Agricultural Engineering · Engineering Sciences · also Design of Machine Elements, Farm Machinery Design
Sizing the parts of farm machines so they do not fail: shafts, keys, joints, springs, bearings, drives and overload devices.
In the exams: GATE AG Section 2 Machine Design (belts, spur gears, shear pins recur); ICAR AICE subject 39 Units 1–2; NCEES PE machinery component design
Machine design is the part of engineering that decides the size, shape and material of each part of a machine so that it does its job for its intended life without breaking, yielding, wearing out or deflecting too much. For an agricultural engineer the parts are familiar: the power take-off (PTO) shaft, the keys in a thresher drive, the bolts and welds of a cultivator, tine springs, rotavator bearings, chaff-cutter V-belts, gearbox gears and the shear bolt meant to break before anything expensive does.
Farm machines are a hard case for the designer. Loads from soil, crop and stones are variable, shock-like and poorly known in advance; machines work in dust and mud and must be cheap. Most farm machine failures are fatigue failures at a notch, weld toe or keyway, not simple overloads.
This manual follows a standard course in design of machine elements, from the design process and failure theories to overload devices and load measurement. Formulas are those of standard textbooks such as Shigley and Bhandari; where books or national standards differ, the text says so.
Worldwide the subject is examined mostly by calculation. GATE Agricultural Engineering repeatedly asks for belt tension and power, spur gear strength and shear pin size; the ICAR PhD entrance (subject 39) and the ASRB NET treat it at postgraduate depth; the NCEES PE Agricultural and Biological Engineering exam tests machinery component design using supplied references; and the Philippine ABE board lists machine design within power and machinery.
Design of a machine element follows a repeated loop. First the need is defined: what the part must do, the loads and speeds, the life, the environment and the cost limit. Then the designer chooses a mechanism and a layout, works out the forces on each element from statics and dynamics, selects a material, and sizes the element so that the stresses and deflections stay within safe limits. The design is then checked for manufacture, assembly, maintenance and safety, and usually refined several times. For farm machines the load analysis is the weakest step, because soil and crop forces vary with soil, moisture, speed and depth; designers rely on measured field loads, published draft data such as the ASABE machinery data standard, and service factors. Using standard shafts, keys, bolts, bearings, belts and chains lowers cost and keeps spares available locally. Preferred numbers based on the Renard geometric series (R5, R10, R20, R40, with step ratios of about 1.58, 1.25, 1.12 and 1.06) are used to choose standard sizes so that a range of products covers the need with few sizes.
Most machine elements are made of steel or cast iron. Plain carbon steels are named by carbon content: in the AISI/SAE system a steel such as 1045 is a plain carbon steel with about 0.45 per cent carbon; the Indian Standard system writes a similar steel as 45C8, where 45 is one hundred times the carbon percentage and 8 is ten times the manganese percentage. Low-carbon steels (below about 0.3 per cent carbon) are ductile, easy to weld and used for frames, brackets and sheet parts. Medium-carbon steels (about 0.3 to 0.6 per cent) are used for shafts, axles, gears and keys and can be hardened. High-carbon and alloy steels are used for springs, soil-engaging tools, blades and bearings where hardness and wear resistance matter; boron steels are widely used for hardened tillage tools. Grey cast iron is cheap, casts into complex shapes, damps vibration well and is much stronger in compression than in tension; it is brittle and is used for housings, pulleys and frames that carry mainly compressive or low tensile loads. Ductile (spheroidal graphite) iron has much better toughness. Bronze is used for sliding bearings and polymers for light bushes and guards. A ductile material shows large plastic strain before fracture (elongation above about 5 per cent is the usual dividing line); a brittle material fractures with little plastic strain.
The factor of safety (FoS, also written N or n) is the ratio of the failure stress to the working or allowable stress. For a ductile material under static load failure is taken as yielding, so allowable stress equals yield strength divided by FoS. For a brittle material, which does not yield, failure is fracture, so allowable stress equals ultimate strength divided by FoS. Under fluctuating load the failure stress is the endurance limit or fatigue strength, treated in the next chapter. The factor of safety covers uncertainty in loads, in material properties, in the stress analysis, in manufacture and in the consequences of failure. Typical textbook values run from about 1.5 to 2 for steel under well-known steady loads to 3 or 4 for shock, uncertainty or brittle materials; where a code or company rule sets a value, it governs. For farm machinery a service factor is often applied to the nominal load first (for example to account for shock from stones and roots) and a factor of safety applied to the material strength second; the two must not be confused or double-counted. A related idea is the margin of safety, FoS minus one.
