B$.3$ CALCULATE SPRING$-$MASS$-$DAMPER RESPONSE TO INITIAL

DISPLACEMENT

Figure $6$: spring$-$mass$-$damper vibrating system

Consider a spring$-$mass vibrating system with $m=8kg,k=600\frac{N}{m},$

$c=15\frac{N}{\frac{m}{s}}$ as shown in Figure $6.$ The vibrating system is given

an initial displacement $u_0=2mm$ with zero initial velocity. Do the

following:

$(1)$ Display input data

$(2)$ Calculate the natural frequency in $ra\frac{d}{sec{}_n}$ and in $Hz{f}_n$

$(3)$ Calculate the critical damping $c_{cr}$

$(4)$ Calculate the damping ratio $$

$(5)$ Calculate the damped frequency in $ra\frac{d}{sec{}_d}$ and in $Hz{f}_d$

$(6)$ Calculate the damped period of oscillation ${}_d$

$(7)$ Calculate the characteristic$-$equation roots $s_1,s_2$ and the

constants $C_1,C_2$

$(8)$ Calculate the constants $A,B$ and then the constants $C,$

$(9)$ Calculate the duration $t_1$ of one oscillation and the

corresponding position $u_1=u(t_1)$

$(10)$ Calculate the value $u_{0\mathrm{.}02}$ representing $2\%$ of the initial

displacement

$(11)$ Plot the response of the system for $N_{\text{o}\text{s}\text{c}}=10$ oscillations and use

datatips on the plot to find the following:

$($a$)$ the largest response

$($b$)$ the response value calculated at item $(9)$

$($c$)$ the time when the response magnitude has decreased

permanently below $u_{0\mathrm{.}02}$ calculated at item $(10)$

$(12)$ Discuss your results