Agricultural Engineering · Engineering Sciences · also Kinematics and Dynamics of Machinery
How mechanisms move and transmit power: linkages, cams, gears, belts, flywheels, governors, balancing and vibration.
In the exams: GATE AG Sections 2 and 3 (power transmission, belts and gears recur); ICAR AIEEA-PG Unit III
Theory of machines is the study of how the parts of a machine move relative to one another and what forces that motion needs and creates. It has two halves. Kinematics asks only about motion: which way a link can move, how fast a point on it travels, how a cam shapes a follower's lift, what speed ratio a gear train gives. Dynamics adds mass and force: how much energy a flywheel must store, how a governor holds engine speed, what shaking force an unbalanced crank sends into the frame, at what speed a shaft whirls.
For an agricultural engineer the subject is everywhere on the farm. A tractor is a single machine that contains almost every chapter of this manual: a slider-crank engine with a flywheel and a governor, a clutch, a gearbox and a planetary final drive, a bevel differential with a lock, brakes on each rear wheel, and a power take-off shaft driving implements through belts, chains and universal joints. Threshers, reapers, balers and mowers are built from four-bar linkages, cams, eccentrics and belt drives; their failures are very often failures of balance, vibration, belt slip or chain wear.
Exams treat it as a calculation subject. In India, GATE Agricultural Engineering asks belt, gear-train, flywheel and governor numericals under power transmission and machine design, and ICAR AIEEA-PG lists theory of machines by name in Unit III. The US FE exam covers the same kinematics, vibration and energy methods under Dynamics, and licensure exams such as the Philippine agricultural and biosystems engineering board test it inside power and machinery. This manual gives the definitions, the standard equations with the assumptions behind them, and the judgement a practitioner needs. It uses SI units: angular speed ω in rad/s, speed N in revolutions per minute (rpm), with ω = 2πN/60.
A link (or element) is a body, or an assembly of bodies moving as one, that has motion relative to some other body. Two links in contact that constrain each other's relative motion form a kinematic pair. Pairs are classified by the type of contact. Lower pairs have surface contact: the turning (revolute) pair of a pin in a hole, the sliding (prismatic) pair of a piston in a cylinder, the screw pair of a lead screw, and the cylindrical, spherical and planar pairs. Higher pairs have point or line contact: a cam and its follower, two meshing gear teeth, a wheel rolling on a rail. A kinematic chain is a set of links joined by pairs so that relative motion is possible and definite. When one link of a chain is fixed, the chain becomes a mechanism; a mechanism that transmits useful work or force is a machine. The same chain gives different mechanisms depending on which link is fixed; each of these is called an inversion of the chain. Inversion changes the absolute motion of links but not the relative motion between them.
The mobility or degrees of freedom F of a mechanism is the number of independent inputs needed to define the position of every link. A free body in a plane has three degrees of freedom: two translations and one rotation. In a planar mechanism of n links, fixing one link leaves 3(n - 1) freedoms; each lower pair (one-freedom joint) removes two and each higher pair removes one. This gives the Kutzbach criterion F = 3(n - 1) - 2j - h, where j is the number of lower pairs and h the number of higher pairs. F = 1 means a constrained mechanism driven by one input, such as a four-bar linkage (n = 4, j = 4, F = 1). F = 0 means a structure: a triangle of three pinned links gives 3(2) - 2(3) = 0. F less than zero means a statically indeterminate (redundant) structure. Grubler's criterion is the special case for F = 1 with lower pairs only: 3n - 2j - 4 = 0. The criterion can mislead: it ignores special geometry, so a parallelogram linkage with a redundant parallel link has more motion than the count suggests, and a roller follower adds a spurious freedom because the roller can spin freely without affecting the output.
The four-bar chain is the commonest mechanism on farm machines: mower drives, straw walkers, seed-drill agitators, reaper knife drives and three-point hitch geometry all use it. Let s be the shortest link, l the longest and p and q the other two. Grashof's law states that if s + l is less than or equal to p + q, at least one link can make a full revolution relative to the others; if s + l is greater than p + q, no link can fully rotate and every inversion is a triple-rocker. For a Grashof chain, the inversion depends on which link is fixed. Fixing a link adjacent to the shortest gives a crank-rocker: the shortest link rotates fully and the opposite link oscillates. Fixing the shortest link gives a double-crank (drag-link) mechanism. Fixing the link opposite the shortest gives a double-rocker. When s + l equals p + q the chain has change points where all links fall into line and motion becomes uncertain. The transmission angle is the angle between the coupler and the output link; force is transmitted best near 90 degrees, and designers commonly keep it above about 40 degrees. The single slider-crank chain has four links and four pairs, three turning and one sliding. Its inversions are the reciprocating engine and pump (frame fixed), the rotary engine and the Whitworth quick-return mechanism (crank fixed), the oscillating-cylinder engine and the crank-and-slotted-lever quick-return mechanism (connecting rod fixed), and the hand or pendulum pump (slider fixed). The double slider-crank chain gives the elliptical trammel, the Scotch yoke, which produces exact simple harmonic motion, and the Oldham coupling, which joins two parallel shafts with a small lateral offset.
