Question

A cantilever-type micro-accelerometer is designed by scaling down each dimension of a macro-accelerometer by a factor of 100. If the natural frequency of the macro-accelerometer is $\omega$, then the natural frequency of the micro-accelerometer will be

• A. $100\omega$
• B. $10\omega$
• C. $0.1\omega$
• D. $0.01\omega$

Answer 1

The correct option is B $10\omega$

$x=\frac{F{l}^3}{3EI}$

$\therefore F=\frac{x\left(3EI\right)}{l^3}$

$but,F=kx$

$\therefore k=\frac{3EI}{l^3}$ (k =stiffness)

$\omega =\sqrt{\frac{k}{J}}=\sqrt{\frac{3EI}{l^3}}=\sqrt{\frac{3E\times \frac{1}{12}b{t}^3}{l^3}}$

Now,
$b\to 100b$
$t\to 100t$
$l\to 100l$

$\therefore {\omega }^{\prime }=\sqrt{\frac{3E\frac{1}{12}\times \left(100b\right)(100{)}^3{t}^3}{(100{)}^3{l}^3}}=10\omega$