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.
Sam Chak on 23 Dec 2022
Edited: Sam Chak on 23 Dec 2022
Hi @Mustafa Furkan,
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;
Sam Chak on 25 Dec 2022
Edited: Sam Chak on 25 Dec 2022
Unless you implied the author, Prof. Daniel J. Inman made an error in his book, I don't see any problem with the original formula that you supplied (in the image). I have checked the formula a few times. THe code looks the same as in the image.
Perhaps, you can be a little specific about the result or plot that you are expecting.
% t = 0:0.01:20;
% v = (200/20)*t;
% wb = 0.2909*v;
% r = wb/6.303;
r = 0:0.01:10;
ksi = 0.038;
X = 0.025*sqrt((1 + (2*ksi*r).^2)./((1 - r.^2).^2 + (2*ksi*r).^2));
plot(r, X), grid on
More Answers (1)
Sulaymon Eshkabilov on 23 Dec 2022
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.
Nice sim demo: https://www.mathworks.com/matlabcentral/fileexchange/94585-mass-spring-damper-systems
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