Vibration of a cantilever beam

[TW Cantor]

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 I am 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!

 

[TW Cantor]

Oh, and I forgot to say that the pipe is vertical and that the velocity and deceleration is in the x axis

 

[SteamKing]

What I know:

Pipe external diameter = 0.1m
Pipe internal diameter = 0.75m

You have the D_ext < D_int

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

Any help or tips would be appreciated!

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.

 

[TW Cantor]

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 I am 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, d_ext = 0.1m
Pipe internal diameter, d_int = 0.075m
Tip mass, m_t = 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 = ((m_t×a)×L³)/(3×E×I)

pipe deflection as a UDL = (((m_pipe×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×ω_n×t)}×sin(ω_d×t + Φ)

where C is a constant determined from initial conditions, ω_n is the natural frequency, ω_d 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

 

[Dr.D]

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.