A tensile test gives strength under a single stress, but real parts carry combined stresses. Theories of failure predict when a combined state reaches failure. The maximum principal stress theory (Rankine) says failure occurs when the largest principal stress reaches the uniaxial strength; it suits brittle materials. The maximum shear stress theory (Guest or Tresca) says a ductile material yields when the maximum shear stress reaches half the yield strength in tension; so shear yield strength is taken as 0.5 Syt. For plane stress with bending stress sigma and shear stress tau this gives an equivalent stress of the square root of (sigma squared plus 4 tau squared). The distortion energy theory (von Mises, also Huber–Hencky) says yielding occurs when the distortion strain energy per unit volume reaches that at yield in a tensile test; it gives shear yield strength of Syt divided by root 3, about 0.577 Syt, and an equivalent stress of the square root of (sigma squared plus 3 tau squared). Distortion energy agrees best with tests on ductile metals; maximum shear stress is slightly conservative and simple, which is why shaft codes use it. In terms of principal stresses, Tresca uses the largest difference between principal stresses, and von Mises uses the square root of (sigma1 squared minus sigma1 sigma2 plus sigma2 squared) in plane stress.
Any abrupt change of section, such as a shoulder, groove, keyway, hole or thread, raises the local stress above the nominal stress calculated by simple formulas. The theoretical stress concentration factor Kt is the ratio of the peak stress to the nominal stress; it depends only on geometry and is read from charts (Peterson's charts are the standard source). A sharper notch (smaller fillet radius relative to the section) gives a higher Kt. Under static load a ductile material yields locally at the notch and the load redistributes, so stress concentration is usually ignored for ductile parts under steady load; for brittle materials, and for all materials under fatigue loading, it must be included. Not all materials feel the full theoretical effect in fatigue. Notch sensitivity q, between 0 and 1, measures how much: the fatigue stress concentration factor is Kf = 1 + q (Kt - 1). When q is 0 the material is insensitive and Kf = 1; when q is 1 it is fully sensitive and Kf = Kt. High-strength steels are more notch-sensitive than soft steels, and cast iron has low notch sensitivity because its graphite flakes already act as internal notches. Designers reduce it with generous fillets, relief grooves, sled-runner keyway ends and by keeping holes and welds out of high-stress regions.
Fatigue is failure under repeated or fluctuating stress at a level well below the static strength. It starts as a small crack at a point of high local stress or at a surface flaw, grows a little with each cycle, and ends in sudden fracture of the remaining section. The fracture surface typically shows smooth beach marks around the origin and a rough final fracture zone. Fatigue behaviour is described by the S–N curve, stress amplitude against number of cycles to failure. Steels and some other ferrous metals show a knee, beyond about one to ten million cycles, below which failure does not occur however many cycles are applied; this stress is the endurance limit. Aluminium and most non-ferrous alloys have no true endurance limit, so a fatigue strength at a stated number of cycles is used. For wrought steel the rotating-beam endurance limit of a polished specimen is commonly estimated as about half the ultimate tensile strength, up to a cap (Shigley uses 700 MPa for steels with ultimate strength above about 1400 MPa). The endurance limit of a real part is lower than that of the polished specimen. It is corrected by modifying factors (Marin factors) for surface finish, size, type of load, temperature and reliability, and by the fatigue stress concentration factor, applied either to the stress or to the strength but not to both. Rough, forged or corroded surfaces reduce it greatly.
A fluctuating stress is described by its mean stress, sigma m = (sigma max + sigma min)/2, and its amplitude, sigma a = (sigma max - sigma min)/2. Fully reversed stress has zero mean; repeated stress runs from zero to a maximum. A tensile mean stress reduces the amplitude a part can safely carry. On a plot of amplitude against mean stress, the endurance limit Se sits on the amplitude axis. The Soderberg line joins Se to the yield strength Syt on the mean axis; the design equation is sigma a / Se + sigma m / Syt = 1 / N. The Goodman line joins Se to the ultimate strength Sut: sigma a / Se + sigma m / Sut = 1 / N. The Gerber parabola passes through the same end points but curves outward: N sigma a / Se + (N sigma m / Sut) squared = 1. Soderberg is the most conservative and guards against yielding; Goodman is widely used and conservative; Gerber fits test data for ductile materials best, so with Goodman and Gerber a separate first-cycle yield check (sigma a + sigma m below Syt / N) is made. In the stress-concentration form, the alternating stress is multiplied by Kf; practice varies on whether to apply Kf to the mean stress for ductile materials. When the same part sees blocks of different stress levels, the Palmgren–Miner rule sums the life fractions: the sum of n over N for each block reaching one indicates failure. Miner's rule is simple but only approximate.
A shaft is a rotating member, usually of circular section, that transmits power. An axle carries bending but no torque, and a spindle is a short shaft. Torque follows from power and speed: T = P / omega = 60 P / (2 pi n), with P in watts, n in revolutions per minute and T in newton metres. A PTO shaft transmitting 15 kW at 540 rpm carries about 265 N m. Bending comes from pulley and sprocket tensions, gear tooth forces and the weight of attached parts, found by drawing bending-moment diagrams in the horizontal and vertical planes and combining them as the square root of the sum of squares at each section. The critical section is where moment, torque and stress concentration combine worst. The shear stress from torque in a solid round shaft is tau = 16 T / (pi d cubed); the bending stress is sigma = 32 M / (pi d cubed). For a hollow shaft with ratio k of inner to outer diameter, both are divided by (1 - k to the fourth). A hollow shaft is stronger and stiffer than a solid shaft of the same mass, which is why hollow sections are used where weight matters.
