# Homework Help: 2nd order Laplace transforms

1. Jul 20, 2013

### elijah78

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

Solve the DE for y(t) with the IC's
y(0)=20.8m/s and y'(0)=0

if the input is a step function scaled by the desired velocity Vo.
vd(t)=Vou(t).
Assume the desired velocity Vo=27.8m/s

2. Relevant equations

y''(t) + (D/M)y'(t) + (K/M)y(t) = (K/M)vd(t)

M = 1,000kg
D = 100kg/s
K = controller gain
y(t) = output velocity
vd is the input function

3. The attempt at a solution

So I'm Laplace transforming the whole 2nd order equation and I end up with a mess. The next problem is to find the optimal controller gain K for a desired response.

My Laplace transform of the 2nd order equation is:

Y(s) = [ KVo + Ms2y(0) + Dsy(0) ] / [ s2 + (D/M)s + (K/M) ]

if in fact i'm doing it correctly, here is where i am stuck.

Last edited: Jul 20, 2013
2. Jul 20, 2013

### elijah78

plugging in the variables into Y(s) i get:

Y(s) = [ 27.8K + 20800s2 + 2080s ] / [ s2 + .1s + .001K ]

3. Jul 20, 2013

### Staff: Mentor

This Laplace Transform doesn't look correct to me. Please show your work. Please show your Laplace Transform expressions for y'', y', and vd(t).

Chet

4. Jul 20, 2013

### elijah78

Laplace Transform expressions for y'', y', and vd(t).

y''(t):

s2Y(s) - sy(0) - y'(0)

+

(D/M)y'(t):

(D/M)(sY(s) - y(0))

+

(K/M)y(t):

(K/M)Y(s)

=

(K/M)vd(t):

(K/M)(Vo/s)

then I factor out Y(s) and then solve for Y(s). that's when i get that ugly fraction up there that i don't know what to do with. i know how to do simple inverse transforms with tables and partial fractions.

Last edited: Jul 20, 2013
5. Jul 20, 2013

### Staff: Mentor

It looks like the expressions above are correct, but there were algebra errors in solving for Y(s). Please try again. It looks like you are missing a factor of Ms in the denominator.

6. Jul 20, 2013

### elijah78

ok so this time i got:

Y(s) = [ (KVo/Ms) + y'(0) + sy(0) + (D/M)y(0) ] / [ s2 + (D/M)s + (K/M) ]

7. Jul 20, 2013

### Staff: Mentor

Good. Don't forget that y'(0) = 0.
Now, the next step is to manipulate this into a form that is a linear combination of some of the transforms in your tables. Start out by factoring y(0)/s out of the numerator.

Chet

8. Jul 20, 2013

### elijah78

if it wasn't for that darn K. so:

(y(0)/s)[ (KVo/My(0)) + s2 + (D/M)s ] / [ s2 + (D/M)s + (K/M) ]

9. Jul 21, 2013

### Staff: Mentor

No problem. I'm going to retype what you have:

$$y(s)=\frac{y(0)}{s}\frac{(s^2+(D/M)s+\frac{KV_0}{My(0)})}{(s^2+(D/M)s+\frac{K}{M})}$$

Check out the terms in parenthesis in the numerator and denominator. Does this suggest something you can do algebraically to simplify things?

10. Jul 21, 2013

### elijah78

the only option i can see would be to pull 1 s out of the first 2 terms of each polynomial. if the K weren't there i could factor the polynomial using quadratic.

11. Jul 22, 2013

### Staff: Mentor

Suppose you wrote the term in parenthesis in the numerator as
$$\left(s^2+(D/M)s+\frac{KV_0}{My(0)}\right)=\left(s^2+(D/M)s+\frac{K}{M}\right)+\left(\frac{KV_0}{My(0)}-\frac{K}{M}\right)$$

Then the Laplace Transform would become:

$$y(s)=\frac{y(0)}{s}\left(1+\frac{\left(\frac{KV_0}{My(0)}-\frac{K}{M}\right)}{(s^2+(D/M)s+\frac{K}{M})}\right)$$

This should be pretty easy to invert.

Chet

12. Jul 29, 2013

### elijah78

A big big big thank you to you guys for your help. i ended up with an A on the project and a B for the semester. Thank you!