Propeller shaft, universal joints and slip joints

Why it matters

In a front-engine, rear-wheel-drive or four-wheel-drive vehicle, the gearbox output and the rear axle are at different heights, are not exactly in line, and move relative to each other as the suspension works. The propeller shaft, its universal joints and its slip joint must carry full first-gear torque across that gap at speeds of several thousand rev/min without vibrating. Speed fluctuation of Hooke's joints and whirling of long shafts are classic exam calculations and real causes of driveline vibration.

Key ideas

• Propeller shaft (drive shaft). A tube, usually of seamless or welded steel, sometimes aluminium alloy or carbon-fibre composite, joining the gearbox (or transfer case) output to the final drive. A hollow section is used because the outer material carries most of the torsional and bending stress, so a tube is much stiffer and stronger per kilogram than a solid bar. The shaft is dynamically balanced (small balance weights welded on).
• Two-piece shafts. On long-wheelbase vehicles a single tube would have too low a whirling (critical) speed, so the shaft is split into two or more sections with a rubber-mounted centre bearing and an extra universal joint.
• Hooke's (cardan, cross) universal joint. Two yokes at right angles joined by a cross (spider) whose four trunnions run in needle-roller bearings. It transmits torque between shafts meeting at an angle α. It is not a constant-velocity joint: when the input turns at constant speed, the output speed fluctuates twice per revolution between ω·cos α and ω / cos α. The fluctuation grows rapidly with angle, causing torsional vibration, noise and bearing wear; working angles are kept small (typically a few degrees).
• Cancelling the fluctuation. With two Hooke's joints, the speed variation is cancelled at the output if (1) the two joint angles are equal and (2) the intermediate shaft's two yokes lie in the same plane ("in phase"); the intermediate shaft itself still fluctuates. This is why driveline angles are laid out carefully in a Hotchkiss drive.
• Constant-velocity (CV) joints. The balls or rollers of a CV joint lie in the plane bisecting the angle between the shafts, so input and output speeds are always equal. Rzeppa (ball-type) joints handle large angles (around 45°) and are used at the wheel end of front-wheel-drive half-shafts; tripod and double-offset joints allow plunge and are used at the inboard end. Double-cardan joints (two Hooke's joints in one housing with a centring device) are used on some 4×4 propeller shafts.
• Flexible (rubber) couplings ("doughnuts", Giubo) allow small angles and damp vibration without lubrication.
• Slip joint. A splined sleeve (yoke) sliding on a splined shaft, normally at the gearbox end, lets the shaft length change as the axle moves up and down (the axle swings on its own arc while the shaft swings on another) and as the drivetrain moves on its mountings. It must slide freely under torque; a dust boot and grease keep it so. Without it, suspension movement would push the joints and bearings axially.
• Whirling (critical speed). A rotating shaft has a natural bending frequency. If its speed reaches that frequency, small unbalance makes it whirl violently. The maximum propeller-shaft speed (top vehicle speed × final-drive ratio) must stay well below the first critical speed – a common design margin is 20–50%; take your design code's value. Critical speed rises with diameter and falls with the square of length, which is why long shafts are split.

Formulas

ω₂ / ω₁ = cos α / (1 − sin²α · cos²θ)

• Hooke's joint velocity ratio. ω₁, ω₂: input and output angular speeds (rad/s); α: angle between the shafts; θ: angle turned by the input yoke from the position where it lies in the plane of the two shafts.

ω₂,max = ω₁ / cos α, ω₂,min = ω₁ · cos α, fluctuation (ω₂,max − ω₂,min)/ω₁ = sin α · tan α

• Maximum at θ = 0° and 180°, minimum at 90° and 270°.

τ = 16 · T · D / [π · (D⁴ − d⁴)]

• τ: maximum torsional shear stress in a hollow shaft (Pa); T: torque (N·m); D, d: outer and inner diameters (m).

ω_c = (π / L)² · √(E · I / m'), N_c = 60 · ω_c / (2π)

• First whirling speed of a uniform shaft simply supported at its ends. ω_c: critical speed (rad/s); N_c (rev/min); L: length between joints (m); E: Young's modulus (Pa); I = π (D⁴ − d⁴)/64: second moment of area (m⁴); m' = ρ · π (D² − d²)/4: mass per unit length (kg/m).

