B$.5$ OVERDAMPED SPRING$-$MASS$-$DAMPER SYSTEMS

Consider a spring$-$mass$-$damper system with natural frequency

$f_n=3Hz$ and various values of the damping ratio $.$ The system is

given an initial displacement $u_0=10mm$ with zero initial velocity.

Do the following:

$(1)$ display input data

$(2)$ calculate the rad$/$s frequency ${}_n$

$(3)$ Assume overdamped conditions $>1$ and develop the solution

as follows:

$($a$)$ write the expressions of the roots $s_1,s_2$ in terms of ${}_n$ and $$

$($b$)$ write the solution $u(t)$ in terms of $s_1,s_2$ and constants $C_1,C_2$

$($c$)$ write the expression of the constants $C_1,C_2$ in terms of initial

displacement $u_0$ and initial velocity $u_0^{}$

$($d$)$ take $=2\mathrm{.}5$ and calculate the numerical values of $s_1,s_2$ and

$C_1,C_2$

$($e$)$ plot the response $u(t)$ for $t=0,$dots,$3$ seconds

$(4)$ Assume critically damped conditions $=1$ and do the following:

$($a$)$ write the expressions of the two partial solutions $u_1(t),u_2(t)$

in terms of ${}_n$

$($b$)$ write the expression of the complete solution in terms of ${}_n$

and $C_1,C_2$

$($c$)$ write the expressions of the constants $C_1,C_2$ in terms of

initial displacement $u_0$ and initial velocity $u_0^{}$

$($d$)$ calculate the numerical values of $C_1,C_2$

$($e$)$ plot the response $u(t)$ for $t=0,$dots,$3$ seconds

$(5)$ Graduate Work: derive the ODE for $=1$ and verify that the

two solutions $u_1(t)$ and $u_2(t)$ from $(4)($a$)$ satisfy the ODE

$(6)$ Graduate Work: some race$-$car manufacturers design for

critical damping whereas others use overdamped design.

Discuss the pros and cons of the two options