$2$ a$.$ FIGURE Q$2$a shows a spring$-$mass$-$damper system. The mass, $m,$ damping

constant, $c$ and spring constant, $k$ are set to $1kg,4N\frac{s}{m}$ and $4\frac{N}{m},$

respectively. The equation of motion of the system is described by:

$x^{}+4x^{}+4x=0$

i$.$ Determine the free$-$vibration response of the system when the initial

displacement, $x_o$ is $0$ m and initial velocity, $x_o^{}$ is $1\frac{m}{s}.$

$[10$ marks$]$

ii$.$ Explain the changes that could be observed in the free$-$vibration

response of the system when the initial displacement, $x_0,$ initial velocity,

$x_o^{},$ mass, $m$ and damping constant, $c$ remain the same, while the spring

constant, $k$ is changed as follows:

$($a$)k=8\frac{N}{m}.$

$($b$)k=3\frac{N}{m}.$