# Relation between natural frequency and the thickness

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?

• What are the testing results? – Solar Mike Mar 14 at 4:04
• Actually, the modal testing itself is used for analysing the relation between the capacity of a lithium ion battery and its internal structure through non-destructive method. As the battery is charged, the thickness is higher. Every 10% of charging capacity, the modal testing is applied on the surface of the battery pouch. The result shows, as the capacity increases, the natural frequency is increasing too, although in around 85% of charging capacity, it decreases again – elluthfi Mar 14 at 6:54
• This is the accelerance result of the vibration testing data that I process using GNU Octave: imgur.com/sCJz05h and this is when I plot the natural frequency from the accelerance peak: imgur.com/a/mWPRddR – elluthfi Mar 14 at 15:27

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.

• Thank you, but what is I, L, b, and h itself? – elluthfi Mar 14 at 6:58
• @elluthfi, I will edit my answer to explain those variables – kamran Mar 14 at 7:04

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.