Resonant frequency calculation
Resonant frequency calculation
(OP)
Hi All, I 'd like to estimate the resonant frequency of thin metal ( aluminium ) plates of small dimensions e.g. 1 X 0.5 x 0.1 mm . Can anyone suggest an approximate formula and units? If i wanted to get a resonant frequency in GHz , what would the dimensions need to be ? My thanks in advance. Opticsman.





RE: Resonant frequency calculation
Not building radar absorbent material are we?
Cheers
Greg Locock
RE: Resonant frequency calculation
What mechanical modes are you interested in: longitudinal, transverse, torsional? What medium is your beam located or attached to, i.e. what are boundary conditions?
To get criticals in the ghz range(?) you'll be exciting a lot of bulk and surface waves, so you'll have to be more specific about the excitation (i.e spatial distribution).
If microwave excitation is involved, then you may need to start with Maxwells equations and the exact mechanical equations with whatever form of magneto-electric/elastic coupling that you plan to focus upon. Typically in that frequency range you'll also need to include thermal excitation of your modes.
RE: Resonant frequency calculation
RE: Resonant frequency calculation
fn = [1/(2 pi)][3.5156 / L^2] sqrt( EI /rho )
where
L^2 = length squared
E = elastic modulus
I = area moment of inertia of the cross section
rho = mass/length
Now assume that the material, width, and thickness are constant. The natural frequency is inversely proportional to length squared.
Tom Irvine
www.vibrationdata.com
RE: Resonant frequency calculation
I am not sure about the factor 1/2pi. From Den Hartog, for a cantilever beam, we have:
freq=3.52 * {E*I/[(mass/length)*length^3]}^0.5
(mass/length is referred to as mu1 )
which is derived as:
(noting that a quarter cosine wave is a staring point, but not an exact solution)
y=y0(1-cos*pi*x/(2L)) carrying on and substituing into potential and kinetic energy integrals, we find:
Potential = Pi^4*E*I*y0^2/(64*l^3)
Kinetic = mu1*omega^2*y0^2*L*(3/4-2/pi)
Equate to find:
omega= pi^2/[8*(3/4-2/pi)^0.5]*[E*I/(mu1*L^4)]
equals"
3.66/L^2 * (E*I/mu1)^0.5; the coefficient which goes to 3.55 at an exact solution.
So, what of the 1/2pi factor?
(I jsut happened to be looking for confirmation of this myself!)
Best regards,
Bill
RE: Resonant frequency calculation
The first equation at the top, I accidentally wrote length^3 when it should read length^4:
freq=3.52 * {E*I/[(mass/length)*length^4]}^0.5
of course, you can move the length out from under the radical, and get:
3.52/L^2 * [E*I/mu1]^0.5
RE: Resonant frequency calculation
RE: Resonant frequency calculation
I see now--that Den Hartog uses omega as "circular frequency" in this case---so of course, 1/(2pi) to convert to cycles/sec.
And of course I now see it in the equations, too---like you say, no conversions from radians due to the transcendentals....
Best regards
Bill
RE: Resonant frequency calculation
I'm using the Formula that Tom Irvine pointed out however I'm having some issues with Units..
Units are as follows...
E=lb/in^2
I=in^4
rho= lb/in
If I use the formula this gives me the square root of in^3. Can someone help me out here?
Thanks,
Juan
RE: Resonant frequency calculation
E has dimensions M/(LT^2).
rho has dimensions M/L.
I has dimesions L^4
fn has dimensions (1/L^2) * sqrt[ (M/(LT^2) * (L^4) * (L/M) ]
= (1/L^2) * sqrt(L^4/T^2)
= 1/T
M
--
Dr Michael F Platten
RE: Resonant frequency calculation
I am confusing Force pounds with mass pounds. This is exactly my problem, thanks!
That's what I get for learning only the metric system all the way through school and using and American for development efforts.
This information is really helpful for me to determine the dimensions for my instrument!!!
Juan
RE: Resonant frequency calculation
Tunalover