7/24/2019 Isolated Building System Using Substructure Method

1/1

Dear Anastasia,

Below is what I did for your question. I hope that this is going to help you.

Jer-Fu (Jeff) Wang @ https://www.researchgate.net/profile/Jer-Fu_Wang

Sep. 9, 2015

EQUATIONS OF MOTION OF ISOLATED STRUCTURAL SYSTEM

Lets consider an n-degree-of-freedom planar building with an isolation system at its

bottom. Define pM , pC , and pK as the nn mass, damping, and stiffness matrices

of the fixed superstructure, respectively; bm is the mass of building base floor; bc and

bk are the damping and stiffness coefficient of the isolation system, respectively. We can

write the equation of motion of the superstructure as

)(txbpppppppp rMxKxCxM (1)

where px is the 1n displacement vector, with respect to the base, of the primary

building and bx is the displacement of the building base with respect to the ground.

Considering the free-body diagram of the superstructure, we can write

)()()(1

bbbb

n

igbppgbb xcxkuxxmuxm ii

(2)

or

)()(1111

2

1

2

2

1

bbbbgbb

n

ip

p

p

p

p

p

p

xcxkuxmm

x

x

x

m

m

m

i

n

(3)

In matrix form, Eq.(2) becomes

)()( bbbbgbtppr xcxkuxm xMI (4)

Combining Eqs. (1) and (4) gets

gtb

p

b

p

b

p

b

p

b

p

tpr

pppu

mxkxcxm

0x0

0Kx

0

0Cx

MI

rMM (5)

Now the equation of motion of the isolated structure is expressed in terms of the

properties of the superstructure and the isolation system! You can easily apply the modal

properties of the superstructure to construct pC and pK . You can also apply the

damping properties of the isolation system to calculate bc .