Contents

Springs and dampers

Ride over a rough road

The whole car driven over a road of stated roughness: an ISO 8608 surface, seven degrees of freedom, the rear wheels a wheelbase behind, and how far each tyre's load swings.

Model as of 0.338Figures from the Formula Student single-car templateOctave check Exact

A tyre's grip follows its load, and on a rough road the load never stands still. This page drives the template over a road of a stated roughness and measures how far the four contact-patch loads swing about their static values, how much the body moves, and how fast the dampers work. It takes the whole car, not a quarter of it, because the rear wheels meet every bump a moment after the front ones, and that delay is what pitches the body.

Seven degrees of freedom

Fig. 1

Seven degrees of freedom: the body’s heave, pitch and roll, and each wheel’s height, on springs, dampers and tyres.

Schematic

The body is one rigid mass that heaves by $z$ , pitches by $θ$ and rolls by $φ$ , and each of the four wheels moves up and down on its own: seven degrees of freedom. Every motion is measured from static equilibrium, so $z = 0$ is the car standing on a flat road and gravity drops out. Pitch is positive nose down and roll positive right side down, as on Axes, signs and units. With $x_{i}$ and $y_{i}$ a corner's position from the centre of gravity, forwards and to the left, the suspension at corner $i$ extends by

d_{i} = z - x_{i} θ + y_{i} φ - z_{u, i}

(1)

and pushes the body up with the corner spring and damper, and with the heave and roll elements on the halves of the axle's motion they see (Springs, bars, heave and roll elements):

F_{i} = - k_{w} d_{i} - c_{w} {\dot{d}}_{i} - k_{h} \frac{d_{i} + d_{j}}{2} - k_{b} \frac{d_{i} - d_{j}}{2} - c_{h} \frac{{\dot{d}}_{i} + {\dot{d}}_{j}}{2} - c_{b} \frac{{\dot{d}}_{i} - {\dot{d}}_{j}}{2}

(2)

$j$: the other wheel of the same axle
$k_{w}, c_{w}$: the corner spring and damper, at the wheel
$k_{h}, c_{h}$: the heave element and its damper, at the wheel
$k_{b}, c_{b}$: the roll element or bar and its damper, at the wheel

The body takes the four forces, and each wheel takes its own and the tyre's:

Show the equations with Symbols This car’s numbers

m_{s} \ddot{z} = \sum_{i} F_{i}, I_{θ} \ddot{θ} = - \sum_{i} x_{i} F_{i}, I_{φ} \ddot{φ} = \sum_{i} y_{i} F_{i}

(3)

m_{s} = 300 - 4 \cdot 8 kg, I_{θ} = 130 k g m^{2}, I_{φ} = 45 k g m^{2}

(3)

m_{u} {\ddot{z}}_{u, i} = - F_{i} + k_{t} (z_{r, i} - z_{u, i})

(4)

$z_{r, i}$: the road's height under wheel $i$
$k_{t}$: the tyre's vertical rate, 110 N/mm

A damper part's curves take the coefficients' place when the car has one, bump and rebound apart, as Dampers reads them; the template damps with coefficients, the same both ways. The fourteen states are integrated by the fourth-order Runge–Kutta method. What the tyre puts on the road is its static share of the car's weight plus its spring's force:

Show the equations with Symbols This car’s numbers

N_{i} = N_{i, 0} + k_{t} (z_{r, i} - z_{u, i}), N_{i} \geq 0

(5)

N_{F L, 0} = 691.6 N, N_{R L, 0} = 779.9 N

(5)

The road

ISO 8608 classes a road by the power spectral density of its height, against spatial frequency $n$ , the number of waves per metre:

Show the equations with Symbols This car’s numbers

Φ (n) = Φ (n_{0}) {(\frac{n}{n_{0}})}^{- w}

(6)

Φ (n_{0}) = 0.000016 m^{3} for class C, n_{0} = 0.1 c y c l e s / m, w = 2

(6)

$n_{0}$: the reference spatial frequency, a 10 m wave
$w$: the waviness: how fast the roughness falls off at shorter waves

The classes run from A, a very good road, to E, a very poor one, and each has four times the last one's $Φ (n_{0})$ , so twice its height. The app builds a road of a class as a sum of waves, each carrying the share of the spectrum in its band:

Show the equations with Symbols This car’s numbers

z_{r} (s) = \sum_{k = 1}^{N} A_{k} \cos (2 π n_{k} s + ψ_{k}), A_{k} = \sqrt{2 Φ (n_{k}) Δ n_{k}}

(7)

N = 400, n = 0.01 \dots 10 c y c l e s / m, z_{r, rms} = 0.0044 m

(7)

$s$: the distance along the road
$n_{k}, Δ n_{k}$: the band's geometric centre and width; the bands are spaced evenly in $\log n$
$ψ_{k}$: the wave's phase, drawn at random from a seeded generator, so one seed is always one road

A cosine of amplitude $A$ has a mean square of $A^{2} / 2$ , which is why $A_{k}$ carries the factor 2. The waves run from 100 m long to 0.1 m: the longest a lap cares about to about the tyre's contact patch. The two tracks are two independent samples of the same class, seeds 1 and 1001. Real roads are alike under both tracks over long waves, so this overstates the roll a long wave puts into the car.

