Short Answer:

Examples of multi-degree-of-freedom (MDOF) systems are those mechanical or structural systems that have more than one independent motion. These systems require two or more coordinates to describe their vibration completely. Common examples include multi-storey buildings, automobiles, aircraft wings, machinery with multiple components, and robotic arms.

In simple words, any system in which different parts can move independently, but still influence each other, is an MDOF system. Such systems are very common in real life and are analyzed to study natural frequencies, mode shapes, and dynamic behavior for safe and efficient design.

Detailed Explanation :

Examples of MDOF Systems

A multi-degree-of-freedom (MDOF) system is a system that requires more than one independent coordinate to describe its motion completely. In such systems, several masses or components can move independently but are connected through springs, dampers, or structural members, causing them to interact with each other.

In real-world engineering, most systems and structures are multi-degree systems because they consist of several components that move or vibrate simultaneously. Understanding examples of MDOF systems helps engineers model and analyze complex vibration behaviors that cannot be explained by a single-degree-of-freedom (SDOF) model.

Below are some common and practical examples of MDOF systems found in engineering applications.

1. Multi-Storey Building

A multi-storey building is one of the best examples of an MDOF system. Each floor of the building can move or vibrate independently when subjected to external forces such as wind or earthquakes.

• Each floor represents one degree of freedom, where its horizontal or vertical displacement can be considered an independent coordinate.
• When an external load (like seismic activity) acts, the floors move in different ways, but their motions are related due to the stiffness of the structural columns and beams connecting them.

Number of DOFs: Equal to the number of floors (for example, a three-storey building has 3 degrees of freedom).

Application: Used in earthquake vibration analysis to determine how each floor responds to ground motion.

2. Automobile (Vehicle Suspension System)

An automobile system is a common mechanical example of a two-degree-of-freedom (2-DOF) system.

• The car body can move vertically (bounce) and rotate about its center of gravity (pitch).
• The suspension springs and shock absorbers connect the car body to the wheels, which also move independently due to road irregularities.

Degrees of Freedom:

1. Vertical motion (bounce of the car body).
2. Angular motion (pitching about the center of gravity).

Application: Used in designing and analyzing vehicle suspensions to ensure ride comfort and stability.

3. Aircraft Wing

The aircraft wing behaves as a multi-degree-of-freedom system because it can deform and vibrate in multiple directions:

• Bending motion: Up and down deflection of the wing.
• Torsional motion: Twisting about the wing’s longitudinal axis.

These two motions are coupled, meaning that bending affects twisting and vice versa.

Degrees of Freedom:

1. Vertical bending.
2. Torsional twisting.

Application: Used in aeroelasticity studies to prevent structural failure due to vibration and flutter during flight.

4. Two-Mass Spring System

A simple two-mass spring system demonstrates the concept of a multi-degree-of-freedom mechanical model.

• It consists of two masses  and  connected by springs and dampers.
• Each mass can move independently, but the motion of one mass affects the other through the connecting spring.

Degrees of Freedom: Two (since two independent displacements  and  are required to describe the motion).

Application: Used to model vibration isolation, machine components, and rotating equipment.

5. Robotic Manipulator (Robot Arm)

A robotic manipulator or robot arm is a complex MDOF system with several joints and links.

• Each joint or link provides one independent motion, such as rotation or translation.
• The total number of degrees of freedom equals the total number of movable joints.

Degrees of Freedom: Usually between 4 and 7 for industrial robots, depending on design.

Application: Used in automation, manufacturing, and precision control systems where coordinated multi-axis movement is required.

6. Bridge Structure

A bridge, especially a suspension or cantilever bridge, behaves as a multi-degree system because different sections can move independently due to vehicle loads or wind forces.

• The bridge deck, cables, and supporting towers all vibrate in different modes such as vertical bending, torsion, and lateral motion.
• Engineers consider these degrees of freedom to analyze vibrations and ensure the bridge remains safe under dynamic loads.

Application: Used in structural vibration analysis to prevent resonance or fatigue failure.

7. Turbine and Shaft System

A rotating turbine or shaft system also represents a multi-degree-of-freedom model.

• The shaft may have several disks mounted along its length, and each disk can vibrate independently in lateral and torsional directions.
• The motion of one disk influences the others through the shaft’s stiffness.

Degrees of Freedom:
Depend on the number of disks or masses considered in the model.

Application: Used in rotor dynamics to study critical speeds and prevent excessive vibrations.

8. Human Body Model

Even the human body can be modeled as an MDOF system for studying biomechanics or vehicle vibration comfort.

• Different parts of the body (head, torso, legs) have independent motions connected by flexible joints.
• When seated in a moving vehicle, the body experiences vibrations that can be analyzed using MDOF principles.

Application: Used in ergonomics, vehicle ride comfort, and biomechanical studies.

9. Machine Tool Structure

In a lathe, milling, or drilling machine, different components such as the tool, spindle, and workpiece vibrate independently when subjected to cutting forces.

• Each component can have multiple vibration modes like bending or torsion.
• The combined system behaves as a multi-degree model with several interacting parts.

Application: Used in machine design to avoid chatter vibrations and improve machining accuracy.

10. Multi-Link Mechanical Systems

Any mechanical linkage system with multiple connected rods or levers, such as in cranes or excavators, forms an MDOF system. Each link can rotate or move independently depending on the applied forces.

Application: Used in kinematics and control systems to simulate real motion.

Importance of Studying MDOF Systems

• To understand coupled vibration behavior in machines and structures.
• To identify natural frequencies and mode shapes for safe operation.
• To design systems with proper damping and stiffness to prevent resonance.
• To predict dynamic response accurately under various loading conditions.

Conclusion

In conclusion, examples of MDOF systems include multi-storey buildings, vehicles, aircraft wings, robotic arms, turbine shafts, and bridges. These systems have more than one independent motion, and their analysis helps in determining natural frequencies, mode shapes, and vibration responses. Since most real-world structures and machines are multi-degree systems, studying them is essential for designing safe, efficient, and stable mechanical and structural systems.