Vibrations in cantilever beams

In summary, the author is trying to find a quick and dirty way of estimating the amount of time it takes for an arm to stop vibrating. He is planning on using the equation x(t)=e^(-ζωt)[Asin(ωt)+Bcos(ωt)]+C to find the time constant. He is also looking for information on how to find the damping values for a cantilever beam made out of aluminium.
  • #1
Feodalherren
605
6
Hi fellow mechanical engineers,

I am designing a rather simple excel program for work that deals with vibrations in robots. Imagine a robot that is made up of linear axes that can move in x,y,z sort of like a 3D printer, take a look at this picture:
https://pasteboard.co/8hvV5vf.png

Focusing on the part that is highlighted in pink, imagine that it is a solid beam. Now imagine that it picks up some mass and then starts moving in the x direction and then stops before it starts moving in the y direction. I need to find a quick and dirty way of estimating the amount of time that it takes for the arm to stop vibrating.

I'm planning on going back to my systems and vibrations textbook for this, more precisely the following equation:

mx''+cx'+kx=F

My hope is to get a rather simple solution of the form

x(t)=e^(-ζωt)[Asin(ωt)+Bcos(ωt)]+C

which I can use to find the the time constant and then use that to find an approximate time for when the vibrations are small enough for the robot to start moving again.

Have I simplified this problem too much? I'm starting to think so since I can't find any information regarding a damping coefficient as a material property, which I just assumed that it was. I was going to ignore damping from air and model the beam as a cantilever beam.

I found this NASA paper on it:
https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19650021096.pdf

which seems to suggest that it's far more complicated than I originally thought. So my question is, is it possible to find the damping values for a cantilever beam made out of aluminium anywhere?
 
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  • #3
Thank you, that was very helpful!
 

Related to Vibrations in cantilever beams

1. What are vibrations in cantilever beams?

Vibrations in cantilever beams refer to the oscillatory motion or movement of a beam that is only supported at one end. These vibrations can be caused by various factors such as external forces, material properties, and structural design.

2. How do vibrations affect cantilever beams?

Vibrations can have both positive and negative effects on cantilever beams. On one hand, they can cause the beam to bend and deflect, which can be useful for certain applications such as in musical instruments. On the other hand, excessive vibrations can lead to structural failure and damage.

3. What factors contribute to vibrations in cantilever beams?

There are several factors that can contribute to vibrations in cantilever beams, including the material properties of the beam, the applied load or force, and the shape and size of the beam. Other factors such as temperature, damping, and external vibrations can also play a role.

4. How can vibrations in cantilever beams be controlled?

There are various methods for controlling vibrations in cantilever beams. One approach is to modify the design of the beam, such as changing its length or cross-sectional area, to reduce its natural frequency. Another method is to add dampers or other devices to absorb or dissipate the vibrations.

5. What are the practical applications of studying vibrations in cantilever beams?

Understanding vibrations in cantilever beams is important for a wide range of industries and applications. This knowledge can be applied in the design and construction of buildings, bridges, and other structures to ensure their stability and safety. It is also relevant in fields such as aerospace, mechanical engineering, and materials science.

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