Short Answer:
The difference between SDOF and MDOF systems lies in the number of independent motions or coordinates needed to describe their vibration. A single-degree-of-freedom (SDOF) system has only one independent motion, while a multi-degree-of-freedom (MDOF) system has two or more.
In simple words, an SDOF system represents simple vibration like a mass-spring setup, whereas an MDOF system represents real-world structures such as vehicles, buildings, or machines, where several parts can move independently but interact with each other through springs and dampers.
Detailed Explanation :
Difference between SDOF and MDOF Systems
Vibration systems in mechanical engineering are categorized based on the number of independent displacements or coordinates required to describe their motion. This classification gives rise to Single-Degree-of-Freedom (SDOF) and Multi-Degree-of-Freedom (MDOF) systems.
The main difference between these two systems lies in the complexity of motion, number of independent coordinates, and mathematical representation. While an SDOF system is simple and easy to analyze, an MDOF system represents real-life conditions and provides more accurate vibration analysis of complex structures.
Let us discuss both systems in detail to understand the differences clearly.
Single-Degree-of-Freedom (SDOF) System
A single-degree-of-freedom system is a vibrating system in which the motion can be completely described by a single coordinate or displacement. It means that the entire system moves as a single unit in response to external or internal forces.
Characteristics of SDOF System:
- One Independent Coordinate:
Only one displacement is needed to describe the system’s motion (for example, the vertical motion of a spring-mass system). - Simple Structure:
The system usually consists of one mass, one spring, and one damper. - Equation of Motion:
The motion is governed by a single second-order differential equation:
where
= mass,
= damping coefficient,
= stiffness,
= external force,
= displacement.
- Single Natural Frequency:
The system has only one natural frequency and vibrates in one distinct mode shape. - Examples:
- A mass attached to a spring and damper.
- A simple pendulum.
- A vibrating beam with one end fixed and one end loaded by a single mass.
- Application:
SDOF systems are useful for studying basic vibration principles, damping, resonance, and frequency response.
Multi-Degree-of-Freedom (MDOF) System
A multi-degree-of-freedom system is a system that requires two or more independent coordinates to describe its motion completely. In such systems, different masses or elements can move independently but are connected through springs or dampers, making their motion interdependent.
Characteristics of MDOF System:
- Multiple Independent Coordinates:
Two or more displacements are required to describe the system’s motion (e.g., in a two-mass system, and ). - Complex Structure:
It consists of several masses, springs, and dampers connected together. - Coupled Equations of Motion:
The motion is described by a set of coupled second-order differential equations:
where
= mass matrix,
= damping matrix,
= stiffness matrix,
= displacement vector.
- Multiple Natural Frequencies and Mode Shapes:
An MDOF system has multiple natural frequencies (ω₁, ω₂, ω₃, …) and corresponding mode shapes that describe how each part vibrates. - Examples:
- Multi-storey buildings under earthquake forces.
- Automobile suspension (bounce and pitch motion).
- Aircraft wings (bending and torsion).
- Two-mass spring-damper systems.
- Application:
Used to analyze complex mechanical systems, structures, and machines where vibration occurs in several interconnected parts.
Comparison Between SDOF and MDOF Systems
| Feature | SDOF System | MDOF System |
| Definition | System with one independent motion | System with two or more independent motions |
| Number of Coordinates | One | Two or more |
| Equations of Motion | Single differential equation | Set of coupled differential equations |
| Natural Frequency | Only one | Multiple (depends on number of DOFs) |
| Mode Shape | Only one mode shape | Multiple mode shapes corresponding to each frequency |
| Complexity | Simple and easy to solve | Complex and needs matrix methods |
| Energy Distribution | Entire energy stored in one mode | Energy distributed among several modes |
| Examples | Mass-spring-damper, pendulum | Multi-storey building, car, aircraft wing, two-mass system |
(Note: The table is presented here in text form for clarity, not as a visual table.)
Physical Interpretation
The physical difference between SDOF and MDOF systems lies in how motion and energy are shared among components:
- In an SDOF system, the entire structure vibrates as a single unit.
- In an MDOF system, different parts can move independently, and the total response is the sum of all mode vibrations.
Thus, MDOF systems represent more realistic behavior of actual mechanical and structural systems where motion occurs in several directions or parts simultaneously.
Practical Examples
- SDOF Examples:
- Simple pendulum.
- Mass-spring-damper system.
- Vibrating cantilever with one concentrated mass.
- MDOF Examples:
- Multi-storey building (each floor represents one degree of freedom).
- Vehicle suspension system (bounce and pitch motion).
- Rotor shaft with multiple discs.
These examples show how SDOF systems are idealized, while MDOF systems closely model real-world applications.
Importance of Distinguishing Between SDOF and MDOF
- Helps engineers decide the appropriate modeling method for vibration analysis.
- Allows prediction of natural frequencies to avoid resonance.
- Useful in machine design, building safety, and vehicle suspension design.
- Provides understanding of how vibration energy is distributed among multiple components.
Conclusion
In conclusion, the main difference between SDOF and MDOF systems is that an SDOF system requires only one coordinate to describe motion, while an MDOF system needs two or more. The SDOF system is simple, with one natural frequency and one mode of vibration, whereas an MDOF system is complex, having multiple natural frequencies and mode shapes. While SDOF systems are useful for basic vibration analysis, MDOF systems represent real-world mechanical and structural systems more accurately, making them essential for engineering design and safety analysis.