Change of variable?

Preben Alsholm — Thu, 31 May 2012 10:54:02 Z

There are a couple of obvious problems:

(1) Presumably you want to use the package T, so you need to do with(T); or use S:=T:-sin(t);

(2) You wrote DS:=diff(DS,t): This would make DS=0 and subsequently DDS=0. You probably meant DS:=diff(S,t):

However, the inhomogeneous term is irrelevant when it comes to the angular eigenfrequency omega. That one is
(1/2)*sqrt(4*c[0]*m[1]-d[0]^2)/m[1] for the ODE as written. And as d[0] is small it can be neglected so it is roughly sqrt(c[0]/m[1]) as you say.

Notice that even after doing with(T); the result of dsolve(diff(x(t),t,t)+x(t)=0); is x(t) = _C1*sin(t)+_C2*cos(t), which means it is not expressed in terms of the T-versions but the global versions, which are available as :-sin and :-cos. Try
res:=dsolve(diff(x(t),t,t)+x(t)=0);
eval(res,t=360);
subs({:-sin = sin,:-cos=cos},res);
eval(%,t=360);

You may want to avoid redefining sin and cos and consider changing variables in the ODE, i.e. using another time tau:

PDEtools:-dchange({t=tau*k,x(t)=X(tau)},ODE,[tau,X]);
#And then pick k to fit your needs.
#If you use the angle in degrees as time tau, then you have tau = t*omega*180/Pi, so you should pick
k = Pi/(omega*180).
An additional comment. Why not just write s:=x; v:=D(x): a:=(D@D)(x):

MaplePrimes - answers and comments on Question, How do i switch between time and angle?

Change of variable?