For two points A and B on the same rigid link, the velocity of B relative to A is perpendicular to the line AB and has magnitude ω x AB, where ω is the angular velocity of the link. This single fact drives graphical velocity analysis. In a velocity polygon, the fixed pivots collapse to one pole point; the velocity of each moving point is drawn from the pole, and the relative velocity of two points on a link is drawn perpendicular to that link. Once the polygon is closed, the angular velocity of any link is its relative velocity divided by its length. The instantaneous centre (IC) method is an alternative. At any instant a link in plane motion is turning about some point, fixed or not, and the velocity of every point on it equals ω times its distance from that point, directed perpendicular to the line joining them. A mechanism of n links has n(n - 1)/2 instantaneous centres. Some are found by inspection: the pin of a turning pair is an IC; for a sliding pair the IC is at infinity perpendicular to the slide; for pure rolling it is at the contact point. The rest are found with the Aronhold-Kennedy theorem of three centres: the three ICs of any three links in plane motion lie on one straight line. A four-bar mechanism therefore has six ICs, four at the pins and two found with Kennedy's theorem. The angular velocity ratio of the output to the input link of a four-bar equals the inverse ratio of the distances of their fixed pivots from the IC of the two links.
The acceleration of B relative to A on a rigid link has two components. The centripetal (radial) component is ω² x AB, directed from B towards A. The tangential component is α x AB, perpendicular to AB, where α is the angular acceleration of the link. A crank rotating at constant speed has no tangential acceleration, only the centripetal ω²r towards the crank centre. When a point slides along a link that is itself rotating, as a slider moves along the slotted lever of a quick-return mechanism, a third term appears: the Coriolis component of acceleration, 2ωv, where ω is the angular velocity of the rotating link and v the sliding velocity of the point relative to it. Its direction is found by rotating the sliding velocity vector through 90 degrees in the sense of ω. The Coriolis component is zero if either the link does not rotate or the slider does not slide relative to it. Forgetting it, or halving it, are the classic errors in these problems.
For an engine or reciprocating pump with crank radius r, connecting rod length l and ratio n = l/r, measure the crank angle θ from inner dead centre. The piston displacement from inner dead centre is x = r[(1 - cos θ) + (n - √(n² - sin²θ))], which for usual values of n is well approximated by x = r[(1 - cos θ) + sin²θ/(2n)]. Differentiating at constant crank speed ω gives the piston velocity v = ωr[sin θ + sin 2θ/(2n)] and the piston acceleration a = ω²r[cos θ + cos 2θ/n]. At inner dead centre (θ = 0) the acceleration is ω²r(1 + 1/n), and at outer dead centre (θ = 180 degrees) it is ω²r(1 - 1/n) in magnitude, so the obliquity of the connecting rod makes the motion unsymmetrical. The first term, ω²r cos θ, is the primary part and the second, ω²r cos 2θ/n, is the secondary part, which matters for balancing. The angular velocity of the connecting rod is ω_c = ω cos θ / √(n² - sin²θ), close to ω cos θ/n; it is greatest at the dead centres and zero when the crank is at 90 degrees. Maximum piston velocity occurs slightly before the crank is at 90 degrees for the outward stroke, not at 90 degrees, because of obliquity.
A cam is a higher-pair element that gives a follower a prescribed motion, usually reciprocating or oscillating, from a rotating shaft. Radial (disc) cams move the follower perpendicular to the camshaft; cylindrical (drum) cams move it parallel to the shaft. Followers are classified by the contact surface (knife-edge, roller, flat-faced or mushroom, and spherical), by their motion (reciprocating or oscillating) and by their line of motion (radial, when the follower axis passes through the cam centre, or offset). The base circle is the smallest circle drawn from the cam centre to the cam profile. The trace point is the point on the follower used to generate the pitch curve, and the prime circle is the smallest circle to the pitch curve. The lift or stroke is the greatest follower movement; dwell is a period when the follower is stationary while the cam turns. The pressure angle is the angle between the direction of follower motion and the common normal at the contact point; a large pressure angle pushes the follower sideways and can jam a translating follower, so it is commonly limited to about 30 degrees for translating roller followers, and it is reduced by enlarging the base circle.
Let h be the lift, θo the cam angle turned during the outstroke (in radians) and ω the cam speed. Uniform velocity motion lifts the follower at constant speed hω/θo but has theoretically infinite acceleration at the start and end of the stroke, so it is used only with rounded-off ends at low speed. Simple harmonic motion (SHM) follows s = (h/2)[1 - cos(πθ/θo)]. Its maximum velocity is πhω/(2θo), at mid-stroke, and its maximum acceleration is π²hω²/(2θo²), at the ends of the stroke. Uniform acceleration and retardation (parabolic) motion accelerates the follower uniformly for the first half of the stroke and retards it uniformly for the second half. Its maximum velocity is 2hω/θo and its acceleration is constant at 4hω²/θo², the smallest peak acceleration of the standard motions, but the acceleration still changes suddenly at the start, middle and end. Cycloidal motion follows s = h[θ/θo - sin(2πθ/θo)/(2π)]. Its maximum velocity is 2hω/θo and its maximum acceleration is 2πhω²/θo², at one quarter and three quarters of the stroke; acceleration is zero at both ends, so there is no sudden jump. For this reason cycloidal motion is preferred for high-speed cams. The force needed to keep a follower on the cam during the retardation part of the stroke is supplied by a spring; if spring force is less than the follower mass times its retardation, the follower leaves the cam (jump).
