Normal frequencies of 2 pendula connected with a massless rigid rod

[smithg86]

Homework Statement

I need to figure out the normal frequencies (or eigenfrequencies) of a system of two simple pendula (call them A and B), connected with by a massless rigid rod at an arbitrary distance from the pivot point or the mass. That is, the pendula are not connected at their masses, but at some point along the strings.

Homework Equations

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The Attempt at a Solution

My textbook only deals with the case where the pendula are connected at their masses. But, to find the normal frequencies of that system, they first solved a system differential equations, which described the position of A and B as a function of time. Also, they assumed the angle made with the vertical was small enough so sin ($$\vartheta$$) $$\approx$$ $$\vartheta$$. Further, they assumed that the amplitudes of the oscillations were small enough so the motion of the pendula could be looked at in the horizontal direction only.

I was able to follow the derivation of the problem in the textbook, but I'm having trouble picturing how the system would operate. Do you consider the part of the system above the rigid rod to be a single pendulum, with two strings? And the parts below the rigid rod are considered regular pendula, but attached to a moving structure?

Can someone give me a hint?

Oh, and I'm pretty sure there are only 2 normal frequencies: optical and acoustic...

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edit:
Also, the coupling rod is positioned such that there is no twisting of the system. it is positioned high up enough along the strings so this effect can be ignored.

 

[Gokul43201]

What do you mean by "twisting"? Do you just mean that the entire setup is always coplanar, or something else?

PS: Also, please write down the original question EXACTLY as it was given to you.

 

[smithg86]

What do you mean by "twisting"? Do you just mean that the entire setup is always coplanar, or something else?

PS: Also, please write down the original question EXACTLY as it was given to you.

Gokul,

This problem arose during a lab. Let me explain:

Two identical simple pendula hung from a support rod, about 4-5 inches apart from each other. If set in motion, the motion of pendula A did not affect the motion of pendula B, and vice versa. The length of the pendula strings is known.

A lightweight (almost massless) rigid coupling rod was attached to the two pendula strings: the two pendula were now coupled. The coupling was attached at the same position along each of the strings. The height of the coupling could be varied throughout the experiment.

The initial conditions of two normal modes were known: (1) equal displacement of the pendula in the same direction; (2) equal but opposite displacement of the pendula. In both cases, the pendula start at rest. The frequency of oscillation was recorded for various positions of the coupling along the strings.

Regarding the 'twisting', we were instructed: "If you put the bar too low (lower than about 1/2 of the length of the strings) you will find that there is substantial coupling to another normal mode of oscillation, the torsional mode."

For notation purposes, the distance from the coupling to the top of the pendula was called d.

Regarding the motion of the pendula:
-the plane in which the coupling rod and the support rod lie is the Y-Z plane, where the Z-axis is 'up'
-the plane in which the pendula swing is the X-Z plane. For simplicity, we assume the pendula only experience horizontal motion. There is no motion in the Z-direction.

Forgive me, I misstated the question in my first post. I need to show that the frequency of one of the normal modes is constant w.r.t. changing d. I also need to show that the frequency of the second normal mode is related to the sq root of d. I assumed that finding a general formula for the normal frequencies of the system was the best way to show what was required.

 

[Gokul43201]

Looks like you have 6 parameters and 1 constraint equation (from the rod). You must realize that torsion is necessarily an important mode for the lengths of string above the connecting rod. For the sections below the rod, you could possibly approximate the motion to be restricted to XZ plane (but it would be unjustifiable to apply the same approximation for the string above the rod).

Each of the two sections of string above the rod can described using two angles each $(\theta_1, \phi_1)$ and $(\theta_2, \phi_2)$. The constraint equation that reduces these 4 variables to 3 independent co-ordinates is one that sets the distance between the lower ends of these two upper sections equal to the length of the rod. Then, you have two additional angles (in the XZ plane) to describe the lower sections of string relative to the rod-attachment points.

That's 5 independent co-ordinates - what a pain!