Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.153
k � spring constant � force producing unit deflection, lb/in
v o � initial velocity of mass, in/s
e� � F o/k � displacement under static load, in
A closed solution is possible if the integral can be evaluated.
Assume, for example, the mass is subjected to a suddenly applied force F o that
remains constant (Fig. 5.111a). If y o and v o are initially zero, the displacement y
of the mass at any time t can be obtained from the integral in Eq. (5.274) by setting
ƒ(�) � 1:
t
y � e�� � sin �(t � �) d� � e�(1 � cos �t) (5.275)
0
This equation indicates that the dynamic load factor D � 1 � cos �t. It has a
maximum value D m � 2 when t � �/�. Figure 5.111b shows the variation of
displacement with time.
Multidegree Systems. A multidegree lumped-mass system may be analyzed by
the modal method after the natural frequencies of the normal modes have been
determined (Art. 5.18.2). This method is restricted to linearly elastic systems in
which the forces applied to the masses have the same variation with time. For other
cases, numerical analysis must be used.
In the modal method, each normal mode is treated as an independent one-degree
system. For each degree of the system, there is one normal mode. A natural frequency
and a characteristic shape are associated with each mode. In each mode,
the ratio of the displacements of any two masses is constant with time. These ratios
define the characteristic shape. The modal equation of motion for each mode is
j
gƒ(t) � Fr� 2
rn
dAn 2
r�1
2 n n j
dt 2 � Wr�rn r�1
� � A � (5.276)
FIGURE 5.111 Harmonic motion. (a) Constant force applied to an undamped onedegree
system, such as the one in Fig. 5.110a. (b) Displacements vary with time like
a cosine curve.