The spring that connects two equal pendulums in a conventional coupled pendulums demonstration is replaced here by a thread, with a small weight in the middle of it. A horizontal pen, attached to this weight, writes the trace of the weight on a vertical piece of paper. The (equal) bobs (1.5 kg) are suspended from the ceiling with nylon threads (250 cm, i.e. T ~ 3.15 sec), 120 cm apart; the connecting thread (160 cm) is attached to the pendulums (with clips) 90 cm above the weights; it carries a weight (15 g; paperclips in a plastic cup) to which a fineliner pen is attached. The pendulums swing in the same plane (rather than in perpendicular planes for uncoupled pendulums: lissajous figures). The normal modes of the system (pendulums in phase or with 180 deg phase-difference, respectively) are represented by horizontal or vertical straight lines on the paper.
Harmonogram: Eventually, pronounced thick lines form 9 nested parallellograms with alternating orientations. The ovals inscribed in a parallellogram degenerate with time to a straight line over one or the other diagonal of the parallellogram (representing the situation with one or the other bob at rest). In passing a diagonal the sense of rotation of the pen is reversed. Each parallellogram corresponds to 36 periods of the system (~ 113 sec).
Timecuts at 0:10, 0:20, 0:31, 0:50, 1:01, 1:11, 1:22
Different graphs can be obtained by changing the parameters used here. In particular, the two pendulums can be made unequal.

The graph is essentially the same as the harmonogram of uncoupled pendulums, in parametric form: ..................................... ........................x = p*sin(w1*t) ; y = p*sin(w2*t),................................
with w2 - w1 = delta , delta/w much smaller than 1, ................. (w1/w2 = n1/n2, with n1 and n2 natural numbers, give simple lissajou curves; p is a frictional damping factor, p = exp(-A*t); 'w' means omega). This is rather surprising, since the experimental setup is quite different: the pendulums swing in the same plane (rather than in perpendicular planes), and 'feel' each other through a connecting spring, with springconstant k (F=k*L).
The explanation of this similarity of graphs makes use of normal mode analysis and some (simple) geometry:
Solving the equations of motion of the two pendulums for the normal coordinates gives .....................................................
x1 + x2 = sin (w1*t)........w1 = SQR(g/l)
.................................................................

x1 - x2 = sin (w2*t).........w2 = SQR(g/l + 2*k/M)

(g=acceleration of gravity, l=length of pendulums, x1 and x2 are the horizontal departures from equilibrium to the right, respectively, M is the mass of the bobs, k is an effective force constant (see below).
The thread, with total lenght 2*l2, carrying the mass m and the pen, forms an isosceles triangle, with base
2*l3 and sides l2 It's height
h = SQR(l2^2 - l3^2)
so, to first order:
dh = -(l3/h)*dl3 = -(l3/h)*( x2 - x1 )/2 (=y) ; ....................
x =( x2 + x1)/2;


(neglecting higher order terms), so x and y are proportional to the normal coordinates of the system, and behave as sin(w1*t) and sin(w2*t).(xy: orthogonal frame in graph).
An effective forceconstant can be derived :
The horizontal forcecomponent exerted by the string (carrying the weight m2) on the pendulum,
.....................f = m2 * g * (((l2/(l3-x1))^2 - 1)^(-0.5) ,
can be plotted as f(x1), together with the line m1*g*x1/l. The tangent at the intersection of the line with the curve (equilibrium) gives the forceconstant k = 0.5 N/m (?).

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