CHAPTER II. ON APPLIED DYNAMICS.

§ 83. _Laws of Motion._--The action of a machine in transmitting _force_ and _motion_ simultaneously, or performing _work_, is governed, in common with the phenomena of moving bodies in general, by two "laws of motion."

_Division 1. Balanced Forces in Machines of Uniform Velocity._

§ 84. _Application of Force to Mechanism._--Forces are applied in units of weight; and the unit most commonly employed in Britain is the _pound avoirdupois_. The action of a force applied to a body is always in reality distributed over some definite space, either a volume of three dimensions or a surface of two. An example of a force distributed throughout a volume is the _weight_ of the body itself, which acts on every particle, however small. The _pressure_ exerted between two bodies at their surface of contact, or between the two parts of one body on either side of an ideal surface of separation, is an example of a force distributed over a surface. The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body considered as a whole, the _resultant_ of the distributed force, determined according to the principles of statics, and considered as acting in a _single line_ and applied at a _single point_, may, for the occasion, be substituted for the force as really distributed. Thus, the weight of each separate piece in a machine is treated as acting wholly at its _centre of gravity_, and each pressure applied to it as acting at a point called the _centre of pressure_ of the surface to which the pressure is really applied.

§ 85. _Forces applied to Mechanism Classed._--If [theta] be the _obliquity_ of a force F applied to a piece of a machine--that is, the angle made by the direction of the force with the direction of motion of its point of application--then by the principles of statics, F may be resolved into two rectangular components, viz.:--

Along the direction of motion, P = F cos [theta] \ (49) Across the direction of motion, Q = F sin [theta] /

If the component along the direction of motion acts with the motion, it is called an _effort_; if _against_ the motion, a _resistance_. The component _across_ the direction of motion is a _lateral pressure_; the unbalanced lateral pressure on any piece, or part of a piece, is _deflecting force_. A lateral pressure may increase resistance by causing friction; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resistance. Resistances are distinguished into _useful_ and _prejudicial_, according as they arise from the useful effect produced by the machine or from other causes.

§ 86. _Work._--_Work_ consists in moving against resistance. The work is said to be _performed_, and the resistance _overcome_. Work is measured by the product of the resistance into the distance through which its point of application is moved. The _unit of work_ commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a _foot-pound_.

Work is distinguished into _useful work_ and _prejudicial_ or _lost work_, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance.

§ 87. _Energy: Potential Energy._--_Energy_ means _capacity for performing work_. The _energy of an effort_, or _potential energy_, is measured by the product of the effort into the distance through which its point of application is _capable_ of being moved. The unit of energy is the same with the unit of work.

When the point of application of an effort _has been moved_ through a given distance, energy is said to have been _exerted_ to an amount expressed by the product of the effort into the distance through which its point of application has been moved.

§ 88. _Variable Effort and Resistance._--If an effort has different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length [Delta]s of the corresponding portion of the path of the point of application; the sum

[Sigma] · P[Delta]s (50)

is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by

[int]P ds. (51)

Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.

The work done by a machine can be actually measured by means of a dynamometer (q.v.).

§ 89. _Principle of the Equality of Energy and Work._--From the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other. Now from the laws of statics it is known that, in order that a system of forces applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero,--products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite.

In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time _the energy exerted is equal to the work performed_.

The symbolical expression of this law is as follows: let efforts be applied to one or any number of points of a machine; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds; let resistances be overcome at one or any number of points of the same machine; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the given interval of time by ds´; then

[Sigma] · P ds = [Sigma] · R ds´. (52)

The lengths ds, ds´ are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.

§ 90. _Static Equilibrium of Mechanisms._--The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds´ were made, it becomes P ds/dt = R ds´/dt, that is, Pv = Rv´, which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:--

A four-bar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the configuration shown, both loads being supposed to act vertically? Find the instantaneous centre O_(bd), and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos [theta] and R cos [phi]. Let the mechanism have a small motion; then, for the instant, the link b is turning about its instantaneous centre O_(bd), and, if [omega] is its instantaneous angular velocity, the velocity of the point x is [omega]r, and the velocity of the point y is [omega]s. Hence, by the principle just stated, P cos [theta] × [omega]r = R cos [phi] × [omega]s. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to P_p = R_q, which shows that the ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.

The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force f_s, acting on the link s, to find the force f_t acting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:--

(1) Find the three virtual centres O_(rs), O_(rt), O_(st), which must be three points in a line.

(2) Resolve f_s into two components, one of which, namely, f_q, passes through O_(rs) and may be neglected, and the other f_p passes through O_(st).

(3) Find the point M, where f_p joins the given direction of f_t, and resolve f_p into two components, of which one is in the direction MO_(rt), and may be neglected because it passes through O_(rt), and the other is in the given direction of f_t and is therefore the force required.

This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of his _Mechanics of Machinery_. Another general solution of the problem is given in the _Proc. Lond. Math. Soc._ (1878-1879), by the same author. An example of the method of solution stated above, and taken from the _Mechanics of Machinery_, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force f_s acting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres O_(rs), O_(st) are at the respective axes of the wheels r and t, and the centre O_(rt) divides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.

The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, "On the application of graphic methods to the determination of the efficiency of machinery" (_Trans. Roy. Soc. Ed._, vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This self-strained frame is called the _dynamic frame_ of the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.

§ 91. _Efficiency._--The _efficiency_ of a machine is the ratio of the _useful_ work to the _total_ work--that is, to the energy exerted--and is represented by

[Sigma]·R_u ds´ [Sigma]·R_u ds´ [Sigma]·R_u ds´ U --------------- = --------------------------------- = --------------- = ---. (53) [Sigma]·R ds´ [Sigma]·R_u ds´ + [Sigma]·R_p ds´ [Sigma]·P ds E

R_u being taken to represent useful and R_p prejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.

§ 92. _Power and Effect._--The _power_ of a machine is the energy exerted, and the _effect_ the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.

The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.

§ 93. _Modulus of a Machine._--In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances arc generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of the _lost_ work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the _modulus_ of the machine. The general form of the modulus may be expressed thus--

E = U + [phi](U, A) + [psi](A), (54)

where A denotes some quantity or set of quantities depending on the form, arrangement, weight and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form of

E = (1 + A)U + B, (55)

where A and B are _constants_, depending on the form, arrangement and weight of the mechanism. The efficiency corresponding to the last equation is

U 1 --- = -----------. (56) E 1 + A + B/U

§ 94. _Trains of Mechanism._--In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train--each section being treated as a machine, driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident that _the efficiency of the whole train is the product of the efficiencies of its sections_.

§ 95. _Rotating Pieces: Couples of Forces._--It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of the _moment_ of the _couples of forces_ acting on it, and of the angular velocity. The ordinary British unit of moment is a _foot-pound_; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work.

If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.

A couple is said to be _right_ or _left handed_ with reference to the observer, according to the direction in which it tends to turn the body, and is a _driving_ couple or a _resisting_ couple according as its tendency is with or against that of the actual rotation.

Let dt be an interval of time, [alpha] the angular velocity of the piece; then [alpha]dt is the angle through which it turns in the interval dt, and ds = vdt = r[alpha]dt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, then

P ds = Pv dt = Pr[alpha] dt = M[alpha] dt (57)

is the energy exerted by the couple M in the interval dt; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.

§ 96. _Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece._--In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the _equivalent_ force or couple applied to some other point or piece; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.

These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.

§ 97. _Balanced Lateral Pressure of Guides and Bearings._--The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.

