, the vector sum of all the moments will be zero. Two like forces on opposite sides of the axis have opposite moments which must have opposite senses. By taking the origin first at one unknown force and then at the other and drawing the vector polygon of the moments, the moments of the required kinetic reactions can be found. These determine the forces themselves since the lever arms I are known. The work may be. checked by constructing the force polygon corresponding to Fig. 2, ii. The method of procedure will now be illustrated. Example. Two cranks at right angles are of 10 inch radii, the distance between them being 1.5 ft The mass at each crank weighs 320 lbs. Find the balance masses at 1 ft. outside of each crank.
Take LP = 1, then 'mu' = 8.33. Fig. 3 shows a perspective view of shaft as a free body; H, V and H', V' are the horizontal and vertical components of the required forces. H and H' will evidently point away from the reader. For equilibrium V 4- V'=8.33, H =8.33