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manpreet Best Answer 2 years ago

I am currently investigating how a hacksaw blade's time period of oscillation changes when I add mass to the end of it or when I change the length it is clamped at.

I found the following equation from an IB worksheet:

T 2 = 16 M π 2 x 3 b E d 3

Where:

$T$ is the time period for one oscillation;
$M$ is the mass of the cantilever;
$x$ is the length of the cantilever;
$b$ is the breadth of the cantilever and $d$ is the thickness of the cantilever;
and $E$ is the stiffness of the cantilever.

I've looked around the internet and asked my teachers, but I haven't been able to derive this equation from first principles or the equations given in my syllabus.

If anyone knows where this comes from or would like to try and work it out, I would be very greatful.

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manpreet 2 years ago

I was able to derive a similar equation, but not the exact one you mention.

The differential equation of a beam can be stated as

E I \partial 4 u ( z , t ) \partial z 4 + ρ A \partial 2 u ( z , t ) \partial t 2 = 0

where: $E$ is the elastic modulus, $I$ is the section area moment, $ρ$ is the mass density, $A$ is the section area and $u (z, t)$ is the deformation as a function of position $z$ and time $t$ .

In your case,

ρ A I = M x = REPLY 0 views 0 likes 0 shares Facebook Twitter Linked In WhatsApp

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Derive equation for a cantilever in SHM

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Answers (2)

manpreet Best Answer 2 years ago

manpreet 2 years ago

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