Building Design and Construction Handbook - Merritt - Ventech!

5.156 SECTION FIVE
2 Wdy
� Rm � k(ym � y) � Foƒ(t) (5.281)
2
g dt
Suppose, for example, the one-degree undamped system in Fig. 5.109a behaves
in accordance with the bilinear resistance function of Fig. 5.112a and is subjected
to a suddenly applied constant load (Fig. 5.112b). With zero initial displacement
and velocity, the response in the first stage (y � y e), according to Eq. (5.281), is
y � e�(1 � cos �t ) (5.282)
1
dy � e�� sin �t (5.283)
1
dt
Equation (5.275) also indicates that displacement y e will be reached at a time t e
such that cos �t e � 1 � y e/e�.
For convenience, let t 2 � t � t e be the time in the second stage; thus, t 2 � 0at
the start of that stage. Since the condition of the system at that time is the same as
at the end of the first stage, the initial displacement is y e and the initial velocity
e�� sin �t e.
The equation of motion of the second stage is
2 Wdy
� Rm � F o
(5.284)
2
g dt
The solution, taking into account initial conditions for y e � y � y m is
g 2
y � (Fo � R m)t2 � e��t2 sin �te � y e
(5.285)
2W
Maximum displacement occurs at the time
W�e�
tm � sin �t e
(5.286)
g(R � R )
m o
and can be obtained by substituting t m in Eq. (5.285).
The third stage, unloading after y m has been reached, can be determined from
Eq. (5.281) and conditions at the end of the second stage. The response, however,
is more easily found by noting that the third stage consists of an elastic, harmonic
residual vibration. In this stage the amplitude of vibration is (R m � F o)/k, since
this is the distance between the neutral position and maximum displacement, and
in the neutral position the spring force equals F o. Hence, the response can be
obtained directly from Eq. (5.275) by substituting y m � (R m � F o)/k for e�, because
the neutral position, located at y � y m � (R m � F o)/k, occurs when �t 3 � �/2,
where t 3 � t � t e � t m. The solution is
Rm � Fo Rm � Fo
y � ym � � cos �t 3
(5.287)
k k
Response in the three stages is shown in Fig. 5.112c. In that diagram, however,
to represent a typical case, the coordinates have been made nondimensional by
expressing y in terms of y e and the time in terms of T, the natural period of vibration.
(J. M. Biggs, ‘‘Introduction to Structural Dynamics,’’ and R. Clough and J. Penzien,
‘‘Dynamics of Structures,’’ McGraw-Hill Book Company, New York; D. G.
Fertis and E. C. Zobel, ‘‘Transverse Vibration Theory,’’ The Ronald Press Company,