For a ductile shaft under steady bending and torsion, the maximum shear stress theory gives tau max = (16 / pi d cubed) times the square root of (M squared + T squared). The quantity root (M squared + T squared) is called the equivalent twisting moment Te. The maximum principal stress theory gives an equivalent bending moment Me = one half of (M + root (M squared + T squared)), used with sigma = 32 Me / (pi d cubed). The distortion energy theory gives an equivalent moment of root (M squared + 0.75 T squared) used with the allowable normal stress. The ASME code for transmission shafting uses the maximum shear stress theory with shock and fatigue factors: tau max = (16 / pi d cubed) root ((Kb M) squared + (Kt T) squared), where Kb and Kt (in some books Km and Kt) are combined shock and fatigue factors for bending and torsion; for gradually applied loads on a rotating shaft typical values are about 1.5 and 1.0, rising for sudden or heavy shock loads. The same code sets the allowable shear stress for commercial steel shafting at 0.30 times yield strength or 0.18 times ultimate strength, whichever is less, and reduces it by 25 per cent where there is a keyway. For fluctuating loads a fatigue criterion such as Shigley's DE-Goodman is better. Diameters are then rounded up to a standard size.
A shaft can be strong enough and still fail in service because it twists or bends too much. The angle of twist is theta = T L / (G J), with J = pi d to the fourth over 32 for a solid shaft and G about 80 GPa for steel. Transmission shafts are commonly limited to about a quarter of a degree per metre of length, and line shafts to around 1 degree over a length of 20 diameters, but limits vary by application and textbook. Lateral deflection is limited at gears (to keep teeth in proper mesh) and at bearings (to keep the slope within what the bearing tolerates). A rotating shaft also has a critical (whirling) speed at which small unbalance produces large deflection; operating speeds are kept well away from it. On farm machines the main shafts are the tractor PTO stub and the implement driveline, the rotor shafts of rotavators, threshers and choppers, and the ground-wheel drive shafts of seed drills. Their loads fluctuate strongly, so they fail mostly by fatigue at shoulders, keyway ends, shear-pin cross-holes or welds; good fillets, no welding on shafts and correct alignment prevent most of these.
A key is a piece of steel inserted between shaft and hub to transmit torque. A sunk key sits half in a keyway in the shaft and half in the hub. For a rectangular sunk key the common textbook proportions are width about d/4 and thickness about two-thirds of the width; a square key has width equal to thickness, about d/4. Standard tables (for example ISO or national standards) give key sizes for ranges of shaft diameter and should be used in practice. A key fails either by shear across its width or by crushing on its side faces. In shear, the torque capacity is T = tau times w times l times d/2; in crushing, with half the key height bearing on each side, T = sigma c times (h/2) times l times d/2. A square key is equally strong in shear and crushing when the allowable crushing stress is twice the allowable shear stress. The required length is the larger of the two calculated lengths, and is often kept at least equal to the shaft diameter. Other types are the taper (gib-head) key, which also locks the hub axially; the feather key, which lets the hub slide along the shaft; the Woodruff key, a semicircular disc suited to tapered shafts and light drives; and the saddle key, which holds by friction only. Splined shafts, which are in effect many integral keys, carry more torque with less stress concentration and allow axial sliding; the tractor PTO stub and telescoping PTO drive shafts use splines. PTO dimensions and speeds are fixed by standards such as ISO 500 so that implements fit any tractor.
A coupling joins two shafts end to end. Rigid couplings need accurate alignment. A muff or sleeve coupling is a hollow cylinder keyed to both shaft ends; a split-muff (clamp) coupling is clamped by bolts and can be fitted without moving the shafts. A flange coupling has a flange keyed to each shaft, and the flanges are bolted together; torque passes through the bolts in shear (or by friction if the bolts are tightened enough). If there are n bolts of diameter db on a pitch circle of diameter D, and each has allowable shear stress tau, the torque capacity is T = n times (pi/4) db squared times tau times D/2. The hub, the flange at the hub junction and the key are also checked. A protected flange coupling has a rim covering the bolt heads. Flexible couplings accept small misalignment and damp shock. The bushed-pin flexible coupling carries torque through pins fitted with rubber bushes; the bushes are checked for bearing pressure and the pins for bending and shear. The Oldham coupling joins two parallel shafts with a small lateral offset through a floating disc with perpendicular tongues. The Hooke's joint (universal joint) connects shafts whose axes intersect at an angle. A single Hooke's joint does not give constant velocity: with a constant input speed and shaft angle alpha, the output speed varies twice per revolution between the input speed times cos alpha and the input speed divided by cos alpha. For a PTO drive shaft with two joints, uniform output is obtained when the angles at the two joints are equal and the yokes on the intermediate shaft lie in the same plane (in phase). Wrong phasing or unequal angles cause vibration, noise, and early failure of joints and gearboxes. Every PTO drive shaft must be shielded; unguarded shafts cause serious entanglement injuries.
