Angle of Twist & Power Transmission
Predict how far a shaft rotates under load — and connect torque to the horsepower or kilowatts flowing through it.
A car's driveshaft spins at thousands of revolutions per minute, delivering power from the engine to the wheels. Under that load the shaft twists slightly — not enough to see, but enough to matter. If the twist is too large, gears misalign, vibrations grow, and efficiency drops. Engineers predict that twist before the shaft is ever built, using the angle-of-twist formula — and they connect it directly to the power the shaft must carry.
The angle-of-twist formula
Just as axial loading stretches a bar by δ = PL/AE, torsion twists a shaft by an angle φ. The derivation combines the definition of shear strain (γ = ρ dφ/dx) with Hooke's law for shear (τ = Gγ) and the torsion formula (τ = Tρ/J). Integrating along the length gives:
The product GJ is called the torsional rigidity. It has units of N·m². To get φ in radians, ensure T is in N·m, L is in metres, G is in Pa (N/m²), and J is in m⁴. Mixing millimetres and metres is the most common source of errors in hand calculations. A useful sanity check: for steel shafts of ordinary size, the twist angle is usually small — a few degrees at most.
Stepped shafts
If a shaft changes diameter along its length — a stepped shaft — the torque is the same in every segment (for a single applied torque), but J changes. The total angle of twist is the sum of the twists of each segment:
φ_total = Σ (T_i L_i) / (G_i J_i)
The segment with the smaller diameter dominates the total twist because J scales with the fourth power of diameter. A single narrow section can make an otherwise thick shaft surprisingly flexible in torsion.
Power, torque, and rotational speed
Power is the rate of doing work. For a rotating shaft, a torque T acting through an angle θ does work W = Tθ. Differentiating with respect to time gives the power:
For a fixed power, torque is inversely proportional to speed. A high-speed motor delivering 100 kW at 3000 rpm produces roughly half the torque of a low-speed motor delivering the same 100 kW at 1500 rpm. This is why gearboxes exist: they trade speed for torque (or vice versa) while conserving power (minus efficiency losses).
- J_AB = π/32 × 40⁴ = 2.513 × 10⁵ mm⁴ = 2.513 × 10⁻⁷ m⁴.
- J_BC = π/32 × 60⁴ = 1.272 × 10⁶ mm⁴ = 1.272 × 10⁻⁶ m⁴.
- φ_AB = (1000)(1.0) / (80×10⁹ × 2.513×10⁻⁷) = 1000 / 20 104 = 0.0497 rad = 2.85°.
- φ_BC = (1000)(1.0) / (80×10⁹ × 1.272×10⁻⁶) = 1000 / 101 760 = 0.00983 rad = 0.563°.
- φ_total = 0.0497 + 0.00983 = 0.0595 rad = 3.41°.
- Power: ω = 2π × 1500 / 60 = 157.1 rad/s. P = 1000 × 157.1 = 157 100 W = 157 kW.
- Sanity check: the smaller-diameter segment contributes about 5× more twist than the larger one, consistent with J scaling as d⁴.
- J = π/32 × 40⁴ = 2.513 × 10⁵ mm⁴ = 2.513 × 10⁻⁷ m⁴.
- φ = (800)(1.5) / (80×10⁹ × 2.513×10⁻⁷) = 1200 / 20 104 = 0.0597 rad.
- φ = 0.0597 × 180/π = 3.42°.
- T = 9550 × P(kW) / N(rpm) = 9550 × 30 / 1800 = 159.2 N·m.
- Check: ω = 2π × 1800 / 60 = 188.5 rad/s. P = 159.2 × 188.5 = 30 000 W = 30 kW.
Check your understanding
- Angle of twist: φ = TL/(GJ) (radians).
- Stepped shafts: total twist is the sum of segment twists.
- Power transmission: P = Tω, with ω = 2πN/60 for N in rpm.
- For the same power, higher speed means lower torque.
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