# Vibrations in cantilever beams

## Main Question or Discussion Point

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 text book 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?

• 