Gears transmit motion positively, without slip, between shafts. Spur gears join parallel shafts with teeth parallel to the axis. Helical gears join parallel shafts with teeth on a helix; they run more quietly and carry more load but create an axial thrust, which double-helical (herringbone) gears cancel. Bevel gears join intersecting shafts, usually at right angles; spiral bevel and hypoid gears are used in tractor and vehicle final drives. A worm and worm wheel join non-intersecting perpendicular shafts and give large reductions in one step; low-lead worms can be self-locking. The pitch circle is the imaginary circle that rolls without slip on the pitch circle of the mating gear. With T teeth on pitch diameter d, the module is m = d/T in millimetres, the circular pitch is pc = πm, and in inch practice the diametral pitch is P = T/d (teeth per inch), with pc x P = π. Mating gears must have the same module and pressure angle. For standard full-depth involute teeth, the addendum is one module and the dedendum 1.25 modules; the difference is the clearance. The law of gearing says the common normal at the point of tooth contact must always pass through the fixed pitch point on the line of centres; then the velocity ratio stays constant. Involute teeth are almost universal because the pressure angle is constant and a small change in centre distance does not spoil the velocity ratio.
The path of contact is the length of the line of action along which a pair of teeth is in contact. It is found from the addendum circles: path = √(Ra² - R²cos²φ) + √(ra² - r²cos²φ) - (R + r) sin φ, where R and r are the pitch radii, Ra and ra the addendum radii and φ the pressure angle. The arc of contact equals the path of contact divided by cos φ. The contact ratio, the average number of tooth pairs in contact, is the arc of contact divided by the circular pitch, or the path of contact divided by the base pitch πm cos φ. It must exceed one, and designers usually want at least about 1.2 for smooth running. Interference occurs when the tip of a tooth tries to contact the non-involute flank of the mating tooth, inside its base circle. It happens when a pinion with few teeth meshes with a large gear or a rack; it undercuts and weakens the pinion teeth. For a pinion meshing with a rack with addendum equal to one module, the minimum number of teeth to avoid interference is 2/sin²φ: 31.9, so 32 teeth, for 14.5 degrees and 17.1, so 18 teeth, for 20 degrees (17 is often quoted as the practical figure).
The velocity ratio of a gear pair is N1/N2 = T2/T1 = d2/d1: speeds are inversely proportional to tooth numbers. For external gears the direction reverses; an internal gear turns the same way as its pinion. In a simple gear train every shaft carries one gear, and the overall ratio depends only on the first and last gears; intermediate idlers change the direction of rotation (an odd number of idlers keeps the direction of the last gear the same as the first) but not the ratio. In a compound train some shafts carry two gears keyed together, so the train value is the product of the driver teeth divided by the product of the driven teeth: N_last/N_first = (T1 x T3 x T5 ...)/(T2 x T4 x T6 ...). A reverted train has the first and last gears on the same axis, as in lathe back gears and many gearboxes. The centre distances of the two stages must then be equal: r1 + r2 = r3 + r4, which for gears of the same module means T1 + T2 = T3 + T4. The worm drive ratio is the number of worm-wheel teeth divided by the number of worm starts, so a double-start worm and a 40-tooth wheel give 20:1.
In an epicyclic train at least one gear axis moves: planet gears are carried on an arm (carrier) that rotates about the central axis while they mesh with a sun gear and often an internal ring (annulus). The tabular method is the safest way to solve them. Step one: fix the arm and give the sun +1 revolution; write each gear's turns using the fixed-axis ratios. Step two: multiply the row by an unknown x. Step three: add y revolutions to every element, including the arm, which amounts to turning the whole train as a solid. The total for each element is then known in terms of x and y, and the two conditions of the problem (for example ring fixed and sun speed given) fix x and y. The same result comes from the relative-speed formula (N_ring - N_arm)/(N_sun - N_arm) = -T_sun/T_ring. For a simple sun-planet-ring train with equal module, T_ring = T_sun + 2T_planet. With the ring fixed, sun as input and carrier as output, the reduction is N_sun/N_arm = 1 + T_ring/T_sun: a 20-tooth sun and 80-tooth ring give 5:1, a common arrangement in tractor final drives. Torques follow from equilibrium and energy. The three external torques on a train sum to zero, and without losses input power equals output power, so T_in ω_in + T_out ω_out = 0; the fixing torque on the held member is whatever balances the other two.
