Case 5: vibration measuring instrument.

1] Seismic instrument [vibrometer]

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

Experimental value

R = 0 to 0.3 undamped accelerometer

R = 0.3 to 0.7 damped

R = 1 velometer

R = 3 to 7 vibrometer.

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

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

$\frac{Z.w_n}{y.w} = \frac{r}{\sqrt{(1-r^2)^2 + (2 \xi r)^2}}$

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

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

$\frac{w_n^2.7}{w^2.y} = \frac{1}{\sqrt{(1-r^2)^2 + (2 \xi r)^2}}$

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