Short Answer:

A single-degree-of-freedom (SDOF) system is a mechanical system that can move or vibrate in only one independent direction or coordinate. Its motion can be described completely by a single variable, such as displacement, velocity, or acceleration. Examples include a simple pendulum or a mass-spring-damper system, where only one coordinate defines the entire motion.

In an SDOF system, the system’s behavior under external forces like vibration or impact can be easily analyzed. Such systems are widely used to study fundamental vibration principles, including natural frequency, damping, and resonance, which help in designing stable and safe machines and structures.

Detailed Explanation:

Single-Degree-of-Freedom (SDOF) System

A single-degree-of-freedom (SDOF) system is a simplified model used in vibration analysis and mechanical system dynamics. It describes systems where the motion is completely defined by a single coordinate or variable. This means that the system can only move in one direction, and all other movements depend on this single variable. The main purpose of studying SDOF systems is to understand the basic behavior of vibrating systems, which can later be extended to more complex systems with multiple degrees of freedom.

An SDOF system generally consists of three main components — mass, spring, and damper. These three elements together define how the system vibrates when disturbed from its equilibrium position. The mass represents the object’s inertia, the spring provides restoring force proportional to displacement, and the damper offers resistance to motion, reducing vibration amplitude with time.

Components of SDOF System

1. Mass (m):
   The mass represents the object that moves due to vibration. It stores kinetic energy during motion. In most mechanical systems, mass is a concentrated body like a weight attached to a spring.
2. Spring (k):
   The spring provides the restoring force that brings the mass back to its equilibrium position. This force is proportional to the displacement of the mass and follows Hooke’s Law, , where  is the stiffness of the spring and  is the displacement.
3. Damper (c):
   The damper opposes the motion of the mass by providing a force proportional to the velocity of the motion. It helps reduce the amplitude of vibration over time. The damping force can be represented as , where  is the damping coefficient and  is velocity.

These three elements together make a mass-spring-damper system, which is the most common example of an SDOF model.

Equation of Motion

The motion of an SDOF system is governed by Newton’s Second Law of Motion. The sum of all forces acting on the mass equals the mass multiplied by its acceleration:

Where:

•  = mass of the system
•  = damping coefficient
•  = stiffness of the spring
•  = displacement
•  = external applied force

This equation helps in analyzing how the system behaves under various types of excitation, such as free vibration (no external force) or forced vibration (external periodic force).

Types of Vibration in SDOF Systems

1. Free Undamped Vibration:
   In this case, there is no damping, and the system oscillates freely after being disturbed. The motion is purely sinusoidal with a constant amplitude and a natural frequency given by:

where  is the natural frequency in radians per second.

1. Free Damped Vibration:
   Here, damping is present, and the vibration amplitude gradually decreases with time. The system eventually comes to rest due to energy loss in the damper.
2. Forced Vibration:
   When an external force acts on the system, it vibrates at the frequency of that force. If this frequency matches the natural frequency, resonance occurs, leading to large amplitudes that may cause damage to the system.

Importance of SDOF System

Studying an SDOF system is crucial because it helps engineers understand the basic principles of vibration behavior in machines, vehicles, buildings, and structures. Most complex systems can be simplified into several SDOF systems for easier analysis. By learning about the natural frequency, damping, and response characteristics, engineers can design systems that minimize unwanted vibration, noise, and mechanical failure.

SDOF models are widely used in:

• Design of mechanical and structural systems to avoid resonance.
• Dynamic analysis of machines such as engines and turbines.
• Seismic analysis of buildings and bridges.
• Testing materials for vibration resistance.

Conclusion

A single-degree-of-freedom (SDOF) system is the simplest model for studying vibrations and dynamic behavior of mechanical systems. It is characterized by one independent coordinate and consists mainly of mass, spring, and damper components. Understanding SDOF systems helps engineers analyze motion, control vibrations, and ensure the stability and safety of mechanical structures. Although simple, SDOF models form the foundation for analyzing complex multi-degree systems used in real engineering applications.