§ 98. _Friction. Unguents._--The most important kind of resistance in machines is the _friction_ or _rubbing resistance_ of surfaces which slide over each other. The _direction_ of the resistance of friction is opposite to that in which the sliding takes place. Its _magnitude_ is the product of the _normal pressure_ or force which presses the rubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in § 14, as the _coefficient of friction_, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. The _total pressure_ exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its _obliquity_, or inclination to the common perpendicular of the surfaces, is the _angle of repose_ formerly mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, f the coefficient of friction, and [phi] the angle of repose; then

f = tan [phi] \ (58) R = fN = N tan [phi] = T sin [phi] /

Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie, James Wood, D. Rankine and others. The most complete and elaborate experiments are those of Morin, published in his _Notions fondamentales de mécanique_, and republished in Britain in the works of Moseley and Gordon.

The experiments of Beauchamp Tower ("Report of Friction Experiments," _Proc. Inst. Mech. Eng._, 1883) showed that when oil is supplied to a journal by means of an oil bath the coefficient of friction varies nearly inversely as the load on the bearing, thus making the product of the load on the bearing and the coefficient of friction a constant. Mr Tower's experiments were carried out at nearly constant temperature. The more recent experiments of Lasche (_Zeitsch, Verein Deutsche Ingen._, 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant. For further information on this point and on Osborne Reynolds's theory of lubrication see BEARINGS and LUBRICATION.

§ 99. _Work of Friction. Moment of Friction._--The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.

When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called the _moment of friction_. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.

To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and f the coefficient of friction. Then the moment of friction of du is fprdu;

the total moment of friction is f [integral] pr·du; \ and the work performed in a unit cf time in overcoming friction, > (59) when the angular velocity is [alpha], is [alpha]f [int] pr·du. /

It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication, and also by conditions of strength and stiffness.

§ 100. _Total Pressure between Journal and Bearing._--A single piece rotating with a uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it--which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data:--

The direction of the effort. The direction of the useful resistance. The weight of the piece and the direction in which it acts. The magnitude of the useful resistance. The radius of the bearing r. The angle of repose [phi], corresponding to the friction of the journal on the bearing.

And there are required the following:--

The direction of the reaction of the bearing. The magnitude of that reaction. The magnitude of the effort.

Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be called _the given force_.

The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.

Also, let AAA, fig. 128, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle [phi] with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r · sin [phi]. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.

Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in § 7.

The work lost in overcoming the friction of the bearing is the same as that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.

In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the "given force") ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.

An investigation of the forces acting on a bearing and journal lubricated by an oil bath will be found in a paper by Osborne Reynolds in the _Phil. Trans._ pt. i. (1886). (See also BEARINGS.)

§ 101. _Friction of Pivots and Collars._--When a shaft is acted upon by a force tending to shift it lengthways, that force must be balanced by the reaction of a bearing against a _pivot_ at the end of the shaft; or, if that be impossible, against one or more _collars_, or rings _projecting_ from the body of the shaft. The bearing of the pivot is called a _step_ or _footstep_. Pivots require great hardness, and are usually made of steel. The _flat_ pivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the _intensity_ of the pressure is

p = N/[pi]r²; (60)

and, introducing this value into equation 59, the _moment of friction of the flat pivot_ is found to be

(2/3)fNr (61)

or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.

The friction of a _conical_ pivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.

The moment of friction of a _collar_ is given by the formula--

r³ - r´³ (2/3)fN --------, (62) r² - r´²

where r is the external and r´ the internal radius.

In the _cup and ball_ pivot the end of the shaft and the step present two recesses facing each other, into which art fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.

It appears that the rapidity with which a rubbing surface wears away is proportional to the friction and to the velocity jointly, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Schiele has invented a pivot which preserves its original figure by wearing equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends:--

The rapidity of wear of a surface measured in an _oblique_ direction is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX in T; PT = PY × _secant obliquity_, and this is to be a constant quantity; hence the curve is that known as the _tractory_ of the straight line OX, in which PT = OR = constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a tracing-point, pencil or pen. Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid along the axis from O towards X, P will be drawn after it from R towards C along the tractory. This curve, being an asymptote to its axis, is capable of being indefinitely prolonged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing. The moment of friction of "Schiele's anti-friction pivot," as it is called, is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under the same pressure.

Records of experiments on the friction of a pivot bearing will be found in the _Proc. Inst. Mech. Eng._ (1891), and on the friction of a collar bearing ib. May 1888.

§ 102. _Friction of Teeth._--Let N be the normal pressure exerted between a pair of teeth of a pair of wheels; s the total distance through which they slide upon each other; n the number of pairs of teeth which pass the plane of axis in a unit of time; then

nfNs (63)

is the work lost in unity of time by the friction of the teeth. The sliding s is composed of two parts, which take place during the approach and recess respectively. Let those be denoted by s1 and s2, so that s = s1 + s2. In § 45 the _velocity_ of sliding at any instant has been given, viz. u = c ([alpha]1 + [alpha]2), where u is that velocity, c the distance T1 at any instant from the point of contact of the teeth to the pitch-point, and [alpha]1, [alpha]2 the respective angular velocities of the wheels.

Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above equation becomes

/ 1 1 \ u = cv ( --- + --- ). \r1 r2 /

To apply this to involute teeth, let c1 be the length of the approach, c2 that of the recess, u1, the _mean_ volocity of sliding during the approach, u2 that during the recess; then

c1v / 1 1 \ c2v / 1 1 \ u1 = --- ( --- + --- ); u2 = --- ( --- + --- ) 2 \r1 r2 / 2 \r1 r2 /

also, let [theta] be the obliquity of the action; then the times occupied by the approach and recess are respectively

c1 c2 -------------, -------------; v cos [theta] v cos [theta]

giving, finally, for the length of sliding between each pair of teeth,

c1² + c2² / 1 1 \ s = s1 + s2 = ------------- ( --- + --- ) (64) 2 cos [theta] \r1 r2 /

which, substituted in equation (63), gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure.

For inside gearing, if r1 be the less radius and r2 the greater, 1/r1 - 1/r2 is to be substituted for 1/r1 + 1/r2.

§ 103. _Friction of Cords and Belts._--A flexible band, such as a cord, rope, belt or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whether effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and proportional to the normal pressure between them.

Let T1 be the tension of the free part of the band at that side _towards_ which it tends to draw the pulley, or _from_ which the pulley tends to draw it; T2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; [theta] the ratio of the length of that arc to the radius of the pulley; d[theta] the ratio of an indefinitely small element of that arc to the radius; F = T1 - T2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc d[theta]; f the coefficient of friction between the materials of the band and pulley.

Then, according to a well-known principle in statics, the normal pressure at the elementary arc d[theta] is Td[theta], T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF = fTd[theta]. Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT = dF = fTd[theta]; which equation, being integrated throughout the entire arc of contact, gives the following formulae:--

T1 \ hyp log. -- = f^[theta] | T2 | | T1 > (65) -- = ef^[theta] | T2 | | F = T1 - T2 = T1(1 - e - f^[theta]) = T2(ef^[theta] - 1) /

When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the _mean_ tension remains unchanged. Its value is given by this formula--

T1 + T2 ef^[theta] + 1 ------- = ----------------- (66) 2 2(ef^[theta] - 1)

which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.

The equations 65 and 66 are applicable to a kind of _brake_ called a _friction-strap_, used to stop or moderate the velocity of machines by being tightened round a pulley. The strap is usually of iron, and the pulley of hard wood.

Let [alpha] denote the arc of contact expressed in _turns and fractions of a turn_; then

[theta] = 6.2832a \ (67) ef^[theta] = number whose common logarithm is 2.7288fa /

See also DYNAMOMETER for illustrations of the use of what are essentially friction-straps of different forms for the measurement of the brake horse-power of an engine or motor.

