2.2.3. Types of Damping Models

The most common assumption in structural dynamics. The damping force is proportional to the velocity:

F = c · u˙(t)

Where:

• c = damping coefficient
• u˙(t) = velocity of the mass

Viscous damping is a model where the damping force, which resists the motion of a structure, is directly proportional to the velocity of that motion. This means that the faster the structure is moving, the more force is needed to slow it down. The constant 'c' in the formula represents the damping coefficient, which quantifies how much resistance there is for a given velocity. This model is widely used in engineering because it's simple and reflects many real-world scenarios well.

Think of viscous damping like the friction you feel when sliding your hand against a table. The faster you slide your hand, the more resistance you feel. The table provides a consistent amount of friction that increases with speed, similar to how the damping force increases with the velocity of the vibrating structure.

Damping due to friction between two surfaces:

F = µN

Where:

• µ = coefficient of friction
• N = normal force

This type of damping is non-linear and results in constant energy loss per cycle of motion.

Coulomb damping occurs when two surfaces rub against each other, creating resistance due to friction. Here, 'F' is the force of damping, 'µ' is the coefficient that represents how much friction there is between the surfaces, and 'N' is the normal force pressing the surfaces together. Unlike viscous damping, the energy loss in Coulomb damping is constant during each cycle of motion, making it a non-linear process. This means that as the surfaces move, they lose energy at a steady rate regardless of how fast they are moving.

Imagine pushing a box across the floor. The harder you push (the more force you apply), the more friction you have to overcome. Once the box starts sliding, it loses energy at a constant rate because of the friction between the box and the floor, no matter how fast you're pushing it – just like how Coulomb damping works.

Observed in materials that exhibit inelastic behavior. Energy is dissipated as the material goes through stress-strain cycles. Common in reinforced concrete and masonry.

Hysteretic damping occurs in materials that do not return to their original shape after being deformed, such as reinforced concrete or masonry. When these materials are subjected to cyclic loading (like during an earthquake), they absorb and dissipate energy as they deform. This energy dissipation during each cycle of loading and unloading is what characterizes hysteretic damping. It is crucial for understanding how structures behave during seismic events because it allows them to absorb energy without suffering catastrophic failure.

Consider a rubber band. If you stretch it repeatedly and let it go, it returns to its original shape each time. But if you stretch a piece of clay, it deforms and doesn't fully return to its shape. The clay dissipates energy as it yields to the force, demonstrating hysteretic behavior. This is similar to how certain materials in buildings absorb energy during seismic activity.

Structural damping involves multiple phenomena, including minor energy losses at joints. Radiation damping, though small, involves energy being transferred from the structure into the supporting soil.

Structural damping encompasses various mechanisms through which energy is lost within a structure, often due to minor movements at connections, joints, or cracks. As the structure vibrates, some of its energy is lost in these areas. Radiation damping, on the other hand, refers to the energy that is not only dissipated within the structure but also transferred to the foundation and surrounding soil. While this effect may be small, it is important in understanding how a building can lose vibrational energy to the earth beneath it.

Think of structural damping as a shaky bridge. As cars pass over, some energy escapes at the joints due to slight movements, similar to how vibrations can be lost at the joints of a guitar. Radiation damping is like vibrations from a speaker that also travel through walls, creating sound waves that escape into the building materials and foundation as they dissipate into the surrounding environment.

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