The appropriate way of creating Campbell Diagram

[JimLin]

I would like to discuss different ways of creating the Campbell Diagram (from FEA results) used in dynamics. If you have frequency at bench (70F and zero speed) and frequency at the max speed (with temperature), you can either connect these two points directly by a straight line, or you can also use formula with second order ( f = a + b * w^2) or a square root of sum of square. Anyone wants to share experience ?

 

[AlephZero]

It depends what physics is involved in changing the frequencies. If gyroscopic/coriolis forces are important, your quadratic formula doesn't represent the physics.

If the gyro effects are negligible but the stiffness changes with stress (e.g. the frequencies of flexible blades on a rigid rotor) a better formula would be to say that $K = K_e + \omega^2 K_\sigma$ where $K_e$ iis the elastic stiffness and $K_\sigma$ the stiffness from the internal stresses. Since $f = \sqrt{ K/M }$ (approximately, assuming the mode shapes don't change much) this leads to the relationship $f = \sqrt{a + b\omega^2}$.

If the temperature changes you might want to scale $K_e$ for the temperature corresponding to different speeds as well - in other words $a$ is some function of $\omega$, not a constant.

Gven the number of assumptions in all this, it may be simpler just to run the analysis at several speeds across the range.

 

[JimLin]

Thanks. I agree with you. I am talking about component dynamics and there is no gyro included.
How do you come up with this equation f=sqrt (a+ b*ω^2) ? Is it listed in any document ?
To my understanding, it is actually one of the formulas used by many industries.

a = f0^2 b = (f_max^2 - f0^2) / ω_max^2
where f0 is the freq at bench and f_max is the freq at max speed, ω_max is the max speed

 

[AlephZero]

How do you come up with this equation f=sqrt (a+ b*ω^2) ? Is it listed in any document ?

My earler post derives it from assumptions about the way the stiffness changes with rotation speed, and the assumption that the vibration mode shapes don't chage with speed.

It has been around for a very long time. I can't remember where I first came across it. Here's a reference from the 1960s (see eq 4). http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890068582_1989068582.pdf

 

[JimLin]

Thanks for reply. That article emphasized on the stiffening due to CF load. In real engine test, the softening effect due to temperature (especially for turbomachinery) actually was built in the stiffness ( where you wrote K2= k1 + K3*w^2). That's why some of the frequencies of some natural modes actually decrease with speed; while some others increase with speed.

The correlation using another formula f = a + b*w^2 is pretty good too (for component dynamics). Nice to talk to you.