Left track, seed 1Right track, seed 1001

Fig. 2

ISO C under the two tracks: independent samples of the same roughness, 4.4 mm RMS on the left.

Measured

The wheelbase delay

A rear wheel reaches each point of the road a wheelbase after the front wheel on its side:

Show the equations with Symbols This car’s numbers

τ = \frac{L}{v}

(8)

τ = \frac{1600 mm}{15 m / s} = 106.7 ms

(8)

A wave whose length is $2 L$ , $2 L / 3$ , $2 L / 5$ and so on lifts one axle as it drops the other, and pitches the body without heaving it. A wave of length $L$ , $L / 2$ and so on lifts both together and heaves it without pitching. This is the wheelbase filter, and the reason a ride study needs the whole car: four quarter cars side by side would never pitch.

What a ride study reads

Front left: static 692 N, 107 N RMS over the runRear left: static 780 N, 109 N RMS

Fig. 3

The left tyres’ loads swing about their static values; the rear meets the front’s road 107 ms later.

Measured

The first number is how far each tyre's load swings about its static value, as a root mean square over the run:

Show the equations with Symbols This car’s numbers

Δ N_{rms} = \sqrt{{(N_{i} - N_{i, 0})}^{2}}

(9)

Δ N_{rms, F L} = 106.9 N = 15.5 % of N_{F L, 0}

(9)

A tyre's grip rises with its load, but less than in proportion (Slip and the Magic Formula). A tyre whose load swings up and down therefore has less grip on average than one held at its mean, and the load variation is the ride number that reaches the lap time. On this road the template's front right tyre came down to 102 N, against a static 692 N.

The others say how hard the car works. The body's vertical acceleration was 0.54 m/s² RMS, unweighted: a comfort weighting is defined for a seated person, and this model has no seat. The body pitched at 1.74 °/s RMS and rolled at 3.07 °/s. The front left damper moved at 33.5 mm/s RMS and spent 88 % of the run below 50 mm/s, which is why the low-speed part of a damper's curve is the part to set first (Dampers).

What the model assumes

The body is rigid and the chassis does not twist. Springs and motion ratios are linear, taken at the static ride. A heave element with a gap or a rate of its own along its travel is the exception: the run follows it.
Each tyre is a linear spring with a point contact under the wheel's centre. It does not span a short bump the way a real contact patch does. Its spring acts both ways in the motion; only the load it reports stops at zero, so a run in which a wheel spends long in the air is past what the model can say.
The car runs straight at a steady speed, with no aerodynamic load and no load transfer from accelerating or cornering.
The two tracks are independent samples of the same class.
The car starts at rest on a flat road, and each wheel steps onto the rough one where it begins: the front wheels at once, the rear ones a wheelbase later. The run's numbers include that start.

In the app

Vehicle Dynamics › Ride over road runs it. Under "Define the road and run" the road is ISO A, C or E, a single bump or a step, with the speed, the duration, the dampers (the car's or a trial damper) and the seed; "Drive it" runs it. The panel opens on the run this page shows: ISO C at 15 m/s for 4 s, seed 1.

"Run outcome" gives the worst wheel-load variation, equation (9) as a share of the static load on the worst corner; the body acceleration; the pitch and roll rates; and the share of the run below 50 mm/s on the front left damper. It draws the damper velocity distribution, rebound positive, and the contact-patch loads over time. "Body motion" plots heave, pitch and roll, with pitch positive nose up.

Check it yourself

The script builds both roads from their seeds, drives the template over them with fourth-order Runge–Kutta as the app does, and reads off the same numbers.

rough_road.mwith example_springs.m, example_car.m, pd_road.m, pd_ride.m, pd_compare.mExact: agrees to machine precision

>> rough_road

  At 15 m/s the rear wheels meet the front wheels' road 106.7 ms later.

  Quantity                                This script   PaceDynamics       Diff
  -----------------------------------------------------------------------------
  left road, RMS elevation, mm                4.36917        4.36917   8.1e-14% 
  integrator step, ms                               1              1         0% 
  FL static load, N                           691.605        691.605   1.6e-14% 
  FR static load, N                           691.605        691.605   1.6e-14% 
  RL static load, N                           779.895        779.895         0% 
  RR static load, N                           779.895        779.895         0% 
  FL load RMS, N                              106.862        106.862         0% 
  FL load RMS / static                       0.154513       0.154513   1.8e-14% 
  FL lightest load, N                         388.035        388.035         0% 
  FR load RMS, N                              100.438        100.438   2.8e-14% 
  FR load RMS / static                       0.145224       0.145224   3.8e-14% 
  FR lightest load, N                         101.931        101.931   3.3e-13% 
  RL load RMS, N                              108.816        108.816   9.1e-14% 
  RL load RMS / static                       0.139527       0.139527     8e-14% 
  RL lightest load, N                         467.335        467.335   1.2e-14% 
  RR load RMS, N                              100.981        100.981   1.5e-13% 
  RR load RMS / static                        0.12948        0.12948   1.5e-13% 
  RR lightest load, N                         143.348        143.348         0% 
  body vertical acceleration RMS, m/s^2       0.541224       0.541224   1.2e-13% 
  pitch rate RMS, deg/s                        1.7418         1.7418   5.1e-14% 
  roll rate RMS, deg/s                        3.07097        3.07097         0% 
  FL damper speed RMS, mm/s                   33.4925        33.4925   4.2e-14% 
  FL share below 50 mm/s                     0.875031       0.875031         0% 
  FR damper speed RMS, mm/s                   31.0958        31.0958   1.4e-13% 
  FR share below 50 mm/s                     0.910022       0.910022         0% 
  RL damper speed RMS, mm/s                    35.333         35.333     1e-13% 
  RL share below 50 mm/s                     0.864534       0.864534         0% 
  RR damper speed RMS, mm/s                   32.2961        32.2961         0% 
  RR share below 50 mm/s                     0.908523       0.908523         0% 

