MAE 5130: VISCOUS FLOWS

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MAE 5130: VISCOUS FLOWS. Tuned Mass Damper Example October 5, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk. CITIGROUP BUILDING IN MANHATTAN. One of largest skyscrapers in NYC 59 floors, 915-foot (279 m) tall

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MAE 5130: VISCOUS FLOWS

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MAE 5130: VISCOUS FLOWS

Tuned Mass Damper Example

October 5, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

CITIGROUP BUILDING IN MANHATTAN
• One of largest skyscrapers in NYC
• 59 floors, 915-foot (279 m) tall
• To help stabilize building, a tuned mass damper was placed at its top
• Weighs 400 tons and has a volume of 255 cubic feet (7 m³)
• Designed to counterbalance effects of wind by making the building sway
• Concrete block that slides on a thick layer of oil and converts the kinetic energy of the building into friction
• Mass reduces building's movement from wind deflection by 40-50%
• Citigroup Center was the first skyscraper in the United States to feature a tuned mass damper
PROBLEM STATEMENT AND QUESTIONS
• Bearing is fixed to building, which oscillates, while block stays approximately stationary as the building oscillates horizontally under it, and exerts on the building a horizontal damping force in the direction opposite to the building’s velocity. A block of mass, M, per unit depth into the page is supported on two oil bearings of length, L, the oil being pumped from the center reservoir out via the bearings and around again. The pump is adjusted so that the bearing gap is maintained at a fixed value h, such that h << L. The reservoir width is a, and again h << a. The oil has a density r and a very high viscosity m. In your analysis, you may assume that the flow inside the bearings is highly viscous, i.e. inertia free.
• Given quantities are: r, m, h, L, a, M, g, V(t)

QUESTIONS:

• At an instant when the rightward horizontal velocity of the building relative to the mass is V(t), obtain expressions for:
• P, the required pressure in the reservoir
• Q’, the required oil volume flow rate through the pump, per unit depth
• F’, the instantaneous leftward force exerted by the oil on the building, per unit depth
• Suppose that we come up with a specific TMD design where h/L=10-4 and the maximum Reynolds number based on h and mean flow speed in the gap is or order unity. What additional conditions, if any, need to be satisfied for your answer to part 1 to be applicable?