§ 104. _Stiffness of Ropes._--Ropes offer a resistance to being bent, and, when bent, to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.

The _work lost_ in pulling a given length of rope over a pulley is found by multiplying the length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again.

The following empirical formulae for the stiffness of hempen ropes have been deduced by Morin from the experiments of Coulomb:--

Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in inches, n = 48d² for white ropes and 35d² for tarred ropes; r the _effective_ radius of the pulley in inches; T the tension in pounds. Then

n \ For white ropes, F = --- (0.0012 + 0.001026n + 0.0012T) | r | > (68) n | For tarred ropes, F = --- (0.006 + 0.001392n + 0.00168T) | r /

§ 105. _Friction-Couplings._--Friction is useful as a means of communicating motion where sudden changes either of force or velocity take place, because, being limited in amount, it may be so adjusted as to limit the forces which strain the pieces of the mechanism within the bounds of safety. Amongst contrivances for effecting this object are _friction-cones_. A rotating shaft carries upon a cylindrical portion of its figure a wheel or pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid cone accurately turned to fit the hollow cone. This frustum is made always to turn along with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or withdrawn from, the hollow cone by means of a lever. When the cones are pressed together or engaged, their friction causes the pulley to rotate along with the shaft; when they are disengaged, the pulley is free to stand still. The angle made by the sides of the cones with the axis should not be less than the angle of repose. In the _friction-clutch_, a pulley loose on a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw; this hoop has short projecting arms or ears. A fork or _clutch_ rotates along with the shaft, and is capable of being moved longitudinally by a handle. When the clutch is moved towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to communicate its rotation to the pulley by friction. There are many other contrivances of the same class, but the two just mentioned may serve for examples.

§ 106. _Heat of Friction: Unguents._--The work lost in friction is employed in producing heat. This fact is very obvious, and has been known from a remote period; but the _exact_ determination of the proportion of the work lost to the heat produced, and the experimental proof that that proportion is the same under all circumstances and with all materials, solid, liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of work which produces a British unit of heat (or so much heat as elevates the temperature of one pound of pure water, at or near ordinary atmospheric temperatures, by 1° F.) is 772 foot-pounds. This constant, now designated as "Joule's equivalent," is the principal experimental datum of the science of thermodynamics.

A more recent determination (_Phil. Trans._, 1897), by Osborne Reynolds and W. M. Moorby, gives 778 as the mean value of Joule's equivalent through the range of 32° to 212° F. See also the papers of Rowland in the _Proc. Amer. Acad._ (1879), and Griffiths, _Phil. Trans._ (1893).

The heat produced by friction, when moderate in amount, is useful in softening and liquefying thick unguents; but when excessive it 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. For modern methods of forced lubrication see BEARINGS.

§ 107. _Rolling Resistance._--By the rolling of two surfaces over each other without sliding a resistance is caused which is called sometimes "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 relatively to each other. The following are approximate values of the arm in decimals of a foot:--

Oak upon oak 0.006 (Coulomb). Lignum vitae on oak 0.004 " Cast iron on cast iron 0.002 (Tredgold).

§ 108. _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 may be 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._

§ 109. _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.

§ 110. _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 centrifugal force exerted on the axis of rotation.[3]

_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.

§ 111. _Centrifugal Couples of a Rotating Body._--Besides the tendency (if any) of the combined centrifugal forces of the particles of 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.

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 counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.

§ 112.* _Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis._--The method here briefly explained is taken from a paper by W. E. Dalby, "The Balancing of Engines with special reference to Marine Work," _Trans. Inst. Nav. Arch._ (1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa²r/g lb., where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O

(1) [Sigma]F = 0 \ (a) (2) [Sigma]Fa = 0 /

The plane through O to which the shaft is perpendicular is called the _reference plane_, because all the transferred forces act in that plane at the point O. The plane through the radius of the weight containing the axis OX is called the _axial plane_ because it contains the forces forming the couple due to the transference of F to the reference plane. Substituting the values of F in (a) the two conditions become

a² (1) (W1r1 + W2r2 + W3r3 + ...)--- = 0 g a² (b) (2) (W1a1r1 + W2a2r2 + ... )--- = 0 g

In order that these conditions may obtain, the quantities in the brackets must be zero, since the factor a²/g is not zero. Hence finally the conditions which must be satisfied by the system of weights in order that the axis of rotation may be a permanent axis is

(1) (W1r1 + W2r2 + W3r3) = 0 (2) (W1a1r1 + W2a2r2 + W3a3r3) = 0 (c)

It must be remembered that these are all directed quantities, and that their respective sums are to be taken by drawing vector polygons. In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r. For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r. The magnitude of the vector is the product War. The conditions (c) may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon, and, second, that the sum of the vectors War must form a closed polygon. The general problem in practice is, given a system of weights attached to a shaft, to find the respective weights and positions of two balance weights or counterpoises which must be added to the system in order to make the shaft a permanent axis, the planes in which the balance weights are to revolve also being given. To solve this the reference plane must be chosen so that it coincides with the plane of revolution of one of the as yet unknown balance weights. The balance weight in this plane has therefore no couple corresponding to it. Hence by drawing a couple polygon for the given weights the vector which is required to close the polygon is at once found and from it the magnitude and position of the balance weight which must be added to the system to balance the couples follow at once. Then, transferring the product Wr corresponding with this balance weight to the reference plane, proceed to draw the force polygon. The vector required to close it will determine the second balance weight, the work may be checked by taking the reference plane to coincide with the plane of revolution of the second balance weight and then re-determining them, or by taking a reference plane anywhere and including the two balance weights trying if condition (c) is satisfied.

When a weight is reciprocated, the equal and opposite force required for its acceleration at any instant appears as an unbalanced force on the frame of the machine to which the weight belongs. In the particular case, where the motion is of the kind known as "simple harmonic" the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin. Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system. The balance weights found as part of the equivalent revolving system when reciprocated by their respective crank pins form the balance weights for the given reciprocating system. These conditions may be exactly realized by a system of weights reciprocated by slotted bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is usually effected through a connecting rod, as in the case of steam engines. In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin. The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion. The balance weights are to be separately calculated for each system, the one set being added to the crank shaft as revolving weights, and the second set being included with the reciprocating weights and operated by a properly placed crank on the crank shaft. Balance weights added in this way to a set of reciprocating weights are sometimes called bob-weights. In the case of locomotives the balance weights required to balance the pistons are added as revolving weights to the crank shaft system, and in fact are generally combined with the weights required to balance the revolving system so as to form one weight, the counterpoise referred to in the preceding section, which is seen between the spokes of the wheels of a locomotive. Although this method balances the pistons in the horizontal plane, and thus allows the pull of the engine on the train to be exerted without the variation due to the reciprocation of the pistons, yet the force balanced horizontally is introduced vertically and appears as a variation of pressure on the rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep this variation of rail pressure within safe limits. The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the connecting rod l increases. A more accurate though still approximate expression for the force on the frame due to the acceleration of the piston whose weight is W is given by

W / r \ --- [omega]² r ( cos [theta] + --- cos 2[theta] ) g \ l /

The conditions regulating the balancing of a system of weights reciprocating under the action of accelerating forces given by the above expression are investigated in a paper by Otto Schlick, "On Balancing of Steam Engines," _Trans, Inst. Nav. Arch._ (1900), and in a paper by W. E. Dalby, "On the Balancing of the Reciprocating Parts of Engines, including the Effect of the Connecting Rod" (ibid., 1901). A still more accurate expression than the above is obtained by expansion in a Fourier series, regarding which and its bearing on balancing engines see a paper by J. H. Macalpine, "A Solution of the Vibration Problem" (ibid., 1901). The whole subject is dealt with in a treatise, _The Balancing of Engines_, by W. E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing are to be found in the _Transactions_ of the Institution of Naval Architects.

