Submitted by: Submitted by Robie
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Pages: 2
Category: Science and Technology
Date Submitted: 08/26/2016 06:12 AM
FREE VIBRATIONS, UNDAMPED SYSTEMS
m
k
f
x
f = kx
Single-degree-of freedom system
f
x
SHEAR BUILDING MODEL LUMPED MASS MODEL
f, x
f, x
Column Stiffness:
f = (3EI/L3) x f = (12EI/L3) x
CAUSES OF VIBRATION
* Initial displacement
* Initial velocity
* External excitation force
When there is no external excitation force (i.e. vibration is produced by an initial displacement or initial velocity of the system), the vibration is called “FREE VIBRATION”.
FORCES IN FREE VIBRATION
* Inertial force, FI = m a = m dv/dt = m d2x/dt2
* Elastic force, FS = k x
* Damping force, FD = c v = c dx/dt
x, v, a
m
FI
FD
FS
EQUILIBRIUM
F = 0 FI + FD + FS = 0
ma + cv + kx = 0
Free vibration, undamped ma + kx = 0
m d2x/dt2 + k x = 0
Let 2 = k/m x” + 2x = 0
D2 + 2 = 0 D = ± i
Therefore, x = A eit + B e-it
From Euler relations sin a = (eia + e-ia) / 2 eia = sin a + cos a
cos a = (eia - e-ia) / 2 e-ia = sin a - cos a
x = C1 sin t + C2 cos t
From initial conditions (boundary conditions), x(t=0) = xo , v(t=0) = vo
xo = C1 (0) + C2 (1) C2 = xo
dx/dt = C1 cos t - C2 sin t vo = C1 (1) – (0)
* C1 = vo /
Therefore, the equation of motion is,
vo
x
x(t) = (vo/ sint + xo cos t
T
xo
t
= natural frequency (radians/sec)
f = natural cyclic frequency (cycles/sec or Hertz, Hz) = / 2
T = natural period (sec) T = 2 T = 2 f
FYI: a(t) = d2x/dt2 = - C12 sint - C22 cost = - vo sint - xo 2 cost
v(t) = dx/dt = vo cos t - xo sin t