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Damping Effects

Run the Computer Simulation

To understand how important damping is, run the computer simulation, once with the default values and once with no damping. Compare the results to see just what effect damping has on building vibrations, remembering to compare the scales on the y axes of the two graphs. Note how much more each floor shakes, and how each never stabilizes as in the damped simulation, when all damping is removed. To see even more clearly the effect of damping, you can "turn off" the sine sweep by making the beginning and ending frequency the same. Damping makes the vibration regular and smaller compared to undamped oscillation.

As previously noted, earthquakes can be particularly disastrous to buildings through resonance. However, most structures have some amount of damping, which helps curbs resonance. Damping of buildings occurs in incredibly diverse ways and can incorporate many different kinds of mechanisms. For example, the following are sources of damping:

• Air: The viscous drag of the structure moving through air.
• Columns: May have some visco-elasticity.
• Structural Yielding: The energy absorbed by yielding parts of a structure can significantly reduce vibrations in an earthquake.
  • Severe earthquakes cause structural elements to give way, adding a significant amount of damping.
  • Properly designed structures will yield in a controlled fashion. (A further discussion about building design for earthquakes is located at Exploratorium: Life Along the Faultline.)

Because damping is so complicated, and because its effects are often secondary compared to inertial (mass) and stiffness properties, it is common to first assume that the damping forces increase in proportion to the velocity . In addition, it is quite convenient and not uncommon to describe the damping of the structural system by a damping matrix that is proportional to the mass matrix and the stiffness matrix, , where the constants and are proportionality constants. This type of damping is called proportional damping. The units of the elements of the damping matrix are [Force/Length/Time]. The units of the constant is [1/Time] and the units of the constant is [Time]. For our laboratory model, we were happy to find that the damping can be very accurately described by a simple diagonal matrix

Since the mass matrix is also diagonal, we call this type of damping mass-proportional damping. Physically, this type of damping matrix indicates that most of the damping applied to the structure is due to the viscous drag of the structure moving through the air. We call this aerodynamic damping. You can think of this kind of damping as a set of viscous "dash-pots" connecting each mass to a stationary point. Including the effects of damping, the matrix equations of motion is written

Just about any structural system that is subjected to earthquake ground accelerations, , may be written in the form of equation (12).

Quick Quiz: If the damping matrix were would it be mass proportional to a diagonal mass matrix of the form

Yes
No