1

Two Story Building free vibrations

2 1.62k

m

A1

m1

m2

.62A1

Mode II

A1

m1

m2

1.62A1

Mode I

1 0.62k

m

General Free Vibration Response

2 1.62k

m

Mode II

A1m1

m2

.62

A1

A1m1

m2

1.62

A1

Mode I +

1 0.62k

m

1 1 1 1 1 2 2

2 1 1 1 1 2 2

( ) sin sin

( ) 0.62sin 1.62sin

I II

I II

u t A t A t

u t A t A t

4 unknowns: 4 IC

1 1 1 2, , ,I IIA A 1 10 1 10 2 20 2 20(0) , (0) , (0) , (0)u u u v u u u v

2

More floors:

m1

m2

m3

m4

m1

m2

m3

m4

u1(t)

u2(t)

u3(t)

u4(t)

k1

k2

k2

k3

1 1 1 2 2 1

2 2 2 2 3 3 2

3 3 3 3 4 4 3

4 4 4 4 4

0

0

0

0

m u k k k u

m u k k k k u

m u k k k k u

m u k k u

1 1 1 0i i i i i i i imu k u u k u u

ki+1(ui+1- ui)

ui(t)

ki (ui- ui-1)

Find 4 natural frequencies,

Mode shapes

m1

m2

m3

m4

m1

m2

m3

m4

u1(t)

u2(t)

u3(t)

u4(t)

k1

k2

k2

k3

1 1 1 2 2 1

2 2 2 2 3 3 2

3 3 3 3 4 4 3

4 4 4 4 4

K:Stiffness Matrix

0

0

0

0

M: Mass Matrix

m u k k k u

m u k k k k u

m u k k k k u

m u k k u

2 2det 0, 0A M K M K

2 2det 0, 0A 1 H 1 H

3

They look like this (equal k’s, equal m’s)

1 0.35k

m

2

k

m 3 1.5

k

m 4 1.9

k

m

Mode Shape Orthogonality

0j iA A i j M

2 0i i iA A M K2

2

2

0

0

0

i j i j i

T T

i i j i j

i i j i j

A A A A

A A A A

A A A A

M K

M K

M K

2 0j j jA A M K

2 0j i j i jA A A A M K

(1)

(2)

2 2 0i j j iA A M