Tractors, Power Tillers and Power Transmission

Tractor weight transfer and traction

Traction on a two-wheel drive (2WD) tractor depends on the weight carried by the driving rear axle, not on the total weight. The normal load on the driven wheels governs the available thrust, so a tractor whose weight sits mostly on the front axle can have poor traction even when its total weight is large.

On cohesionless soil such as pure sand, the coefficient of traction is taken as \tan\phi, where \phi is the angle of internal friction. The maximum thrust is then F_{max} = \mu W_r, with W_r the load on the driven axle. The coefficient can also be back-calculated as \mu = F_{max}/W_r from a measured thrust. Because the thrust is linear in W_r, added ballast on the driven wheels raises the thrust by \mu\,\Delta W, provided weight transfer is neglected. The percentage change in thrust is simply the percentage change in driven load in this case.

When the tractor pulls an implement, the drawbar force acts at a height above the ground and shifts load between the axles. Taking moments about the front wheel contact point, the load transferred onto the rear axle is \Delta W_r = P\,d/L, where P is the pull, d is the perpendicular distance from the front contact point to the line of pull, and L is the wheelbase. The static axle loads do not enter this moment balance. The traction calculation must therefore use the dynamic rear load, which is the static rear load plus the transferred load.

Longitudinal stability requires the dynamic front-axle load to stay above about 20 percent of the total weight, the usual rule of thumb. Below this value the tractor tends to rear up under pull. The remedy is ballast on the front axle, not on the rear axle.

Gross traction is the tractive effort at the wheel periphery, GT = T/r, where T is the axle torque and r is the rolling radius. Net traction NT is gross traction minus rolling resistance, and the coefficient of net traction is NT/W_d, with W_d the dynamic load on the driven wheels. Slip reduces the actual forward speed to (1-s) of the theoretical speed. Drawbar power is NT times actual speed, and axle power is GT times theoretical speed. Tractive efficiency is their ratio, \eta_t = (1-s)\,NT/GT. This efficiency falls as slip grows.

• Dynamic weight — the load on an axle while the tractor is working, including weight transfer. e.g. The rear dynamic load is the static rear load plus the transfer caused by the pull.
• Coefficient of net traction — net traction divided by the driven dynamic load. e.g. A net traction of 9 kN on 30 kN of driven load gives 0.30.

Rolling resistance of rigid wheel

A rigid wheel on a hard, level surface loses little energy to deformation, and its rolling resistance is written as R = f_r W, where f_r is the coefficient of rolling resistance and W is the normal load. On an incline of angle \theta, the resistance has two parts. The rolling term acts normal to the surface and is reduced only by the small factor \cos\theta. The gravity component W\sin\theta acts along the slope and opposes motion. For practical slopes the rolling term changes very little, so the extra resistance on an upslope is essentially W\sin\theta. The rolling resistance on the incline is therefore increased, not decreased, relative to the level case.

For a rigid wheel on a non-deformable surface with no slip, the forward speed equals the wheel circumference times the wheel rotational speed, V = \pi D N_w, with consistent units. The total speed reduction from engine to wheel is the product of all gear ratios in series, such as the transmission, differential and final drive. Engine speed divided by wheel speed therefore gives the product of the stage ratios, and the unknown ratio is found by division.

• Coefficient of rolling resistance — the ratio of rolling resistance to normal load, f_r = R/W. e.g. A 2 kN load with 0.05 coefficient gives 0.1 kN of resistance on level ground.
• Grade resistance — the gravity component W\sin\theta along a slope. e.g. On a 10 degree slope, W\sin 10^\circ is added to the rolling resistance.

Tyre axis system (SAE)

In the SAE tyre axis system, the x-axis lies in the wheel plane along the direction of travel, the y-axis is the wheel spin axis, and the z-axis is vertical through the tyre contact patch. Each moment is named after the axis it acts about. The moment about the x-axis is the overturning moment, the moment about the y-axis is the rolling resistance moment, and the moment about the z-axis is the aligning torque. The aligning torque tends to return a steered wheel to its straight-ahead orientation.

• Aligning torque — the self-aligning moment about the vertical axis of the tyre. e.g. A steered tyre with a positive aligning torque tends to turn back toward straight-ahead travel.

Three-point hitch mast position

A tractor three-point hitch carries an implement through two lower links and a central top link. The top link is called the mast. For a tractor-mounted mouldboard plough, the mast is kept nearly vertical during ploughing. This setting keeps the plough beam and the draft line in the correct geometry so that the plough works at the intended depth. Inclining the mast appreciably, or aligning it with the direction of travel or with the rear axle, is not the correct ploughing setting.

• Top link (mast) — the upper member of the three-point hitch that controls the implement's inclination. e.g. Lengthening the top link tilts the implement's upper attachment point.

Tractor drawbar power and transmission efficiency

The thrust developed at the ground by the driving wheels equals the wheel torque divided by the rolling radius, F = T_w/r. The wheel torque is the engine torque multiplied by the overall gear ratio and by the transmission efficiency, T_w = T_e\, i\, \eta. Combining these gives F = T_e\, i\, \eta / r. Because the transmission efficiency is below unity, the ground thrust is always lower than the ideal value obtained with no losses. The rolling radius and the efficiency enter the expression in the same way, so they can cancel when they have the same numerical value.

• Overall gear ratio — engine speed divided by wheel speed, equal to the product of the stage ratios. e.g. Two stages of 3:1 and 4:1 give an overall ratio of 12:1.

Double-acting hydraulic cylinder speed and force

In a double-acting cylinder, extension forces oil onto the full piston area, A_p = \pi D^2/4. Retraction forces oil onto the annular area on the rod side, A_r = \frac{\pi}{4}(D^2 - d^2), because the rod occupies part of the face. For the same pump flow Q, the piston velocity is v = Q/A. The smaller annular area therefore gives a higher piston velocity on retraction than on extension. The force developed on retraction is smaller, since it is the pressure times the smaller annular area.