For a machine foundation, given weight $=60kN,$ spring constant $=11,000k\frac{N}{m},$ and $c=200kN-\frac{s}{m},$ determine

$($a$)$ whether the system is overdamped, underdamped, or critically damped,

$($b$)$ the logarithmic decrement, and

$($c$)$ the ratio of two successive amplitudes.

Solution

From Eq$.(2\mathrm{.}47),$

$c_c=2\sqrt[\phantom{2}]{km}=2\sqrt[\phantom{2}]{11,000(\frac{60}{9\mathrm{.}81})}=518\mathrm{.}76kN-\frac{s}{m}$

$\frac{c}{c_c}=D=\frac{200}{518\mathrm{.}76}=0\mathrm{.}386<mn>1</mn>$

Hence, the system is underdamped.

$2.$ From Eq$.(2\mathrm{.}70),$

$=\frac{2D}{\sqrt[\phantom{2}]{1-D^2}}=\frac{2(0\mathrm{.}386)}{\sqrt[\phantom{2}]{1-(0\mathrm{.}386{)}^2}}=2\mathrm{.}63$

Again, from Eq$.(2\mathrm{.}70),$

$\frac{Z_n}{Z_{n+1}}=e^{}=e^{2\mathrm{.}63}=13\mathrm{.}87$