§ 113.* _Centrifugal Whirling of Shafts._--When a system of revolving masses is balanced so that the conditions of the preceding section are fulfilled, the centre of gravity of the system lies on the axis of revolution. If there is the slightest displacement of the centre of gravity of the system from the axis of revolution a force acts on the shaft tending to deflect it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to revolve stably, this force must be balanced at any instant by the elastic resistance of the shaft to deflexion. To take a simple case, suppose a shaft, supported on two bearings to carry a disk of weight W at its centre, and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction. Neglecting the mass of the shaft itself, when the shaft rotates with an angular velocity a, the centrifugal force Wa²e/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say. The elastic resistance evoked by this deflexion is proportional to the deflexion, so that if c is a constant depending upon the form, material and method of support of the shaft, the following equality must hold if the shaft is to rotate stably at the stated speed--

W ---(y + e)a² = cy, g

from which y = Wa²e/(gc - Wa²).

This expression shows that as a increases y increases until when Wa² = gc, y becomes infinitely large. The corresponding value of a, namely [root]gc/W, is called the _critical velocity_ of the shaft, and is the speed at which the shaft ceases to rotate stably and at which centrifugal whirling begins. The general problem is to find the value of a corresponding to all kinds of loadings on shafts supported in any manner. The question was investigated by Rankine in an article in the _Engineer_ (April 9, 1869). Professor A. G. Greenhill treated the problem of the centrifugal whirling of an unloaded shaft with different supporting conditions in a paper "On the Strength of Shafting exposed both to torsion and to end thrust," _Proc. Inst. Mech. Eng._ (1883). Professor S. Dunkerley ("On the Whirling and Vibration of Shafts," _Phil. Trans._, 1894) investigated the question for the cases of loaded and unloaded shafts, and, owing to the complication arising from the application of the general theory to the cases of loaded shafts, devised empirical formulae for the critical speeds of shafts loaded with heavy pulleys, based generally upon the following assumption, which is stated for the case of a shaft carrying one pulley: If N1, N2 be the separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is neglected when that of the other is under consideration, then the resulting speed of whirl due to both causes combined may be taken to be of the form N1N2 [root][(N²1 + N1²)] where N means revolutions per minute. This form is extended to include the cases of several pulleys on the same shaft. The interesting and important part of the investigation is that a number of experiments were made on small shafts arranged in different ways and loaded in different ways, and the speed at which whirling actually occurred was compared with the speed calculated from formulae of the general type indicated above. The agreement between the observed and calculated values of the critical speeds was in most cases quite remarkable. In a paper by Dr C. Chree, "The Whirling and Transverse Vibrations of Rotating Shafts," _Proc. Phys. Soc. Lon._, vol. 19 (1904); also _Phil. Mag._, vol. 7 (1904), the question is investigated from a new mathematical point of view, and expressions for the whirling of loaded shafts are obtained without the necessity of any assumption of the kind stated above. An elementary presentation of the problem from a practical point of view will be found in _Steam Turbines_, by Dr A. Stodola (London, 1905).

§ 114. _Revolving Pendulum. Governors._--In fig. 131 AO represents an upright axis or spindle; B a weight called a _bob_, suspended by rod OB from a horizontal axis at O, carried by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle rotates, the bob, being made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at O in the direction OB, and then it revolves steadily in a circle. This combination is called a _revolving_, _centrifugal_, or _conical pendulum_. Revolving pendulums are usually constructed with _pairs_ of rods and bobs, as OB, Ob, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point O 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 considered 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 = z; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W : F :: z : y; that is to say, y/z = F/W = ya²/g; consequently

z = g/[alpha]² (69)

Or, if n = [alpha] 2[pi] = [alpha]/6.2832 be the number of turns or fractions of a turn in a second,

g 0.8165 ft. 9.79771 in. \ z = -------- = ---------- = ----------- > (70) 4[pi]²n² n² n² /

z is called the _altitude of the pendulum_.

If the rod of a revolving pendulum be jointed, as in fig. 132, 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 O, 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, for further particulars of which see STEAM ENGINE.

_Division 3. Working of Machines of Varying Velocity._

§ 115. _General Principles._--In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced. Also, in order that the forces acting on each piece of a machine may be always exactly balanced, it is necessary that the velocity of that piece should be uniform.

An excess of the effort exerted on any piece, above that which is necessary to balance the resistance, is accompanied with acceleration; a deficiency of the effort, with retardation.

When a machine is being started from a state of rest, and brought by degrees up to its proper speed, the effort must be in excess; when it is being retarded for the purpose of stopping it, the resistance must be in excess.

An excess of effort above resistance involves an excess of energy exerted above work performed; that excess of energy is employed in producing acceleration.

An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery.

When a machine undergoes alternate acceleration and retardation, so that at certain instants of time, occurring at the end of intervals called _periods_ or _cycles_, it returns to its original speed, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other; and at the end of each cycle the principle of the equality of energy and work stated in § 87, with all its consequences, is verified exactly as in the case of machines of uniform speed.

At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, as applied to _direct deviation_, or acceleration and retardation.

§ 116. _Energy of Acceleration and Work of Retardation for a Shifting Body._--Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time [Delta]t let its velocity be increased at a uniform rate of acceleration from v1 to v2. The rate of acceleration will be

dv/dt = const. = (v2 - v1)[Delta]t;

and to produce this acceleration a uniform effort will be required, expressed by

P = w(v2 - v1)g[Delta]t (71)

(The product wv/g of the mass of a body by its velocity is called its _momentum_; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)

To find the _energy_ which has to be exerted to produce the acceleration from v1 to v2, it is to be observed that the _distance_ through which the effort P acts during the acceleration is

[Delta]s = (v2 + v1)[Delta]t/2;

consequently, the _energy of acceleration_ is

P[Delta]s = w(v2 - v1) (v2 + v1)/2g = w(v2² - v1²)2g, (72)

being proportional to the increase in the square of the velocity, and _independent of the time_.

In order to produce a _retardation_ from the greater velocity v2 to the less velocity v1, it is necessary to apply to the body a _resistance_ connected with the retardation and the time by an equation identical in every respect with equation (71), except by the substitution of a resistance for an effort; and in overcoming that resistance the body _performs work_ to an amount determined by equation (72), putting Rds for Pas.

§ 117. _Energy Stored and Restored by Deviations of Velocity._--Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration; so that that energy may be considered as _stored_ during the acceleration, and _restored_ during the retardation, in a manner analogous to the operation of a reciprocating force (§ 108).

Let there be given the mean velocity V = ½(v2 + v1) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v2 - v1, and the extreme velocities v1, v2, which that body must have, in order alternately to store and restore an amount of energy E. By equation (72) we have

E = w(v2² - v1²)´2g

which, being divided by V = ½(v2 + v1), gives

E/V = w(v2 - v1)/g;

and consequently

v2 - v1 = gE/Vw (73)

The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the motion of the body, is

(v2 - v1)V = gE/V²w. (74)

§ 118. _Actual Energy of a Shifting Body._--The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is

wv²/2g. (75)

This is called the _actual energy_ of the motion of the body, and is half the quantity which in some treatises is called vis viva.

The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.

§ 119. _Principle of the Conservation of Energy in Machines._--The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in § 89 for machines of uniform speed.

