# Homework Help: Mass Spring Damper Transfer Function

1. Nov 27, 2013

### TW Cantor

1. The problem statement, all variables and given/known data

The translational system in the first attachment represents the rear/front suspension of a car of mass 1262kg. The distance between the axles is 2.41m and the distance between the centre of mass and the front axle is 1.22m.

I am told that:
m2 = 40 kg
c1 = 800 Nm/s
k1 = 11370 Nm
k2 = 190000 Nm

All the following questions are for the front suspension

1) Find the sprung mass

2) Find the expression for the damping force between m1 and m2

3) Find the expression for Newton's second law for the motion of the first mass m1

4) Find the stiffness matrix for the system

5) Find the transfer function from input Xr to ouput X1

6) Find the poles of the transfer function

2. Relevant equations

3. The attempt at a solution

1) the centre of mass is 1.22m from the front axle so the weight distribution is found as follows:
100*1.22/2.41 = 50.62% of the weight on the rear wheels and therefore 49.38% on the front wheels.

the front sprung mass is then:
(49.38*1262)/(2*100) = 311.57 kg

2) to find the forces acting on m1 I drew a free body diagram, which is shown in the second attachment.

from this i found that the damping force is defined as:
c1*(ẋ1-ẋ2)

3) again using the free body diagram for m1 i found the expression for its motion as:
m1*ẍ1 = [-k1*(x1-x2)-c1*(x1-x2)]

4) using the free body diagram i found expressions for both m1 and m2 and came up with the following expressions for each:

m1*ẍ1 + c1*ẋ1 + k1*x1 = c1*ẋ2 + k1*x2

m2*ẍ2 + c1*ẋ2 + x2*(k1+k2) = c1*ẋ1 + k1*x1

Completing a laplace transform on each of these expressions gives:

(m1*s2 + c1*s + k1)*X1 = (c1*s + k1)*X2

(m2*s2 + c1*s + k1+k2)*X2 = (c1*s + k1)*X1

Putting this into matrix form gives:
m1*s2 + c1*s + k1, c1*s + k1;

c1*s + k1, m2*s2 + c1*s + k1+k2;

*

X1
X2

(i dont know how to put this into a matrix form)

I am told that this is equal to:

0
k2*Xr

5) I then assume to find the transfer function i would treat the stiffness matrix from part 4) as a simultaneous equation. Doing this i get:

X1*(m1*s2 + c1*s + k1) + X2*(c1*s + k1) = 0

X1*(1) + X2*(m2*s2 + c1*s + k1+k2) = k2*Xr

Combining the two equations i get an expression containing Xr and X1 but rearranging it for X1/Xr to get the transfer function. However the expression i get seems a bit unlikely.

If I have gone wrong at any point please let me know, I know how to get the final question im just not convinced i've got everything right so far. Any help would be brilliant!!

Thanks

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Last edited: Nov 27, 2013