Question 2.6.1: This example illustrates how an independent measurement of a......

From equation (2.93), the amplitude of the measured values of acceleration \left|\omega_n^2 z(t)\right| is related to the actual values of acceleration |\dot{y}(t)| by

\omega_n^2 z(t)=\frac{-1}{\sqrt{\left(1-r^2\right)^2+(2 \zeta r)^2}} \ddot{y}(t)      (2.93)

\frac{\left|\omega_n^2 z(t)\right|}{|\ddot{y}(t)|}=\frac{1}{\sqrt{\left(1-r^2\right)^2+(2 \zeta r)^2}}=\frac{10\ m / s ^2}{9.8\ m / s ^2}=1.02

Rewriting this expression yields one equation in \zeta and r:

\left(1-r^2\right)^2+(2 \zeta r)^2=0.96

A second expression in \zeta and r can be obtained from the definition of the damped natural frequency:

\frac{\omega_b}{\omega_d}=\frac{\omega_b}{\omega_n} \frac{1}{\sqrt{1-\zeta^2}}=r \frac{1}{\sqrt{1-\zeta^2}}=\frac{628\ rad / s }{628\ rad / s }=1

Thus r=\sqrt{1-\zeta^2}, providing a second equation in \zeta and r. This can be manipulated to yield \zeta^2=\left(1-r^2\right), which when substituted with the preceding expression for r and \zeta yields

\zeta^4+4 \zeta^2\left(1-\zeta^2\right)=0.96

This is a quadratic equation in \zeta^2 :

3 \zeta^4-4 \zeta^2+0.96=0

This quadratic expression yields the two roots \zeta=0.56, 1.01. Using \zeta=0.56, the damping constant is \left(\sqrt{1-\zeta^2}=0.83, \omega_n=\omega_d / \sqrt{1-\zeta^2}=758.0\ rad / s \right)

c=2 m \omega_n \zeta=2(0.01)(758.0)(0.56)=8.49\ N \cdot s / m

Similarly, the stiffness in the accelerometer is

k=m \omega_n^2=(0.01)(758.0)^2=5745.6\ N / m