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10.10 A three-story building is modeled as shown in Fig. P10.10. The stiffness matrix and the mass matrix for the structure are, respectively, 800 -800 K = -800 2400 –1600 kips/in.. 0 -1600 4000 100 M =|0 20 kip-sec²/in. Lo0 2

10.10 A three-story building is modeled as shown in Fig. P10.10. The stiffness matrix and the mass matrix for the structure are, respectively, 800 -800 K = -800 2400 –1600 kips/in.. 0 -1600 4000 100 M =|0 20 kip-sec²/in. Lo0 2

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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10.10 A three-story building is modeled as shown in Fig. P10.10. The stiffness matrix and the mass matrix for the structure are, respectively, 800 -800 K= -800 2400 –1600 kips/in.. 0 -1600 4000 [1007 M = 0 20 kip-sec²/in. L00 2 One procedure for defining a system viscous damping matrix C that leads to a diagonal generalized damping matrix C is to employ Rayleigh damping, which is defined by C = a,M + a;K (10.76) This is also called proportional damping, since the damping matrix is proportional to a linear combination of the mass matrix and the stiffness matrix. The constants a, and a, can be chosen to produce specified modal damping factors for two selected modes. Combining Eqs. 10.74, 10.75, and 10.76, we get (a) Determine the Rayleigh damping coefficients ao and a, in Eq. 10.76 such that the 3-DOF system has damping facrors š, = 52 = 0.01 in its first two modes. (b) What is the resulting value of the damping factor §, of the third mode? u, m, = 1 kip-sec²fin. k, = 800 kips/in. m2 = 2 kz = 1600 kips/in. m3 = 2 kg = 2400 kips/in. Figure P10.10
Transcribed Image Text:10.10 A three-story building is modeled as shown in Fig. P10.10. The stiffness matrix and the mass matrix for the structure are, respectively, 800 -800 K= -800 2400 –1600 kips/in.. 0 -1600 4000 [1007 M = 0 20 kip-sec²/in. L00 2 One procedure for defining a system viscous damping matrix C that leads to a diagonal generalized damping matrix C is to employ Rayleigh damping, which is defined by C = a,M + a;K (10.76) This is also called proportional damping, since the damping matrix is proportional to a linear combination of the mass matrix and the stiffness matrix. The constants a, and a, can be chosen to produce specified modal damping factors for two selected modes. Combining Eqs. 10.74, 10.75, and 10.76, we get (a) Determine the Rayleigh damping coefficients ao and a, in Eq. 10.76 such that the 3-DOF system has damping facrors š, = 52 = 0.01 in its first two modes. (b) What is the resulting value of the damping factor §, of the third mode? u, m, = 1 kip-sec²fin. k, = 800 kips/in. m2 = 2 kz = 1600 kips/in. m3 = 2 kg = 2400 kips/in. Figure P10.10
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