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## Natural frequency and natural period of structure

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 (= dx2/dt ).

Combined equation 1 and 2, the equation become

m (dx2/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]

### Natural period of structure

T = 1/f  =2p (m/k)1/2     [Eq. 7]

Example of a building with column fixed at top and column

k = 12EI/L3       [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 ] = 2pm1/2L3/2/(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.

### Natural period of structure based on ASCE 7

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

T = CT hn0.75       [Eq.10]

where

hn is the height of building above base.

CT is building period coefficient.

### Damping of structure

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 (dx2/dt) + c (dx/dt)+ kx = 0       [Eq.11]

where c is damping coefficient.

### Critical damping

The roots of equation 11 is

q = -c/2m ± [(c/2m)2 - w2]1/2       [Eq.12]

When [(c/2m)2 - w2] = 0

c = 2mw  = cc      [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.

### Overdamping

When c >  cc = 2mw  is called overdamping

An overdamping structure will not vibrate.

### Underdamping

when the term [(c/2m)2 - w2] is less than zero, it is called underdamping.  Equation 11 becomes

q = -c/2m ± [-1]1/2 wd     [Eq.14]

where wd = w2 - (c/2m)2 = w [1 - c/2mw)2] is damped circular natural frequency.

Since cc = c/2mw is critical damping,

wd = w [1 - (c/cc)2]1/2 = w [1 - z2]1/2     [Eq.15]

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

wd » w    [Eq.16]

In practice, w is used for wd for simplicity.

### Damped natural period

Damped natural period is determined as

Td = 1/ fd =2p/wd » 2p/w  = T    [Eq.17]

The rate of decrease

d = ln (Xn+1/Xn ) » 2pz