Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.159
Eq. (5.289) and is rapidly damped out. Hence, the free part is called the transient
response, and the forced part, the steady-state response. The maximum value of
the dynamic load factor for the steady-state response D m is called the dynamic
magnification factor. It is given by
1
Dm � (5.296)
2 2 2 2 2
�(1 � � /� ) � (2��/� )
2 2
With damping, then, the peak values of Dm occur when � � � �1 � � /� and
are approximately equal to �/2�. For example, for 10% critical damping.
�
Dm � � 5
0.2�
So even small amounts of damping significantly limit the response at resonance.
Coulomb Damping. For a one-degree system with Coulomb damping, the equation
of motion for free vibration is
2 Wdy
� ky � �F ƒ
(5.297)
2
g dt
where F is the constant friction force and the positive sign applies when the veƒ
locity is negative. If initial displacement is y o and initial velocity is zero, the response
in the first half cycle, with negative velocity, is
� �
F F
ƒ ƒ
y � yo � cos �t � (5.298)
k k
equivalent to a system with a suddenly applied constant force. For the second half
cycle, with positive velocity, the response is
� � � �
Fƒ � Fƒ
y � �yo�3 cos � t � � (5.299)
k � k
If the solution is continued with the sign of F changing in each half cycle, the
ƒ
results will indicate that the amplitude of positive peaks is given by y o � 4nF /k,
ƒ
where n is the number of complete cycles, and the response will be completely
damped out when t � ky oT/4F , where T is the natural period of vibration, or
ƒ
2�/�.
Analysis of the steady-state response with Coulomb damping is complicated by
the possibility of frequent cessation of motion.
(S. Timoshenko, D. H. Young, and W. Weaver, ‘‘Vibration Problems in Engineering,’’
4th ed., John Wiley & Sons, Inc., New York; D. D. Barkan, ‘‘Dynamics
of Bases and Foundations,’’ McGraw-Hill Book Company; W. C. Hurty and M. F.
Rubinstein, ‘‘Dynamics of Structures,’’ Prentice-Hall, Englewood Cliffs, N.J.)
5.18.6 Approximate Design for Dynamic Loading
Complex analysis and design methods seldom are justified for structures subject to
dynamic loading because of lack of sufficient information on loading, damping,