When a vehicle turns, the outer driving wheel travels a longer path than the inner one. If both were fixed to one axle, one would have to slip, which scuffs tyres, wastes power, damages turf and makes steering hard. The differential lets the two half-shafts turn at different speeds while both are driven. The common automotive and tractor form is the bevel-gear differential. The input pinion drives a crown wheel (ring gear) bolted to a cage (differential case). The cage carries two or more bevel planet pinions on a cross pin, and these mesh with two bevel side (sun) gears splined to the left and right half-shafts. Because the side gears are equal, the train is an epicyclic train with a ratio of -1 when the cage is held, and it follows that the cage speed is the average of the two side-gear speeds: N_left + N_right = 2N_cage. On a straight road the planet pinions do not spin on their pin and the whole assembly turns as one. On a turn, the planets spin, slowing one side gear and speeding the other by the same amount. If one wheel is held still, the other turns at twice the cage speed. The crown wheel and pinion also provide a fixed reduction, the final drive ratio, equal to crown-wheel teeth divided by pinion teeth; many tractors add a planetary or spur final reduction after the differential.
An ordinary open differential divides torque equally between the two half-shafts, ignoring internal friction, whatever their speeds. That is its weakness in the field: the torque each wheel can take is limited by the wheel with the least grip, so when one rear tyre spins in a wet patch the other wheel receives no more torque than the spinning one, and the tractor stops. Tractors therefore carry a differential lock, usually a dog or multi-plate clutch that locks the cage to one half-shaft so that both wheels turn together. It must be disengaged before turning, because a locked axle resists steering and can cause loss of control, especially at speed or on slopes. Turning geometry gives the speed split. If the vehicle's centreline follows a circle of radius R and the track width (distance between wheel centres) is w, the outer wheel runs on radius R + w/2 and the inner on R - w/2, so their speeds are in that ratio. For example, with R = 6 m and w = 1.5 m, the outer wheel turns 6.75/5.25 = 1.29 times as fast as the inner.
Belts and ropes transmit power between shafts by friction; they are cheap, quiet, absorb shock and suit large or uncertain centre distances, which is why threshers, chaff cutters, pumps and combine drives use them. Neglecting slip and belt thickness, the belt speed is the same on both pulleys, so N1 d1 = N2 d2 and the velocity ratio N2/N1 = d1/d2. Belt thickness t is allowed for by measuring to the belt's mid-thickness: N2/N1 = (d1 + t)/(d2 + t). Slip, expressed as a percentage s of the speed that should have been transmitted, reduces the driven speed: N2/N1 = (d1/d2)(1 - s/100); with slip s1 between driver and belt and s2 between belt and follower the combined factor is (1 - s1/100 - s2/100) to a first approximation. Creep is a different effect: the belt stretches more on the tight side than the slack side, so it slides slightly on the pulleys even without gross slip, and the follower speed falls a little. For an open belt, the belt turns the same way on both pulleys and the length is L = π(r1 + r2) + 2x + (r1 - r2)²/x, with x the centre distance. For a crossed belt, which reverses the direction, L = π(r1 + r2) + 2x + (r1 + r2)²/x. The angle of contact (lap) on the smaller pulley of an open drive is θ = π - 2 sin⁻¹[(r1 - r2)/x] radians; on a crossed drive it is π + 2 sin⁻¹[(r1 + r2)/x] on both pulleys. Slip starts on the pulley with the smaller value of μθ, usually the smaller pulley.
For a flat belt about to slip over an arc of contact θ with coefficient of friction μ, the tight-side and slack-side tensions satisfy T1/T2 = e^(μθ), with θ in radians. For a V-belt or rope in a groove of total angle 2β, the wedge action multiplies the normal force, and T1/T2 = e^(μθ cosec β). With a 40-degree groove (β = 20 degrees), cosec β is about 2.92, so a V-belt grips roughly as if μ were three times larger; this is why V-belts carry more power in less space than flat belts. The power transmitted is P = (T1 - T2)v, with v the belt speed in m/s. At speed, the belt's own mass m per metre length needs a centripetal force, which adds a centrifugal tension Tc = mv² to both sides equally. It does not change the effective pull T1 - T2, but it uses up part of the allowable tension: if Tmax is the greatest safe tension, the tight side available for power is T1 = Tmax - Tc. The power is P = (Tmax - mv²)(1 - 1/e^(μθ))v. Differentiating shows that power is greatest when Tc = Tmax/3, that is at v = √(Tmax/(3m)); above that speed, more speed gives less power. Tmax is the allowable stress times the cross-sectional area. A belt is installed with an initial tension T0; for a belt that obeys Hooke's law, T0 = (T1 + T2)/2, or (T1 + T2 + 2Tc)/2 when centrifugal tension is included.
Roller chains give a positive drive without slip at moderate speeds and are used on seed drills, planters, combine feeders, conveyors and power-tiller transmissions. The pitch p is the distance between adjacent roller centres. A sprocket with T teeth has pitch circle diameter d = p/sin(180°/T). The chain speed is v = pTN/60 m/s for p in metres and N in rpm, and the velocity ratio is N1/N2 = T2/T1. Because the chain wraps the sprocket as a polygon, not a circle, its linear speed rises and falls once per tooth as each link engages: this chordal (polygonal) action gives a speed variation of 1 - cos(180°/T) of the maximum, so it is large for small sprockets and small for large ones. The length of a chain is expressed in pitches, and an offset (cranked) link is needed if the number of links is odd. Chains need lubrication and correct slack; as pins and bushes wear, the chain's pitch grows, it rides higher on the teeth and eventually jumps them, so chain elongation is checked in service.
