Relation between natural frequency and the thickness

Question

If I have a solid material, let's say two bricks, one of which is thicker and I do modal/vibration testing. What would be the relation between thickness and natural frequency of any of them, is there any equation that can explain it, or does it depend on the material itself?

Answer 1

We know for undamped harmonic vibration

$\omega= angular velocity \space, L=length of brick \ , m=mass \ , \\ I= \text{second moment of area of the brick section } \ , \ h= \text{the height of the brick section} $

$ \omega = \sqrt {k/m }$

And $ k=I/L $

$ I =bh^3/12$

So everything else being equal the thicker brick will have larger " I " hence larger k, therefore, a shorter period and larger natural frequency.

Answer 2

Natural frequencies depend upon stiffness and mass. These two properties depend on the material characteristics and geometric configurations.

In general:

1. Increasing stiffness, the natural frequencies are increased
2. Increasing mass, the natural frequencies are reduced

With geometry things are a bit different. Because it will depend how you distribute mass and stiffness in your object (body)

For instance, thinking of an long and thin bar:

1. Increasing its length will decrease its natural frequencies
2. Increasing its cross sectional area will increase the natural frequencies

Answering your question, if you reduce the thickness of your block, such as it will look more like as a paper sheet, then I guess your natural frequency will be reduced.