[Search for users] [Overall Top Noters] [List of all Conferences] [Download this site]

Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

630.0. "The propeller and the toroid" by EAGLE1::DANTOWITZ (Ho, ho, ho ... (-:|>* ) Wed Dec 17 1986 13:48


	Given a toroid and an airplane propeller I would like
	to plot the path of the propeller as it rotates and moves
	through the toroid. 

	Assumptions:

	The circular cross-sections of the toroid have a radius R.
	The propeller has a length of 2*R-delta.  (delta is a small
	number, this permits the propeller to be situated within
	the toroid.)

        At time t=0 the propeller has a two starting coordinates, C0 and
	C0', these represent the end points of the propeller.

	Let the propeller rotate PR radians as it moves TR radians AROUND the
	inside of the toroid.

	Provide a set of equations for Ct and Ct', the end points of the
	propeller at time t.

	David
T.RTitleUserPersonal
Name		DateLines
630.1A tad more information might facilitate thingsNOBUGS::AMARTINAlan H. MartinWed Dec 17 1986 15:1620
Not that I'm about to solve the problem, but I think a successful solution
will involve inventing at least two more pieces of information.  You might
want to invent names for them, rather than making people come up with
them and making the solutions slightly harder to compare.

1.  The radius of the torus, measured from the center of the hole to the
center of a circular cross-section.  (Two times pi times this number
will determine the distance that the propeller's hub travels in one
trip around the torus).  Without this information, you couldn't distinguish
between a normal sized donut and a garden hose wrapped around the Earth.

2.  The angular speed of the propeller.  Either the spin or the "forward"
speed will suffice, since they are related by TR and PR.  Without knowing
this, I could state that Ct=C0,Ct'=C0' is a solution, by assuming that
the angular speeds are zero.

Also, do you have a preference for the kind of coordinate system you want
the answer expressed in?  Spherical or cylindrical coordinates centered on
the toroid's axis are probably easier than rectangular coordinates.
				/AHM/THX
630.2CLT::GILBERTeager like a childWed Dec 17 1986 15:5323
    Let the toroid be centered at the origin (in cartesian coordinates),
    sortof in the xy-plane.

    The circular cross-section of the toroid is R, the distance from
    the origin to the center of this cross-section is S.

    At time t, the center of the propeller is at (S sin(Kt+L),S cos(Kt+L),0).

    At time t, the end-points of the propeller are at:

	(S sin(Kt+L) +/- R sin(Mt+N) cos(Kt+L),
    	 S cos(Kt+L) +/- R sin(Mt+N) sin(Kt+L),
    	             -/+ R cos(Mt+N))

    Or the same thing, with the -/+ replaced by +/-, since the propellor
    might be turning counter-clockwise.

    Since the propeller rotates PR radians as it moves TR radians around
    the inside of the toroid, we have M/K = PR/TR.

    The initial conditions impose constraints on L and N.  These can be
    solved by a couple of inverse trigonometric functions (hint: use
    the z-coordinate first).