A turning-moment (crank-effort) diagram plots the torque on the crankshaft against crank angle. For a single-cylinder four-stroke engine, one cycle occupies 720 degrees (4π radians): the power stroke gives a large positive torque, while the suction, compression and exhaust strokes give smaller, partly negative torques. The area under the diagram is the work done per cycle, and the mean torque is that work divided by the cycle angle; for a steady load, the mean resisting torque equals the mean engine torque. Where the engine torque exceeds the mean, the excess energy speeds up the crankshaft; where it falls short, the crankshaft slows. The maximum fluctuation of energy ΔE is the difference between the greatest and least values of the cumulative excess energy over the cycle, found by adding the areas above and below the mean line in turn. The coefficient of fluctuation of energy is ΔE divided by the work done per cycle. The coefficient of fluctuation of speed is Cs = (ω1 - ω2)/ω, where ω1 and ω2 are the greatest and least speeds and ω = (ω1 + ω2)/2 is the mean. A given ΔE is absorbed with a speed change determined by the flywheel's moment of inertia.
The kinetic energy of a flywheel is E = ½Iω², with I = mk², m the mass and k the radius of gyration. The change of energy between the greatest and least speeds is ΔE = ½I(ω1² - ω2²) = Iω(ω1 - ω2) = Iω²Cs = mk²ω²Cs. This is the flywheel equation: given ΔE from the turning-moment diagram and the allowed Cs, it gives the required I. For a rim-type flywheel most of the inertia is in the rim, so k is taken as the mean rim radius; for a solid disc of radius R, I = mR²/2. Because energy stored goes with ω², a flywheel on a high-speed shaft needs far less mass than one on a slow shaft: halving the speed needs four times the inertia for the same ΔE and Cs. The limit is rim strength. A thin rotating rim carries a hoop stress σ = ρv², where ρ is the density and v the rim speed, independent of the rim cross-section; so the permissible stress sets a maximum rim speed, commonly quoted as about 25 to 30 m/s for cast-iron flywheels in classical texts, with steel allowing more. For a punching or shearing press driven by a small motor, the flywheel supplies most of the energy for the short punching stroke and the motor restores it during the rest of the cycle. The energy needed per hole is found from the punching force and the plate thickness; the motor supplies energy at a steady rate over the whole cycle; the flywheel supplies the shortfall during the stroke, which is ΔE. A flywheel smooths speed within a cycle; it does not control mean speed over many cycles, which is the governor's job.
A governor keeps the mean speed of an engine within limits as the load changes, by adjusting the fuel (in a diesel, the fuel delivered by the injection pump) or the charge (in a spark-ignition engine, the throttle). If load increases, speed falls, the governor senses the fall and increases fuel until the engine produces the new torque; if load falls, it reduces fuel. Farm tractors rely on this: at a fixed throttle setting the governor holds engine and power take-off speed nearly steady while ploughing load varies. Centrifugal governors use rotating masses (balls or flyweights) whose outward movement with speed lifts a sleeve connected to the fuel control. They are subdivided into pendulum types such as the Watt governor, gravity-loaded types such as the Porter and Proell governors, and spring-loaded types such as the Hartnell, Hartung, Wilson-Hartnell and Pickering governors. A variable-speed (all-speed) governor holds any speed selected by the throttle lever and is usual on tractors; a minimum-maximum (idle and maximum) governor controls only idle and maximum speed, leaving the driver to control the range between, and is usual on road vehicles.
In the Watt governor, a ball of mass m hangs on an arm pivoted on the spindle axis and rotates at ω. Balancing the centrifugal effect against gravity gives the height h of the cone, from the plane of the balls to the point where the arms meet on the axis: h = g/ω². With N in rpm, h = 895/N² metres, so at 60 rpm h = 0.249 m and at 100 rpm h = 0.089 m. The height becomes too small to be useful above about 60 to 80 rpm, which is why the Watt governor is obsolete. The Porter governor adds a heavy central load M on the sleeve. With equal arm and link lengths and no friction, N² = (m + M)/m x 895/h: the central load raises the speed at which a given height is reached and increases the governor's force, so it can work at practical speeds. Friction at the sleeve adds or subtracts F from Mg and makes the governor insensitive over a band of speeds. The Hartnell governor uses a spring on the sleeve and bell-crank levers carrying the balls. If the ball arm is x and the sleeve arm y, then for equilibrium (ignoring the balls' weight and arm obliquity), the centrifugal force F = mω²r satisfies 2Fx = (S + Mg)y, where S is the spring force; the sleeve lift equals the change of ball radius times y/x, and the spring stiffness is the change in spring force divided by the sleeve lift.
