| Posted on Monday, 01 March, 2004 - 01:05 pm: |
This is the equation of the motion of a pendulum if air resistance is ignored: d2q/dt2=-g/l×sinq Using this, is it possible to find an equation relating the period of the pendulum to q, without approximating sinq to q?
| Posted on Monday, 01 March, 2004 - 01:49 pm: |
Yes, I think there is (don't quote me on that), but I'd have to look it up.....
Matt.
| Posted on Monday, 01 March, 2004 - 02:42 pm: |
I think what you are referring to is a complete elliptic integral of the first kind. Have a look here .
Andre
| Posted on Monday, 01 March, 2004 - 09:49 pm: |
are you aware of the usual approach: let sinx ~= x if x is small. andre's approach will use a function that that you can only compute numerically - it's not "nice" like 3x + 1 or something. but then neither is sinx!!
| Posted on Monday, 01 March, 2004 - 10:24 pm: |
You can get most of the way through this as follows: Note that
|
|
. q |
and q. Solving this, and using initial conditions of q = a,
|
. q | =0 |
at t=0 gives:
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| Posted on Tuesday, 02 March, 2004 - 08:59 am: |
It cannot be solved explicitly. Taking the square root of both sides and integrating (assume that at t=0, theta=0; at t=P/4, theta=alpha; where P is the period of the pendulum) we get the expression for P in terms of a complete elliptic integral of the first kind. That's the best you can do.
That said, it is possible to evaluate this integral to any desired accuracy very rapidly by virtue of its connection to the Arithmetic-Geometric Mean interation.
Andre
| Posted on Tuesday, 02 March, 2004 - 02:13 pm: |
While you cannot solve the full equation explicitly, you can obtain an explicit approximation by taking more terms in the approximation of sin x; eg
, or
.These give you a higher order approximation to the true behaviour of the pendulum.
| Posted on Wednesday, 03 March, 2004 - 09:04 am: |
The full solution result (for the period of a pendulum) is not quite as bad as it looks (although it's a bit involved along the way).
First write down the equation of motion of the pendulum:
Now integrate once as described above
The period of the pendulum is 4 times the time for theta to go from 0 to alpha, so
Through a couple of substitutions (see here , eq (13) ..(26)) we can rewrite this as
where
All this seems to lead nowhere practical, however it turns out that K(k) can be readily evaluated numerically:
Where M is a function (the arithmetic geometric mean iteration) defined iteratively as follows:
Define a0 =1, b0 =(1-k2 )1/2
;
Repeating until the difference between an and bn is small enough. The result is then an . Convergence is very rapid.
Andre
| Posted on Wednesday, 03 March, 2004 - 10:41 am: |
Thanks a lot, Andre!