_In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed._

§ 120. _Actual Energy of Circular Translation--Moment of Inertia._--Let a small body of the weight w undergo translation in a circular path of the radius [rho], with the angular velocity of deflexion [alpha], so that the common linear velocity of all its particles is v = [alpha][rho]. Then the actual energy of that body is

wv²/2g = w[alpha]²p²/2g. (76)

By comparing this with the expression for the centrifugal force (w[alpha]²p/g), it appears that the actual energy of a revolving body is equal to the potential energy Fp/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.

The product wp²/g, by which the half-square of the angular velocity is multiplied, is called the _moment of inertia_ of the revolving body.

§ 121. _Flywheels._--A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity.

The principles according to which variations of angular velocity store and restore energy are the same as those of § 117, only substituting _moment of inertia_ for _mass_, and _angular_ for _linear_ velocity.

Let W be the weight of a flywheel, R its radius of gyration, a2 its maximum, a1 its minimum, and A = ½([alpha]2 + [alpha]1) its mean angular velocity. Let

I/S = ([alpha]2 - [alpha]2)/A

denote the _unsteadiness_ of the motion of the flywheel; the denominator S of this fraction is called the _steadiness_. Let e denote the quantity by which the energy exerted in each cycle of the working of the machine alternately exceeds and falls short of the work performed, and which has consequently to be alternately stored by acceleration and restored by retardation of the flywheel. The value of this _periodical excess_ is--

e = R²W ([alpha]2² - [alpha]1²), 2g, (77)

from which, dividing both sides by A², we obtain the following equations:--

e/A² = R²W/gS \ >. (78) R²WA²/2g = Se/2 /

The latter of these equations may be thus expressed in words: _The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness._

In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60; and when great steadiness is required S = from 100 to 150.

The periodical excess e may arise either from variations in the effort exerted by the prime mover, or from variations in the resistance of the work, or from both these causes combined. When but one flywheel is used, it should be placed in as direct connexion as possible with that part of the mechanism where the greatest amount of the periodical excess originates; but when it originates at two or more points, it is best to have a flywheel in connexion with each of these points. For example, in a machine-work, the steam-engine, which is the prime mover of the various tools, has a flywheel on the crank-shaft to store and restore the periodical excess of energy arising from the variations in the effort exerted by the connecting-rod upon the crank; and each of the slotting machines, punching machines, riveting machines, and other tools has a flywheel of its own to store and restore energy, so as to enable the very different resistances opposed to those tools at different times to be overcome without too great unsteadiness of motion. For tools performing useful work at intervals, and having only their own friction to overcome during the intermediate intervals, e should be assumed equal to the whole work performed at each separate operation.

§ 122. _Brakes._--A brake is an apparatus for stopping and diminishing the velocity of a machine by friction, such as the friction-strap already referred to in § 103. To find the distance s through which a brake, exerting the friction F, must rub in order to stop a machine having the total actual energy E at the moment when the brake begins to act, reduce, by the principles of § 96, the various efforts and other resistances of the machine which act at the same time with the friction of the brake to the rubbing surface of the brake, and let R be their resultant--positive if _resistance_, _negative_ if effort preponderates. Then

s = E/(F + R). (79)

§ 123. _Energy distributed between two Bodies: Projection and Propulsion._--Hitherto the effort by which a machine is moved has been treated as a force exerted between a movable body and a fixed body, so that the whole energy exerted by it is employed upon the movable body, and none upon the fixed body. This conception is sensibly realized in practice when one of the two bodies between which the effort acts is either so heavy as compared with the other, or has so great a resistance opposed to its motion, that it may, without sensible error, be treated as fixed. But there are cases in which the motions of both bodies are appreciable, and must be taken into account--such as the projection of projectiles, where the velocity of the _recoil_ or backward motion of the gun bears an appreciable proportion to the forward motion of the projectile; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw or other propeller bears a very considerable proportion to the velocity of the water moved forwards and sideways by the ship. In cases of this kind the energy exerted by the effort is _distributed_ between the two bodies between which the effort is exerted in shares proportional to the velocities of the two bodies during the action of the effort; and those velocities are to each other directly as the portions of the effort unbalanced by resistance on the respective bodies, and inversely as the weights of the bodies.

To express this symbolically, let W1, W2 be the weights of the bodies; P the effort exerted between them; S the distance through which it acts; R1, R2 the resistances opposed to the effort overcome by W1, W2 respectively; E1, E2 the shares of the whole energy E exerted upon W1, W2 respectively. Then

E : E1 : E2 \ W2(P - R1) + W1(P - R2) P - R1 P - R2 | :: ----------------------- : ------ : ------ >. (80) W1W2 W1 W2 /

If R1 = R2, which is the case when the resistance, as well as the effort, arises from the mutual actions of the two bodies, the above becomes,

E : E1 : E2 \ :: W1 + W2 : W2 : W1 /, (81)

that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights; and they receive accelerations inversely proportional to their weights, according to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a system of bodies do not affect the motion of their common centre of gravity.

For example, if the weight of a gun be 160 times that of its ball 160/161 of the energy exerted by the powder in exploding will be employed in propelling the ball, and 1/161 in producing the recoil of the gun, provided the gun up to the instant of the ball's quitting the muzzle meets with no resistance to its recoil except the friction of the ball.

§ 124. _Centre of Percussion._--It is obviously desirable that the deviations or changes of motion of oscillating pieces in machinery should, as far as possible, be effected by forces applied at their centres of percussion.

If the deviation be a _translation_--that is, an equal change of motion of all the particles of the body--the centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then

W dv P = --- · -- (82) g dt

is the unbalanced effort required.

If the deviation be a rotation about an axis traversing the centre of gravity, there is no centre of percussion; for such a deviation can only be produced by a _couple_ of forces, and not by any single force. Let d[alpha] be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then ½Id([alpha]^2) = I[alpha] d[alpha] is the variation of the body's actual energy. Let M be the moment of the unbalanced couple required to produce the deviation; then by equation 57, § 104, the energy exerted by this couple in the interval dt is M[alpha] dt, which, being equated to the variation of energy, gives

d[alpha] R²W d[alpha] M = I-------- = --- · --------. (83) dt g dt

R is called the radius of gyration of the body with regard to an axis through its centre of gravity.

Now (fig. 133) let the required deviation be a rotation of the body BB about an axis O, not traversing the centre of gravity G, d[alpha] being, as before, the deviation of angular velocity to be produced in the interval dt. A rotation with the angular velocity [alpha] 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 [alpha]. OG, OG being the perpendicular distance between the two axes. Hence the required deviation may be regarded as compounded of a deviation of translation dv = OG·d[alpha], to produce which there would be required, according to equation (82), a force applied at G perpendicular to the plane OG--

W d[alpha] P = --- · OG · -------- (84) g dt

and a deviation d[alpha] 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 value is

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.

§ 125.* _To find the moment of inertia of a body about an axis through its centre of gravity experimentally._--Suspend 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)

R^2 = CG × OG = (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/g × R², can be computed.

