Solving driven differential equations

[danj303]

Homework Statement

The suspension system of a car is designed so that it is a damped system described by

z'' + 2z' + z = f(t)

where z is the vertical displacement of the car from its rest position. The car is driven over a (smooth!) road which has "catseye" embedded in the road surface every d metres. Assume that each time the car passes over a catseye, the suspension system receives a unit impulse. Thus, if the car is driven with velocity v m/s over K + 1 catseyes then

f(t) = (see attachment)


(a) Find z(t) when z(0) = z'(0) = 0.


(b) Plot the solutions for K = 10 and K = 50 with v = 10 What is the effect of a larger K? Fix K = 50. Let v = 10 and plot the solutions for d = 5 and d = 10 on the same axes. What is the effect of the distance between the cateyes? Let d = 10 and plot the solutions for v = 10 and v = 20. What is the effect of speed?

Homework Equations

on attachment

 

[gabbagabbahey]

Homework Statement

The suspension system of a car is designed so that it is a damped system described by

z'' + 2z' + z = f(t)

where z is the vertical displacement of the car from its rest position. The car is driven over a (smooth!) road which has "catseye" embedded in the road surface every d metres. Assume that each time the car passes over a catseye, the suspension system receives a unit impulse. Thus, if the car is driven with velocity v m/s over K + 1 catseyes then

f(t) = (see attachment)


(a) Find z(t) when z(0) = z'(0) = 0.


(b) Plot the solutions for K = 10 and K = 50 with v = 10 What is the effect of a larger K? Fix K = 50. Let v = 10 and plot the solutions for d = 5 and d = 10 on the same axes. What is the effect of the distance between the cateyes? Let d = 10 and plot the solutions for v = 10 and v = 20. What is the effect of speed?

Homework Equations

on attachment

What have you tried?

 

[danj303]

Ive got no idea where to start. Thats the problem. I would assume f(t) needs to be expressed in terms of heaviside step functions and then the differential equation solved but I am not sure.

 

[gabbagabbahey]

Ive got no idea where to start. Thats the problem. I would assume f(t) needs to be expressed in terms of heaviside step functions and then the differential equation solved but I am not sure.

Have you learned the method of using Laplace transforms to solve ODEs? If so, try that...

 

[danj303]

Ok that's all well and good but can you explain a little bit further. We have covered laplace transforms but only recently and I am trying to work thorugh some problems to understand how it works. Thanks

 

[gabbagabbahey]

Ok that's all well and good but can you explain a little bit further. We have covered laplace transforms but only recently and I am trying to work thorugh some problems to understand how it works. Thanks

The first step is always to take rthe Laplace transform of both sides of the ODE...what do you get when you do that?

 

[danj303]

The left side is easy if you set it to equal zero. The laplace transform is

s^2*Y(s) + 2s*Y(s) + Y(s) = 0

but I am not sure about the right hand side with the sum from 0 to K of the dirac delta function.

 

[gabbagabbahey]

The left side is easy if you set it to equal zero. The laplace transform is

s^2*Y(s) + 2s*Y(s) + Y(s) = 0

but I am not sure about the right hand side with the sum from 0 to K of the dirac delta function.

Do you know how to take the Laplace transform of a single delta function? If so, just use the fact that the Laplace transform is linear:

$$\mathcal{L}\left[g_1(t)+g_2(t)\right]=\mathcal{L}\left[g_1(t)\right]+\mathcal{L}\left[g_2(t)\right]$$

 

[danj303]

Im not sure about that part. Can you elaborate.

 

[gabbagabbahey]

Im not sure about that part. Can you elaborate.

You're not sure how to take the LT odf a delta function? Or you aren't sure how to add a bunch together?

 

[danj303]

Im not sure how to take the laplace transform of δ(t-kd/v)

 

[gabbagabbahey]

Im not sure how to take the laplace transform of δ(t-kd/v)

It's probably in your textbook, so you might try opening that up and reading part of it ...or, you can use the definition of Laplace transform and the definition of delta function and calculate it directly...

 

[danj303]

Alright after some time I get the laplace transform of the Right hand side as e^-(kd/_v*s)

and then sloving the entire equation for Y(s) I get

Y(s) = ^e-(kd/_v*s)/_s²+2s+1

But does this take into account the sumation and then how do I take the inverse laplace transform to solve the DE

 

[vela]

No, it doesn't take into account the summation. The Laplace transform of the righthand side is

$$L[f(t)] = L\left[\sum_{k=0}^K \delta(t-kd/v)\right] = \sum_{k=0}^K L[\delta(t-kd/v)]$$

where the last step uses the linearity of the Laplace transform. What you calculated is the inside of the last summation.

The exponential factor merely indicates a delay. To find the inverse Laplace transform, you temporarily neglect it, find the inverse transform of the rest, and then time-shift the answer. For example, suppose X(s)=e^2s/(s+1). To find x(t), you first find the inverse Laplace transform of 1/(s+1), which is e^-t. Now you time-shift the answer by 2, so x(t)=e^-(t-2)u(t-2).

 

[danj303]

I find

z(t) = t e^-(t-kd/_v) Heaviside(t-^kd/_v)

The only thing I am not sure about is fitting the summation back in. How do I do this??

 

[gabbagabbahey]

I find

z(t) = t e^-(t-kd/_v) Heaviside(t-^kd/_v)

Close, but shouldn't your $t$ out front really be $\left(t-\frac{kd}{v}\right)$?

The only thing I am not sure about is fitting the summation back in. How do I do this??

Not only is the Laplace transform linear, so is its inverse. That means if [tex]Z(s)=\sum_{k=0}^K g_k(s)[/itex] , then

$$z(t)=\mathcal{L}^{-1}\left[\sum_{k=0}^K g_k(s)\right]=\sum_{k=0}^K \mathcal{L}^{-1}\left[g_k(s)\right]$$