T-Pain Is On A Boat: Re: [Electric Boats] Re: Innovation and Resonance

On 24/05/2013, at 6:20 AM, "cire" <cirejay@hotmail.com> wrote:

-
> From: Julian Webb
> To: electricboats@yahoogroups.com
> Sent: Thursday, May 23, 2013 3:11 AM
> Subject: Re: [Electric Boats] "Batteries Not Included"
> SNIP...

> - To cut a long story short unless the importance of electrical and mechanical resonance in systems is appreciated, (which has no place, constant or symbol in any traditional calculations or formulas that I have ever seen) they cannot be understood or replicated.
>

Looks like its understood to me

$y p = A t cos (5 t) + B t sin (5 t) + C cos (t) + D sin (t) .$

Working out the first and second differentials:

$y' p y'' p = = A cos (5 t) + B sin (5 t) - 5 A t sin (5 t) + 5 B t cos (5 t) - C sin (t) + D cos (t), - 10 A sin (5 t) + 10 B cos (5 t) - 25 A t cos (5 t) - 25 B t sin (5 t) - C cos (t) - D sin (t) .$

On substituting into the equation we expect all the tcos(5t) and tsin(5t) terms to cancel:

$- 10 A sin (5 t) + 10 B cos (5 t) + 24 C cos (t) + 24 D sin (t) = 2 sin (5 t) + cos (t) .$

We can see that both B=0 and D=0, therefore we don't need these terms in the particular solution. If we had been clever and thought ahead we may have been able to foresee this and save ourselves some work. This sort of insight should hopefully come with practice; if in doubt include all the terms. From above we can also see A=−15and C=124 giving us:

$y p = 1 24 cos (t) - t 5 cos (5 t),$

and full solution:

$y = y c + y p = A cos (5 t) + B sin (5 t) + 1 24 cos (t) - t 5 cos (5 t) .$