Short Answer:
A single-degree-of-freedom (SDOF) system is one that can move or vibrate in only one direction and whose motion can be described by a single coordinate. Several practical systems behave approximately as SDOF systems.
Examples include a spring–mass system, a simple pendulum, a vehicle suspension system, a vibrating beam or floor, and a rotating shaft with a disk. In each of these systems, the motion of the mass or body is defined by only one independent displacement, making them ideal for simple vibration analysis.
Detailed Explanation :
Examples of SDOF Systems
A single-degree-of-freedom (SDOF) system is the simplest type of mechanical vibration system in which only one coordinate is required to describe its motion. It may be translational (motion in a straight line) or rotational (angular motion). SDOF systems are often used as simplified models to study the basic vibration behavior of more complex systems in engineering.
In such systems, motion occurs due to the interaction of mass, stiffness, and damping elements. The equation of motion for all SDOF systems can be written as:
where is the mass, is the damping coefficient, is the stiffness, and is the external force. The examples given below illustrate this principle clearly.
- Spring–Mass System
This is the most basic and common example of a single-degree-of-freedom system. It consists of a mass attached to a spring of stiffness , either vertically or horizontally. When the mass is displaced and released, it vibrates up and down due to the restoring force of the spring.
- The displacement of the mass defines the motion of the entire system, meaning one coordinate is enough.
- If damping is added (like a dashpot), it becomes a spring–mass–damper system, still considered SDOF.
- The equation of motion is:
Applications: Used to model shock absorbers, vibration isolators, and basic mechanical oscillators.
- Simple Pendulum
A simple pendulum consists of a small bob of mass suspended from a fixed point by a string of length . When displaced by a small angle and released, it oscillates back and forth due to the force of gravity.
- The motion is described by one coordinate — the angular displacement .
- The restoring force is proportional to , and for small angles, .
The equation of motion is:
The natural frequency of a simple pendulum is:
Applications: Used in clocks, seismometers, and motion studies where small angular vibrations are considered.
- Vehicle Suspension System
Each wheel of a vehicle can be modeled as a single-degree-of-freedom system. The vehicle suspension consists of a spring (for stiffness) and a damper (for energy dissipation) supporting the mass of the vehicle body.
- The vertical motion of the vehicle body above a wheel is represented by one coordinate — the displacement of the body relative to its equilibrium position.
- The spring provides restoring force, and the damper absorbs shock.
The equation of motion is:
where represents the road excitation.
Applications: Used to design comfortable and stable vehicle suspensions that minimize vibration and shock.
- Cantilever Beam (First Mode)
A cantilever beam fixed at one end and free at the other behaves approximately as a single-degree-of-freedom system when it vibrates in its first mode.
- The free end’s displacement represents the entire beam’s motion.
- The stiffness is determined by the beam’s material and geometry.
- The motion is governed by a second-order differential equation similar to a spring–mass system.
This model is used to analyze deflection, resonance, and vibration characteristics in structures like airplane wings, building floors, and machinery arms.
Applications: Useful in structural vibration studies, especially when analyzing the fundamental (lowest) mode of vibration.
- Rotating Shaft with Disk (Torsional Vibration)
In torsional vibration, a shaft acts as the spring (providing torsional stiffness), and a disk mounted on the shaft represents the mass moment of inertia. When the shaft is twisted and released, it oscillates back and forth about its axis.
- The angular displacement defines the system’s motion.
- The restoring torque is proportional to the angle of twist.
The equation of motion is:
where is the mass moment of inertia, is torsional damping, and is torsional stiffness.
Applications: Found in engines, turbines, and drive shafts where rotational vibration analysis is important.
- Building or Structural Model
A single-story building subjected to lateral motion, such as an earthquake or wind load, can be modeled as a single-degree-of-freedom system.
- The building’s roof mass acts as the vibrating body, the columns provide stiffness, and damping comes from material and structural friction.
- The lateral displacement of the top floor represents the entire motion.
The vibration of such a model helps predict how real structures will respond to dynamic loads.
Applications: Widely used in civil engineering to study building vibration during earthquakes.
Importance of SDOF Examples in Engineering
- Simplified Analysis:
SDOF models help understand fundamental vibration behavior without complex mathematics. - Foundation for Multi-DOF Systems:
Real systems are often made of multiple SDOF systems combined, so studying individual examples builds the foundation for complex vibration study. - Design Optimization:
Helps engineers predict resonance conditions and optimize stiffness and damping for safe operation. - Experimental Testing:
SDOF systems are commonly used in laboratory experiments to determine natural frequencies, damping ratios, and stiffness constants.
Conclusion
Single-degree-of-freedom systems are simple yet powerful models used to study vibration characteristics in mechanical and structural engineering. Examples include the spring–mass system, pendulum, vehicle suspension, cantilever beam, and rotating shaft. Each example represents motion described by a single coordinate, making analysis straightforward and efficient. Understanding these examples helps engineers predict dynamic behavior, control vibration, and design systems that operate safely and smoothly under dynamic loads.