§ 126.* _To find the force competent to produce the instantaneous acceleration of any link of a mechanism._--In many practical problems it is necessary to know the magnitude and position of the forces acting to produce 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 OF MATERIALS (q.v.). Let BK (fig. 134) be any link moving in any manner in a plane, and let G be its centre of gravity. Then its motion may be analysed into (1) a translation of its centre of gravity; and (2) a rotation about an axis through its centre of gravity perpendicular to its plane of motion. Let [alpha] be the acceleration of the centre of gravity and let A be the angular acceleration about the axis through the centre of gravity; then the force required to produce the translation of the centre of gravity is F = W[alpha]/g, and the couple required to produce the angular acceleration about the centre of gravity is M = IA/g, W and I 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 the crank shall revolve uniformly. Let the acceleration of the two points B and K therefore be supposed known. The problem is now to find the acceleration [alpha] and A. Take any pole O (fig. 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 : gb. 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 generality are taken in fig. 135, so that the centre of gravity G is not in the line joining them. First find the value of R experimentally. Then produce the given 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 take a point Z on the line OY so that YG × 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 _Valve and Valve Gear Mechanisms_, by W. E. Dalby (London, 1906), where the inertia stresses brought upon the several links of a Joy 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 of 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) moves with uniform velocity, so that if [alpha] is its angular velocity and r its radius, the acceleration is [alpha]²r in a direction along the crank arm from the crank pin to the centre of the shaft. Thus the acceleration of one point K is known completely. The acceleration of a second point, usually taken at the centre of the crosshead 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,[4] discovered also independently by Kirsch.[5] But probably the most convenient is the construction due to G. T. Bennett[6] which is as follows: Let OK be the crank and KB the connecting rod. On the connecting rod take a point L such that KL × KB = 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_.

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 rod being ascertained, F can be immediately calculated. To find a point in its line of action, take a point Q on the rod such that KG × GQ = R², R having been determined experimentally by the method of § 125; join G with O and through Q draw a line parallel to BO to cut GO in Z. Z is a point in the line of action of the resultant force F; hence through Z draw a line parallel to Og. The force F acts in this line, and thus the problem is completely solved. The above construction for Z is a corollary of the general theorem given in § 127.

§ 129. _Impact._ Impact or collision is a pressure of short duration exerted between two bodies.

The effects of impact are sometimes an alteration of the distribution of actual energy between the two bodies, and always a loss of a portion of that energy, depending on the imperfection of the elasticity of the bodies, in permanently altering their figures, and producing heat. The determination of the distribution of the actual energy after collision and of the loss of energy is effected by means of the following principles:--

I. The motion of the common centre of gravity of the two bodies is unchanged by the collision.

II. The loss of energy consists of a certain proportion of that part of the actual energy of the bodies which is due to their motion relatively to their common centre of gravity.

Unless there is some special reason for using impact in machines, it ought to be avoided, on account not only of the waste of energy which it causes, but from the damage which it occasions to the frame and mechanism. (W. J. M. R.; W. E. D.)

FOOTNOTES:

[1] In view of the great authority of the author, the late Professor Macquorn Rankine, it has been thought desirable to retain the greater part of this article as it appeared in the 9th edition of the _Encyclopaedia Britannica_. Considerable additions, however, have been introduced in order to indicate subsequent developments of the subject; the new sections are numbered continuously with the old, but are distinguished by an asterisk. Also, two short chapters which concluded the original article have been omitted--ch. iii., "On Purposes and Effects of Machines," which was really a classification of machines, because the classification of Franz Reuleaux is now usually followed, and ch. iv., "Applied Energetics, or Theory of Prime Movers," because its subject matter is now treated in various special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil Engine, and fully developed in Rankine's The Steam Engine and Other Prime Movers (London, 1902). (Ed. _E.B._)

[2] Since the relation discussed in § 7 was enunciated by Rankine, an enormous development has taken place in the subject of Graphic Statics, the first comprehensive textbook on the subject being _Die Graphische Statik_ by K. Culmann, published at Zürich in 1866. Many of the graphical methods therein given have now passed into the textbooks usually studied by engineers. One of the most beautiful graphical constructions regularly used by engineers and known as "the method of reciprocal figures" is that for finding the loads supported by the several members of a braced structure, having given a system of external loads. The method was discovered by Clerk Maxwell, and the complete theory is discussed and exemplified in a paper "On Reciprocal Figures, Frames and Diagrams of Forces," _Trans. Roy. Soc. Ed._, vol. xxvi. (1870). Professor M. W. Crofton read a paper on "Stress-Diagrams in Warren and Lattice Girders" at the meeting of the Mathematical Society (April 13, 1871), and Professor O. Henrici illustrated the subject by a simple and ingenious notation. The application of the method of reciprocal figures was facilitated by a system of notation published in _Economics of Construction in relation to framed Structures_, by Robert H. Bow (London, 1873). A notable work on the general subject is that of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and a discussion of the subject of reciprocal figures from the special point of view of the engineering student is given in _Vectors and Rotors_ by Henrici and Turner (London, 1903). See also above under "_Theoretical Mechanics_," Part 1. § 5.

[3] This is a particular case of a more general principle, that _the motion of the centre of gravity of a body is not affected by the mutual actions of its parts_.

[4] J. 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).

[5] Prof. Kirsch, "Über die graphische Bestimmung der Kolbenbeschleunigung," _Zeitsch. Verein deutsche Ingen_. (1890), p. 1320.

[6] Dalby, _The Balancing of Engines_ (London, 1906), app. 1.

MECHANICVILLE, a village of Saratoga county, New York, U.S.A., on the west bank of the Hudson River, about 20 m. N. of Albany; on the Delaware & Hudson and Boston & Maine railways. Pop. (1900), 4695 (702 foreign-born); (1905, state census), 5877; (1910) 6,634. It lies partly within Stillwater and partly within Half-Moon townships, in the bottom-lands at the mouth of the Anthony Kill, about 1-1/2 m. S. of the mouth of the Hoosick River. On the north and south are hills reaching a maximum height of 200 ft. There is ample water power, and there are manufactures of paper, sash and blinds, fibre, &c. From a dam here power is derived for the General Electric Company at Schenectady. The first settlement in this vicinity was made in what is now Half-Moon township about 1680. Mechanicville (originally called Burrow) was chartered by the county court in 1859, and incorporated as a village in 1870. It was the birthplace of Colonel Ephraim Elmer Ellsworth (1837-1861), the first Federal officer to lose his life in the Civil War.

MECHITHARISTS, a congregation of Armenian monks in communion with the Church of Rome. The founder, Mechithar, was born at Sebaste in Armenia, 1676. He entered a monastery, but under the influence of Western missionaries he became possessed with the idea of propagating Western ideas and culture in Armenia, and of converting the Armenian Church from its monophysitism and uniting it to the Latin Church. Mechithar set out for Rome in 1695 to make his ecclesiastical studies there, but he was compelled by illness to abandon the journey and return to Armenia. In 1696 he was ordained priest and for four years worked among his people. In 1700 he went to Constantinople and began to gather disciples around him. Mechithar formally joined the Latin Church, and in 1701, with sixteen companions, he formed a definitely religious institute of which he became the superior. Their Uniat propaganda encountered the opposition of the Armenians and they were compelled to move to the Morea, at that time Venetian territory, and there built a monastery, 1706. On the outbreak of hostilities between the Turks and Venetians they migrated to Venice, and the island of St Lazzaro was bestowed on them, 1717. This has since been the headquarters of the congregation, and here Mechithar died in 1749, leaving his institute firmly established. The rule followed at first was that attributed to St Anthony; but when they settled in the West modifications from the Benedictine rule were introduced, and the Mechitharists are numbered among the lesser orders affiliated to the Benedictines. They have ever been faithful to their founder's programme. Their work has been fourfold: (1) they have brought out editions of important patristic works, some Armenian, others translated into Armenian from Greek and Syriac originals no longer extant; (2) they print and circulate Armenian literature among the Armenians, and thereby exercise a powerful educational influence; (3) they carry on schools both in Europe and Asia, in which Uniat Armenian boys receive a good secondary education; (4) they work as Uniat missioners in Armenia. The congregation is divided into two branches, the head houses being at St Lazzaro and Vienna. They have fifteen establishments in various places in Asia Minor and Europe. There are some 150 monks, all Armenians; they use the Armenian language and rite in the liturgy.

