A critically damped door-closing spring–mass–damper becomes overdamped as its spring softens. The new damping ratio is 1.2 and all other parameters remain fixed. Find original/new spring-stiffness ratio to 2 decimals.
Damping ratio is $\zeta=c/(2\sqrt{km})$. With fixed c,m, $\zeta_{new}/\zeta_{old}=\sqrt{k_{old}/k_{new}}$. Since the original ratio is 1, stiffness ratio is $1.2^2=1.44$. Spring softening raises the damping ratio even though the damping coefficient has not changed.
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