chapter  5
78 Pages

Modal Analysis

Complex vibrating systems usually consist of components that possess distributed energy-storage and energy-dissipation 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. A distributed (continuous) vibrating system may be approximated (modeled) by an

appropriate set of lumped masses that are 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 lumped-parameter 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 method of modal analysis. The method is based on the fact that these idealized systems (models) have preferred frequencies (or natural frequencies) and geometric configurations (or natural mode shapes), 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 analysis techniques. Modal analysis is an important tool in the analysis, diagnosis, design, and control of

vibration. In some systems, mechanical malfunction or failure can be attributed to the excitation of their ‘‘preferred’’ types of motion such as modal vibrations and resonances. By modal analysis it is possible to establish the extent and locations of severe vibrations in a system. For this reason, it is a powerful diagnostic tool (see Chapter 4). 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 (see Chapter 11), which are based on modal analysis. By sensitivity analysis methods using a ‘‘modal’’ model, it is possible to determine what degrees of freedom (DoF) of a mechanical system are most sensitive to the addition or removal of mass elements 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 techniques of modal analysis, the dynamic characteristics of the overall system can be determined from the subsystem information. This approach has several advantages, including: (a) subsystems can be developed by different methods such as

or other bled to obtain the overall model; (b) the analysis of a high-order system can be reduced to several lower-order analyses; and (c) the design of a complex system can be done 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 valuable tool in the process of design and development of mechanical systems. Modal control (see Chapter 12), which is a control technique that employs modal analysis, is quite effective in the vibration control of complex mechanical systems.