1. **Stating the problem:** We have a mechanical system with a mass (kereta) connected to springs $K_1$, $K_2$, and $K_3$, and a damper $C_1$. The goal is to derive the mathematical model relating displacement $x$ and force $F$, then transform it into the Laplace domain to find the transfer function $G(s) = \frac{X(s)}{F(s)}$. 2. **Modeling the system:** - The mass $m$ (kereta) is acted upon by force $F$. - Springs $K_1$ and $K_2$ are in series, so their equivalent spring constant is: $$K_{12} = \frac{K_1 K_2}{K_1 + K_2}$$ - Spring $K_3$ and damper $C_1$ are in parallel, so their combined force is: $$F_{K3C1} = K_3 x + C_1 \dot{x}$$ 3. **Writing the equation of motion:** Using Newton's second law: $$m \ddot{x} = F - F_{K12} - F_{K3C1}$$ where $F_{K12} = K_{12} x$ (force from series springs). So: $$m \ddot{x} + C_1 \dot{x} + (K_3 + K_{12}) x = F$$ 4. **Laplace transform:** Taking Laplace transform assuming zero initial conditions: $$m s^2 X(s) + C_1 s X(s) + (K_3 + K_{12}) X(s) = F(s)$$ 5. **Transfer function $G(s)$:** Solving for $X(s)/F(s)$: $$G(s) = \frac{X(s)}{F(s)} = \frac{1}{m s^2 + C_1 s + (K_3 + \frac{K_1 K_2}{K_1 + K_2})}$$ **Final answer:** $$G(s) = \frac{1}{m s^2 + C_1 s + K_3 + \frac{K_1 K_2}{K_1 + K_2}}$$ This transfer function relates the input force $F(s)$ to the displacement output $X(s)$ of the mass in the Laplace domain.