# How quarter car modeling on MATLAB?

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I am trying to model the quarter suspension model in Inman's engineering vibration book in MATLAB, but I have not made any progress. The vehicle will move on the sinusoidal path as in the figure and the height of the road is 5 cm.

I want to plot the largest vibration amplitude of the vehicle for the speed range v=[0 200] km/h and the total vibration for r=0.3 for the time interval t=[0 20] s. The average vehicle weight will be M=1000 kg. I predict the spring stiffness to be K=140 kN/m, damping coefficient c=900 Ns/m.

##### 2 Comments

Image Analyst
on 22 Dec 2022

### Accepted Answer

Sam Chak
on 23 Dec 2022

Edited: Sam Chak
on 23 Dec 2022

I prefer the simulation-based numerical solution over the formula-based analytical solution in the example because there is no graph provided in your image.

By Newton's 2nd law, we should get

where the disturbance force caused by the wavy road surface is given by

.

Thus, the model can be rewritten as

,

where the road surface is given by . I believe you can use calculus to find .

The following is based on the info in Example 2.4.2 (provided by you).

tspan = linspace(0, 20, 20001);

x0 = [0 0]; % initial rest condition

[t, x] = ode45(@quarterCar, tspan, x0);

plot(t, x(:,1)), grid on, xlabel('Time, [seconds]'), ylabel('Deflection, [meter]')

title('Deflection experienced by the car')

% Maximum deflection experienced by the car (same as in Example 2.4.2)

dmax = max(x(:,1)) - min(x(:,1))

% Quarter Car suspension model

function xdot = quarterCar(t, x)

xdot = zeros(2, 1);

% Example 2.4.2

m = 1007; % mass of car

c = 2000; % damping coefficient

k = 4e4; % stiffness coefficient

h = 1; % road height in cm

v = 20; % car velocity (fixed speed)

% Original Problem

% m = 1000; % mass of car

% c = 900; % damping coefficient

% k = 140e3; % stiffness coefficient

% h = 5; % road height in cm

% v = (200/20)*t; % car velocity ramp up from 0 to 200 km/h in 20 seconds

% Standard formulas and state-space model (1st-order ODE form)

wb = 0.2909*v; % formula given in Example 2.4.2

a = (h/2)/100; % sinusoidal amplitude in m

d = k*a*sin(wb*t) + c*a*wb*cos(wb*t); % disturbance due to wavy road surface

A = [0 1; -k/m -c/m]; % state matrix

B = [0; 1/m]; % input matrix

xdot = A*x + B*d;

end

##### 12 Comments

### More Answers (1)

Sulaymon Eshkabilov
on 23 Dec 2022

Hi,

Your system equation is: M*ddx+C*dx+K*x = F(t) and you are studying a forced vibration.

There are a few different ways how to model and solve this spring-mass-damper system.

(1) To obtain an analytical solution, use a symbolic math function: dsolve()

(2) To obtain an analytical solution, use symbolic math functions: laplace(), ilaplace()

(3) To obtain a numerical solution, use ode solvers: ode45, ode23, ode113, etc

(4) To obtain a numerical solution, use Simulink modelling: ode45, ode23, ode113, etc

(5) To obtain a numerical solution, write a code using Euler, Runge-Kutta and other methods

(6) To obtain a numerical solution, use control toolbox functions: tf(), lsim()

There are a few nice sources how to model and solve this spring-mass-damper system.

(6) https://www.mathworks.com/matlabcentral/fileexchange/95288-mass-spring-damper-1-dof

Nice sim demo: https://www.mathworks.com/matlabcentral/fileexchange/94585-mass-spring-damper-systems

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