$(30$ points$)$ Derive the frequency equation for the longitudinal vibration of the systems shown in the Figure $1.$

$[$Ans: $($a$)[$:$tan(\frac{l}{c})=\frac{\text{A}\text{E}\text{c}(M_1+M_2)}{A^2{E}^2-{}^2{M}_1{M}_2{c}^2}\}.$

Ans:$(b)tan(\frac{l}{c})=\frac{\text{A}\text{E}\text{c}(k_1+k_2)}{k_1{k}_2{c}^2-A^2{E}^2{}^2}.$

$]=[\frac{\text{A}\text{E}\text{c}({}^{2}M-k)}{[A^2{E}^2+Mk{c}^2]}.$

Figure $1$: Bars with different end conditions.