A governor is sensitive if a small change of speed produces a large sleeve movement. Sensitiveness is measured by the range of speed divided by the mean speed, (N1 - N2)/N, where N1 and N2 are the maximum and minimum equilibrium speeds; the smaller this ratio, the more sensitive the governor. A governor is stable if for each speed within the working range there is only one radius of rotation at which the balls are in equilibrium, which requires the radius to increase as speed increases. A governor is isochronous if the equilibrium speed is the same at every radius: the range of speed is zero, the governor is infinitely sensitive. Watt and Porter governors cannot be isochronous, because their equilibrium height depends on speed; a Hartnell governor can be made nearly so by spring choice. Isochronism is undesirable in practice because the governor then hunts: the sleeve rushes to one extreme at the slightest speed change, the fuel supply overshoots, and speed oscillates above and below the mean. Hunting is made worse by too sensitive a governor and by friction or lag; it is cured by less sensitivity, damping (a dashpot) or a larger flywheel. The effort of a governor is the mean force on the sleeve for a given fractional change of speed, and its power is the work done at the sleeve, effort times lift; both express its ability to overcome friction in the fuel linkage. For engines, governor regulation (speed droop) is the drop from no-load to full-load governed speed expressed as a percentage; texts and test codes differ on whether the denominator is the no-load, full-load or rated speed, so the definition must be stated with the number.
A mass m at radius r rotating at ω exerts a centrifugal force mω²r on its shaft, rotating with the shaft. The bearings feel it as a rotating load that shakes the frame and loads the bearings in fatigue; because it rises with the square of speed, a small unbalance that is harmless at low speed becomes serious at high speed. A single mass is balanced by a mass on the opposite side in the same plane with an equal mr product: m_b r_b = m r. Several masses in one plane are balanced when the vector sum of their mr products is zero; the balancing mass supplies the closing side of the force polygon. This is static balance: the rotor stays in any position when placed on knife-edges. When the masses lie in different planes along the shaft, their forces can sum to zero while their moments about a point do not, and the rotor rocks end to end in its bearings: it is statically balanced but dynamically unbalanced. Complete (dynamic) balance needs both the sum of mr and the sum of mrl to be zero, where l is the distance of each plane from a reference plane. Any unbalance can be corrected by adding or removing mass in two chosen planes, which is how balancing machines work: taking moments about one correction plane eliminates the other and gives one balancing mass directly. International standards (ISO 1940-1, now ISO 21940-11) express permissible residual unbalance as a balance quality grade G, the product of permissible eccentricity and angular speed in mm/s; G6.3 is the grade commonly applied to general machinery parts, and the exact grade for a part must be taken from the standard.
The reciprocating parts of an engine or reciprocating pump (piston, rings, pin and about one-third of the connecting rod mass, by a common rule of thumb for dividing the rod) have acceleration ω²r(cos θ + cos 2θ/n). The force needed to accelerate them, m ω²r(cos θ + cos 2θ/n), reacts on the frame as an unbalanced shaking force along the line of stroke. The primary part, mω²r cos θ, varies at crank frequency and has the same form as the line-of-stroke component of a mass m rotating at the crank pin. The secondary part, mω²r cos 2θ/n, varies at twice crank frequency, like a mass m/n at radius r rotating at 2ω, and its peak is n times smaller than the primary. Balancing a fraction c of the reciprocating mass with a mass opposite the crank reduces the unbalanced force along the line of stroke to (1 - c)mω²r cos θ but introduces a new unbalanced force cmω²r sin θ perpendicular to it. A compromise fraction is therefore chosen, and single-cylinder engines are usually left with some shaking in one direction or the other.
In a multi-cylinder engine the primary and secondary forces and couples of all cylinders add vectorially. Each cylinder's primary force can be represented by a mass rotating at the crank angle and each secondary force by a mass m/n rotating at twice the crank angle, so the rotating-mass methods apply: for primary balance, the sums of the primary forces and couples must be zero; for secondary balance, the same sums taken with doubled crank angles must be zero. The in-line four-cylinder engine with cranks at 0, 180, 180 and 0 degrees (the usual layout of many tractor engines) is in primary force and primary couple balance, because the cranks are symmetrical about the centre. At doubled angles all four cranks point the same way (0, 360, 360, 0), so the secondary forces add: the unbalanced secondary force is 4mω²r cos 2θ/n, at twice engine speed. A six-cylinder in-line engine is in balance for primary and secondary forces and couples, one reason for its smoothness.
The simplest vibrating system is a mass m on a spring of stiffness k. Displaced and released, it oscillates at the natural circular frequency ωn = √(k/m) rad/s, or fn = ωn/(2π) Hz. If the spring is loaded by the weight of the mass, the static deflection δ = mg/k, so ωn = √(g/δ) and fn = 0.4985/√δ Hz with δ in metres; a static deflection of 1 mm gives about 15.8 Hz and 10 mm about 5 Hz. Springs in parallel add stiffness, k = k1 + k2; springs in series add flexibility, 1/k = 1/k1 + 1/k2. A shaft or beam carrying a mass behaves as a spring: a cantilever with end load has stiffness 3EI/L³ and a simply supported beam with central load 48EI/L³. The same idea governs torsional vibration: a disc of polar moment of inertia I on a shaft of torsional stiffness q = GJ/L has ωn = √(q/I). In a two-rotor system (for example an engine flywheel and a driven machine rotor joined by a shaft) there is a node, a section that does not twist, located so that the two rotors vibrate with the same frequency: the distances from the node are inversely proportional to the rotor inertias, l_A I_A = l_B I_B.
