An excerpt from my thermowell calculation spreadsheet follows:
Well Natural Frequency per ASME PTC 19.3 Fig. 1.1 (Equation (1)) w should be rad/s
fn=Kf/U**2 sqrt E/y fn 70 F= Hz 205.3649401
fn is natural frequency of the well at temperature in Hz fn 1050 F= Hz 188.525015
U is the insertion length of the thermowell in inches (was L) U Length= in. 10
E is the modulus of elasticity of the well material at temperature in psi E= psi 28000000
Kf is a well design constant for length and element diameter from table 1.4 Kf= 2.09
y is the specific weight of the well material at temperature, lb/in3 y= lb/in3 0.29
fw is the wake frequency in Hz
Element diameter dia. In 0.25
Well material Type 304 SSt
Obtain E from Metal Properties, y from ASME section VIII
This is in my instructions to self but it looks better in Excel.
Yeah, thanks for the suggestion and solutions provided, but I am still wondering the micromechanism of these two.
I think the material crystalline strcture will determine the resonant frequency, since this is related to the Young's modulus and which is dominantely determined by the binding energy of the atomics.
But the resonant frequencies are also related to the geometry and size, etc. of the material specimen.
But these are all the phenonmenom that demonstrate these two are related, I want to get the information about "why they are expressed so", it helps~
The natural frequency is a function of stiffness and mass. No more paramaters apply.
Stiffness however is determined by both shape and modulus of elasticy. Hence, using a material with another modulus will result in a shift in frequency.
You are right about E being a function of the lattice properties, I seem to rememeber you can work out the upper limit of stiffness values for materials by looking at the bond stiffness.
Ah, here we go, Chapter 6 in "Engineering Materials" by Ashby and Jones. Physical basis of Young's modulus.
As to the link between stiffness density and speed of planar vibration in the material then you need to look at the derivation of the single dimension wave equation.
Since the atoms in solid materials are constantly vibrating at very high frequencies and with relatively small amplitudes. And the vibrations of adjacent atoms are coordinating with the atomic bonding. These vibrations are coordinated in such a way that traveling lattice waves are produced, which might be thought the elastic wave or simple sound waves, and that propagates the crystal at the velocity of sound.
