In the previous tutorial we have seen how you can create a 1d model of typical spring damper and simulate it dynamically either by using the differential equations or by transfer function through SolidThinking Activate (or by Simulink).
Here we will see how to simulate a simplified suspension system of a quarter car model.
The system diagram:
m2 – Sprung mass
m1 – Unsprung mass
k2 – Suspension stiffness
k1 – Wheel stiffness
b2 – Damping coefficient of suspension
x2 – Displacement of sprung mass
x1 – Displacement of unsprung mass
xr – Displacement of road
u1 – Excitation force on unsprung mass
f1 = k2 * (x2 – x1)
f2 = b2 * (x2’ – x1’)
f3 = k1 * (x1 – r)
By applying Newton’s second law of motion:
m2 * x2” = -f1 –f2
- m2 * x2” = – k2*(x2 –x1) –b*(x2’ – x1’)
- x2” = (b/m2)*x1’ – (b/m2)*x2’ – (k2/m2)*x2………………………………1
m1*x1” = f1 + f2 – f3 –u1
- m1*x1” = k2*(x2-x1) + b2*(x2’-x1’) – kw(x1 – r) – u1
- x1” = (b/m1)*x2’ – (b/m2)*x1’ + (k2/m1)*x2 – [(k2 – k1)/m1]*x1 + (kw*r)/m1 – u1/m1 …………2
Let’s go to SolidThinking Activate (or Matlab Simulink for that matter). Since I have already explained the basics here, I am avoiding the details.
You will end up creating the following model to capture the above two system equations:
Following input values I have considered for running the simulation:
And, the initial conditions for the integration blocks are:
x1’(0) = 0.05
x1(0) = 0.15
x2’(0) = 0.03
x2(0) = 0.05
For input oscillation, I have considered a step generator of following profile:
Step time – 2
Initial value – 0
Final value – 1
And, finally i have run the simulation for 30 sec.
To view the required outputs, I have added two main scopes, namely Scope and Scope_1 for viewing the displacements of the unsprung mass and sprung mass respectively.
Dynamic displacements of the sprung and unsprung mass are: