Engineering Homework problem - Pendulum oscillatory system

[aligator11]

Homework Statement

Mechanical Vibrations, #Lagrange #eigenfrequency

Relevant Equations

Lagrange equations, eigenfrequency equation for the given system

Hi All,

Anyone willing to help out in explaining what eigenfreuqncy for this oscilatory system, would be? Also if anybody knows the equation to calulate this stuff please, if you're willing to share I'd be greatful!

Thanks, regards.

 

[BvU]

Hi,

what have you done yourself, so far ? And what is present in your toolbox to help solve this exercise ?

Did you notice the limitation to small oscillations ? Did you make use of it ?

https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

 

[aligator11]

Truthfully I'm not in possession of any course materials from vibration mechanisms. What I do have are educative videos, lectures provided by my lecturer and notes which I keep track of. I know that sucks but the rest I have to find out on my own which makes me anxious as I'm not sure sometimes if I'm even looking correctly.

So far I know how to calculate a 1DoF natural frequency with given parameters. For the MDoF system I have frankly no clue what should I even look for. Unfortunately, my professor hands out these Microsoft form tests before each lecture, but I'm not certain if I will learn anything during the lecture too, therefore I came here for help.

Cheers.

 

[aligator11]

Also, the only potentially valid equation which I managed to find is as follows:

 

[BvU]

hands out these Microsoft form tests

Funny sheets, with a question 1 and then a question 5.
Funny they make a "no gravity field" limitation too.

Did you notice I asked a few questions ? They were aimed at you !

Let's start with 1:
What is the small angle approximation for a pendulum ?

 

[aligator11]

1. Small oscilations = vibrations were values of cos/sin(angles) may be approximated to angles [rad]. For cos(phi)=1, sin(phi)/tan(phi)=phi

 

[BvU]

So what does that leave for question 1 ?

 

[aligator11]

the first box should be checked instead of the 2nd one

 

[BvU]

I agree that box 2 can be discarded ($g = 0$) and for box 2 and 3 the small angle approximations $\sin x = x, \ \cos x = 1$ have not been implemented.

Explain why you think that box one is better than the alternative ...

 

[aligator11]

The difference between boxes 4 and 1 is the change in angular displacement for PE, where it varies with (+) and (-) sign for each answer. I believe that the correct solution should be one containing the element with (phi1-phi2) as we are superimposing one displacement onto the second one. That means that by exciting the first pendulum the impulse through the spring continues to flow until it reaches the second pendulum. The motion then becomes for the given moment unstable as the second pendulum is excited and the spring receives an impulse that is then directed in the negative direction (-phi2).

 

[aligator11]

the first box should be checked instead of the 2nd one

so I meant box 4 not box 1

 

[BvU]

I agree. The spring energy depends on $\Delta \phi$, not on the sum.