Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.143
load that produces a unit deflection at midspan; thus, k � 48EI/L 3 , where E is the
modulus of elasticity, psi; I the moment of inertia, in 4 ; and L the span, in, of the
beam. The idealized mass equals W/g, where W is the weight of the load, lb, and
g is the acceleration due to gravity, 386 in/s 2 .
Also, a single mass on a spring (Fig. 5.108d) may represent the rigid frame in
Fig. 5.108c. In that case, k � 2 � 12EI/h 3 , where I is the moment of inertia, in 4 ,
of each column and h the column height, in. The idealized mass equals the sum of
the masses on the girder and the girder mass. (Weight of columns and walls is
assumed negligible.)
The spring and mass in Fig. 5.108b and d form a one-degree system. The degree
of a system is determined by the least number of coordinates needed to define the
positions of its components. In Fig. 5.108, only the coordinate y is needed to locate
the mass and determine the state of the spring. In a two-degree system, such as
one comprising two masses connected to each other and to the ground by springs
and capable of movement in only one direction, two coordinates are required to
locate the masses.
If the mass with weight W, lb, in Fig. 5.108 is isolated, as shown in Fig. 5.108e
it will be in dynamic equilibrium under the action of the spring force � ky and the
inertia force (d 2 y/dt 2 )(W/g). Hence, the equation of motion is
2 Wdy
� ky � 0 (5.235)
2
g dt
where y � displacement of mass, in, measured from rest position. Equation (5.235)
may be written in the more convenient form
The solution is
2 2
dy kg dy 2
� y � � � y � 0 (5.236)
2 2
dt W dt
y � A sin �t � B cos �t (5.237)
where A and B are constants to be determined from initial conditions of the system,
and
kg
� � (5.238)
�W
is the natural circular frequency, rad/s.
The motion defined by Eq. (5.237) is harmonic. Its natural period, s, is
Its natural frequency, Hz, is
�
2� W
T � � 2� (5.239)
� kg