768 MECHANICS [APPLIED MECHANICS. is prejudicial, by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their temperature so high as to set fire to neighbouring combustible matters. Excessive heating is prevented by a constant and copious supply of a good unguent. The elevation of temperature produced by the friction of a journal is sometimes used as an experimental test of the quality of unguents. When the velocity of rubbing is about 4 or 5 feet per second, the elevation of temperature has been found by some recent experiments to be, with good fatty and soapy unguents, 40 to 50" Fahr. ; with good mineral unguents, about 30. 116. Rolling Resistance. By the rolling of two surfaces over each other without sliding a resistance is caused which is called some times "rolling friction," but more correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces rela tively to each other. The following are approximate values of the arm in decimals of a foot : O;ik upon oak 0-006 (Coulomb). Lignum vitas on oak 0-004 ,, Cast iron on cast iron 002 (Tredgold). 117. Reciprocating Forces Stored and Restored Energy. When a force acts on a machine alternately as an effort and as a resistance, it may be called a reciprocating force. Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls ; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise ; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it. Thus that amount of energy is not lost, but has its operation deferred ; and it is said to be stored when the weight is lifted, and restored when it falls. In a machine of which each piece is to move with a uniform velocity, if the effort and the resistance be constant, the weight of each piece must be balanced on its axis, so that it may produce lateral pressure only, and not act as a reciprocating force. But if the effort and the resistance be alternately in excess, the uniformity of speed may still be preserved by so adjusting some moving weight in the mechanism that when the effort is in excess it maybe lifted, and so balance and employ the excess of effort, and that when the resistance is in excess it may fall, and so balance and overcome the excess of resistance, thus storing the periodical excess of energy, and restoring that energy to perform the periodical excess of work. Other forces besides gravity may be used as reciprocating forces for storing and restoring energy, for example, the elasticity of a spring or of a mass of air. In most of the delusive machines commonly called "perpetual motions," of which so many are patented in each year, and which are expected by their inventors to perform work without receiving energy, the fundamental fallacy consists in an expectation that some reciprocating force shall restore more energy than it has been the means of storing. Division 2. Deflecting Forces. 118. Deflecting Force for Translation in a Curved Path. In machinery, deflecting force is supplied by the tenacity of some piece, such as a crank, which guides the deflected body in its curved path, and is unbalanced, being employed in producing deflexion, and not in balancing another force. 119. Centrifugal Force. Sec p. 682, 35, and p. 701, 119. 120. Rectangular Resolution of Centrifugal Force. See p. 701, 117 and 119. 121. Centrifugal Force of a Rotating Body. The centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body. The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder ; but these forces balance each other, and do not affect the resultant cen trifugal force exerted on the axis of rotation. 1 If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing. Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration, and increased friction, and a tendency of the shafts to jump out of their bearings. 122. Centrifugal Couples of a Rotating Body. Besides the tend ency (if any) of the combined centrifugal forces of the particles of 1 This is a particular case of a more general principle, that the motion of tlie centre of gravity of a body is not affected by the mutual actions of its parts, for the proof of winch see p. 718, 179. a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis (see p. 732, 237). It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft, and tend to make it leave its bearings. The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counter poises which are placed between the spokes of the driving-wheels of locomotive engines. 123. Revolving Pendulum Governors. In fig. 32 AO represents an upright axis or spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at 0, carried by the vertical axis. When the spindle is at rest the bob hangs close to it ; when the spindle rotates, the bob, be ing made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at in the direction OB, and then it revolves steadily in a circle. This combination is called a revolving, centrifugal, or conical pendulum. Re volving pendulums are usually con structed with pairs of rods and bobs, as OB, 06, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point may balance each other. In finding the position in which the bob will revolve with a given angular velocity a, for most practical cases connected with machinery the mass of the rod may be con sidered as insensible compared with that of the bob. Let the bob be a sphere, and from the centre of that sphere draw BH = y perpendicular to OA. Let OH = 3 ; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W : F : : 2 : y ; that is to say, 1L = w = .^L ; consequently W Or, if = = be the number of turns or fractious of a turn 2ir 6 -2832 in a second, g _ 0-8165 foot ^9 -79771 inches )
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s is called the altitude of the pendulum. If the rod of a revolving pendulum be jointed, as in fig. 33, not to a point in the vertical axis, but to the end of a projecting arm C, the position in which the bob will revolve will be the same as if the rod were jointed to the point 0, where its prolongation cuts the vertical axis. A revolving pendulum is an essential part of most of the contrivances called governors, for regulating the speed of prime movers. The earlier kinds of governors act on the prime mover by the variations of their alti tude. Thus in Watt s steam-engine governor the rods, through a combination of levers and linkwork C, c, D, d (fig. 32), act on a lever EF, which acts upon the throttle-valve for the admission of steam so as to enlarge or contract its opening when the speed becomes too small or too great. In a more recent kind of governors invented by the Messrs Siemens, which may be called differential governors, the regulation of the prime mover is effected by means of the difference between the velocity of a wheel driven by it and that of a wheel regulated by a re volving pendulum. Fig. 34 illustrates this class of governors. A is a vertical dead-centre or fixed shaft, about which the after-men tioned pieces turn ; C is a pulley driven by the prime mover, and fixed to a bevel-wheel, which is seen below it ; E is a bevel-wheel similar to the first, and having the same apex. To this wheel are hung the bobs B, of which there are _ A I FI B B F s
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