Answer Happy

[VW1c] (0.0) 5=N 4=N-1 Consider a compound pendulum consisting of N identical masses which are connected by light strings of length a, and which can swing freely in the 2-y plane under the influence of gravity. We label the masses starting from 0 at the bottom to N - 1 at the top. For convenience we also add an extra fixed mass labelled by N at the top of the chain where it is fixed to the ceiling. The figure shows the N = 5 case. The position of mass n € {0, 1,...,N} relative to the origin at the top of the chain is in = {;i+ynj. We will use the hor- izontal displacements {20,..., IN-1} as our coordinates. 3 2 2 31 0 For small amplitude oscillations, meaning In| <a, the equation of motion of the chain is given by än = 2 (n({n+1 – 22n + In-1) + (In+1 – <n)] n = 0, 1,...,N – 1. for a
This can be derived using the same procedure as you used for the chain of masses coupled by springs. Of course, we can write these equations in matrix form and find the normal modes of the discrete chain by calculating the eigenvectors and eigenvalues of the matrix. Here we will consider the continuum limit of this system, in which the pendulum becomes a continuous hanging chain/rope. In particular, we want to determine the frequencies and shape functions for the modes of the chain. Below is a figure showing the first four modes. These can be generated by applying a periodic driving force to the top of chain which is in resonance with one of the modes. The total length of the pendulum is L = aN. In the continuum limit we take N + 0 and a + 0) in such a way that L = aN remains constant. We then expect the masses to lie on a smooth curve described by a function e(y, t) where y € (0, L) is vertical height measured from the bottom end of the pendulum when it is hanging in equilibrium. In fact, we will identify In = x(y = na, t).
In general we can either think of y = na as the height of mass n, or as the position of mass n as measured along the curved chain. Within the small amplitude approximation we have already made there is no difference between these two viewpoints. (a) Rewrite the equation of motion so that it applies to the function z(y,t) at a fixed point y = na. The n-label should no longer appear anywhere in the equation. (b) Take the continuum limit of the equation you found above to obtain the chain's equation of motion as a partial differential equation for x(y, t). (c) We are interested in the normal mode solutions of this equation. If we proceed via separation of variables we can write z(y,t) = f(y)g(t). As usual we will find that g(t) = A cos(wt + o) with w the mode frequency. Use this to derive an ordinary differential equation for f(y), the shape function of the mode. (d) The differential equation you found above is quite famous, but needs to be rewritten slightly to bring it into its standard form. Perform a change of variables from y to z=avy with a = 2w/Vg and show that the equation becomes [df + + 1] df + dz 0. dz2 Hint: It is easier to start with the given form and work your way back to your equation. Begin by expressing and in terms of derivatives to y.
The differential equation above is clearly a special case of the Bessel differential equation, with solutions J, and Yo. However, since Yo (2) diverges at z = 0, we must exclude it on physical grounds. We therefore conclude that f = Jo. In terms of the original coordinates our solution for f(y) is f(3) = Jo (2 Vy/9) However, this is not the full story since we have not yet taken into account the boundary condition that the top of the chain is held fixed. This condition will restrict the possible values of w and lead to a discrete set of mode frequencies, just as we found for the string. (e) Impose the boundary condition and provide an expression for Wn, the frequency of the n'th mode, in terms of the various quantities we have defined. Note: You can insert your solution for wn into the Mathematica file VW1.nb to see whether the boundary condition is satisfied. You do not need to submit this file. The shape of the modes should be a good match for what was seen in the photos above.