Finding position equation from differential equation for damped two-spring mass system

[stunner5000pt]

Ok the system which is doing the oscillations is described here

http://scienceworld.wolfram.com/physics/SpringsTwoSpringsandaMass.html

however, that differential equation over there gives the equation of motion. What i need is the differential equation for the position of the mass

ALso given to me is the length of the left spring L1 and the right spring L2. Find th equation of motion

$$m \frac{d^2 y}{dt^2} = -b\frac{dy}{dt}-(k_{1}+k_{2})y$$

now if this the equatio of moti0n then the equation of posotion would be given by the integral of this??
so then i sub v = dy/dt and solve?? and assume a new function like $$e^{\lambda t}$$ for the function y??

 

[quasar987]

The system described on the Wolfram page assumes no friction. The equation of motion you wrote has a friction term in it. Please copy/paste the question exactly how it is asked.

 

[stunner5000pt]

Let k1 be the spring constant for the left spring L1 be its length
Let k2 be the psirng constant for the right spring L2 be its length
Let m be the mass
Let b the damping coefficient
Write a second order differential equation for the position at time t
Thats how the question is worded
Refer to the wolfram page for the diagram only

 

[dextercioby]

Yes,then everything is okay.

Daniel.

 

[stunner5000pt]

Yes,then everything is okay.

Daniel.

what do you mean?

 

[vincentchan]

let $y=Ae^{\lambda t}$
just want to remind you lambda and A are not real

the real part of y is your solution...
if you don't want to deal with complex number...you can try $Ae^{at}sinbt+Be^{at}cosbt$

 

[stunner5000pt]

ok i cna solve it but for the lambda i'd get for $$y = Ae^{\lambda t}$$

$$\lambda = = \frac{-Ab \pm \sqrt{A^2 b^2 -4Am(k_{1}+k_{2})}}{2Am}$$
ok so i got the lambda nad all i solved the bloody thing

but how does the lambda relate to the length of the springs and stuff??
because the book gives an answer
$$m \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + (k_{1}+k_{2})y = k_{1}L_{1} - k_{2} L_{2}+k_{2}$$

how'd they get that?

 

[dextercioby]

The last equation u posted is incorrect.You may want to post the correct one.

Daniel.

 

[stunner5000pt]

which one is wrong?

this one?? I ttyped it straight out from the textbook $$m \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + (k_{1}+k_{2})y = k_{1}L_{1} - k_{2} L_{2}+k_{2}$$

or the other one in the last post

 

[dextercioby]

Yes,that one.Do you see why it's wrong...?

Daniel.

 

[stunner5000pt]

fidning the value of lambda fives me the function y and y is the function of position correct?

Is ther any initial condition i have to asume in thsi situation ??

 

[vincentchan]

Your lambda is wrong (just a minor mistake, almost right)... try it again.
if your lambda are distinct you will have 2 solution:
$$y_1 = A_+e^{\lambda_+ t}$$
$$y_2 = A_-e^{\lambda_- t}$$
are your answer...
and any linear combination of y1 and y2 is your answer,too... therefore, the general form of your solution is:
$$y = A_1 y_1 + A_2 y_2$$

edit: y,A,lambda are complex number... for this type of question, complex number is a little bit easier to work with...

if you want a real solution, use $e^{ix} = cosx+isinx$ and a real initial condition substitude into y, since you have 2 unknown in y, you need 2 equation for the initial condition: ie. y(0)=y_0, y'(0)=y'_0...

 

[dextercioby]

I was contesting the equation you posted and said to have copied from the book.Maybe it was a typo in the book.

Daniel.