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carlj0nes
 one year ago
Whoever can answer this the best I will send you 10 owlbucks.
Can someone please explain how we could use this differential equation to model an undamped oscillator.
(d^2 x)/(dt^2 )10 dx/dt+34x=0
Then if we were to extend this differential equation into a forced damped oscillator, explain how you could model such an oscillator using a forced second order differential equation.
The description must be commensurate to a level 5 qualification.
carlj0nes
 one year ago
Whoever can answer this the best I will send you 10 owlbucks. Can someone please explain how we could use this differential equation to model an undamped oscillator. (d^2 x)/(dt^2 )10 dx/dt+34x=0 Then if we were to extend this differential equation into a forced damped oscillator, explain how you could model such an oscillator using a forced second order differential equation. The description must be commensurate to a level 5 qualification.

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Need help @carlj0nes

phi
 one year ago
Best ResponseYou've already chosen the best response.11I believe the damping term is associated with the dx/dt term find the roots to the "characteristic equation", and examine the discriminant to classify the behavior of the system

phi
 one year ago
Best ResponseYou've already chosen the best response.11Your question is not clear to me. You can start with the physics of , for example, a spring that you pull and then release. Making some simple assumptions, you can derive a differential equation that models the behavior of the spring. You can introduce damping to model how the spring slows down. You will get an equation that looks like the one you posted.

carlj0nes
 one year ago
Best ResponseYou've already chosen the best response.0The original question was. A Damped oscillation with no external forces can be modelled by the equation I previously posted. Where x(in mm) is the amplitude of the oscillation at time t seconds. The initial amplitude of the oscillation is 2mm (i.e. when t=0) and the velocity is 0mm/s. By using the information provided above, determine the particular solution of the differential equation. Just to confirm I have already completed this part of the question but I need to do the question posted for a higher grade

phi
 one year ago
Best ResponseYou've already chosen the best response.11an undamped harmonic oscillator would have this equation \[ \frac{ d^2x }{ dt^2 }+ \cancel{10\frac{ dx }{ dt }}+34x = 0 \]

phi
 one year ago
Best ResponseYou've already chosen the best response.11btw, are you sure it is 10 dx/dt ? that 10 leads to e^5t term which grows (as opposed to being damped) with time.

carlj0nes
 one year ago
Best ResponseYou've already chosen the best response.0This is what I have so far (d^2x)/(dt^2)10 dx/dt +34x=0 r ^2 −10r+34=0 (r−5)^ 2 −25+34=0 (r−5) ^2 =−9 (r−5)=±√−9 =±√9 √ −1 r−5=±3i r=5±3i so x(t)=ae ^(5+3i)t +be ^(5−3i)t x(t)=e ^5t (asin(3t)+bcos(3t))= e^ 5t (ksin(3t+θ)) −e 5t ≤e 5t sin(3t+θ)≤e 5t

carlj0nes
 one year ago
Best ResponseYou've already chosen the best response.0not sure if it is correct but I am after a explanation as well

phi
 one year ago
Best ResponseYou've already chosen the best response.11I would stop at x(t)=e ^5t (asin(3t)+bcos(3t)) and use the initial conditions to find a and b at t=0, x= 2 so \[ 2 = e^{5 \cdot 0} ( a \sin(3\cdot 0) + b \cos(3\cdot 0)\\ 2= 1 \cdot(0 + b\cdot 1) \\ b=2 \]

phi
 one year ago
Best ResponseYou've already chosen the best response.11now find dx/dt (which is the velocity), so we can use the second initial condition v=0 at t=0 \[ \frac{d}{dt} x(t)= \frac{d}{dt} \left(e^{5t} (a \sin(3t)+b\cos(3t))\right) \]

phi
 one year ago
Best ResponseYou've already chosen the best response.11should replace b with 2 and dx/dt with 0 (i.e. v=0) in that, and solve for a

phi
 one year ago
Best ResponseYou've already chosen the best response.11you should get \[ \frac{d}{dt} \left(e^{5t} (a \sin(3t)+b\cos(3t))\right) = 0 \\ \left(\frac{d}{dt} e^{5t}\right) (a \sin(3t)+b\cos(3t)) + e^{5t} \frac{d}{dt} (a \sin(3t)+b\cos(3t)) = 0 \]

phi
 one year ago
Best ResponseYou've already chosen the best response.11or \[ 5 e^{5t} (a \sin(3t) + 2 \cos(3t)) + e^{5t} ( 3a \cos(3t) 6\sin(3t) ) = 0 \] now plug in t=0

phi
 one year ago
Best ResponseYou've already chosen the best response.11to "force" the oscillator, set the equation equal to some function of t \[ \frac{ d^2x }{ dt^2 }+ 10\frac{ dx }{ dt }+34x = f(t) \] for example, let f(t)= sin(t)

phi
 one year ago
Best ResponseYou've already chosen the best response.11***A Damped oscillation with no external forces can be modelled by the equation I previously posted.*** Again, I would double check the 10 coefficient. Real oscillations grow smaller, not grow larger with time.

carlj0nes
 one year ago
Best ResponseYou've already chosen the best response.0@phi thanks for your help but can someone explain how we could use this differential equation to model an undamped oscillator.

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.0http://openstudy.com/study#/updates/558c1099e4b0c2c04901546b

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.0if it was +10dx/dt was damping \[x(t)=e^{5t}(asin(3t)+bcos(3t))\] dw:1435827570430:dw

carlj0nes
 one year ago
Best ResponseYou've already chosen the best response.0The question I was given said 10. But it does make sense to be +10 Maybe it was a typo. Out of interest we know what is b as that = 2 but what does a=

carlj0nes
 one year ago
Best ResponseYou've already chosen the best response.0I have seen my tutor today and he said the original is correct for damped but for undamped it does change to +10

phi
 one year ago
Best ResponseYou've already chosen the best response.11**I have seen my tutor today and he said the original is correct for damped but for undamped it does change to +10*** Can you post a screen snapshot of the original material? This make sense only if the original equation was written e.g as \[ \frac{d^2 x}{dt^2 } = 10 \frac{d x}{dt } 34x \] or the leading coefficient of the d^2/dt^2 term is negative.

phi
 one year ago
Best ResponseYou've already chosen the best response.11**Out of interest we know what is b as that = 2 but what does a= *** See the above posts where b is found, and (most!) of the work is done to find "a" Can you give it a try to find a?
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