Page 51 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

48 Chapter 1 Fundamentals oF Vibration
1.6 Vibration analysis procedure

A vibratory system is a dynamic one for which the variables such as the excitations (inputs)
and responses (outputs) are time dependent. The response of a vibrating system generally
depends on the initial conditions as well as the external excitations. Most practical vibrat-
ing systems are very complex, and it is impossible to consider all the details for a mathe-
matical analysis. Only the most important features are considered in the analysis to predict
the behavior of the system under specified input conditions. Often the overall behavior of
the system can be determined by considering even a simple model of the complex physi-
cal system. Thus the analysis of a vibrating system usually involves mathematical model-
ing, derivation of the governing equations, solution of the equations, and interpretation of
the results.

Step 1: Mathematical Modeling. The purpose of mathematical modeling is to represent
all the important features of the system for the purpose of deriving the mathematical (or
analytical) equations governing the system’s behavior. The mathematical model should
include enough details to allow describing the system in terms of equations without mak-
ing it too complex. The mathematical model may be linear or nonlinear, depending on
the behavior of the system’s components. Linear models permit quick solutions and are
simple to handle; however, nonlinear models sometimes reveal certain characteristics of
the system that cannot be predicted using linear models. Thus a great deal of engineering
judgment is needed to come up with a suitable mathematical model of a vibrating system.
Sometimes the mathematical model is gradually improved to obtain more accurate
results. In this approach, first a very crude or elementary model is used to get a quick insight
into the overall behavior of the system. Subsequently, the model is refined by including
more components and/or details so that the behavior of the system can be observed more
closely. To illustrate the procedure of refinement used in mathematical modeling, consider
the forging hammer shown in Fig. 1.16(a). It consists of a frame, a falling weight known
as the tup, an anvil, and a foundation block. The anvil is a massive steel block on which
material is forged into desired shape by the repeated blows of the tup. The anvil is usually
mounted on an elastic pad to reduce the transmission of vibration to the foundation block
and the frame [1.22]. For a first approximation, the frame, anvil, elastic pad, foundation
block, and soil are modeled as a single-degree-of-freedom system as shown in Fig. 1.16(b).
For a refined approximation, the weights of the frame and anvil and the foundation block
are represented separately with a two-degree-of-freedom model as shown in Fig. 1.16(c).
Further refinement of the model can be made by considering eccentric impacts of the tup,
which cause each of the masses shown in Fig. 1.16(c) to have both vertical and rocking
(rotation) motions in the plane of the paper.

Step 2: Derivation of Governing Equations. Once the mathematical model is available,
we use the principles of dynamics and derive the equations that describe the vibration of
the system. The equations of motion can be derived conveniently by drawing the free-body
diagrams of all the masses involved. The free-body diagram of a mass can be obtained by
isolating the mass and indicating all externally applied forces, the reactive forces, and the
inertia forces. The equations of motion of a vibrating system are usually in the form of a set