# Calculating the natural frequency of a piping system

## Main Question or Discussion Point

I'm calculating the natural frequency of a piping system that is spanning between two points, the formula is the square route of various terms, including the effective axial force in the piping. This is taken from a design code, see below:

f = C * SQRT[(E.I/M.L^4)(1+Seff/Pcr+C.(d/D)^2]

C = Constant dependant on boundary conditions of span
E = Youngs modulus
I = Second moment of area (piping)
M = Mass
L = Span length
Seff = Axial force in piping
d = deflection
D = piping diameter

The axial force in the piping is negative, due to the temperature and pressure, and I'm trying to perform a conservative calculation assuming the piping is fully restrained. This results in a negative term in the square root. Is it possible to have a negative natural frequency?

The natural frequency is coming out as -0.5 i Hz (i being sqrt(-1)).

Any help or guidance appreciated. Thanks

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Just some quick observations:

I would say that the interpretation of the result probably depends on the assumptions made when deriving that forumla.

It could be that an imaginary frequency could indicate an overdamped system which does not oscillate, but instead decays towards its equilibrium state (or diverges unphysically away from the equilibrium state, depending on the sign). It may not be possible to say just from looking at that formula.

Either you have made a mistake, or you need to study the assumptions of the derivation of that formula to understand what the imaginary frequency means in your context.

yeah i've been doing some reading over night, i think it might be due to the effective force being overly conservative. there will be 'feed in' to it, which will reduce the compressive force. thanks