Steel Designers Manual - TheBestFriend.org

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Steel Designers' Manual - 6th Edition (2003)
L.Tn=
Fig. 12.2 Simple harmonic motion
Fundamentals of dynamic behaviour 357
time t (seconds)
Y COS Wnt
The amplitude of vibration Y is the peak displacement of the mass relative to its
static equilibrium position. Vibration amplitudes are sometimes referred to as root
T 2
mean square (rms) quantities, where yrms = ( 1/ T)÷Ú ( 0 y dt)
, and T is the total time
over which the vibration is considered. For continuous simple harmonic motion yrms = Y/√2.
12.2.3 Damped free vibration
Energy is always dissipated to some extent during vibration of real structures. Inclusion
of the damping force in the free vibration equation of motion leads to
Mÿ + Cy · + Ky = 0 (12.3)
The solution to this equation for a lightly damped system when the mass is initially
displaced by Y and then released is
-xwn
t
y = Ye cos ( wdt -f)
(12.4)
The motion takes the form shown in Fig. 12.3(a), and it can be seen that the vibration
amplitude decays exponentially with time. The rate of decay is governed by the
amount of damping present.
If the damping constant C is sufficiently large then oscillation will be prevented
and the motion will be as in Fig. 12.3(b).The minimum damping required to prevent
overshoot and oscillation is known as critical damping, and the damping constant
for critical damping is given by Co = 2√(KM), where K and M are the stiffness and
mass of the system.
Practical structures are lightly damped, and the damping present is often
expressed as a proportion of critical. In Equation (12.4)
C
x = critical damping ratio =
Co
The phase shift angle f is small when the damping is small.