Normal frequencies and normal modes of a multi-part system

In summary, the problem at hand involves a thin rod suspended by two identical vertical springs attached to the ceiling. The system is constrained to move in one vertical plane, and the goal is to find the normal frequencies and modes of small oscillations. The suggested coordinates for this system are r, φ, and \alpha, with r and φ representing the position of the rod's center of mass relative to a fixed point between the springs, and \alpha representing the angle of tilt of the rod. The potential energy includes both gravitational and spring potentials, and the lengths of the springs can be determined by finding the coordinates of their respective ends in the suggested coordinate system.
  • #1
the_kid
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Homework Statement


***This is problem 11.29 in Taylor's Classical Mechanics***
A thin rod of length 2b and mass m is suspended by its two ends with two identical vertical springs (force constant k) that are attached to the horizontal ceiling. Assuming that the whole system is constrained to move in just the one vertical plane, find the normal frequencies and normal modes of small oscillations. [Hint: It is crucial to make a wise choice of generalized coordinates. One possibility would be r, φ, and [itex]\alpha[/itex], where r and φ specify the position of the rod's CM relative to an origin half way between the springs on the ceiling, and [itex]\alpha[/itex] is the angle of tilt of the rod. Be careful when writing down the potential energy.)


Homework Equations





The Attempt at a Solution


Right now, I'm just trying to set up the Lagrangian for this system, but the potential is giving me some problems. I recognize that there is a gravitational potential and a spring potential. I'm attempting to find the positions of the ends of the rod relative to some fixed point; however, I'm not sure what fixed point I should choose. Ultimately, I'm trying to find the lengths of the springs in terms of r, φ, and [itex]\alpha[/itex]. I'm getting a little frustrated with the trig and trying things out. Could someone point me in the right direction?
 
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  • #2
Using the suggested coordinate system, write down the (xR, yR) coordinates of the right end of the rod. The top of the right spring is located at (b, 0). The distance between those two points is the length of the right spring. You can do the same thing for the left spring with the top end connected to the ceiling at (-b, 0).
 

1. What are normal frequencies and normal modes?

Normal frequencies and normal modes are properties of a multi-part system that describe the natural oscillations of the system. Normal frequencies refer to the specific frequencies at which the system naturally vibrates, while normal modes refer to the corresponding patterns of motion.

2. How are normal frequencies and normal modes calculated?

Normal frequencies and normal modes can be calculated using mathematical equations that take into account the mass, stiffness, and geometry of each component in the system. These calculations can also be performed using computer simulations.

3. Why are normal frequencies and normal modes important in studying multi-part systems?

Normal frequencies and normal modes provide valuable insight into the behavior and stability of a multi-part system. By understanding the natural vibrations and patterns of motion, scientists can better predict how the system will respond to external forces and make improvements to its design.

4. Can normal frequencies and normal modes change over time?

Yes, normal frequencies and normal modes can change over time due to factors such as changes in environmental conditions, wear and tear on the system, or modifications to its components. These changes can be observed and analyzed to gain a better understanding of the system's behavior.

5. How are normal frequencies and normal modes used in real-world applications?

Normal frequencies and normal modes have various applications in engineering and science, such as in the design of bridges, buildings, and other structures to ensure their stability and safety. They are also used in the field of acoustics to study sound and vibration, and in the study of molecular and atomic vibrations in chemistry and physics.

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