2.2. Concept of Damping

Damping refers to the mechanism by which vibrational energy is dissipated in a dynamic system. Without damping, a structure would continue to vibrate indefinitely once disturbed. In reality, due to internal friction, air resistance, and material deformation, energy is lost over time, leading to the decay of motion.

Damping is a crucial concept in engineering that describes how energy from vibrations in a system is gradually lost. Imagine you have a swing; if you give it a push, it will swing back and forth. Without any resistance or air slowing it down, it would continue swinging forever. However, in the real world, air resistance and friction at the swing’s joints slow it down over time, causing it to eventually come to a stop. This is similar to how structures behave when they are vibrated, such as during an earthquake. If there's no damping, the vibrations would theoretically go on forever. But because of damping, these vibrations decrease until they vanish.

Think of a car going over a bumpy road. The tires and suspension of the car help absorb the bumps and vibrations, allowing for a smoother ride. This is a form of damping at work, where energy from the bumps is dissipated through the car's suspension system, preventing continuous jarring and shaking.

Damping in civil structures arises from:

• Material Damping: Energy loss due to internal friction in materials (concrete, steel).
• Frictional Damping: At interfaces or joints where slipping may occur.
• Structural Damping: Caused by non-linear behavior at connections and cracks.
• Foundation Damping: Due to interaction between the foundation and soil.

Different sources contribute to damping in structures. Material damping happens when the material's internal structure, like concrete or steel, absorbs some energy due to internal friction. Frictional damping occurs where two surfaces meet and slide against each other, like when a door hinges. Structural damping is related to how different parts of a structure, such as connections and cracks, behave under stress; these non-linear behaviors also contribute to how energy is absorbed. Lastly, foundation damping relates to how the foundation interacts with the ground and the soil around it, allowing more energy to dissipate.

Consider a box filled with jellybeans shaken on a table. Some jellybeans will squish and absorb energy, similar to how materials absorb energy in damping. The jiggling of the box is akin to vibrations in a structure, and the way the jellybeans squish is like how different sources of damping work to reduce and absorb that energy.

2.2.3.1 Viscous Damping: 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

2.2.3.2 Coulomb (Dry Friction) Damping: 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.

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

2.2.3.4 Structural and Radiation Damping: 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.

There are different models for understanding how damping works in structures. Viscous damping assumes that the resistance to motion is proportional to how fast something is moving, like how fluid dampers operate in a car's shock absorbers.

Coulomb damping focuses on the friction between two surfaces, like the brakes on a car; this damping is less predictable because it doesn't change with speed, resulting in a constant energy loss each time the object moves over the surface.

Hysteretic damping happens in materials that do not return to their original shape after being stressed. For instance, when reinforced concrete is used in buildings, it dissipates energy through repeated stress cycles.

Finally, structural damping accounts for energy loss at structural connections and interactions with the soil, which also absorbs some energy but tends to be less significant.

Think of a bicycle going down a hill. The brakes apply friction (Coulomb damping), making the bike slow down. If you're riding over bumpy terrain, the bicycle's suspension system absorbs shock (viscous damping), which works similarly to how damping is modeled in structures. The frame of the bike might flex and twist under weight (hysteretic damping), while the contact between the tires and the ground helps slow the bike down further (structural damping).

The damping ratio ζ is a non-dimensional quantity used to express the level of damping:

ζ = c / (2√(km)) Where:

• c = damping coefficient
• k = stiffness
• m = mass

Values of damping ratio:

• ζ = 0: Undamped
• 0 < ζ < 1: Underdamped (typical in structures)
• ζ = 1: Critically damped
• ζ > 1: Overdamped

For buildings:

• Steel structures: ζ ≈ 2%−3%
• Concrete structures: ζ ≈ 4%−7%

The damping ratio, represented by ζ, helps us understand how well a structure can dissipate energy. If ζ equals 0, it means there's no damping, and the structure would vibrate indefinitely. Values between 0 and 1 indicate underdamping, which is commonly seen in structures; they lose energy over time but still vibrate noticeably. When ζ is equal to 1, the system is critically damped, which is ideal, allowing it to return to equilibrium without overshooting. If ζ is greater than 1, the system is overdamped, meaning it returns to its position slowly and without oscillation.

For practical purposes, typical damping ratios are often estimated: steel structures might have a damping ratio around 2% to 3%, while concrete structures can range from 4% to 7%.

You can think of damping ratio like a car's shock absorbers. If the shocks are too soft (underdamped), the car bounces around too much. If they are just right (critically damped), the car smooths out quickly without excessive bouncing. If they are too stiff (overdamped), the car takes a long time to settle back down after hitting bumps, impacting ride comfort. Similarly, in buildings, having the right amount of damping ensures stability and comfort during events like earthquakes.

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