Cantilever Beam, Tapered in breadth, uniform thickness, natural frequency

[jazzkiwi]

Hey there
I need to find the natural frequency of a Cantilever Beam. The beam is tapered in breadth, but has a uniform thickness. So basically the end goal is an equation for f as a function of the angle of the side cut.

Clamped at one end
L=constant
B=changes though the length of the beam and is dependent on the angle from the z axis (along the beam)
H=constant (thickness)
E constant,
So obviously the second moment of area will be a function of x or angle.

The end equation will be used in MATLAB to for a plot angle vs frequency

Any pointers, or a solution would be most helpful,
Thanks for your time

 

[Unrest]

Just an incomplete idea -

Find an equivalent system of a massless beam with a point mass at the end. Then use the stiffness of the beam, and the point mass as a simple spring-mass system and get the fundamental frequency from $$\omega = \sqrt{\frac{k}{m}}$$

 

[AlephZero]

If I understood your description correctly, your "beam" is the same shape as a slice cut out of a circular disk with a hole in the middle.

Formulas for the vibration of disks are well known, though you might have to find the derivations and extend them yourself to the case with a hole in the middle. The math involves Bessel functions.

You would need to take Poisson's ratio = 0, so there would be no circumferential strain in the solution for the full disk, and cutting it into "slices" would not make any difference. The simplest beam theories don't depend on Poisson's ratio at all, so this should work out OK.