This paper might help https://link.springer.com/article/10.1007/s40030-0...

If you want you can use beam elements say in free fea demo software like Strand7 see the website and the demo link at the top there for downloading it

There is an analytical method called the theory of receptances which can be used to estimate or calculate the overall response of two simple systems that are joined together. Sadly that is everything I can remember about it, it has been of no use to me in my career.

Idly thinking about it, the frequency will obviously be higher than that of a 9m 10cm pole, and may be higher than that of a 9m 15cm pole. It will be lower than a 6m 10cm pole.

Cheers

Greg Locock


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R.D Blevins Formulas for dýnamics, acoustics and vibration. Table 4.9. Alternatively one can derive an approximation using the static deflection, delta, as a f=1/(2*pi)*(g/delta)^(1/2).

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There is an analytical method called the theory of receptances which can be used to estimate or calculate the overall response of two simple systems that are joined together.

Assuming receptance is the (frequency-dependent) ratio force to displacement, then I don't think this will work, because the boundary conditions in this geometry are not defined by just a force and displacement... they also include a slope and a moment. Maybe I am misunderstanding your suggested approach.

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. Alternatively one can derive an approximation using the static deflection, delta, as a f=1/(2*pi)*(g/delta)^(1/2).

As you said an approximation. It would be exact for massless beam supporting lumped mass attached at the location where we calculate the static deflection. Obviously a distributed-mass beam with no attached mass is a long way from that.

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Here's my roadmap for a solution:
Harris’ Shop and Vib Handbook 6th ed table 7.4 gives the general solution for modeshape of a distributed mass uniform beam based on Euler Bernouli model:
wi(x) = Ai * cos(Ki*x) + Bi * sin(Ki*x) + Ci * cosh(Ki*x) + Di * sinh(Ki*x)
where Ki = (rho_i*Ai*w^2/(Ei*Ii))^(1/4)

you can find the slope by differentiating (with respect to x) once, and the moment by differentiating twice with suitable factor E*I, and the shear by differentiating one more time.

There are two different regions: i=1, i=2
There are 8 coefficients:A1,B1,C1,D1,A2,B2,C2,D2
There are 9 unknowns: A1,B1,C1,D1,A2,B2,C2,D2,w

There are two boundary conditions at the fixed end (displacement=0, slope=0).
There are two boundary conditions at the free end (moment = 0, shear = 0).
There are four boundary conditions at the interface of the two (continuity of displacement, slope, moment and shear)
So we have 8 boundary conditions. It seems to be one boundary condition short...
... but actually the modeshape is only defined to within an arbitrary scaling factor. So we can arbitrarily assign one of the coefficiencts as 1.0 (a magnitude scaling factor 1.0)
That would give us the requisite 9 equations in 9 unknowns.
Good luck with the algebra to get a closed form solution, though...
It would probably be easier to solve above formulation numerically for your specific geometry and material properities. For that matter there are a large number of numerical ways to solve a problem like this.

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Blevins sounds like the way to go for an analytical solution.




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(2B)+(2B)' ?

Here's my quick attempt at numerical solution using a spreadsheet I had handy.

I used steel properties
rho = 7750.4 kg/m^3
E = 2.30974E+11 n/m^2

my results were that the first three Natural frequencies (hz) are:
1.6 7.5 18.3

I haven't double checked it.
you might want to double check it against some other method, or Greg's bounding scenarios.





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(2B)+(2B)' ?

• https://files.engineering.com/getfile.aspx?folder=87502a6b-758e-4756-8dfb-9

Material ? wall thicknesses?


What is the detail of the transition from Ø15 > Ø10 cm ?
Something like this ? -
https://www.sitepro1.com/store/cart.php?m=product_...

I think any kind of an adapter "bushing" is likely to have pretty low stiffness, maybe even lower than the pipe mounts in the link above ?

Thanks for the responses!,

I should have also said, the assumed pole's density is 7800Kgm^3 so its not at risk of buckling under its own weight, and assumed elastic modlus 2x10^11 (trying to keep it simple). I'll download Strand7 and see if i can get it on there

Tmoose asked about details of the transition. I assumed rigid transition. Obviously it makes a difference.

With your new values E and rho, I calculate the first three natural frequencies in hz:
1.5 7.0 17.0
I used a transfer matrix method and Euler Bernoulli beam model.

I believe you should get the same results with any approach that uses the same assumptions and the Euler Bernoulli model. Obviously recreating similar results independently would give more confidence.

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