A cotter joint connects two rods that carry axial load, tension or compression, where relative rotation is not needed and the joint must be quick to take apart. A cotter is a flat wedge-shaped bar passing through slots in the two parts. The taper of the cotter (commonly quoted as between 1 in 48 and 1 in 24) draws the parts together and keeps them tight; a small clearance is left in the slots. Common types are the socket-and-spigot joint, the sleeve-and-cotter joint and the gib-and-cotter joint used with rectangular straps. A socket-and-spigot cotter joint can fail in many ways, and design checks each: tension in the rod at its full diameter; tension in the spigot across the slot; tension in the socket across the slot; shear of the cotter, which is in double shear so the load is P = 2 b t tau with b the cotter width and t its thickness; crushing between the spigot and the cotter, P = sigma c d2 t, where d2 is the spigot diameter; crushing between the socket and cotter; and shear of the spigot end and the socket end behind the slot.
A knuckle joint connects two rods in tension where a small angular movement between them is needed. One rod ends in an eye and the other in a fork; a pin passes through both. Uses include tie rods, valve rods and implement linkages. The failure modes are: tension in the rod; shear of the pin, which is in double shear so that P = 2 times (pi/4) d squared times tau; crushing of the pin against the eye and against the fork; bending of the pin, treated as a beam loaded by the eye and supported by the fork prongs; and tension, shear and crushing of the eye and the fork. In practice the pin diameter comes out close to the rod diameter; standard textbooks give proportions such as an eye outer diameter of about twice the pin diameter. The pin is held by a collar and taper pin or a split pin, and wears, so it should be replaceable. Implement hitch and top-link pins are the same clevis-and-pin idea, loaded in double shear and bending; their securing clips matter as much as their size.
Threaded fasteners hold most farm machines together because they can be removed for service. ISO metric threads are named by nominal diameter and pitch, such as M16 by 2. The strength of a bolt in tension depends on the tensile stress area, which lies between the root (minor) area and the nominal area; standard tables give it (for example about 157 square millimetres for M16, 245 for M20 and 353 for M24 coarse threads). Bolt strength grades are marked on the head: ISO property class 8.8, for instance, means an ultimate strength of about 800 MPa and a yield of about 80 per cent of that. A bolt is tightened to an initial tension (preload) so that the parts stay clamped under service load. The torque needed is approximately T = K Fi d, where Fi is preload, d the nominal diameter and K a nut factor that depends on friction; about 0.2 is the usual value for dry steel threads and it is lower when the threads are lubricated. Torque control gives only a rough preload because friction varies. A bolt that is loaded in tension should be sized on the stress area with a suitable factor of safety, and if it also carries torsion from tightening, that is included by the chosen allowable stress or checked separately.
When an external tensile load P is applied to a preloaded joint, the bolt and the clamped parts act as springs in parallel. With bolt stiffness kb and member stiffness km, the joint constant is C = kb / (kb + km). The bolt load becomes Fb = Fi + C P, and the clamping force on the members becomes Fm = Fi - (1 - C) P. Because members are usually much stiffer than the bolt, C is small and the bolt feels only a fraction of the external load. The joint separates when Fm reaches zero, at P0 = Fi / (1 - C); after that the bolt carries the whole load. This is why preload is useful: it keeps gasketed joints sealed, stops separation, prevents loosening and, most importantly, reduces the fluctuation of bolt stress under a fluctuating load, which greatly improves fatigue life. A long, slender bolt (lower kb), or a bolt of uniform strength whose shank is turned down to about the thread root diameter, takes a smaller share of the fluctuating load and stores more energy under shock. Thread roots are severe stress raisers; rolled threads have better fatigue strength than cut threads.
Brackets often carry a load that does not pass through the centroid of the bolt group. For a load P in the plane of the joint at eccentricity e from the centroid of n equal bolts, each bolt carries a direct (primary) shear P/n in the direction of the load, and a secondary shear from the moment P e, proportional to its distance r from the centroid and acting perpendicular to that radius: F = P e r / (sum of r squared). The two are added as vectors, and the most heavily loaded bolt is the one where they are closest in direction. For a bracket where the load is out of the joint plane, the bolts see tension from the overturning moment, larger for bolts farther from the tilting edge. Farm machines vibrate, so bolts loosen. Locking methods include adequate preload (the most effective), lock nuts, castle nuts with split pins, prevailing-torque nuts with a deformed or nylon insert, spring washers and thread-locking adhesives.
