From: israel@math.ubc.ca (Robert Israel) Subject: Re: Rotating chain Date: 21 Jan 1999 01:03:48 GMT Newsgroups: sci.math Keywords: Shape of a rotating chain with top end fixed In article <781fqp$9ak$1@mail.pl.unisys.com>, "qaz" writes: |> It appears that a chain, suspended from one point, can be "twirled around" |> in such a way that it is always in a vertical plane, and that plane rotates |> at a uniform rate about a vertical axis through the point of suspension. |> There seem to be several stable shapes that it can assume, including ones |> that intersect the axis of rotation several times. What is known about these |> curves? The full nonlinear equation for the rotating chain is a rather complicated thing, but let's look at a linear approximation which should be good when the chain is almost vertical. Describe the chain by a function x = x(y) where x is the distance from the vertical axis and y the distance up the axis, 0 <= y <= L where L is the length of the chain (it's convenient to consider the point of attachment of the chain as y = L). The tension in the chain at y is approximately T = p g y where p is the density of the chain, and for equilibrium we need (T x')' = p w^2 x where w is the angular velocity. Thus we get the second order DE y x" + x' + k x = 0 where k = w^2/g. It has a basis of solutions J_0(2 sqrt(k y)) and Y_0(2 sqrt(k y)) where J_0 and Y_0 are the zero-order Bessel functions of first and second kinds. However, since Y_0 has a logarithmic singularity at the origin, the only solutions which are finite at y=0 are c J_0(2 sqrt(k y)). Now we want x(L)=0, i.e. 2 sqrt(k L) must be one of the zeros of J_0. Thus we have a certain discrete set of values of k L = w^2 L/g at which there are nonzero equilibrium solutions. The n'th value (n = 1, 2, 3, ...) corresponds to a solution that intersects the axis n times (including the point of attachment). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2