1. **Problem Statement:** Understand the basic example of vibration analysis in mechanical systems, such as a mass-spring-damper system. 2. **Formula Used:** The equation of motion for a damped single degree of freedom system is given by: $$m\ddot{x} + c\dot{x} + kx = F(t)$$ where $m$ is mass, $c$ is damping coefficient, $k$ is stiffness, $x$ is displacement, and $F(t)$ is external force. 3. **Explanation:** This formula models how the system vibrates when subjected to forces. The terms represent inertia, damping, and restoring force respectively. 4. **Example:** Consider a mass $m=2$ kg, damping $c=4$ Ns/m, stiffness $k=50$ N/m, and no external force $F(t)=0$. 5. **Intermediate Work:** The characteristic equation is: $$m\lambda^2 + c\lambda + k = 0$$ Substitute values: $$2\lambda^2 + 4\lambda + 50 = 0$$ 6. **Solve for $lambda$:** $$\lambda = \frac{-4 \pm \sqrt{4^2 - 4 \times 2 \times 50}}{2 \times 2} = \frac{-4 \pm \sqrt{16 - 400}}{4} = \frac{-4 \pm \sqrt{-384}}{4}$$ 7. **Interpretation:** Since the discriminant is negative, the system is underdamped and exhibits oscillatory motion with exponential decay. 8. **Natural Frequency and Damping Ratio:** $$\omega_n = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = 5$$ $$\zeta = \frac{c}{2\sqrt{km}} = \frac{4}{2 \times 5 \times 2} = 0.2$$ 9. **Conclusion:** The system vibrates at a damped natural frequency with damping ratio 0.2, typical in vibration analysis. This example illustrates fundamental vibration analysis concepts in vibration engineering.