See _Vita del servo di Dio Mechitar_ (Venice, 1901); E. Boré, _Saint-Lazare_ (1835); Max Heimbucher, _Orden u. Kongregationen_ (1907) I. § 37; and the articles in Wetzer u. Welte, _Kirchenlexicon_ (ed. 2) and Herzog, _Realencyklopädie_ (ed. 3), also articles by Sargisean, a Mechitharist, in _Rivista storica benedettina_ (1906), "La Congregazione Mechitarista." (E. C. B.)

MECKLENBURG, a territory in northern Germany, on the Baltic Sea, extending from 53° 4´ to 54° 22´ N. and from 10° 35´ to 13° 57´ E., unequally divided into the two grand duchies of Mecklenburg-Schwerin and Mecklenburg-Strelitz.

MECKLENBURG-SCHWERIN is bounded N. by the Baltic Sea, W. by the principality of Ratzeburg and Schleswig-Holstein, S. by Brandenburg and Hanover, and E. by Pomerania and Mecklenburg-Strelitz. It embraces the duchies of Schwerin and Güstrow, the district of Rostock, the principality of Schwerin, and the barony of Wismar, besides several small enclaves (Ahrensberg, Rosson, Tretzeband, &c.) in the adjacent territories. Its area is 5080 sq. m. Pop. (1905), 625,045.

MECKLENBURG-STRELITZ consists of two detached parts, the duchy of Strelitz on the E. of Mecklenburg-Schwerin, and the principality of Ratzeburg on the W. The first is bounded by Mecklenburg-Schwerin, Pomerania and Brandenburg, the second by Mecklenburg-Schwerin, Lauenburg, and the territory of the free town of Lübeck. Their joint area is 1130 sq. m. Pop. (1905), 103,451.

Mecklenburg lies wholly within the great North-European plain, and its flat surface is interrupted only by one range of low hills, intersecting the country from south-east to north-west, and forming the watershed between the Baltic Sea and the Elbe. Its highest point, the Helpter Berg, is 587 ft. above sea-level. The coast-line runs for 65 m. along the Baltic (without including indentations), for the most part in flat sandy stretches covered with dunes. The chief inlets are Wismar Bay, the Salzhaff, and the roads of Warnemünde. The rivers are numerous though small; most of them are affluents of the Elbe, which traverses a small portion of Mecklenburg. Several are navigable, and the facilities for inland water traffic are increased by canals. Lakes are numerous; about four hundred, covering an area of 500 sq. m., are reckoned in the two duchies. The largest is Lake Müritz, 52 sq. m. in extent. The climate resembles that of Great Britain, but the winters are generally more severe; the mean annual temperature is 48° F., and the annual rainfall is about 28 in. Although there are long stretches of marshy moorland along the coast, the soil is on the whole productive. About 57% of the total area of Mecklenburg-Schwerin consists of cultivated land, 18% of forest, and 13% of heath and pasture. In Mecklenburg-Strelitz the corresponding figures are 47, 21 and 10%. Agriculture is by far the most important industry in both duchies. The chief crops are rye, oats, wheat, potatoes and hay. Smaller areas are devoted to maize, buckwheat, pease, rape, hemp, flax, hops and tobacco. The extensive pastures support large herds of sheep and cattle, including a noteworthy breed of merino sheep. The horses of Mecklenburg are of a fine sturdy quality and highly esteemed. Red deer, wild swine and various other game are found in the forests. The industrial establishments include a few iron-foundries, wool-spinning mills, carriage and machine factories, dyeworks, tanneries, brick-fields, soap-works, breweries, distilleries, numerous limekilns and tar-boiling works, tobacco and cigar factories, and numerous mills of various kinds. Mining is insignificant, though a fair variety of minerals is represented in the district. Amber is found on and near the Baltic coast. Rostock, Warnemünde and Wismar are the principal commercial centres. The chief exports are grain and other agricultural produce, live stock, spirits, wood and wool; the chief imports are colonial produce, iron, coal, salt, wine, beer and tobacco. The horse and wool markets of Mecklenburg are largely attended by buyers from various parts of Germany. Fishing is carried on extensively in the numerous inland lakes.

In 1907 the grand dukes of both duchies promised a constitution to their subjects. The duchies had always been under a government of feudal character, the grand dukes having the executive entirely in their hands (though acting through ministers), while the duchies shared a diet (_Landtag_), meeting for a short session each year, and at other times represented by a committee, and consisting of the proprietors of knights' estates (_Rittergüter_), known as the _Ritterschaft_, and the _Landschaft_ or burgomasters of certain towns. Mecklenburg-Schwerin returns six members to the Reichstag and Mecklenburg-Strelitz one member.

In Mecklenburg-Schwerin the chief towns are Rostock (with a university), Schwerin, and Wismar the capital. The capital of Mecklenburg-Strelitz is Neu-Strelitz. The peasantry of Mecklenburg retain traces of their Slavonic origin, especially in speech, but their peculiarities have been much modified by amalgamation with German colonists. The townspeople and nobility are almost wholly of Saxon strain. The slowness of the increase in population is chiefly accounted for by emigration.

_History._--The Teutonic peoples, who in the time of Tacitus occupied the region now known as Mecklenburg, were succeeded in the 6th century by some Slavonic tribes, one of these being the Obotrites, whose chief fortress was Michilenburg, the modern Mecklenburg, near Wismar; hence the name of the country. Though partly subdued by Charlemagne towards the close of the 8th century, they soon regained their independence, and until the 10th century no serious effort was made by their Christian neighbours to subject them. Then the German king, Henry the Fowler, reduced the Slavs of Mecklenburg to obedience and introduced Christianity among them. During the period of weakness through which the German kingdom passed under the later Ottos, however, they wrenched themselves free from this bondage; the 11th and the early part of the 12th century saw the ebb and flow of the tide of conquest, and then came the effective subjugation of Mecklenburg by Henry the Lion, duke of Saxony. The Obotrite prince Niklot was killed in battle in 1160 whilst resisting the Saxons, but his son Pribislaus (d. 1178) submitted to Henry the Lion, married his daughter to the son of the duke, embraced Christianity, and was permitted to retain his office. His descendants and successors, the present grand dukes of Mecklenburg, are the only ruling princes of Slavonic origin in Germany. Henry the Lion introduced German settlers and restored the bishoprics of Ratzeburg and Schwerin; in 1170 the emperor Frederick I. made Pribislaus a prince of the empire. From 1214 to 1227 Mecklenburg was under the supremacy of Denmark; then, in 1229, after it had been regained by the Germans, there took place the first of the many divisions of territory which with subsequent reunions constitute much of its complicated history. At this time the country was divided between four princes, grandsons of duke Henry Borwin, who had died two years previously. But in less than a century the families of two of these princes became extinct, and after dividing into three branches a third family suffered the same fate in 1436. There then remained only the line ruling in Mecklenburg proper, and the princes of this family, in addition to inheriting the lands of their dead kinsmen, made many additions to their territory, including the counties of Schwerin and of Strelitz. In 1352 the two princes of this family made a division of their lands, Stargard being separated from the rest of the country to form a principality for John (d. 1393), but on the extinction of his line in 1471 the whole of Mecklenburg was again united under a single ruler. One member of this family, Albert (c. 1338-1412), was king of Sweden from 1364 to 1389. In 1348 the emperor Charles IV. had raised Mecklenburg to the rank of a duchy, and in 1418 the university of Rostock was founded.

