Design of propeller shaft and rear axle

Why it matters

In a front-engine, rear-wheel-drive vehicle the propeller shaft carries gearbox output torque to the final drive, and the rear axle carries it to the wheels while supporting the vehicle's weight. The propeller shaft is limited as much by whirling (critical speed) as by strength, and the axle shaft's loading depends on whether it is semi-floating or full-floating. These are standard automotive design calculations.

Key ideas

Propeller shaft

• A thin-walled steel tube (sometimes aluminium or carbon-fibre composite), with a universal joint at each end and a sliding spline (slip joint) to allow for the change in length as the axle moves on its springs.
• Design torque is the largest torque it can see: maximum engine torque × first-gear ratio (or, if smaller, the torque that makes the rear wheels slip). Torque is highest in first gear, but speed is highest in top gear.
• Strength: torsional shear in a hollow shaft, τ = 16T·d_o / (π·(d_o⁴ − d_i⁴)). A hollow tube gives much more torsional strength and stiffness per kilogram than a solid bar.
• Critical (whirling) speed: a long rotating shaft resonates in bending when its speed equals its first natural frequency. For a uniform shaft simply supported at its ends, ω_c = (π/L)²·√(E·I/(ρ·A)). The maximum operating speed should stay well below it (a margin of about 1.2–1.5 is common; take the rule from your data book). Critical speed rises with tube diameter and falls with the square of length, so long vehicles use a two-piece shaft with a centre bearing, or larger-diameter or composite tubes.
• Universal joint: a single Hooke's joint at angle α makes the output speed fluctuate twice per revolution between ω·cos α and ω/cos α. Using two joints with equal angles and correctly phased yokes cancels the fluctuation.
• Dynamic balance is essential: an unbalanced tube vibrates and loads the joints and gearbox bearings.

Rear axle

• The final drive (bevel or hypoid pinion and crown wheel) multiplies torque by the final drive ratio, and the differential lets the two wheels turn at different speeds in a turn while sharing torque equally.
• The axle housing (banjo type or split type) carries the vehicle weight to the wheels through the springs and acts as a beam.
• Semi-floating axle: the wheel bearing sits between the axle shaft and the housing near the wheel. The shaft carries torque plus bending from the vehicle weight, side forces and tractive forces. Cheap and light; used in cars and light vehicles.
• Three-quarter floating: the bearing is on the housing, inside the hub; the shaft takes torque and some bending from side forces.
• Full-floating: the hub runs on two bearings on the housing; the shaft carries only torque and can be withdrawn without removing the wheel. Used on trucks and buses.

Formulas

Propeller shaft design torque: T = T_e,max·i₁ (or wheel-slip limit) Axle shaft torque (each side): T_a = T_e,max·i₁·i_f / 2 Wheel-slip torque limit per wheel: T_slip = μ·W·r_w Hollow shaft: τ = 16·T·d_o / (π·(d_o⁴ − d_i⁴)) Critical speed: ω_c = (π/L)²·√(E·I / (ρ·A)), N_c = 60·ω_c / (2π) I = π·(d_o⁴ − d_i⁴) / 64, A = π·(d_o² − d_i²) / 4 Hooke's joint: ω_max = ω / cos α, ω_min = ω·cos α Semi-floating axle shaft: T_e = √(M² + T_a²), d³ = 16·T_e / (π·τ)

Symbols: T_e,max = maximum engine torque (N·m); i₁ = first-gear ratio; i_f = final drive ratio; μ = tyre-road friction coefficient; W = wheel load (N); r_w = wheel rolling radius (m); τ = shear stress (MPa); d_o, d_i = outer and inner diameters; L = shaft length between joints (m); E = Young's modulus (Pa); I = second moment of area (m⁴); ρ = density (kg/m³); A = cross-sectional area (m²); ω_c, N_c = critical speed (rad/s, rpm); α = joint angle; M = bending moment at the bearing (N·m); T_e = equivalent twisting moment (N·m).

Worked examples

Example 1 (standard). A car engine gives 200 N·m maximum torque and the first-gear ratio is 3.5. The propeller shaft is a steel tube, 60 mm outside and 54 mm inside diameter, 1.4 m long (E = 210 GPa, ρ = 7850 kg/m³). Top gear is direct and the engine reaches 5500 rpm. Check the shear stress and the critical speed.

1. T = 200 × 3.5 = 700 N·m = 700 000 N·mm.
2. τ = 16 × 700 000 × 60 / (π × (60⁴ − 54⁴)) = 48.0 MPa, acceptable for a steel tube.
3. I = π(0.060⁴ − 0.054⁴)/64 = 2.19 × 10⁻⁷ m⁴; A = π(0.060² − 0.054²)/4 = 5.37 × 10⁻⁴ m².
4. ω_c = (π/1.4)² × √(210 × 10⁹ × 2.19 × 10⁻⁷ / (7850 × 5.37 × 10⁻⁴)) = 525.6 rad/s, so N_c ≈ 5020 rpm.
5. The shaft would reach 5500 rpm in top gear, above its critical speed. Use a two-piece shaft or a larger-diameter (or composite) tube.

Example 2 (GATE level). A semi-floating axle shaft of the same car: final drive ratio 4.1, wheel load 6 kN at a 100 mm overhang from the bearing to the wheel centre, μ = 0.8, rolling radius 0.3 m, permissible shear stress 100 MPa. Find the shaft diameter at the bearing (maximum shear stress theory).

1. Torque from engine: T_a = 200 × 3.5 × 4.1 / 2 = 1435 N·m; wheel-slip limit μ·W·r_w = 0.8 × 6000 × 0.3 = 1440 N·m. Use T = 1435 N·m.
2. Bending at the bearing: M = 6000 × 0.1 = 600 N·m.
3. T_e = √(600² + 1435²) = 1555 N·m.
4. d³ = 16 × 1 555 000 / (π × 100), so d = 42.9 mm; take d = 45 mm (before adding side-force cases and fatigue factors).

Common mistakes

• Designing the propeller shaft for top-gear torque; the maximum torque is in first gear (unless wheel slip limits it).
• Forgetting the critical-speed check, which often decides the tube diameter and length.
• Calling √(M² + T²) the equivalent bending moment; it is the equivalent twisting moment.
• Assuming a full-floating axle shaft carries bending; it carries only torque.
• Forgetting that a single Hooke's joint makes the output speed fluctuate.
• Using total engine torque on each axle shaft without halving it at the differential.

For GATE ME

This topic feeds general questions: hollow versus solid shafts in torsion, critical (whirling) speed of shafts, Hooke's joint speed ratios, and combined bending and torsion. Practise expressing critical speed through static deflection (ω_c = √(g/δ) for a single disc) and through the uniform-shaft formula.

Quick check

1. Engine torque 150 N·m, first gear 3.8. Propeller shaft design torque?
2. If shaft length is doubled, how does the critical speed change?
3. A Hooke's joint runs at 1000 rpm with α = 20°. Maximum output speed?
4. Which axle type carries only torque in the shaft?
5. Final drive 4, first gear 3, engine torque 100 N·m. Torque on each axle shaft?

Answers: 1. 570 N·m. 2. It falls to one-quarter. 3. 1000/cos 20° = 1064 rpm. 4. Full-floating. 5. 600 N·m.