Figure 8.30a. Viscoelastic polymer damper. A building damping of about 4% can be attained using these dampers. Buildings equipped with viscoelastic dampers include the World Trade Center, New York, destroyed on Sept 11, 2001, and the Columbia Seafirst Center, Seattle.

in a direction opposite to the motions of the building itself, thereby reducing the building's oscillations.

The mass itself need weigh only a small fraction—0.25 to 0.70%—of the building's total weight, which corresponds to about 1 to 2% of first modal mass. "Tuned" simply means the mass can be adjusted to move in a fundamental period equal to the building's natural period so that it will be more effective in counteracting the building oscillations. In addition to the initial tuning when it is first installed, the TMD may be fine-tuned as the building period changes with time. The period may increase as the building occupancy changes, as nonstructural partitions are added, or as elements contributing nonstructural stiffness "loosen-up" after initial wind storms.

Thus a TMD may be considered as a small damped mass of single-degree-of-system riding "piggy-back" atop a building. Although its mass is a small fraction of the building's mass, its vibration characteristics are adjusted to mimic those of the building's. For example, if a tall building sways, say, 24 in. to the right at a fundamental frequency of 0.16 Hz, the TMD is designed to move to the left at the same frequency.

The idea of using the inertia of a floating mass to tame the sway of a tall building is not entirely new. In fact, the invention of the TMD as an energy-dissipative vibration absorber is credited to Frahm, who developed the concept in 1909. The theory was later described by Den Hertog in his classic textbook in 1956, and since then has been applied in automotive and aircraft engines to reduce vibrations. Since the wind force-time relationship is not harmonic (sinusoidal), the basic ideas developed by Den Hartog have been modified in building applications to account for the random nature of wind.

When activated during windstorms, the TMD becomes free-floating by rising on a nearly frictionless film of oil. To dissipate energy, the TMD must be allowed to move with respect to the building. In the earlier TMDs installed in tall buildings, spring-like devices connecting the mass to the building pull the building back to center, as the building sways away from its equilibrium position. The mass is also connected to the building with a damping device, in the form of a hydraulic actuator, which is controlled to provide a predetermined percentage of critical damping. This limits the lateral displacements of the mass relative to the building.

The TMD's advantages become academic in a power failure. It needs electricity to work and if that's lost in a heavy windstorm, when the TMD would most be needed, it wouldn't work. So it is advisable to have the TMD wired to an emergency power system.

During a major wind storm, the mass will move in relation to the building some 2 to 5 ft. The system is controlled to activate when a predetermined building lateral acceleration occurs. This motion is registered on an accelerometer and, if the allowable limit is reached, the mass is activated automatically.

Figure 8.30b. Tuned mass damper for Citicorp Tower, New York: (1) building elevation; (2) plan; (3) first-mode response; (4) TMD atop the building. Citicorp Tower, New York

The Citicorp Tower (shown in schematic view in Fig. 8.30b) consists of a unique structural system of perimeter-braced tubes elevated on four 112-ft-high columns and a central core. It rises approximately 914 ft above grade. The tower is square in cross section with plan dimensions of approximately 157 by 157 ft. The top 140-ft portion of the tower slopes downward from north to south.

The TMD designed for the building consists of a concrete block 29 x 29 x 9 ft that weighs 410 tons (820 kips). It is attached to the buiding with two nitrogen-charged pneumatic spring devices and two hydraulic actuators that are controlled to provide damping to the TMD and linearize the "springs." One set counters north-south building dynamic motion and the other set counters east-west motion. The spring stiffness, and thereby the TMD frequency, is adjusted (tuned) by changing the pneumatic pressure. It also has an antiyaw device to prevent twisting of the block, and snubbers to prevent excessive motion of the block.

The TMD is capable of a 45-in. operating stroke in each orthogonal direction. The operating period is adjustable independently in each axis. The mass block is supported with twelve 22-in.-diameter pressure-balanced bearings connected to a hydraulic pump.

The block positioned at the building's 63rd floor (780 ft high) represents approximately 2% of first-period modal mass of the building. The motions of the block are controlled by pneumatic devices and servohydraulics resulting in a system that has the characteristics of a spring-mass-damper system, as shown schematically in Fig. 8.30c.

To dissipate energy, the TMD is allowed to move with respect to the building. It is continuously on standby, and is designed to start up automatically whenever the accelerations exceeds a predetermined value. The TMD kicks in whenever the accelerations for two successive cycles of building motion exceed 3 milli-g (1 milli-g = 1/1000 of acceleration due to gravity. Therefore, 3 milli-g corresponds to an acceleration of approximately 1.16 in./sec2).

The system continues to operate as long as building motions continue and stops only a half-hour after the last pair of building cycles for which maximum acceleration is

Figure 8.30c. Schematic view of a TMD operating on top of the Citicorp Center. The TMD consists of a 400-ton concrete block bearing on a thin film of oil. The structural stiffness of the TMD is aided by pneumatic springs tuned to the frequency of the building. The TMD damping system is aided by shock absorbers.

Figure 8.30c. Schematic view of a TMD operating on top of the Citicorp Center. The TMD consists of a 400-ton concrete block bearing on a thin film of oil. The structural stiffness of the TMD is aided by pneumatic springs tuned to the frequency of the building. The TMD damping system is aided by shock absorbers.

greater than 0.75 milli-g. The TMD provides the building with an effective structural damping of about 4% of critical. This is a significant increase above the inherent damping estimated to be just under 1% of critical. Since wind-induced accelerations of a building are approximately proportional to the inverse of the square root of the damping, when in operation the TMD reduces the building sway oscillations by over 40%.

The Citicorp TMD is installed on the 63rd floor. At this elevation, the building may be represented by a single-degree-of-freedom system with a modal mass of 40,000 kips resonating biaxially at a 6.8-sec period with a critical damping factor of 1%. The TMD is designed with a moving mass of 820 kips, biaxially resonant with a period of 6.7 seconds plus or minus 20%, and an adjustable damping of 8 to 14% of critical. Observe that the moving mass represents approximately 2% of the first-period modal mass, which typically corresponds to about 0.6 to 0.7% of the total mass. John Hancock Tower, Boston, MA

The TMD for the John Hancock Mutual Life Insurance Co.'s glass-clad landmark in Boston is somewhat different from that for Citicorp Tower. It was added as an afterthought to prevent occupant discomfort. Second, Hancock Tower is rectangular in plan and consists of moment frames unlike Citicorp's diagonally braced frame (Fig. 8.30d). Because of the building's shape, location, and vibration properties, its dynamic wind response is mainly in the east-west direction and in torsion about its vertical axis. There is a TMD near each end of an upper floor. They are tuned to a vibration period of approximately 7.5 sec. The total east-west moving mass represents about 1.4% of the building first-mode generalized mass, while in the twist direction the moving masses represents about 2.1% of the building's

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