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Natural frequency of spring is fn = 1/(2*pi) * sqrt(k/m), where k = spring constant and m is the mass.

Ok, so I have a spiral spring which means that spring is coiled couple of times. Its spring constant is (according to Hartog's Mech. Vibrations), k = E*I/l, E=Elastic Modulus, I = Inertia of cross section and l = length. All good at this point.

Now I straighten the coil so I get an cantilevered beam. Its natural frequency is (according Hartog):

f = 1/(2*pi) * sqrt((3*E*I)/(l^3*m))

So assume we have a beam with 1 mm width and 10 mm height. Lets plug couple of values:

E=22*10^9 Pa, I = b^3*h*1/12, b=1*10^-3 m, h= 10*10^-3 m, l = 40*10^-3 m

m = 0.01 kg (just some value)

f (spiral spring) = 1 Hz

f (cantilevered beam) = 46.6 Hz (??)

Why is the natural frequency of the cantilever beam so much higher than the spiral spring? Am I using wrong equations?