What is a single-degree-of-freedom system?

Short Answer:

A single-degree-of-freedom (SDOF) system is a simple mechanical system that can move or vibrate in only one direction or along one coordinate. Its motion can be completely described by a single variable, such as displacement, velocity, or acceleration.

In simple words, an SDOF system has only one independent motion. A basic example is a spring–mass–damper system, where a mass attached to a spring vibrates up and down along a straight line. The entire vibration behavior of this system can be studied using one equation of motion involving one coordinate only.

Detailed Explanation :

Single Degree of Freedom System

A single-degree-of-freedom (SDOF) system is the simplest form of a dynamic system used in vibration analysis. The term “degree of freedom” refers to the number of independent coordinates or variables required to describe the motion of a system completely. Therefore, an SDOF system has only one coordinate that defines its motion.

In other words, if the position or displacement of one point of the system can represent the entire system’s motion, then it is a single-degree-of-freedom system. These systems are often used to model mechanical vibrations of simple machines, components, or structures because they allow easy analysis and understanding of vibration behavior.

Basic Components of a Single-Degree-of-Freedom System

A typical single-degree-of-freedom system consists of three essential components:

  1. Mass (m):
    This represents the inertia of the system or the body that tends to resist motion when an external force acts upon it. It stores kinetic energy during vibration.
  2. Spring (k):
    The spring provides a restoring force proportional to displacement. It stores potential energy when deformed and tries to bring the mass back to its equilibrium position.
  3. Damper (c):
    The damper dissipates energy in the form of heat or friction, reducing the amplitude of vibration over time. It represents energy loss in the system.

When these three elements are combined — mass, spring, and damper — the resulting system is known as a spring–mass–damper system, which is the most common example of an SDOF system.

Equation of Motion for an SDOF System

To understand how an SDOF system behaves, let us consider a mass  attached to a spring of stiffness  and a damper of damping coefficient . The mass is subjected to an external force  and moves along one direction.

According to Newton’s Second Law of Motion,

The total forces acting on the mass are:

  • Inertial force =
  • Damping force =
  • Restoring (spring) force =
  • External force =

The equation of motion becomes:

where,

  •  = mass of the system (kg)
  •  = damping coefficient (N·s/m)
  •  = stiffness of the spring (N/m)
  •  = displacement of the mass (m)
  •  = external force acting on the mass (N)

This second-order differential equation fully describes the motion of a single-degree-of-freedom system.

Types of Vibrations in an SDOF System

  1. Free Undamped Vibration:
    When the system is displaced and released without any damping or external force, it vibrates freely at its natural frequency. The equation simplifies to:
  1. Free Damped Vibration:
    When damping is present but no external force acts, the vibration gradually dies out due to energy loss.
  1. Forced Vibration:
    When an external periodic force acts on the system, it vibrates at the frequency of the external force.

These three vibration types are fundamental to understanding how systems behave under various conditions.

Physical Behavior of an SDOF System

The motion of an SDOF system can be visualized as oscillations of the mass around an equilibrium position:

  • The spring exerts a restoring force that tries to pull the mass back to the center.
  • The mass resists motion due to its inertia.
  • The damper provides resistance proportional to the velocity of motion, reducing the amplitude over time.

The energy alternates between kinetic energy (due to motion) and potential energy (due to spring deformation) during each cycle. If damping is present, part of this energy is lost, and vibrations gradually stop.

Examples of Single-Degree-of-Freedom Systems

  1. Spring–Mass System:
    A simple mass attached to a spring vibrating vertically or horizontally.
  2. Pendulum (Small Oscillations):
    A simple pendulum swinging back and forth under gravity can be approximated as an SDOF system for small angular displacements.
  3. Vehicle Suspension System:
    The suspension of a car (one wheel assembly) can be treated as an SDOF system when analyzing vertical motion.
  4. Building Floor or Beam:
    A single floor of a building vibrating during an earthquake can be modeled as an SDOF system.
  5. Rotating Shaft:
    A rotating shaft with one disk mounted on it can be treated as an SDOF torsional vibration system.

Applications of SDOF Systems

  1. Vibration Analysis:
    Engineers use SDOF models to study the basic principles of vibration before extending them to complex systems.
  2. Resonance Study:
    Helps predict the conditions under which a system experiences resonance — when external excitation matches its natural frequency.
  3. Machine Design:
    Used to design components such as springs, dampers, and rotating machinery to minimize vibration effects.
  4. Seismic Engineering:
    SDOF systems are used to model and analyze how buildings respond to ground motion during earthquakes.
  5. Testing and Measurement:
    SDOF concepts are used in laboratory experiments to determine natural frequencies, damping ratios, and stiffness of materials.

Advantages of Using SDOF Models

  • Simple to analyze mathematically.
  • Useful for understanding the basic principles of vibration and resonance.
  • Provides a foundation for analyzing more complex multi-degree systems.
  • Allows quick estimation of system response and dynamic behavior.

However, it is important to note that real systems often have multiple degrees of freedom, and the SDOF model provides an approximation to simplify analysis.

Conclusion

A single-degree-of-freedom system is the simplest dynamic model that describes motion using only one coordinate or variable. It consists of a mass, stiffness, and damping element, and its motion can be described by a second-order differential equation. SDOF systems are widely used to understand vibration behavior, resonance, and energy transfer in mechanical and structural systems. Although real systems are often more complex, the SDOF model forms the foundation for vibration analysis and dynamic system design in mechanical engineering.