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When a lateral force is apply to the top of a building, the relation between lateral force P and lateral displacement x, is

P = k x [Eq. 1]

Based on Newton's second law of motion

P = -m a [Eq. 2]

where a is acceleration and is equal to second derivative of time (=
dx^{2}/dt ).

Combined equation 1 and 2, the equation become

m (dx^{2}/dt) + kx = 0
[Eq.3]

The solution of equation is a wave function

x = e ^{±iwt} =
A sin wt
+ B cos wt
[Eq.4]

where A and B are values depend on initial condition, and w is circular natural frequency of structure, and

w= (k/m)^{1/2}
[Eq. 5]

where m is mass of structure, k is stiffness of lateral force resistance elements of structure.

Natural frequency of vibration is

f = w/2p =
k/m)^{1/2} / 2p
[Eq. 6]

T = 1/f =2p (m/k)^{1/2}
[Eq. 7]

Example of a building with column fixed at top and column

k = 12EI/L^{3} [Eq. 8]

Where E is elastic modulus, I is moment of inertia, L is length of column.

The natural period

T = 1/[(k/m)^{1/2}/2p ]
= 2pm^{1/2}L^{3/2}/(12EI)^{1/2}
[Eq. 9]

Based on the equations:

1. For taller building, period T is longer and frequency is smaller.

2. For heavier building (larger mass), period T is longer and frequency is shorter. Concrete building is usually heaver than steel building and has longer period and shorter frequency

3. A rigid structure that has shear walls (larger I) will have shorter period and longer frequency than a moment frame structure. A braced frame will also have shorter period and longer frequency than a moment frame structure.

The equation for calculate approximate fundamental period of building in ASCE 7 is

T = C_{T} h_{n}^{0.75 }
[Eq.10]

where

h_{n} is
the height of building above base.

C_{T} is
building period coefficient.

Equation 10 is a wave function that vibration will continue forever. In reality, the structure has internal resistance that cause the vibration to die out. Assuming that the internal resistance is similar to viscous damping that is a function of velocity of motion, the equation become

m (dx^{2}/dt) + c (dx/dt)+ kx = 0
[Eq.11]

where c is damping coefficient.

The roots of equation 11 is

q = -c/2m ±
[(c/2m)^{2} - w^{2}]^{1/2}
[Eq.12]

When [(c/2m)^{2} - w^{2}]
= 0

c = 2mw
= c_{c} [Eq.13]

The damping is called critical damping. Critical damping is the minimum damping that is sufficient to bring the system to rest without vibration.

When c > c_{c} =
2mw
is called overdamping

An overdamping structure will not vibrate.

when the term [(c/2m)^{2} - w^{2}]
is less than zero, it is called underdamping. Equation 11
becomes

q = -c/2m ±
[-1]^{1/2} w_{d}
[Eq.14]

where w_{d }= w^{2} - (c/2m)^{2 }= w
[1 - c/2mw)^{2}]
is damped circular natural frequency.

Since c_{c} =
c/2mw is
critical damping,

w_{d }= w
[1 - (c/c_{c})^{2}]^{1/2 }= w
[1 - z^{2}]^{1/2}
[Eq.15]

where z =
c/c_{c} is called
damping ratio and is a rate indicates how fast the vibration
decreases. For most structure, z
is less than 20%. Therefore,

w_{d }» w
[Eq.16]

In practice, w is
used for w_{d }for
simplicity.

Damped natural period is determined as

T_{d} = 1/ f_{d }=2p/w_{d }» 2p/w_{ } =
T [Eq.17]

The rate of decrease

d =
ln (X_{n+1}/X_{n} ) » 2pz

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