Derivation of Vibration equation

In summary: The rotor has a mass, m0, and the stator does not. The external force, F, is acting on the system and it causes the rotor to rotate. The stator does not rotate because it does not have a mass.
  • #1
JI567
174
0
Can somebody please tell me what should be present in the free body diagram for this system below. Also please mention the arrow details like which direction each force in the free diagram should be acting in. I really need to understand this concept...
upload_2014-11-11_0-41-54.png
 
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  • #2
You need to draw two FBDs, one for the housing and one for the rotor. Give that a shot and get back to us. It does you no good if I simply work the problem for you.
 
  • #3
I did the diagrams, are they correct?

upload_2014-11-11_7-7-16.png
upload_2014-11-11_7-7-59.png
 
  • #4
No. Remember Newton's Third Law. For each internal force (acting between the two bodies), you should show a force and its reaction, one on each FBD.
 
  • #5
Dr.D said:
No. Remember Newton's Third Law. For each internal force (acting between the two bodies), you should show a force and its reaction, one on each FBD.

I have no idea what you are talking about. Are you talking about normal forces in each diagram acting in opposite direction to the mg? Please draw the additional arrows if that's not it
 
  • #6
In the left-hand FBD, where is the force holding up the rotor?

There are other difficulties as well, but this should get you thinking..
 
  • #7
You mean I need to put a centrifugal force on the left hand FBD as well? Okay alright that's fine...Is this the final FBD diagram then? Are there any other forces? Tell me if its correct or needs more forces. thanks

upload_2014-11-12_15-2-58.png
upload_2014-11-11_7-7-59-png.75317.png
 
  • #8
No, it is not correct. There are additional forces required, and the ones you have are not entirely correct.
 
  • #9
How can there be other forces? Could you explain. I think I have included all my forces, gravitational forces, damping force, spring force, centrifugal force. What else force do you want...unless you mean reaction forces below the Blue ground in opposite directions to the spring and damping force? Its not like I am not trying to solve this...if you see I am missing something then can you just tell me.
 
  • #10
You show a force on the rotor of mo*ro*w^2 (I presume that is a force, although there is no arrow head). You show a force (a line without arrow head) on the block with the same label but in a different direction. How can this be?
 
  • #11
These diagrams has to be right! There can't be anything wrong with this one now. I got the equation of motion and it makes sense. Tell me if you are still not satisfied...Let me know if you are still seeing problems

as the general equation satisfies

M*x"+c*x'+kx = Fcoswt
where F is the external force acting on the system and F = m0*r*w^2 so now I can just write the general equation of motion of the system as
(M+m0)*x"+c*x'+kx = m0*r*w^2 cos wt

upload_2014-11-13_1-33-51.png
upload_2014-11-13_1-35-40.png
upload_2014-11-13_1-37-11.png
 
  • #12
Looks like a step backward. You show on the rotor a force labelled m0*r*w^2 but the reaction to that force does not show on the FBD for the stator. Why is that?
 

1. What is the vibration equation?

The vibration equation is a mathematical model that describes the motion of a vibrating object. It is based on the principles of Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass.

2. How is the vibration equation derived?

The vibration equation is derived using the principles of classical mechanics and differential equations. It involves analyzing the forces acting on the vibrating object, such as gravity and spring forces, and using mathematical techniques to express these forces in terms of the object's displacement, velocity, and acceleration.

3. What are the applications of the vibration equation?

The vibration equation has many practical applications, including in the fields of mechanical engineering, civil engineering, and structural dynamics. It is used to design and analyze structures and machines that are subjected to vibrations, such as bridges, buildings, and vehicles.

4. Are there any limitations to the vibration equation?

While the vibration equation is a powerful tool for analyzing vibrating systems, it does have some limitations. It assumes that the vibrating object is a single, rigid body and that all forces acting on the object can be represented by a single equation. In reality, many vibrating systems are more complex and may require more advanced mathematical models.

5. Can the vibration equation be solved for any type of vibration?

The vibration equation can be solved for simple harmonic motion, which is a type of vibration where the restoring force is proportional to the displacement of the object. However, for more complex types of vibration, such as random or non-harmonic motion, the equation may need to be modified or more advanced techniques may be required to solve it.

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