The troubles which arose from the rivalry and jealousy of two or more joint rulers incited the prelates, the nobles and the burghers to form a union among themselves, and the results of this are still visible in the existence of the _Landesunion_ for the whole country which was established in 1523. About the same time the teaching of Luther and the reformers was welcomed in Mecklenburg, although Duke Albert (d. 1547) soon reverted to the Catholic faith; in 1549 Lutheranism was recognized as the state religion; a little later the churches and schools were reformed and most of the monasteries were suppressed. A division of the land which took place in 1555 was of short duration, but a more important one was effected in 1611, although Duke John Albert I. (d. 1576) had introduced the principle of primogeniture and had forbidden all further divisions of territory. By this partition John Albert's grandson Adolphus Frederick I. (d. 1658) received Schwerin, and another grandson John Albert II. (d. 1636) received Güstrow. The town of Rostock "with its university and high court of justice" was declared to be common property, while the Diet or _Landtag_ also retained its joint character, its meetings being held alternately at Sternberg and at Malchin.

During the early part of the Thirty Years' War the dukes of Mecklenburg-Schwerin and Mecklenburg-Güstrow were on the Protestant side, but about 1627 they submitted to the emperor Ferdinand II. This did not prevent Ferdinand from promising their land to Wallenstein, who, having driven out the dukes, was invested with the duchies in 1629 and ruled them until 1631. In this year the former rulers were restored by Gustavus Adolphus of Sweden, and in 1635 they came to terms with the emperor and signed the peace of Prague, but their land continued to be ravaged by both sides until the conclusion of the war. In 1648 by the Treaty of Westphalia, Wismar and some other parts of Mecklenburg were surrendered to Sweden, the recompense assigned to the duchies including the secularized bishoprics of Schwerin and of Ratzeburg. The sufferings of the peasants in Mecklenburg during the Thirty Years' War were not exceeded by those of their class in any other part of Germany; most of them were reduced to a state of serfdom and in some cases whole villages vanished. Christian Louis who ruled Mecklenburg-Schwerin from 1658 until his death in 1692 was, like his father Adolphus Frederick, frequently at variance with the estates of the land and with members of his family. He was a Roman Catholic and a supporter of Louis XIV., and his country suffered severely during the wars waged by France and her allies in Germany.

In June 1692 when Christian Louis died in exile and without sons, a dispute arose about the succession to his duchy between his brother Adolphus Frederick and his nephew Frederick William. The emperor and the rulers of Sweden and of Brandenburg took part in this struggle which was intensified when, three years later, on the death of Duke Gustavus Adolphus, the family ruling over Mecklenburg-Güstrow became extinct. At length the partition Treaty of Hamburg was signed on the 8th of March 1701, and a new division of the country was made. Mecklenburg was divided between the two claimants, the shares given to each being represented by the existing duchies of Mecklenburg-Schwerin, the part which fell to Frederick William, and Mecklenburg-Strelitz, the share of Adolphus Frederick. At the same time the principle of primogeniture was again asserted, and the right of summoning the joint _Landtag_ was reserved to the ruler of Mecklenburg-Schwerin.

Mecklenburg-Schwerin began its existence by a series of constitutional struggles between the duke and the nobles. The heavy debt incurred by Duke Charles Leopold (d. 1747), who had joined Russia in a war against Sweden, brought matters to a crisis; the emperor Charles VI. interfered and in 1728 the imperial court of justice declared the duke incapable of governing and his brother Christian Louis was appointed administrator of the duchy. Under this prince, who became ruler _de jure_ in 1747, there was signed in April 1755 the convention of Rostock by which a new constitution was framed for the duchy. By this instrument all power was in the hands of the duke, the nobles and the upper classes generally, the lower classes being entirely unrepresented. During the Seven Years' War Duke Frederick (d. 1785) took up a hostile attitude towards Frederick the Great, and in consequence Mecklenburg was occupied by Prussian troops, but in other ways his rule was beneficial to the country. In the early years of the French revolutionary wars Duke Frederick Francis I. (1756-1837) remained neutral, and in 1803 he regained Wismar from Sweden, but in 1806 his land was overrun by the French and in 1808 he joined the Confederation of the Rhine. He was the first member of the confederation to abandon Napoleon, to whose armies he had sent a contingent, and in 1813-1814 he fought against France. In 1815 he joined the Germanic Confederation (Bund) and took the title of grand duke. In 1819 serfdom was abolished in his dominions. During the movement of 1848 the duchy witnessed a considerable agitation in favour of a more liberal constitution, but in the subsequent reaction all the concessions which had been made to the democracy were withdrawn and further restrictive measures were introduced in 1851 and 1852.

Mecklenburg-Strelitz adopted the constitution of the sister duchy by an act of September 1755. In 1806 it was spared the infliction of a French occupation through the good offices of the king of Bavaria; in 1808 its duke, Charles (d. 1816), joined the confederation of the Rhine, but in 1813 he withdrew therefrom. Having been a member of the alliance against Napoleon he joined the Germanic confederation in 1815 and assumed the title of grand duke.

In 1866 both the grand dukes of Mecklenburg joined the North German confederation and the _Zollverein_, and began to pass more and more under the influence of Prussia, who in the war with Austria had been aided by the soldiers of Mecklenburg-Schwerin. In the Franco-German War also Prussia received valuable assistance from Mecklenburg, Duke Frederick Francis II. (1823-1883), an ardent advocate of German unity, holding a high command in her armies. In 1871 the two grand duchies became states of the German Empire. There was now a renewal of the agitation for a more democratic constitution, and the German Reichstag gave some countenance to this movement. In 1897 Frederick Francis IV. (b. 1882) succeeded his father Frederick Francis III. (1851-1897) as grand duke of Mecklenburg-Schwerin, and in 1904 Adolphus Frederick (b. 1848) a son of the grand duke Frederick William (1819-1904) and his wife Augusta Carolina, daughter of Adolphus Frederick, duke of Cambridge, became grand duke of Mecklenburg-Strelitz. The grand dukes still style themselves princes of the Wends.

See F. A. Rudloff, _Pragmatisches Handbuch der mecklenburgischen Geschichte_ (Schwerin, 1780-1822); C. C. F. von Lützow, _Versuch einer pragmatischen Geschichte von Mecklenburg_ (Berlin, 1827-1835); _Mecklenburgische Geschichte in Einzeldarstellungen_, edited by R. Beltz, C. Beyer, W. P. Graff and others; C. Hegel, _Geschichte der mecklenburgischen Landstände bis 1555_ (Rostock, 1856); A. Mayer, _Geschichte des Grossherzogtums Mecklenburg-Strelitz 1816-1890_ (New Strelitz, 1890); Tolzien, _Die Grossherzöge von Mecklenburg-Schwerin_ (Wismar, 1904); Lehsten, _Der Adel Mecklenburgs seit dem landesgrundgesetslichen Erbvergleich_ (Rostock, 1864); the _Mecklenburgisches Urkundenbuch_ in 21 vols. (Schwerin, 1873-1903); the _Jahrbücher des Vereins für mecklenburgische Geschichte und Altertumskunde_ (Schwerin, 1836 fol.); and W. Raabe, _Mecklenburgische Vaterlandskunde_ (Wismar, 1894-1896); von Hirschfeld, _Friedrich Franz II., Grossherzog von Mecklenburg-Schwerin und seine Vorgänger_ (Leipzig, 1891); Volz, _Friedrich Franz II._ (Wismar, 1893); C. Schröder, _Friedrich Franz III._ (Schwerin, 1898); Bartold, _Friedrich Wilhelm, Grossherzog von Mecklenburg-Strelitz und Augusta Carolina_ (New Strelitz, 1893); and H. Sachsse, _Mecklenburgische Urkunden und Daten_ (Rostock, 1900).