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A seismic instrument with natural frequency of 6Hz is used to measure the vibration of machine running at 120 rpm.
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The instrument gives the reading for the relative displacement of the seismic mass as 0.05mm. Assume damping factor 0.1 and determine the amplitudes of disp. Velocity and accn of the vibrating m/c.

1] Amplitude of vibration = Y = $4.1 \times 10^{-4} \ m$

2] Velocity = wy = $5.02 \times 10^{-3} \ m/sec$

3] Acceleration = $w^2 y = 63.6 \times 10^{-3} \ m/s^2$

$F_n$ = 6Hz

N = 120 rpm

Z = 0.05 mm

$\xi$ = 0.1

$w_n = 2 \pi f_n$

$= 2 \pi \times 6$

$w \pi = 37.69$ r/s

$w = \frac{2 \pi N}{60} = \frac{2 \pi \times 120}{60}$

W = 12.56 r/s

$r = \frac{w}{w_n} = \frac{12.56}{37.69} = 0.33$

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

$\frac{0.05 \times 10^{-3}}{y} = \frac{(0.33)^2}{\sqrt{ (1- 0.33^2)^2 + (2 \times 0.1 \times 0.33)^2}}$

$y = 4.1 \times 10^{-4} \ m$

Velocity = w.y

$= 12.56 \times 4.1 \times 10^{-4}$

$5.14 \times 10^{-3}$ m/s

Acceleration = $w^2. y$

$= (12.56)^2 . 4.1 \times 10^{-4}$

$= 0.064 m/s^2$

$= 64.67 \times 10^{-3} \ m/s^2$