5.4. SDOF System: Formulation and Idealization

A Single Degree of Freedom (SDOF) system is the simplest dynamic model where the motion of the system can be described using a single coordinate, typically lateral displacement.

A Single Degree of Freedom (SDOF) system simplifies dynamic problems to their most basic form. It means that only one type of movement or vibration is being considered, which usually is sideways or lateral movement. This simplification is crucial in analyses because it allows engineers to focus on the primary way a structure behaves when subjected to dynamic forces like seismic activity.

Imagine a swing in a playground. The swing can only move forward and backward on a single path. This motion is akin to an SDOF system where the movement is straightforward, allowing us to predict how the swing will react when pushed. Just like the swing's motion is simple and predictable, so too is the SDOF system in structural analysis.

• Mass (m): Represents inertia. • Stiffness (k): Represents restoring force. • Damping (c): Represents energy dissipation (optional in ideal models). • Displacement (u): Time-dependent motion variable.

An SDOF system comprises several key elements:

1. Mass (m): This represents the object's inertia, or resistance to motion. If you push something heavy, it takes more force to move it than something light.
2. Stiffness (k): This refers to the structure's ability to return to its original shape after being deformed. A stiff structure returns to its original shape quickly, while a flexible one does so slowly.
3. Damping (c): This is a mechanism that dissipates energy in the system, helping reduce the oscillations over time. It might not be included in ideal SDOF models, but it's crucial in real-life applications to control movement reactions.
4. Displacement (u): This is the actual movement experienced by the system, which changes over time due to external forces.

Think of a car on a suspension bridge as an SDOF system. The car’s weight is the mass (m), the bridge's resistance to bending is the stiffness (k), the shock absorbers that limit how much the car bounces are the damping (c), and the actual movement of the car up and down as it drives across is the displacement (u). Understanding how each part works together helps ensure the bridge can safely handle the car's movement.

mu¨(t)+ku(t)=−mu¨(t) g

The equation of motion for an undamped SDOF system describes how the system responds to forces over time. Here, 'mu¨(t)' represents the acceleration experienced by the mass, 'k' is the stiffness which relates to how much restoring force comes into play, and 'u(t)' is the displacement. The equation illustrates that the total force acting on the mass is equal to its mass times its acceleration, minus the force acting on the mass due to the ground motion. This relationship helps predict how the structure will behave when subjected to seismic forces.

Imagine a person jumping on a trampoline. As they jump (apply force), the trampoline stretches thin (displacement), only to whip back (restore the force) when they fall, showcasing how forces interact with mass and acceleration. The equation explains this dynamic interaction, serving as the mathematical representation of how jumpers like them move up and down.

mu¨(t)+cu˙(t)+ku(t)=−mu¨(t) g

In a damped SDOF system, the equation includes damping: 'cu˙(t)'. This term represents resistance that dissipates energy from the system to reduce vibrations. The inclusion of damping allows for a more realistic model of how structures, particularly in buildings, behave. While the mass and stiffness are still critical, damping becomes vital in refining the model by accounting for energy lost during oscillations, making predictions and analyses much more accurate in real-world scenarios.

Think of riding a bicycle on a bumpy road. As you ride, your bike has shock absorbers that help smooth out the ride (damping) by absorbing some of the impact from bumps. The shock absorbers reduce quick jumps and drops, allowing for a smoother ride, similar to how dampers in a structural system reduce violent vibrations during an earthquake.

• Building floors are infinitely rigid in their own plane. • Masses are lumped at floor levels. • Only lateral displacements are considered (for seismic loads). • Linear elastic behavior unless specified otherwise.

The idealization of structures into SDOF systems involves several key assumptions. First, the floors of a building are assumed to be infinitely rigid, meaning they don't bend, ensuring that the mass is concentrated at certain levels. They also focus solely on lateral displacement, ignoring vertical movements which simplifies the analysis. Lastly, the behavior of the materials is considered linear elastic, meaning they return to their original shape after the load is removed, unless otherwise stated. These assumptions help simplify the modeling process but can also limit the accuracy in some cases.

Imagine building a model of a skyscraper using blocks. If you assume the blocks are perfectly rigid, you only need to focus on how the entire model sways side to side in an earthquake instead of worrying about each block bending or moving individually. This idealization makes creating and adjusting your model easier, but remember—this same simplification might overlook how actual buildings (with flexible materials) behave during strong quakes.

--