Curves and Damped harmonic motion 0_o'

I have to find an equation/model for the following:

A vibrating spring has damped harmonic motion- the oscillations decrease over time

-The centre of vibration is 0

-The starting amplitude of vibration is 0.6mm

-after 5 seconds the amplitude decreases to 0.5 mm

-During this time the spring moved forward and backwards 200 times

My attempt-

it will be a sine curve as centre of vibration is 0

The general form will be y=Ae^ax sin bx

200 forwards and backards= 100 cycles

if 100 cycles in 5 seconds then it is 100/5= 20 cycles per second = frequency (f)

T=1/f and T= 2pi/b and found b to be: 40pi

A=0.6? i.e. initial amount

It was 0.6 at 0s

0.5 at 5s

(0.5/0.6)=Ae^5x/Ae^0x

(A's cancel) therefore:

(0.5/0.6)=e^5x/e^0x

e^5x-0x=0.5/0.6

x= -0.036464

Can anyone tell me if I'm on the right track please :(, Im not sure if all of my working out is valid? if not could you please give the correct working out.

Thanks

Mathhelp246

Re: Curves and Damped harmonic motion 0_o'

Hey Mathshelp246.

For this problem, do you need to use some kind of parametric function?

The reason I ask is that typically for a lot of these kinds of problems, you have specific parametric Differential Equations (DE's) that specify say a damped spring with certain parameters.

If you have that model and have been taught it in class, then I suggest you post it here so we can look at it.

Typically you will take this model (which has a number of free variables) and you will set those to either constants or functions and solve for your final value.

Re: Curves and Damped harmonic motion 0_o'

Thanks for the reply chiro =)

Um the question asks to form a model based on your knowledge of curves, amplitude, frequency and period

and all the information that it gives is that what is listed above:

"A vibrating spring has damped harmonic motion- the oscillations decrease over time

-The centre of vibration is 0

-The starting amplitude of vibration is 0.6mm

-after 5 seconds the amplitude decreases to 0.5 mm

-During this time the spring moved forward and backwards 200 times"

I think that it is based on curves like

Ae^-kx sin bx

Ae^-kx cos bx

I think they want it in an exponential model

" The exponential function is an important model of many natural systems, as are the periodic functions sine and cosine"

The combined functions e^ax sin bx and e^ax cos bx are also important in many applications, with many oscillations and vibrations being closely modelled by these funtions. VIbrations can be modelled by y=Ae^-kx sin bx and y=Ae^-kx cos bx, where k is negative - damped harmonic motion"

Re: Curves and Damped harmonic motion 0_o'

It might help to note that $\displaystyle y = e^{-at}sin(bt)$ *not* a function of x.

-Dan