Real systems lose energy through damping. With viscous damping, where the damping force is c times velocity, the critical damping coefficient is cc = 2√(km) = 2mωn, and the damping ratio is ζ = c/cc. If ζ is greater than one the system is overdamped and returns to rest without oscillating; if ζ equals one it is critically damped and returns fastest without overshoot; if ζ is less than one it is underdamped and oscillates with decaying amplitude at the damped frequency ωd = ωn√(1 - ζ²). The logarithmic decrement δ is the natural logarithm of the ratio of two successive amplitudes, δ = ln(x0/x1) = 2πζ/√(1 - ζ²), close to 2πζ for light damping. Over n cycles, δ = (1/n) ln(x0/xn). Measuring the decay of a tapped structure is a practical way of estimating its damping. Coulomb (dry friction) damping, by contrast, reduces amplitude by a constant amount each cycle rather than by a constant ratio. Rubber mounts, hydraulic seat dampers and shock absorbers on tractor seats and cabs use damping to limit the vibration reaching the operator.
When a harmonic force F0 sin ωt acts on a damped spring-mass system, the steady-state amplitude is X = (F0/k) x MF, where the magnification factor is MF = 1/√[(1 - r²)² + (2ζr)²] and r = ω/ωn is the frequency ratio. At resonance, r = 1, MF = 1/(2ζ): with light damping the amplitude becomes very large. For isolation, the quantity that matters is the force transmitted to the foundation, and the transmissibility is TR = √[1 + (2ζr)²]/√[(1 - r²)² + (2ζr)²]. TR is greater than one for r less than √2, equal to one at r = √2, and less than one only when r exceeds √2. So a machine is isolated by mounting it on springs soft enough that its running frequency is well above √2 times the natural frequency; in the isolation region, adding damping actually increases transmissibility, although some damping is needed to pass through resonance safely on start-up. Rotating unbalance mer creates such a force with F0 = meω². A rotating shaft carrying discs whirls violently at its critical (whirling) speed, which equals the natural frequency of transverse vibration of the shaft-rotor system: Nc = 60fn rpm. Above the critical speed, the rotor's centre of mass moves towards the shaft axis and running is smooth again, so machines are run either well below or well above critical speed and are accelerated quickly through it. For shafts with several loads, Dunkerley's empirical formula, 1/fn² = 1/f1² + 1/f2² + ... + 1/fs², gives a slightly low (safe-side) estimate of the lowest natural frequency. Operators on tractors are exposed to whole-body vibration, which is assessed under ISO 2631-1 and limited by regulation in some jurisdictions (for example the European Union's physical agents directive on vibration); seat suspension design is an application of transmissibility.
A brake absorbs kinetic or potential energy by friction and turns it into heat. In a single block (shoe) brake a short block is pressed on a drum of radius r by a lever. If the normal reaction is RN, the friction force is μRN and the braking torque is μRN r. The lever force needed depends on where the fulcrum lies relative to the line of the friction force: taking moments about the fulcrum includes the moment of the friction force, which in one direction of rotation helps apply the brake (the brake is then partly self-energising) and in the other opposes it. If the friction moment alone is enough to apply the brake, the brake is self-locking, which is wanted only in back-stops. For a long shoe covering a large angle 2θ, an equivalent coefficient μ' = 4μ sin θ/(2θ + sin 2θ) is used. In a band brake a flexible steel band lined with friction material is wrapped round the drum; its tensions obey the belt equation T1/T2 = e^(μθ), and the braking torque is (T1 - T2)r. In a simple band brake, with one end of the band on the fulcrum, the lever force is found by taking moments of the other band tension about the fulcrum; a differential band brake attaches both ends to the lever. Band brakes are used in winches and in some tractor and powershift transmissions. Internal expanding shoe brakes and disc brakes are the road-vehicle types; many tractors use wet multi-disc brakes running in the transmission oil, operated independently on the left and right rear wheels, which allows brake-assisted turning and also means both pedals must be latched together for road travel.
A dynamometer measures the torque, and with speed the power, of a machine. Absorption dynamometers absorb the power and dissipate it as heat. The Prony brake clamps friction blocks on a pulley; the reaction arm of length L carries a load W at balance, so the torque is W L and the brake power is P = 2πNT/60. The rope brake wraps a rope round the flywheel with a dead weight W on one end and a spring balance reading S on the other; the torque is (W - S)(D + d)/2, where D is the drum diameter and d the rope diameter, since the effective radius is to the rope centre. Hydraulic (water) dynamometers churn water between a rotor and a stator casing and measure the reaction torque on the casing; eddy-current dynamometers do the same electromagnetically. Hydraulic and eddy-current dynamometers are the kinds used to test tractor engines and power take-off output at testing stations, following national and OECD tractor test codes. Transmission dynamometers measure the torque as it passes to the load without absorbing it: epicyclic-train dynamometers measure the reaction on the planet carrier, belt-transmission dynamometers measure the difference in belt tensions, and torsion dynamometers measure the angle of twist of a calibrated shaft, T = GJθ/L, today usually with strain gauges.
