Q: oscillating cantilever

[Andy Resnick]

I'm having a tough time with a simple problem- the resonant oscillation of a vertical cantilever immersed in a viscous fluid. I have, from Roark's tables, the oscillation frequency for a cantilever with both a concentrated tip load and distributed load:

ω=1.732√[(EI*g)/(WL³+0.236(wL⁴)]

But I don't understand why (other than dimensional arguments) why there is 'g' in the numerator. For example, why is 'g' still relevant for a vertical cantilever immersed in a neutrally-bouyant viscous fluid (should g -> Δρ g?). The AFM literature (that I've found) for immersed tips isn't particularly informative.

Roark's book doesn't provide any derivation information... any hints/tips/references are gratefully appreciated.

 

[SteamKing]

I'm having a tough time with a simple problem- the resonant oscillation of a vertical cantilever immersed in a viscous fluid. I have, from Roark's tables, the oscillation frequency for a cantilever with both a concentrated tip load and distributed load:

ω=1.732√[(EI*g)/(WL³+0.236(wL⁴)]

But I don't understand why (other than dimensional arguments) why there is 'g' in the numerator. For example, why is 'g' still relevant for a vertical cantilever immersed in a neutrally-bouyant viscous fluid (should g -> Δρ g?). The AFM literature (that I've found) for immersed tips isn't particularly informative.

Roark's book doesn't provide any derivation information... any hints/tips/references are gratefully appreciated.

Even though the beam may be neutrally buoyant when immersed, that doesn't mean that gravity has been 'shut off'; it just means that the gravitational force acting on a beam element or whatever has an additional counteracting force which develops due to the displacement of the fluid in which the beam is immersed.

If you don't trust the tables, and I'm not saying you should if the conditions of your problem don't match the conditions for which the table was developed, then you'll have to do a vibration analysis from scratch using first principles.

 

[Dr.D]

For any single degree of freedom system, the square of the natural frequency is
w^2 = k/m
Looking at the equation you gave, there are weights in the denominator, not the required mass value. The g in the numerator converts the weight to the required mass value.