# How to Estimate Vibration Stop Time for a Decelerating Cantilever Pipe?

• TW Cantor
In summary: Considering that most metal structures have a ζ value in the range of 0.01-0.3, your results might not be realistic.In summary, the author has tried 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. To validate his results, he is doing some handcalcs. He has done a static analysis and calculated the maximum deflection of the beam. His results match. He would also like to estimate the time it takes for the pipe to stop vibrating. Someone could help him out here.
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!

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

TW Cantor said:
What I know:

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

You have the Dext < Dint

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.

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, 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

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.

## 1. What is a cantilever beam?

A cantilever beam is a structural element that is supported at only one end, while the other end is free to move. It is commonly used in bridges, buildings, and other structures to support loads and resist bending.

## 2. What causes vibration in a cantilever beam?

Vibration in a cantilever beam is caused by external forces acting on the beam, such as wind or seismic activity. It can also be caused by the natural frequency of the beam being excited by the applied loads.

## 3. How is the natural frequency of a cantilever beam calculated?

The natural frequency of a cantilever beam can be calculated using the formula: f = 1/2π * √(EI/m), where E is the modulus of elasticity, I is the moment of inertia, and m is the mass per unit length of the beam.

## 4. How can vibration in a cantilever beam be reduced?

Vibration in a cantilever beam can be reduced by increasing the stiffness of the beam, adding mass to the free end, or using damping materials to absorb the energy of the vibration. Proper design and construction techniques can also help minimize vibration.

## 5. What are the practical applications of studying the vibration of cantilever beams?

Studying the vibration of cantilever beams is important in structural engineering to ensure the stability and safety of buildings, bridges, and other structures. It is also useful in the design of mechanical and aerospace systems, such as aircraft wings and helicopter blades, to prevent unwanted vibrations that can cause damage or failure.

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