Welding makes permanent joints and is the usual way of building farm machine frames, tool bars and brackets. The two basic welds are the butt weld, where plates are joined edge to edge, and the fillet weld, where plates meet at right angles or overlap and the weld has a roughly triangular section. The strength of a butt weld in tension is P = sigma t t l, with t the plate thickness (throat) and l the weld length; any reinforcement is ignored. For a fillet weld of leg size h, the throat, the smallest section through the weld, is h cos 45 degrees, or 0.707 h, and stresses are calculated on the throat area. A transverse fillet weld carries load perpendicular to the weld; textbooks commonly design it on tensile stress on the throat, so for a single weld P = 0.707 h l sigma t and for a double weld P = 1.414 h l sigma t. A parallel fillet weld carries load along its length in shear: P = 0.707 h l tau for one weld and 1.414 h l tau for two. Real stress in a fillet weld is complex, and codes (AWS in the United States, EN 1993 in Europe, national codes elsewhere) set the allowable stresses and design rules; allowable values depend on electrode and base metal, so they must be taken from the code in use.
When a welded bracket carries an eccentric load, the weld sees a direct shear, P divided by the throat area, and a secondary shear from the moment. For the moment the weld is treated as a line: its polar moment of inertia about the centroid of the weld group, times the throat, gives J, and the secondary stress at a point at distance r is P e r / J. The two stresses are added as vectors at the most stressed point. For a circular fillet weld around a shaft of diameter d under torque T, the shear stress on the throat is tau = 2 T / (pi t d squared), where t is the throat thickness. For a weld under bending, the bending stress is M divided by the section modulus of the throat area. When both bending and shear act, the resultant is taken by combining them as the square root of sigma squared plus 4 tau squared (maximum shear) or as the vector sum, according to the textbook or code used; the same method must be used consistently with that code's allowable stress.
Welded joints are the most frequent site of fatigue cracks on farm implements: cultivator tine brackets, plough beams, sprayer booms and trailer drawbars. Three things cause this. First, the weld toe is a sharp geometric notch. Second, welding leaves tensile residual stresses close to yield in and near the weld. Third, the heat-affected zone next to the weld may be harder and more brittle than the parent metal, and defects such as undercut, porosity, slag inclusions and lack of fusion act as ready-made cracks. Textbooks give fatigue stress concentration factors for welds that rise from a reinforced butt weld, through the toe of a transverse fillet weld, to the end of a parallel fillet weld, which is among the worst. Fatigue design codes classify weld details into categories with their own S–N curves, and for welded steel details the stress range, not the mean stress, largely controls life because residual stresses are already high. Good practice: put welds in low-stress areas, avoid intermittent welds under fatigue, grind toes flush where fatigue matters, and never weld-repair a cracked shaft or axle without a proper procedure.
Springs store energy and control force or motion. On farm machines they hold tines on spring-tine cultivators, reset trip ploughs, press seed-drill furrow openers, tension belts and chains, and load clutches. The helical compression spring is the most common. Its main dimensions are the wire diameter d, the mean coil diameter D, and the number of active coils N. The spring index C = D/d is usually kept between about 4 and 12: small indices are hard to coil and have high curvature stress, large ones tend to tangle and buckle. The wire is loaded mainly in torsion, with shear stress tau = K times 8 P D / (pi d cubed). The factor K accounts for direct shear and for the curvature of the wire. The Wahl factor, K = (4C - 1)/(4C - 4) + 0.615/C, includes both; the shear stress correction factor Ks = 1 + 0.5/C includes only direct shear and is used by some texts for static loads, with Wahl reserved for fatigue. Spring wire (cold-drawn or oil-tempered carbon steel, chromium-vanadium or chromium-silicon alloy, stainless steel) gets stronger as its diameter falls, so allowable stress is read for the wire size.
The deflection of a helical spring is delta = 8 P D cubed N / (G d to the fourth), with G the modulus of rigidity (about 80 GPa for spring steel). The stiffness (rate) is k = P/delta = G d to the fourth / (8 D cubed N). Springs in series share the same force, so 1/k = 1/k1 + 1/k2; in parallel they share the deflection, so k = k1 + k2. Energy stored is one half of P delta. Active coils are fewer than total coils because end coils do not deflect. For plain ends total coils equal active coils; for squared ends add two; for squared and ground ends also add two, and the solid length is about the total number of coils times d. The free length is the solid length plus the maximum working deflection plus a clearance between coils (often about 15 per cent of the maximum deflection). Long, slender compression springs buckle; when free length exceeds about four times the mean diameter they are usually guided on a rod or in a tube. Springs under fluctuating load, such as tine springs, are designed on the Goodman or Soderberg basis in shear, using the torsional endurance limit of the wire; shot peening raises the fatigue strength. Other types: extension springs (hooks are often the weak point), torsion springs (wire in bending) for levers and pickup tines, leaf springs for trailer axles and seats, disc (Belleville) springs for high force in small space, and elastomer mounts for vibration isolation.
Rolling bearings carry load through balls or rollers between inner and outer rings. Deep-groove ball bearings take radial and moderate axial load and are the commonest type on farm machines; angular-contact and tapered roller bearings take combined loads and are used in pairs in wheel hubs and gearboxes; cylindrical roller bearings take heavy radial loads; spherical roller and self-aligning ball bearings tolerate shaft misalignment, useful on long implement shafts. Insert bearings in pressed-steel or cast housings with spherical outer rings and set-screw or eccentric-collar locking are widely used on implements because they are easy to align. Rolling bearings fail in the end by surface fatigue (spalling). Because fatigue life scatters widely, life is stated statistically. The basic rating life L10 is the life in millions of revolutions that 90 per cent of a large group of identical bearings will reach or exceed. The basic dynamic load rating C is the load for which L10 is one million revolutions. The life equation, standardised in ISO 281, is L10 = (C/P) to the power p, with p = 3 for ball bearings and 10/3 for roller bearings; life in hours is L10 times one million divided by (60 n). So doubling the load cuts ball-bearing life to one-eighth.
