Damping with a Mathematical Touch
Auburn Walker
The recent trend in structural design is moving towards taller buildings, longer bridges, and more daring structures. These structures tend to be more sensitive to wind vibrations, vibrations from human use, and soil vibrations. This has led engineers to the use of damping devices as a way to limit the excess motion caused by these vibrations. Therefore, the need for damping devices is rapidly increasing, especially in areas prone to earthquakes and strong winds. “The damping in a mechanical or structural system is a measure of the rate at which the energy of motion of the system is dissipated”(Breukelman, p3). The role that damping plays is very significant and must be considered when designing; when it is not, there will be harmful consequences.
The type and level of appropriate damping varies from situation to situation. It varies with the materials used, the form of the structure, the nature of the subsoil and vibration, and any other components interacting with the material of the structure. Recommended damping percentages for some general cases have already been experimentally determined. Some of these cases include a few various types of steel frames and concrete frames. “For steel and concrete buildings, Hart and Vasudevan offer a systematic method for estimating damping as a function of spectral velocity and modal frequency, based on an analysis of the response of buildings in the San Fernando earthquake” (Dowrick, p267). However, designers will still need to use their own discretion in interpreting these available values based on their specific structural case.
First, an undamped vibration will be discussed. This graph shows what a typical undamped vibration looks like:
[Example Problems in Matlab and Working Model]
The undamped vibration is continuous. When a damper is added to the system, “that damping gradually dissipates the energy initially imparted to the system, and consequently the motion dies out with increasing time” (Boyce and DiPrima, p192).
[Watson]
The motion equation (which is a second order, ordinary differential equation) can be used to analyze these types of vibrations. “The equation of motion has the following form:
where m denotes the mass of vibrating body, c is the viscous damping coefficient, and k stands for the elasticity coefficient” (Osinski, p31). There are a few different forms of this equation. Some forms divide through by m in the beginning and change the x to u (where u is the displacement), which “may be written
from which (for moderate damping)
where
is the damping ratio, and
is the damped circular frequency” (Dowrick, p19). (The damping values that were mentioned earlier to have already been experimentally determined were referring to the value of.) The actual solving of these equations by hand, taking into account the different variables, can get quite lengthy and complicated; therefore, most of the calculations are done by computer programs—such as Matlab.
Sufficient damping for these higher, longer, more elaborate structures is becoming a reality due to the technological advances. Technology is advancing into more electrical equipment, lowering the mass held in buildings. Also, more and more types of damping devices are being created and improved to fit specific cases.
One type of damping device is the tuned mass damper. A tuned mass damper has a secondary mass attached to a structure with a viscous damper and spring system. “The spring and mass are ‘tuned’ so as to have a natural frequency close to that of the primary structure. When properly tuned, the TMD mass oscillates in the opposite direction from the primary structure” (Breukelman, p3). A 3-D view of a pendulum type tuned mass damper is shown here.
[RWDI, p3]
A tuned mass damper designed for the PetronasTowers (at one time the tallest building in the world), in Malaysia, is shown below.
[RWDI, p1) [RWDI, p3]
Some simpler damping devices are rubber pads and stacks of Belleville washers.
Damping is one of the first things to be considered when making building plans. If it is not, insufficient damping is likely to occur. This would lead to excessive vibrations causing faulty machinery performance, shortened life of a structure (due to more stress), and much more. Insufficient damping was one of the causes of the fall of the TacomaNarrowsBridge—there was not enough damping to control the aerodynamic oscillations due to the wind vibrations. “Damping is [also] very important with earthquakes since it dissipates the destructive energy of an earthquake which will help reduce the damage to the building” (Watson, p1). The correct level of damping is found through differential equations and checked by experimentation. This is all an essential part of structural design.
References
[1] Abu-Sitta, Salman and Gould, Phillip. Dynamic Response of Structures to Wind and
Earthquake Loading. New York: John Wiley & Sons, Inc., 1980.
[2] Bruekelman, Brian. “Damping Systems.” RWDI. Technotes
Newsletter Issue 10. 31 July 2002
[3] Breukelman, Brian. “Tuned Mass Dampers – How They Work.” RWDI. Technotes
Newsletter Issue 6. 31 July 2002
[4] Boyce, William and DiPrima, Richard. Elementary Differential Equations and
Boundary Value Problems, Seventh Edition. New York: John Wiley & Sons, Inc., 2001.
[5] “Damping Systems.” RWDI. 30 July 2002
[6] Dowrick, D.J.. Earthquake Resistant Design, A Manual for Engineers and
Architects. New York: John Wiley & Sons, Ltd., 1977.
[7] Dowrick, David. Earthquake Resistant Design, for Engineers and Architects, 2 ed.
New York: John Wiley & Sons, Ltd., 1987.
[8] “Example Problems in Matlab and Working Model.” ME240-Example Problems.
31 July 2002
[9] Osinski, Zbigniew, ed. Damping of Vibrations. Netherlands: A.A.Balkema,
Rotterdam., 1998.
[10] “SDOF Systems: Free Vibration with Viscous Damping.” efunda: engineering
fundamentals. 2002. 31 July 2002
[11] Watson, James. “Damping.” 20 September 2001. 31 July 2002