3.1 Introduction

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Complex vibrating systems usually consist of components that possess distributed energy-storage and

energy-dissipative characteristics. In these systems, inertial, stiffness, and damping properties vary

(piecewise) continuously with respect to the spatial location. Consequently, partial differential equations,

with spatial coordinates (e.g., Cartesian coordinates x; y; z) and time t as independent variables are

necessary to represent their vibration response.

3-1

© 2005 by Taylor & Francis Group, LLC

A distributed (continuous) vibrating system may be approximated (modeled) by an appropriate set of

lumped masses properly interconnected using discrete spring and damper elements. Such a model is

termed lumped-parameter model or discrete model. An immediate advantage resulting from this lumpedparameter

representation is that the system equations become ordinary differential equations. Often,

linear springs and linear viscous damping elements are used in these models. The resulting linear

ordinary differential equations can be solved by the modal analysis method. The method is based on the

fact that these idealized systems (models) have preferred frequencies and geometric configurations (or

natural modes) in which they tend to execute free vibration. An arbitrary response of the system can be

interpreted as a linear combination of these modal vibrations, and as a result its analysis may be

conveniently done using modal techniques.

Modal analysis is an important tool in vibration analysis, diagnosis, design, and control. In some

systems, mechanical malfunction or failure can be attributed to the excitation of their preferred motion

such as modal vibrations and resonances. By modal analysis, it is possible to establish the extent and

location of severe vibrations in a system. For this reason, it is an important diagnostic tool. For the same

reason, modal analysis is also a useful method for predicting impending malfunctions or other

mechanical problems. Structural modification and substructuring are techniques of vibration analysis

and design that are based on modal analysis. By sensitivity analysis methods using a modal model, it is

possible to determine which degrees of freedom (DoFs) of a mechanical system are most sensitive to

addition or removal of mass and stiffness elements. In this manner, a convenient and systematic method

can be established for making structural modifications to eliminate an existing vibration problem, or to

verify the effects of a particular modification. A large and complex system can be divided into several

subsystems which can be independently analyzed. By modal analysis techniques, the dynamic

characteristics of the overall system can be determined from the subsystem information. This approach

has several advantages, including: (1) subsystems can be developed by different methods such as

experimentation, finite element method, or other modeling techniques and assembled to obtain the

overall model; (2) the analysis of a high order system can be reduced to several lower order analyses; and

(3) the design of a complex system can be carried out by designing and developing its subsystems

separately. These capabilities of structural modification and substructure analysis which are possessed by

the modal analysis method make it a useful tool in the design development process of mechanical

systems. Modal control, a technique that employs modal analysis, is quite effective in the vibration

control of complex mechanical systems.

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