A bearing that carries both radial load Fr and axial load Fa is selected on an equivalent dynamic load P = X V Fr + Y Fa, where X and Y are radial and thrust factors from the bearing tables (they depend on the ratio of axial load to static rating and on bearing type) and V is a rotation factor (1 when the inner ring rotates, larger in some older texts when the outer ring rotates). A load application or service factor is applied for shock. Selection then follows: choose the required life in hours from the application, convert to millions of revolutions, find the required C from C = P times L10 to the power 1/p, and choose the smallest bearing from the catalogue with a rating at least that large that fits the shaft. The static load rating C0 is checked when the bearing is stationary or turns very slowly under load, to avoid permanent dents (brinelling). In farm service dust and water are the main enemies, so seals and lubrication often matter more than the rating.
A sliding (journal) bearing supports a shaft in a sleeve or bush. Three lubrication regimes occur. In hydrodynamic (thick-film) lubrication the rotating journal drags oil into a converging wedge and builds pressure that separates the surfaces fully; friction is low and there is no wear. This needs enough speed and viscosity for the load. In boundary lubrication the surfaces touch through thin adsorbed films, as at start-up and at very low speed; friction and wear are higher. Mixed lubrication lies between. The behaviour is described by the bearing characteristic number mu N / p (viscosity times speed over unit load, where p = W / (l d) is the load on the projected area). Friction is lowest at a certain value of this number; below it the film breaks down. Petroff's equation for a lightly loaded, concentric journal gives the friction coefficient f = 2 pi squared (mu N / p) (r / c), with N in revolutions per second and c the radial clearance. The Sommerfeld number, S = (r / c) squared times mu N / p, is the main dimensionless parameter of hydrodynamic design charts such as those of Raimondi and Boyd. Farm machines use many plain bushes (bronze, sintered oil-impregnated bronze, polymers) in the boundary regime, sized on the product of pressure and velocity (the pv value) and on maximum pressure and speed, using limits supplied for each material.
Belt drives are the commonest power transmission on stationary farm machines: threshers, chaff cutters, pumps, cleaners and grinders, and on combines. For a flat belt the ratio of tight-side tension T1 to slack-side tension T2 at the point of slipping is T1/T2 = e to the power (mu theta), where mu is the coefficient of friction and theta the angle of contact in radians on the pulley that governs (normally the smaller one, unless the materials differ). For a V-belt in a groove of included angle 2 beta, the wedge action increases the effective friction, so T1/T2 = e to the power (mu theta / sin beta). Power transmitted is P = (T1 - T2) v, with v the belt speed. At speed a belt also carries centrifugal tension Tc = m v squared, where m is the mass per metre of length; Tc adds equally to both sides and does not help transmit power. With a maximum allowable tension Tmax, the tension available for power is Tmax - Tc, and power is greatest when Tc = Tmax / 3, at a speed v = square root of (Tmax / 3m). The belt also creeps on the pulleys because of elastic stretching, and slips slightly, so the actual speed ratio is a little less than the pulley diameter ratio, typically by about 1 to 3 per cent.
For an open belt between pulleys of diameters D and d at centre distance C, the angle of contact on the small pulley is theta = 180 degrees minus 2 arcsin ((D - d) / (2C)), and on the large pulley 180 degrees plus the same amount. The belt length is L = 2C + pi (D + d) / 2 + (D - d) squared / (4C). In practice V-belts are selected from standard power-rating tables: the designer applies a service factor for the type of driven machine and hours of use, chooses the belt section from the design power and speed, finds the power per belt for the small pulley diameter and speed, corrects it for the angle of contact and belt length, and divides the design power by the corrected rating to get the number of belts. Classical V-belt sections are lettered A to E, and each has a minimum pulley diameter because small pulleys bend the belt severely and shorten its life.
Roller chains give a positive drive without slip, carry high loads at low speed and tolerate dirt reasonably, which is why they drive seed-metering mechanisms, ground wheels, conveyors, feeder houses and combine headers. A chain is described by its pitch p, the distance between roller centres; standard pitches are multiples of one-eighth or one-sixteenth of an inch (12.7 mm, 15.875 mm, 19.05 mm, 25.4 mm and so on). The pitch diameter of a sprocket with z teeth is D = p / sin (180 degrees / z). Chain speed is v = z p n / 60,000 in metres per second with p in millimetres and n in rpm. Because the chain wraps the sprocket as a polygon, its speed rises and falls as each link engages: this chordal action causes speed variation and impact, which falls as the number of teeth rises. Textbooks recommend a small sprocket of at least about 17 teeth for smooth running at moderate speed, fewer only at low speed. Chain length is counted in pitches: Ln = 2C/p + (z1 + z2)/2 + ((z2 - z1)/(2 pi)) squared times p/C, rounded to an even number to avoid an offset link. An even number of links with an odd number of teeth spreads wear evenly. Chains are selected from power-rating tables with service factors; their life is governed by pin and bush wear, which lengthens the chain, so lubrication, alignment and correct slack matter as much as rating.