  All 29 quantities agree with PaceDynamics (largest difference 3.3e-13 %).

The script

%% Ride over a rough road: an ISO 8608 road, the seven-degree-of-freedom car, the tyre loads
%
% Companion to the documentation page "Ride over a rough road". Runs unchanged in GNU Octave
% (7 or later) and MATLAB. Edit the car in example_car.m and the run in example_springs.m
% (car.ride) and run:
%
%   >> rough_road
%
% The road is ISO 8608's displacement PSD, Phi(n) = Phi(n0) (n / n0)^-2, synthesised as a
% sum of sinusoids with seeded random phases (pd_road.m), one road per track. The car is
% a body that heaves, pitches and rolls on its four corners, each wheel on its tyre
% (pd_ride.m), driven at a constant speed: the rear wheels meet the front wheels' road a
% wheelbase later. What a ride study reads off it is how far each tyre's load swings about
% its static share, because grip follows load, and how hard the body and the dampers work.
%
% References: ISO 8608:2016, Mechanical vibration - Road surface profiles - Reporting of
% measured data; D. Cebon, Handbook of Vehicle-Road Interaction (Swets & Zeitlinger, 1999),
% ch. 2; T. D. Gillespie, Fundamentals of Vehicle Dynamics (SAE, 1992), ch. 5.
%
% Part of the PaceDynamics documentation, https://pacedyn.dev/docs
% SPDX-License-Identifier: MIT

[car, ~] = example_car();
[spr, app] = example_springs();
app = app.ride;
run = spr.ride;

%% The road: one per track, independent samples of the same roughness
left = pd_road(run, run.seed);
right = pd_road(run, run.rightSeed);
s = (0:4095) * run.length / 4096;
z = arrayfun(left.at, s);
fprintf('\n  At %g m/s the rear wheels meet the front wheels'' road %.1f ms later.\n', run.speed, car.wheelbase / run.speed);
pd_compare('left road, RMS elevation, mm', 1000 * sqrt(mean(z.^2)), 1000 * app.roadRms);

%% The car over it
out = pd_ride(car, spr.inertia, left, right, run);
pd_compare('integrator step, ms', 1000 * out.step, 1000 * app.step);
corners = {'FL', 'FR', 'RL', 'RR'};
for i = 1:4
  pd_compare([corners{i} ' static load, N'], out.staticLoad(i), app.staticLoad(i));
end

%% What a ride study reads
rms = @(v) sqrt(mean(v.^2));
dev = out.wheelLoad - out.staticLoad;               % each tyre's load about its static share
for i = 1:4
  pd_compare([corners{i} ' load RMS, N'], rms(dev(:, i)), app.loadRms(i));
  pd_compare([corners{i} ' load RMS / static'], rms(dev(:, i)) / out.staticLoad(i), app.loadFraction(i));
  pd_compare([corners{i} ' lightest load, N'], min(out.wheelLoad(:, i)), app.loadMin(i));
end
pd_compare('body vertical acceleration RMS, m/s^2', rms(out.zdd), app.bodyAccelRms);
rate = @(q) [0; diff(q)] / run.dt * 180 / pi;       % deg/s, differenced between samples
pd_compare('pitch rate RMS, deg/s', rms(rate(out.pitch)), app.pitchRateRms);
pd_compare('roll rate RMS, deg/s', rms(rate(out.roll)), app.rollRateRms);
vel = 1000 * out.damperVel;                         % mm/s at the wheel
for i = 1:4
  pd_compare([corners{i} ' damper speed RMS, mm/s'], rms(vel(:, i)), app.damperRms(i));
  pd_compare([corners{i} ' share below 50 mm/s'], mean(abs(vel(:, i)) < 50), app.lowSpeed(i));
end
pd_compare();

Download: rough_road.m · example_springs.m · example_car.m · pd_road.m · pd_ride.m · pd_compare.m. The scripts are MIT licensed.

Ride over a rough road

Seven degrees of freedom

The road

The wheelbase delay

What a ride study reads

What the model assumes

In the app

Check it yourself

Further reading