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Problems on accelerometer. An accelerometer has natural frequency of 80 Hz and a damping co efficient of 8 Ns/m, When attached to vibrating structure,
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It measures an acceleration of $8m/s^2$ and a frequency of 50Hz. True acceleration of structure is $7.5 m/s^2$.

Determine mass and spring const. of the acceleration.

$f_n$ = 80 Hz

$w_n = 2 \pi \times 80 = 502.65$ rad/sec

$c = \frac{8 Ns}{m}$

f = 50 Hz

$w = 2 \pi \times 50 = 314.15$ rad/sec.

Recorded acceleration = $\ddot{z} = 8 \ m/s^2$

True acceleration = $\ddot{y} = 7.5 \ m/s^2$

$\frac{\ddot{z}}{\ddot{y}} = \frac{1}{\sqrt{ (1-r^2)^2 + (2 \xi r)^2}}$

$\frac{8}{7.5} = \frac{1}{\sqrt{ (1 – 0.625^2)^2 + (2 \xi \times 0.625)^2}}$

$\xi = 0.637 = 0.57$

$\xi = \frac{c}{C_c} = \frac{c}{2m w_n}$

$0.637 = \frac{8}{2 \times m \times 502.65}$

M = 0.0124 kg

$w_n = \sqrt{ \frac{k}{m}}$

$502.65 = \sqrt{ \frac{k}{0.0124}}$

K = 3284 N/m

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