Rayleigh frequencies vs eigenvalue frequencies

[scamplogic]

I've been doing some reading up on how best to automate natural frequency calculations for cantilevers of arbitrary and non-uniform cross-section. After reading "Structural Dynamics" by Mario Paz as well as an old British Standard for chimneys, it seems that the Rayleigh method might be what I need, and it's readily applicable to the way my software already functions. Eigenvalues, on the other hand, are much more computationally difficult, and I'll end up with 100 element matrices that need to be combined into a global stiffness matrix. *shudder*

I was wondering if I might be able to get some advice from a resident expert on dynamic analysis as to any glaring problems or limitations with the Rayleigh method that I should be aware of? If I can convince myself that the difference between the two methods is likely to be less than 5-10%, I'll be happy.

 

[Qshake]

Eigenvalues, on the other hand, are much more computationally difficult, and I'll end up with 100 element matrices that need to be combined into a global stiffness matrix. *shudder

The stiffness for a cantilever or inverted pendulum, in 2D, doesn't require a huge stiffness matrix as you suggest.

I understand that you brought up the idea of a non-prismatic section but still such a structural model should be developed to minimize the computational effort without sacrificing accuracy.

For a simple cantilever model I would expect both Rayleigh and Eigenvalue analysis to yield the same results.

Regards,
Qshake

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[JoshPlumSE]

Rayleigh method being something along the lines of:

Frequency = (1/2pi)* sqrt {Sum (f_i*delta_i)/sum(m_i*delta_i^2)}

Rayleigh method is a good hand-calc method that I use to verify the frequencies that I get from an FEM Eigen solution. It is important to have a decent starting force or starting displacement. For something like a cantilever chimney, this is easy... any equivalent static seismic or wind load should give you a good start to calculate a reasonable 1st natural frequency.

The problem with the method comes if you need to look at higher order modes at all. I suspect that if you know your approximate 2nd order mode shape and can create a set of forces that produces a displacement close to that shape then you would also be able to solve for that higher order mode. But, I have never attempted this myself.

 

[scamplogic]

The reason for calculating the first order natural frequency is to determine whether or not it is below 1.0 Hz and therefore is wind sensitive and therefore requiring specific provisions of AS 1170.2 to be applied. Other than seismic actions - which is outside the scope of my software - I'm not aware of any reason that I would need to calculate the higher modes.

 

[GregLocock]

The problem with rayleigh is that it is an upper bound estimate for the first mode. It is agreat approach in general, and in the case of a cantilever your guess at a deflected mode shape is likely to be reasonable. The advice i was given was to use a static deflection shape as one of the options, not just simple polynomials.

However given the cost of good FEA software I think you would be wise to develop your FEA skills in parallel.

Cheers

Greg Locock


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[scamplogic]

The deflection is quite easy to calculate with a good level of accuracy given that I'm using a minimum of 10 segments.

I've compared the outputs of my software to Microstran, Mstower, PoleCAD, and POLO - so far all are within 5-10%. The premise of my software is that the analysis and design of a pole should take less than 10% of the time as one of these larger (and much more expensive) packages, and therefore the user is willing to trade off the accuracy for the large time savings.