What is the in-plane resonance frequency equation for a single-beam cantilever of rectangular cross-section that is fixed on one end and has a mass M attached to the other end?

Below are the only equations that I was able to find and they came from this site:

$f = \frac{1}{2\pi}\sqrt{\frac{k}{M + \rho \cdot t \cdot w \cdot l}}$

$k = \frac{E \cdot w \cdot t^3}{4 \cdot l^3}$

  • f = Resonance frequency (Hz)
  • M = Mass added to the cantilever's free end (kg)
  • $\rho$ = Density of the material that the cantilever is made of (kg/$m^3$)
  • t = Thickenss of cantilever (m)
  • w = Width of cantilever (m)
  • l = Length of cantilever (m)
  • E = Elastic modulus or Young's modulus (kg/$m \cdot s^2$)

The problem is that the site author never stated if these were the equations for in-plane or out-of-plane motion and I couldn't make a deduction based on the provided context; the equations should differ, correct?


2 Answers 2


If you rearrange the formula to use the beam's moment of inertia, then it is fairly obvious is applies to out-of-plane bending, just like formulas for static deflection.

This also makes the formula applicable to any beam cross section, not just a rectangle!

For your rectangular beam, $I = wt^3/12$. Rearrange the formula to use $EI$ instead of the explicit references to $w$ and $t$.

For bending in the other plane, you just have $I = w^3t/12$, of course.

(Note: I seem to have "in plane" and "out of plane" the other way round from Mohan's answer. I'm not going to argue about which is correct - they are not terms that I would normally use anyway! Every beam cross section has two principal axes, defining two planes - arbitrarily choosing one of them and then calling the two directions "in plane" and "out of plane" doesn't seem very logical to me.)

  • $\begingroup$ I think that the "plane" is that which is defined by the width and length; therefore it would follow that moving "in-plane" would mean moving side to side, where as "out-of-plane" would mean moving up and down (i.e., through the plane which is defined by the thickness). I think this terminology was popularized by Stephen D. Senturia in his book on "Microsystem Design". $\endgroup$
    – Landon
    Commented Sep 19, 2018 at 18:48

This is for single degree of freedom vibration(in plane motion) because the stiffness is calculated by assuming the deflection to be inplane.I have attached an image.For motion in the perpendicular plane,the moment of inertia should be calculated about the vertical centroidal axis of the cross section.enter image description here


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