Axial, torsional and pendulum modes of vibration(harmonic motion)

[intellegense]

Homework Statement

Hello,

This is more of a conceptual question rather than a problem with numbers.
I was wondering if anyone new if there is a relationship between axial, tortional and pendulum vibration. The context is a mass hanging off a spring.

ie will the natural frequency of vibration be the same for all 3 types given the same mass and spring constant?

Homework Equations

The Attempt at a Solution

I tried an experiment and my apparatus is simply a mass hanging off the end of a spring. The only materials i had were an old school stop watch and a ruler so i had a lot of difficulty getting any measurements. none fo which were any real use

i know that the natural frequency of axial vibration is

w(n)= sqrt(k/m)

where k= spring constant
m= mass
w(n)= natural frequency

i worked out the spring constant by graphing different masses versus deflection.

I also know the formula for frequency of a pendulum with a mass and a string, but not with a spring.

my common sense tells me that the natural frequency for all three modes of vibration will be the same when using the same spring and mass but i have no proof to back this up.

Ive searched the internet and haven't found anything relevant.

Any hints would be great.

 

[anathema]

Thank you for your interesting question. I can tell you that there is indeed a relationship between axial, torsional, and pendulum vibration. The key factor in determining the natural frequency in all three cases is the stiffness of the system. In your experiment, you correctly identified the spring constant as a crucial factor in determining the natural frequency of axial vibration. Similarly, the stiffness of the string or rod used in torsional vibration and the stiffness of the pendulum string will also play a role in determining the natural frequency in those cases.

In general, the natural frequency of a system is determined by its stiffness and mass. So, if you have the same mass and stiffness in all three cases, the natural frequency should be the same. However, there may be slight variations due to the specific characteristics of the material used in the spring, string, or rod. Additionally, the length and shape of the pendulum string may also affect the natural frequency in that case.

I would suggest conducting further experiments with more precise equipment to get more accurate measurements and confirm your hypothesis. You can also refer to studies and research papers on the topic to see if there are any specific relationships or formulas between the three types of vibration.

I hope this helps. Good luck with your research!

 

[m2003]

Hello,

Thank you for your question. The three modes of vibration you mentioned - axial, torsional, and pendulum - all involve harmonic motion, which is a type of oscillatory motion where the restoring force is proportional to the displacement from equilibrium. In your experiment, the mass hanging from a spring exhibits harmonic motion as it oscillates up and down, so the natural frequency of vibration for this system can be calculated using the formula you provided: w(n) = sqrt(k/m). This frequency will be the same for all three modes of vibration because they all involve the same system - a mass hanging from a spring. However, the amplitudes and patterns of oscillation may differ depending on the mode of vibration. For example, in axial vibration, the mass moves in a straight line, while in torsional vibration, the mass rotates around a fixed axis. In pendulum vibration, the mass swings back and forth in an arc.

In summary, the natural frequency of vibration will be the same for all three modes, but the specific characteristics of each mode will vary. I hope this helps answer your question. If you have any further inquiries, please don't hesitate to ask. Good luck with your experiment!