N_p = (v / r) · (60 / 2π) · i_f

• Propeller-shaft speed (rev/min) at vehicle speed v (m/s), wheel radius r (m) and final-drive ratio i_f, in direct top gear.

P = T · ω

• Power transmitted (W), ω in rad/s.

Worked examples

Example 1 (standard). A Hooke's joint connects two shafts at 15°. The driving shaft turns at a steady 2000 rev/min. Find the maximum and minimum speeds of the driven shaft and the total fluctuation as a percentage of the input speed.

1. cos 15° = 0.9659.
2. N₂,max = 2000 / 0.9659 = 2070.6 rev/min (at θ = 0°, 180°).
3. N₂,min = 2000 × 0.9659 = 1931.9 rev/min (at θ = 90°, 270°).
4. Fluctuation = (2070.6 − 1931.9)/2000 = 0.0694 = 6.94%; check: sin 15° · tan 15° = 0.2588 × 0.2679 = 0.0694. ✓

Answer: N₂ varies between about 1932 and 2071 rev/min, a 6.9% fluctuation, twice per revolution.

Example 2 (GATE level). A one-piece steel propeller shaft (E = 210 GPa, ρ = 7850 kg/m³) is a tube of 76 mm outer and 70 mm inner diameter, 1.40 m long between joints. It must carry 1500 N·m (first-gear torque) and the car's maximum speed is 180 km/h with wheel radius 0.30 m and final drive 3.9 (direct top). Check the shear stress and the critical speed margin.

1. Polar moment: J = π (0.076⁴ − 0.070⁴)/32 = 9.181 × 10⁻⁷ m⁴.
2. τ = T · (D/2) / J = 1500 × 0.038 / 9.181 × 10⁻⁷ = 62.1 MPa – acceptable for a steel tube.
3. I = J/2 = 4.591 × 10⁻⁷ m⁴; m' = 7850 × π (0.076² − 0.070²)/4 = 5.40 kg/m.
4. ω_c = (π/1.40)² × √(210 × 10⁹ × 4.591 × 10⁻⁷ / 5.40) = 5.035 × 133.6 = 672.8 rad/s, i.e. N_c = 6424 rev/min.
5. Maximum shaft speed: N_p = (50/0.30) × (60/2π) × 3.9 = 6207 rev/min.
6. Margin: 6424 / 6207 = 1.035, only 3.5% above the operating speed.

Answer: τ ≈ 62 MPa (adequate), but N_c ≈ 6420 rev/min is only about 3.5% above the maximum shaft speed of about 6210 rev/min – unacceptable. Use a larger-diameter tube, a lighter, stiffer material (aluminium or composite) or a two-piece shaft with a centre bearing.

Common mistakes

• Assuming a single Hooke's joint gives constant velocity; only the average output speed equals the input speed.
• Saying two Hooke's joints always cancel; the angles must be equal and the intermediate yokes in phase.
• Using the solid-shaft formula 16T/(πD³) for a tube, or forgetting the fourth powers in D⁴ − d⁴.
• Mixing rad/s and rev/min in critical-speed checks.
• Thinking the slip joint lets the shaft bend; it only allows length change. Angles are handled by the universal joints.
• Thinking a heavier, thicker-walled tube always raises critical speed; for a tube of given outer diameter, I/m' changes only slowly with wall thickness, while length has a squared effect.

For GATE ME

The favourite question is the Hooke's joint: maximum and minimum output speed, fluctuation, angular positions of maximum velocity, and the conditions for a double Hooke's joint to give uniform output. Torsional stress in hollow shafts and whirling speed of shafts are also tested (these overlap with strength of materials and vibrations). Practise the velocity-ratio formula at different θ and the critical-speed formula with consistent units.

Quick check

1. At what input angles is the output of a Hooke's joint fastest?
2. Shaft angle 20°, input 1000 rev/min: what are the maximum and minimum output speeds?
3. Give the two conditions for a double Hooke's joint to give uniform output.
4. Why are long propeller shafts split into two pieces?
5. What does the slip joint accommodate?

Answers: 1. θ = 0° and 180°. 2. About 1064 and 940 rev/min. 3. Equal angles at both joints, and the intermediate shaft's yokes in the same plane. 4. To raise the whirling speed above the maximum shaft speed. 5. Changes in shaft length as the axle moves with the suspension and the drivetrain moves on its mounts.