Page 72 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.8 mass or inertia elements 69
By expressing Eq. (E.2) as
T = k t u (E.3)
the desired equivalent torsional spring constant k t can be identified as
k t = mgl (E.4)

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1.8 mass or inertia elements
The mass or inertia element is assumed to be a rigid body; it can gain or lose kinetic energy
whenever the velocity of the body changes. From Newton’s second law of motion, the
product of the mass and its acceleration is equal to the force applied to the mass. Work is
equal to the force multiplied by the displacement in the direction of the force, and the work
done on a mass is stored in the form of the mass’s kinetic energy.
In most cases, we must use a mathematical model to represent the actual vibrating
system, and there are often several possible models. The purpose of the analysis often
determines which mathematical model is appropriate. Once the model is chosen, the
mass or inertia elements of the system can be easily identified. For example, consider
again the cantilever beam with an end mass shown in Fig. 1.25(a). For a quick and rea-
sonably accurate analysis, the mass and damping of the beam can be disregarded; the
system can be modeled as a spring-mass system, as shown in Fig. 1.25(b). The tip mass
m represents the mass element, and the elasticity of the beam denotes the stiffness of the
spring. Next, consider a multistory building subjected to an earthquake. Assuming that
the mass of the frame is negligible compared to the masses of the floors, the building can
be modeled as a multi-degree-of-freedom system, as shown in Fig. 1.35. The masses at

m 5 x 5
x 5 m 5
k 5 k 5 x
m 4 4
x 4 m 4
k 4 k 4 x
m 3 3
x 3 m 3
k 3 x
k 3
m 2 2
x 2 m 2
k 2 k 2
m 1 x 1
x 1 m 1
k 1 k 1

(a) (b)

FiGure 1.35 Idealization of a multistory
building as a multi-degree-of-freedom system.