Ride comfort: quarter-car model and human response to vibration

Why it matters

Ride comfort is one of the first things a customer notices and one of the hardest to engineer, because the suspension must also keep the tyres pressed on the road and fit inside limited suspension travel. On Indian roads, with speed breakers, potholes and long stretches of broken surface, poor ride causes fatigue, back problems in commercial drivers and damage to cargo. The quarter-car model is the standard first tool for choosing spring rates and damping.

Key ideas

The quarter-car model. One corner of the vehicle is represented by two masses:

• the sprung mass m_s — the share of body, passengers and load carried by that corner's spring;
• the unsprung mass m_u — wheel, tyre, hub, brake and part of the suspension links and spring. The suspension spring k_s and damper c_s act between them; the tyre is modelled as a stiff spring k_t (its damping is small) between the unsprung mass and the road. The road profile z_r(t) is the input. This gives two degrees of freedom and two natural modes.

Body (bounce) mode. The sprung mass bounces on the suspension spring and tyre in series. The combined ride rate K_R = k_s·k_t/(k_s + k_t) is slightly less than k_s. Passenger cars aim for a body natural frequency of about 1.0–1.5 Hz (sports cars higher, about 1.5–2 Hz) — close to the frequency of walking, which people find natural.

Wheel-hop mode. The unsprung mass vibrates between the suspension spring and the much stiffer tyre at about 10–15 Hz. Large wheel-hop motion means large swings in tyre load, so less grip — this is the road-holding problem.

Damping. The damping ratio ζ = c_s/(2√(K·m_s)) is typically 0.2–0.4 for comfort-oriented cars. Low damping lets the body float and resonate; high damping makes the car feel harsh because more of the road input is transmitted above resonance. Real dampers are tuned with different rates in bump and rebound.

Transmissibility and isolation. For a base-excited single-degree system (body on ride rate), the ratio of body to road amplitude is the transmissibility T. It exceeds 1 near resonance (r = f/f_n ≈ 1) and falls below 1 only when r > √2 — isolation. Adding damping reduces the resonant peak but worsens isolation at high frequency: the central compromise of passive suspension design.

Road input. A road with wavelength λ driven over at speed V excites frequency f = V/λ. Long wavelengths excite body bounce; short ones excite wheel hop and are felt as harshness. Real roads are random, described by a power spectral density.

Design trade-offs. A softer spring lowers body frequency and improves comfort but needs more suspension travel (rattle space) and allows more roll and pitch. A lighter unsprung mass (alloy wheels, aluminium knuckles) improves both comfort and road holding — hence interest in a high sprung-to-unsprung mass ratio (about 8–12 for cars). Active and semi-active suspensions relax the compromise.

Human response to vibration. Sensitivity depends on frequency, direction, amplitude, duration and posture. ISO 2631 (the reference standard; use its weighting tables for actual values) shows that seated people are most sensitive to vertical vibration at about 4–8 Hz (resonance of the abdominal organs and spine) and to horizontal vibration at about 1–2 Hz. Very low frequencies, about 0.1–0.5 Hz, cause motion sickness. Comfort is judged from the frequency-weighted r.m.s. acceleration (m/s²); shocks are better captured by the vibration dose value (VDV). Weighted r.m.s. values below about 0.3 m/s² are generally not uncomfortable, while values above about 1–2 m/s² are very uncomfortable (check the exact bands in the standard).

Pitch and the Olley criteria. Front and rear ends meet a bump at different times. Making the front ride frequency slightly lower than the rear (by roughly 10–20%) lets the rear "catch up", turning pitch into flatter bounce.

Formulas

K_R = k_s·k_t / (k_s + k_t) — ride rate (N/m); k_s suspension spring rate at the wheel, k_t tyre vertical stiffness.

f_b = (1/2π)·√(K_R / m_s) — body (bounce) natural frequency (Hz); m_s sprung mass per corner (kg).

f_h = (1/2π)·√((k_s + k_t) / m_u) — wheel-hop frequency (Hz); m_u unsprung mass per corner (kg).