Spur gears have straight teeth parallel to the axis and transmit power between parallel shafts. The module m is the pitch circle diameter divided by the number of teeth, d = m z, in millimetres; in the inch system the diametral pitch is used instead. The pressure angle, most often 20 degrees today (14.5 degrees in older designs), is the angle between the line of action and the common tangent to the pitch circles. With full-depth 20 degree involute teeth, a pinion needs at least about 18 teeth (17.1 in theory) to avoid interference and undercutting when cut by a rack cutter. The force between teeth acts along the line of action. Its tangential component, which transmits torque, is Pt = 2 T / d = P / v, where v is the pitch-line velocity pi d n / 60,000 in metres per second. The radial (separating) component is Pr = Pt tan phi, and the resultant is Pt / cos phi. These forces load the gear shaft and bearings. The face width b is normally between about 9.5 m and 12.5 m, with 10 m a common textbook choice; wider faces load unevenly if the shafts deflect.
Lewis treated a gear tooth as a cantilever beam of uniform strength loaded at its tip by the whole tangential force, assuming one pair of teeth carries the load. The beam strength of the tooth is Sb = m b sigma b Y, where sigma b is the allowable bending stress and Y is the Lewis form factor, which depends on the number of teeth and the tooth form. For 20 degree full-depth involute teeth, Y is approximately 0.484 - 2.87/z (the older form based on circular pitch, y = 0.154 - 0.912/z, differs by the factor pi). More teeth give a larger Y and a stronger tooth. The weaker of two meshing gears has the smaller product sigma b Y; with the same material the pinion is weaker because it has fewer teeth. For design, the beam strength must exceed the effective load, which is the tangential force multiplied by a service factor Cs for shock and divided by a velocity factor Cv. Barth's velocity factors are commonly used: Cv = 3 / (3 + v) for ordinary cut gears up to about 10 metres per second, 6 / (6 + v) for carefully cut gears up to about 20 metres per second, and 5.6 / (5.6 + square root of v) for precision hobbed or ground gears at higher speed.
Barth's factor is a crude allowance for dynamic effects. Buckingham gave a more rational estimate of the extra dynamic load caused by tooth errors and by the masses of the gears. In SI units his equation is Pd = 21 v (b C + Pt) / (21 v + square root of (b C + Pt)), where Pd is the incremental dynamic load in newtons, v the pitch-line velocity in metres per second, b the face width in millimetres, Pt the tangential force in newtons and C a deformation factor in newtons per millimetre that depends on the tooth error and the elastic moduli of the gear materials. The effective load is then Peff = Cs Pt + Pd, and the beam strength must exceed Peff times the factor of safety. Gear teeth also fail by pitting, a surface fatigue caused by repeated contact stress. Buckingham's wear strength equation is Sw = b Q dp K, where dp is the pinion pitch diameter, Q = 2 zg / (zg + zp) for external gears and K is a load-stress factor that depends on the surface hardness of the two gears. The wear strength must also exceed the effective load; surface hardening raises K. Modern practice uses the AGMA or ISO 6336 methods, which add factors for load distribution, size, surface and reliability. Tractor gears are case-hardened alloy steel, checked for both bending and pitting.
Farm machines meet sudden loads that no reasonable design can carry: a plough share hits a buried rock, a baler plunger jams on a wad of crop, a rotavator blade catches a root, a combine header gathers a stone. The peak torque can be several times the working torque. Designing every part for such peaks would make the machine heavy and costly, so a deliberate weak link is built in. The overload device must pass the normal working load, including its usual peaks, without operating, and must operate before the load reaches the level that would damage the drive, the implement or the tractor. The main types are shear bolts and shear pins, friction slip clutches, ratchet or cam-type overload clutches that disengage and re-engage with a noise that warns the operator, and on soil-engaging tools spring-trip and hydraulic-reset standards. For tillage tools in stony soil, automatic-reset devices save time and bolts; in soil with few obstructions, shear bolts are cheaper. A free-wheel (overrunning) clutch on the PTO line protects the tractor from the inertia of heavy rotating implements such as mowers.