A clutch connects and disconnects a driving shaft and a driven shaft while one is running, so that a load can be picked up gradually and the drive interrupted for gear changes. Positive clutches (jaw or dog clutches) lock by interlocking teeth; they transmit high torque without slip but can only be engaged at low relative speed, and they are used in differential locks and some gearbox selectors. Friction clutches transmit torque by friction between surfaces pressed together by springs, a diaphragm, levers or hydraulic pistons. They include the single-plate clutch, with two friction surfaces (both faces of one driven plate); the multi-plate clutch, which packs more surfaces into a small diameter and is often wet (running in oil) for cooling; the cone clutch; and the centrifugal clutch, in which shoes are thrown outward against a drum as speed rises, used on small engines, chain saws and some power tillers so that the load engages automatically above idle. Many tractors have a dual clutch in which one pedal movement first disengages the transmission and further movement disengages the power take-off, or independent PTO clutches, often wet multi-plate and hydraulically engaged, so that the PTO can run while the tractor is stopped.
For a friction surface of outer radius R1 and inner radius R2 pressed with axial force W, two assumptions give the torque. With uniform pressure, which is realistic for a new clutch with flat, rigid plates, p = W/[π(R1² - R2²)] and the torque per surface is T = (2/3)μW(R1³ - R2³)/(R1² - R2²). After running in, the outer part of the lining, which slides faster, wears faster until the wear is uniform; wear is proportional to pressure times sliding velocity, so p r is constant. With uniform wear the torque per surface is T = μW(R1 + R2)/2, that is, the force acts at the mean radius. For the same W, uniform wear gives a slightly smaller torque than uniform pressure, so designers use it as the safe assumption. Under uniform wear the greatest pressure is at the inner radius, and p_max R2 = C with W = 2πC(R1 - R2). For n pairs of friction surfaces, the torque is multiplied by n. A single-plate clutch with both sides effective has n = 2. A multi-plate clutch with n1 plates on the driving shaft and n2 on the driven shaft has n1 + n2 - 1 pairs of surfaces. For a cone clutch of semi-cone angle α, the normal force on the cone is W/sin α, so the torque under uniform wear is μW R_mean/sin α: a small cone angle gives a large torque for a small axial force but makes the cone hard to disengage. For a centrifugal clutch, each shoe of mass m at radius r presses on the drum with the centrifugal force mω²r minus the spring force S, and the torque is μ(mω²r - S)R times the number of shoes, where R is the drum's inner radius.
Learn the few relationships that carry most marks and derive the rest: v = ωr, T1/T2 = e^(μθ), P = (T1 - T2)v, N1T1 = N2T2, the tabular method for epicyclic trains, ΔE = Iω²Cs, h = 895/N², mω²r for balance, ωn = √(k/m) and the uniform-wear clutch torque. Always convert rpm to rad/s, check whether an angle must be in radians, and write the assumption (uniform wear, no slip, frictionless governor) next to the formula. Draw velocity and acceleration polygons by hand at least once so the Coriolis term makes sense. Take a tractor's power train apart in the workshop and identify every chapter of this manual in it.
A working agricultural engineer uses this subject to specify and troubleshoot drives. You choose pulley diameters and belt sections to give the right thresher or pump speed from an engine or motor, check belt and chain speeds, set belt tension, and calculate PTO-driven implement speeds through gearboxes. You diagnose vibration by frequency: once per revolution usually points to unbalance, twice engine speed to secondary forces in a four-cylinder engine, and a resonance to a mounting that is too stiff or too soft. You read tractor test reports and know what PTO power, governed speed and droop mean, and you size flywheels or balance rotors for small stationary engines and processing machines.
With 25 years of experience the decisions are about systems and risk. You approve transmission layouts for new implements and small tractors, choosing between belts, chains, gears and hydraulics on cost, service life, safety and the skills of local mechanics. You set balancing and vibration acceptance limits for threshers, fans and mills using the international balance-grade and vibration-severity standards, and whole-body vibration limits for operators. You investigate failures such as broken PTO shafts, stripped final drives or cracked frames, separating design faults from misuse, and you write the test procedures that manufacturers and testing institutes follow.
At research level the subject becomes modelling and optimisation. Multibody dynamics models of tractors and implements predict ride vibration, hitch loads and rollover; mechanism synthesis designs linkages for transplanters, weeders and harvest robots to trace required paths; and gear and bearing dynamics models predict noise and life in powershift and continuously variable transmissions. Researchers study active and semi-active seat and cab suspensions, vibration-based condition monitoring of farm machines with machine learning, and energy recovery in hybrid-electric tractors. The core methods of this manual remain the checks used to validate every simulation.
This manual has no diagrams yet, so velocity and acceleration polygons, cam profile layouts, turning-moment diagrams and controlling-force curves are described in words only; students should draw them from a textbook. It does not cover the graphical synthesis of mechanisms (function, path and motion generation, Freudenstein's equation), the complete derivations of the Porter governor with unequal arms and friction or the Proell governor, gyroscopic couples (for example on turning vehicles and rotating implements), balancing of V, radial and opposed engines, multi-degree-of-freedom and continuous-system vibration, universal (Hooke's) joint speed fluctuation, or detailed tooth-strength and belt-selection design, which belongs to machine design. Values such as permissible belt stresses, flywheel rim speeds, pressure-angle limits and balance grades vary by material, standard and manufacturer and must be taken from the governing standard or supplier data for real design.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed by script; the working is in the explanation.
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