ζ = c_s / (2·√(K_R·m_s)) — damping ratio of the body mode (dimensionless); c_s in N·s/m.

f = V / λ — excitation frequency (Hz) from a road wavelength λ (m) at speed V (m/s).

T = √[(1 + (2ζr)²) / ((1 − r²)² + (2ζr)²)] — displacement transmissibility for base excitation; r = f/f_n.

a_peak = (2πf)²·X — peak acceleration (m/s²) for a harmonic motion of amplitude X (m).

F_d = c_s·v, F_s = k_s·x — damper and spring forces (N), v relative velocity (m/s), x deflection (m).

Worked examples

Example 1 (standard). One corner of a car has m_s = 300 kg, m_u = 40 kg, k_s = 22 000 N/m, k_t = 200 000 N/m, c_s = 1800 N·s/m. Find the ride rate, body and wheel-hop frequencies and damping ratio.

1. K_R = k_s·k_t/(k_s + k_t) = 22 000 × 200 000/222 000 = 19 820 N/m.
2. f_b = (1/2π)√(K_R/m_s) = (1/2π)√(66.07) = 8.128/6.283 = 1.29 Hz (spring alone would give 1.36 Hz).
3. f_h = (1/2π)√((k_s + k_t)/m_u) = (1/2π)√(5550) = 74.50/6.283 = 11.9 Hz.
4. ζ = c_s/(2√(K_R·m_s)) = 1800/(2√(5.946×10⁶)) = 1800/4877 = 0.37.

Example 2 (GATE level). The same car (treat the body as a single-degree system on K_R with ζ = 0.37) drives over a road with a sinusoidal profile of wavelength 10 m and amplitude 20 mm. (a) At what speed does body resonance occur? (b) Find the body amplitude and peak body acceleration at 54 km/h.

1. Resonance when V/λ = f_b: V = 1.294 × 10 = 12.9 m/s ≈ 46.6 km/h.
2. At 54 km/h, V = 15 m/s, so f = V/λ = 1.5 Hz and r = 1.5/1.294 = 1.160.
3. 2ζr = 0.856; 1 − r² = −0.3445. T = √[(1 + 0.733)/(0.1187 + 0.733)] = √(1.733/0.852) = 1.43.
4. Body amplitude X = 1.43 × 20 = 28.5 mm — still amplified, because r < √2.
5. a_peak = (2πf)²·X = (9.425)² × 0.0285 = 2.53 m/s² — very uncomfortable; a softer spring or higher speed (r > √2) would isolate.

Common mistakes

• Using the full vehicle mass instead of the sprung mass per corner.
• Ignoring the tyre spring in series, which overestimates body frequency.
• Assuming more damping always improves comfort; it helps only near resonance.
• Expecting isolation at any r > 1. Transmissibility falls below 1 only for r > √2.
• Confusing natural frequency in rad/s with Hz (divide ω by 2π).
• Quoting "1–2 Hz causes motion sickness". Motion sickness is linked to about 0.1–0.5 Hz; 4–8 Hz vertical is the most sensitive band for comfort.

For GATE ME

Expect natural-frequency and damping-ratio calculations, transmissibility from road wavelength and speed, speed at resonance, and conceptual questions on sprung/unsprung mass, isolation and human vibration sensitivity. This topic overlaps heavily with base excitation in vibrations, so practise those formulas in vehicle form.

Quick check

1. k_s = 20 000 N/m, m_s = 320 kg (ignore tyre). Body frequency?
2. At which r does transmissibility fall below 1?
3. Which frequency band of vertical vibration are seated humans most sensitive to?
4. A car at 20 m/s crosses ridges every 5 m. Excitation frequency?
5. Why do engineers try to reduce unsprung mass?

Answers: 1. (1/2π)√62.5 = 1.26 Hz. 2. r > √2 ≈ 1.41. 3. About 4–8 Hz. 4. 4 Hz. 5. It improves road holding and reduces harshness transmitted to the body.