WEBVTT

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Quarter-car response. Synthetic profile; 80 km/h; zero-state start; 100 m run-in.

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MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

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zs, zu: sprung and unsprung displacement (m); r: road input (m). Stiffness / sprung mass: 63.3 and 653 s⁻²; damping / mass: 6 s⁻¹. Unsprung / sprung mass = 0.15; L = evaluation distance (m).

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INPUT · a specified road profile · read the axis units and the numerical values.

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Synthetic profile; 80 km/h; zero-state start; 100 m run-in.

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Computed suspension travel / distance = 3.256 m/km over the 100–300 m evaluation interval.

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OUTPUT · suspension relative velocity · read the axis units and the numerical values.

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Integrate the absolute relative velocity, then divide by distance: R = 3.256 m/km. Signed cancellation is not allowed.

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CALCULATION · substitute the defined inputs; keep units through each intermediate step.

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Profile: r(x) = 0.003 sin(2πx/10) + 0.001 sin(2πx/2) metres

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Time coordinate: t = x / (80/3.6); 200 m takes 9 s seconds

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Suspension travel: ∫ |vs − vu| dt ≈ 0.651269 metres over approximately 200 m

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Normalisation: 1000 × travel / evaluated distance = 3.256345 m/km

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SENSITIVITY · change one declared input · read the axis units and the numerical values.

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Half amplitude: 1.62817 m/km. Multiply both profile amplitudes by 0.5.

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Reference: 3.25635 m/km. Amplitudes 3 mm and 1 mm; wavelengths unchanged.

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Double amplitude: 6.51269 m/km. Multiply both amplitudes by 2; keep speed and coefficients fixed.

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Check: Recompute with a smaller output step and compare the integrated response. Inspect the run-in separately, preserve the actual evaluated distance, and confirm that halving both amplitudes halves the response within solver tolerance.

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ONE ROW · A road profile is a signal before it is a score. What survives when we collapse it to one number?

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PARAMETER MAP · 24461 evaluated cells. The colour limits stay fixed across every row.

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Read one row as a longitudinal profile. Moving between rows scales the same wavelengths; it does not change their spacing. For the linear, zero-state quarter-car calculation above,

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scaling the entire profile by a multiplies suspension travel and the response index by |a|. The 3D height is road elevation, not vehicle acceleration.

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VALUE LANDSCAPE · A 0–20 m close-up retains every half-metre sample. Height is profile elevation; the second axis is amplitude multiplier.

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This is a prescribed profile family, not measured roughness. A different wavelength mix requires solving the vehicle response again; amplitude alone cannot describe an arbitrary road.

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The run-in response is an educational calculation, not a certified IRI result.