Mass Spring Damper Transfer Function

[TW Cantor]

Homework Statement

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 X_r to ouput X₁

6) Find the poles of the transfer function

Homework Equations

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 don't 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*X_r

Combining the two equations i get an expression containing X_r and X₁ but rearranging it for X₁/X_r 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 I am just not convinced I've got everything right so far. Any help would be brilliant!

Thanks

 

[KataKoniK]

for your post! It looks like you have a good understanding of the problem and have made some progress in solving it. However, there are a few areas that could use some clarification and improvement.

1) Your calculation for the sprung mass seems to be correct, but it would be helpful to show the full equation and any units involved.

2) Your expression for the damping force between m1 and m2 is correct, but it would be helpful to show where the values for c1 and ẋ1 and ẋ2 come from.

3) Your expression for Newton's second law for the motion of m1 is also correct, but again it would be helpful to show where the values for m1, ẍ1, k1, and x1 come from.

4) Your approach for finding the stiffness matrix is correct, but it would be helpful to show the full matrix and explain how you got it.

5) Your attempt at finding the transfer function is a bit confusing. It would be helpful to show the full equations and explain how you got them. Also, you should be able to put the stiffness matrix into a proper matrix form by using brackets [ ].

Overall, your understanding of the problem seems to be good, but it would be helpful to show more of your calculations and explain your reasoning in more detail. Keep up the good work!