$2\mathrm{.}24$ A free vibrations test is run to determine the stiffness and damping properties of an elastic element. A $20-kg$ block is attached to the element. The block is displaced $1$ cm and released. The resulting oscillations are monitored with the results shown in Fig. $2-16.$ Determine $k$ and $c$ for this element.

CHAP. $2]$

FREE VIBRATIONS OF $1-$DEGREE$-$OF$-$FREEDOM SYSTEMS

$53$

From Fig. $2-16$ the displacement of the block at the end of the first cycle is $0\mathrm{.}005$ m $.$ The logarithmic decrement is calculated as

$=ln(\frac{0\mathrm{.}01(m)}{0\mathrm{.}005(m)})=0\mathrm{.}693$

from which the damping ratio is determined

$=\frac{}{\sqrt[\phantom{2}]{4{}^2+{}^2}}=\frac{0\mathrm{.}693}{\sqrt[\phantom{2}]{4{}^2+(0\mathrm{.}693{)}^2}}=0\mathrm{.}11$

The damped natural period is determined from Fig. $2-16$ as $0\mathrm{.}06$ s $.$ The damped natural frequency is

${}_d=\frac{2}{T_d}=\frac{2}{0\mathrm{.}06(s)}=104\mathrm{.}7\frac{\text{r}\text{a}\text{d}}{(s)}$

The natural frequency is calculated from

${}_n=\frac{{}_d}{\sqrt[\phantom{2}]{1-{}^2}}=\frac{104\mathrm{.}7\frac{\text{r}\text{a}\text{d}}{(s)}}{\sqrt[\phantom{2}]{1-(0\mathrm{.}11{)}^2}}=105\mathrm{.}3\frac{\text{r}\text{a}\text{d}}{(s)}$

Thus the stiffness and damping coefficient are calculated

$k=m{}_n^2=(20kg)(105\mathrm{.}3\frac{\text{r}\text{a}\text{d}}{(s)}{)}^2=2\mathrm{.}22{10}^5\frac{(N)}{(m)}$

$c=2m{}_n=2(0\mathrm{.}11)(20kg)(105\mathrm{.}3\frac{\text{r}\text{a}\text{d}}{(s)})=4\mathrm{.}63{10}^2\frac{(N)-s}{(m)}$