30.1.2. Single Degree of Freedom (SDOF) Systems

• A simplified model used to study the dynamic response of structures.

Single Degree of Freedom (SDOF) systems are simple models used in engineering to analyze how structures respond to dynamic forces, such as those generated during an earthquake. By focusing on a single degree of freedom, engineers can simplify the complex behaviors of real structures into more manageable mathematical models.

Think of a swing at a playground. When you push it, it moves back and forth in a simple manner, which is similar to how an SDOF system operates. Just like the swing can be analyzed simply by looking at its back-and-forth movement—its only degree of motion—structures can be similarly simplified for analysis.

• SDOF systems are idealized with a mass-spring-damper model: mx¨(t)+cx˙(t)+kx(t)=−mu¨(t) g

In earthquake engineering, SDOF systems are often modeled as mass-spring-damper systems. The equation mx¨(t)+cx˙(t)+kx(t)=−mu¨(t) represents this system, where:

• m is the mass of the structure,
• c represents the damping constant which absorbs energy,
• k is the stiffness of the spring, and
• u¨(t) is the ground acceleration. This equation helps determine how the structure will respond to seismic forces by capturing essential dynamics such as how quickly it accelerates, how fast it moves, and how much it stretches.

Imagine a car suspension system. The car's mass acts like the mass in our equation, the springs work like the spring in the model, and the shocks dampen the movement, preventing excessive bouncing. Just like how the car's suspension system absorbs bumps in the road, the damping in SDOF systems helps manage forces during an earthquake.

• Where: – x¨(t): Relative acceleration – x˙(t): Relative velocity – x(t): Relative displacement – u¨(t): Ground acceleration g

Each term in the SDOF equation has a specific meaning that contributes to understanding the structure's behavior:

• x¨(t) is the acceleration of the structure relative to ground movement, indicating how fast the structure is changing its speed.
• x˙(t) is the velocity of the structure, showing how fast the structure is moving at any moment.
• x(t) is the displacement, which tells us how far the structure has moved from its original position.
• u¨(t) is the ground acceleration caused by an earthquake, and it serves as the external force acting on the structure. By examining these parameters, engineers can predict the dynamic response to seismic events.

Consider a person on a trampoline. When they jump (u¨(t)), the acceleration they feel (x¨(t)) increases and makes them bounce higher, which correlates to their speed on the way up (x˙(t)) and how far they are from a standing position (x(t)). Similarly, in our SDOF model, the interaction between these variables helps describe the motion of structures during earthquakes.

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