UConn Math Club
Oscillations of a Hanging Chain
Michael Rozman (University of Connecticut)
Wednesday, November 1, 2017
In this tak we analyze small-amplitude oscillations of a chain, suspended from a fixed point and free at its lower end. (A chain is just a metaphor for an ideally flexible, uniform, and inextensible slender object.) When the lower end is slightly disturbed, the chain is assumed to oscillate in a vertical plane about its position of equilibrium. We are interested in the frequencies of the oscillations and the shape of the chain.
This problem was first considered by Daniel Bernoulli and Leonhard Euler. Bernoulli submitted his results to the St. Petersburg Academy in 1732. Euler submitted his own version to the Academy in 1735, and returned to the subject 40 years later, in 1774.
We are going to derive the equation of the chain's motions, and present its solution as a definite integral that cannot be expressed in terms of elementary functions. Instead we find the leading term of the asymptotics of this integral, obtaining the frequencies and shapes of the normal modes. An arbitrary motion of the chain can be expressed as a linear combination of the normal modes, since the governing equation is linear.
Comments: Free pizza and drinks!