A shear pin or shear bolt is designed to break, so it is sized with the ultimate shear strength of its material, not with an allowable stress. If a shear bolt joins a hub to a flange at a radius R from the axis and the device must break at a torque Ts, the shear force on the bolt is Ts/R, and for a single shear plane the diameter follows from Ts/R = tau u times pi d squared / 4. If a pin passes diametrically through a shaft of diameter D and a hub, it is sheared at two planes, each at radius D/2, so Ts = 2 times (pi d squared / 4) times tau u times D/2. The breaking torque is chosen above the highest normal working torque, with a margin for normal peaks, and below the torque that would damage the weakest protected component. The ultimate shear strength of steel is often taken as roughly 0.6 to 0.8 of its ultimate tensile strength, but the scatter is large, so pins are best made from a controlled material, often with a notch or reduced neck, and replacements must be of the same specification. Replacing a shear bolt with a stronger bolt or a rod removes the protection and breaks gearboxes. Shear pins can also fail in fatigue after many load peaks below the design breaking torque; a fitted, tight pin in hardened bushes reduces this. A friction slip clutch holds torque by friction between discs clamped by springs, and slips when the torque exceeds a set value. The torque capacity of a clutch with n friction surfaces, axial spring force W and coefficient of friction mu is T = n mu W Rm. Under the uniform-wear assumption, which applies to worn clutches and is slightly conservative, Rm = (R1 + R2)/2. Under the uniform-pressure assumption, for new clutches, Rm = (2/3)(R1 cubed - R2 cubed)/(R1 squared - R2 squared). The spring force sets the slip torque, so springs are adjusted to a specified length. Slip clutches reset themselves but heat up when they slip, and they need regular attention: friction discs can rust or glaze in storage and then transmit far more torque than intended. Most operator manuals therefore ask for the clutch to be slipped (freed) before the season.
Design of farm machines rests on measured loads. The basic sensor is the bonded electrical-resistance strain gauge, a foil grid glued to the part. Its resistance changes in proportion to strain: delta R / R = GF times epsilon, where the gauge factor GF is about 2 for common metal-foil gauges. Because the change is tiny, gauges are wired in a Wheatstone bridge. With one active gauge (quarter bridge) and excitation voltage V, the output is about V times GF times epsilon / 4 for small strains; two active gauges with strains of the same sign in opposite arms (half bridge) double this, and four active gauges (full bridge) give about V GF epsilon. Adjacent arms subtract, so a dummy or active gauge in an adjacent arm cancels temperature effects and unwanted strains. Where principal directions are unknown, a three-gauge rosette is used. Torque on a shaft is measured with gauges at 45 degrees to the axis, where the principal strains in pure torsion are plus and minus tau / (2G); a full bridge of four such gauges cancels bending and axial strains. Signals from rotating shafts leave through slip rings or wireless telemetry; commercial torque transducers and PTO dynamometers do the same job. Forces are measured with load cells of beam, column, shear-web or ring type; the extended octagonal ring transducer is widely used in tillage research to measure draft and vertical forces on a tool. Three-point-hitch dynamometers and instrumented drawbars measure implement draft on tractors. Recorded load spectra feed fatigue design and the setting of overload devices.
A student uses this subject to size the parts of a design project: work out the torque in a thresher drive, choose a shaft diameter from combined bending and torsion, pick a key and check it, select a V-belt set and a bearing from catalogues, and design a shear bolt for the drive. In exams the same steps appear as short calculations, so the student should practise converting power and speed to torque, getting units right (N, mm and MPa together) and knowing which failure theory and which fatigue line each question expects.
An engineer five years into a machinery firm or a testing station meets machine design as failures. A cracked tine bracket, a sheared shaft or a bearing that seizes after one season must be traced to its cause: fatigue at a weld toe, a keyway ending at a shoulder, a mis-phased PTO shaft, a shear bolt replaced by a stronger bolt, or dust entering a seal. The practitioner redesigns the detail, specifies the correct fastener grade and tightening torque, selects bearings and belts from catalogues with the right service factor, and writes the maintenance instruction that keeps overload devices working.
A senior engineer with twenty-five years of experience sets the design rules of a product line: the service factors, load spectra and factors of safety for each class of implement, the standard parts list, and the validation tests. They decide when a field test or a strain-gauge survey is needed before release, balance weight and cost against durability, see that products meet the safety and dimensional standards of each market (PTO, hitch and guarding standards), and review warranty data to find the few details that cause most failures.
A researcher with a doctorate and twenty-five years of experience works on the loads and life of machines: measuring soil and crop loads with instrumented tools and extended octagonal ring transducers, building load spectra and cumulative damage models, testing new materials for wear-resistant soil tools, applying finite element analysis and fatigue codes to welded frames, and developing lighter structures or sensors for condition monitoring. Such work supplies the design data, test codes and standards that practitioners use, and trains the next generation to question assumed loads and factors of safety rather than copy them.
This manual does not reproduce design data tables (Peterson stress concentration charts, Marin factor curves, bearing X and Y factors, V-belt and chain power ratings, AGMA or ISO 6336 gear factors, weld code allowable stresses); questions give those values in the stem. It does not cover helical, bevel or worm gears, brakes, power screws, flywheels (see theory of machines), finite element analysis, or US customary worked examples. Knuckle and cotter joint empirical proportions differ between textbooks and are given only as examples. Allowable stresses, service factors and factors of safety vary by code, country and company and must be taken from the governing standard.
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.
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