Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Vibration of a cantilever beam

  1. Feb 21, 2015 #1
    Hi there!

    Im trying to do an analysis in Abaqus of a cantilever pipe, with a tip mass at the free end, that is decelerating to a stop at 10 m/s2 from 10m/s, causing it to vibrate. To validate my results im doing some handcalcs. I have done a static analysis and calculated the maximum deflection of the beam, and my results match. I would also like to estimate the time it takes for the pipe to stop vibrating, could anyone help me out here?

    What I know:

    Pipe external diameter = 0.1m
    Pipe internal diameter = 0.75m
    Tip mass = 200kg
    Pipe density = 7800 kg/m3
    Young's modulus = 207GPa
    Poissons ratio = 0.3
    damping ratio = 0.1
    Pipe length = 1m

    What I have tried so far:

    Well to work out the beam deflection I summed the effects of the inertia of the tip mass, as well as the inertia of the pipe mass itself. Shown in the two equations below:

    mass deflection = ((mass*acceleration)*length3)/(3*Young's*SecondMoment)

    pipe deflection = (((mass*acceleration)/length)*length4)/(8*Young's*SecondMoment)

    Then to take into account time and the damping coefficient i just multiplied this answer by (1-0.1)time

    Though this seems to simple and my results to not much to those from my simulation.

    Sorry I don't know how to insert my equations as equations!

    Any help or tips would be appreciated!
     
  2. jcsd
  3. Feb 21, 2015 #2
    Oh, and I forgot to say that the pipe is vertical and that the velocity and deceleration is in the x axis
     
  4. Feb 22, 2015 #3

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    You have the Dext < Dint

    Hit the ∑ on the Tool Bar line at the top of the text box. This will give you access to basic math symbols and Greek letters.

    Also, you can make reasonable facsimiles of formulas w/o writing out everything; for example, use E for Young's modulus, I for second moment of area, etc.
     
  5. Feb 22, 2015 #4
    Hi there!
    Im trying to do an analysis in Abaqus of a cantilever pipe, with a tip mass at the free end, that is decelerating to a stop at 10 m/s2 from 10m/s, causing it to vibrate. The pipe is vertical, fixed at the top with the mass at the bottom, and it is moving in the x axis. To validate my results im doing some handcalcs. I have done a static analysis and calculated the maximum deflection of the beam, and my results match. I would also like to estimate the time it takes for the pipe to stop vibrating, could anyone help me out here?

    What I know:

    Pipe external diameter, dext = 0.1m
    Pipe internal diameter, dint = 0.075m
    Tip mass, mt = 200kg
    Pipe density, ρ = 7800 kg/m3
    Young's modulus, E = 207GPa
    Poissons ratio, v = 0.3
    damping ratio, ζ = 0.1
    Pipe length, L = 1m
    Deceleration, a = -10m/s2
    Second moment of area, I = 1.076 * 10-6

    What I have tried so far:

    Well to work out the beam deflection I summed the effects of the inertia of the tip mass, as well as the inertia of the pipe mass itself. Shown in the two equations below:

    mass deflection = ((mt×a)×L3)/(3×E×I)

    pipe deflection as a UDL = (((mpipe×a)/L)×L4)/(8×E×I)

    Then to take into account time and the damping coefficient i just multiplied this answer by (1-ζ)time

    Though this seems to simple and my results to not much to those from my simulation. Looking back at my old notes it seems I would have to do a second order system analysis on the beam. Since it is undamped I guess the appropriate equation would be:

    x(t) = C×e(-ζ×ωn×t)×sin(ωd×t + Φ)

    where C is a constant determined from initial conditions, ωn is the natural frequency, ωd is the damped natural frequency and Φ is the phase shift.

    Does this seem like the right way of going about it? Any ideas on where to start? Thanks
     
  6. Feb 26, 2015 #5
    I really wonder where you got your expression (1-ζ)t to account for damping? Also, your earlier form for this equation suggests that you are using ζ=0.1, an extremely high value for an all metal structure.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Vibration of a cantilever beam
  1. Cantilever beam theory (Replies: 0)

Loading...