Consider this forced translational mass$-$spring$-$damper $($MSD$)$ system: QUESTION $1($continued$)$:

Suppose that there are three desired responses for output $"x(t)"($i$.$e$.$ position of $"M"),$

as indicated, with ZERO initial conditions $($i$.$e$.x(0)=0$ and $x^{}(0)=0).$

For each Part:

Clearly explain why there is $($or is not$)$ a possible combination of spring $,$ or $K_3)$

and external force $,$ or $\{$:$f_3(t))$ which will provide the desired type of output

response.

HINT: You do NOT need to solve any differential equations for this Question.

Part a$)$ Desired output response: $x(t)=A{e}^{-5t}+B{e}^{-3t}+C{e}^{-8t}$

Part b$)$ Desired output response: $x(t)=D+e^{-5t}[H_{1}cos(9t)+H_{2}sin(9t)]$

Part c$)$ Desired output response: $x(t)=D+e^{-4t}[H_{1}cos(2t)+H_{2}sin(2t)]$

The input is the external force $"F(t)"$ and the output is position $"x(t)."$

The transfer function for this system is

$G(s)=\frac{x(s)}{F(s)}=\frac{1}{M{s}^2+Bs+K}$

It is known that $M=1kg,B=8\frac{(N)}{\frac{m}{sec}},$ and there are three possible values of K :

$(K_1=15\frac{(N)}{(m)},K_2=20\frac{(N)}{(m)},K_3=41\frac{(N)}{(m)})$

The only possible external forces $"F(t)"$ have the following Laplace transforms:

$F_1(s)=\frac{70(s)}{s^2+49}($corresponding to external force $"f_1(t)")$

$F_2(s)=\frac{21}{(s)+8}($corresponding to external force $"f_2(t)")$

$F_3(s)=\frac{12}{(s)}($corresponding to external force $"f_3(t)")$