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:**

Where,

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

Where,

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:

m1=1;

m2=1.5;

ks=10;

kw=12.5;

b=0.2;

r=0;

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: