4.1 Introduction

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Often in vibration analysis, it is assumed that inertial (mass), flexibility (spring), and dissipative (damping)

characteristics can be “lumped” as a finite number of “discrete” elements. Such models are termed lumpedparameter

or discrete-parameter systems. Generally, in practical vibrating systems, inertial, elastic, and

dissipative effects are found continuously distributed in one, two, or three dimensions. Correspondingly,

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we have line structures, surface/planar structures, or spatial structures. They will possess an infinite

number of mass elements, continuously distributed in the structure, and integrated with some connecting

flexibility (elasticity) and energy dissipation. In view of the connecting flexibility, each small element of

mass will be able to move out of phase (or somewhat independently) with the remaining mass elements. It

follows that a continuous system (or a distributed-parameter system) will have an infinite number of degrees

of freedom (DoFs) and will require an infinite number of coordinates to represent its motion. In other

words, extending the concept of a finite-degree-of-freedom system as analyzed previously, an infinitedimensional

vector is needed to represent the general motion of a continuous system. Equivalently, a onedimensional

continuous system (a line structure) will need one independent spatial variable, in addition to

time, to represent its response. In view of the need for two independent variables in this case, one for time

and the other for space, the representation of system dynamics will require partial differential equations

(PDEs) rather than ordinary differential equations (ODEs). Furthermore, the system will depend on the

boundary conditions as well as the initial conditions.

Strings, cables, rods, shafts, beams, membranes, plates, and shells are example of continuous members.

In special cases, closed-form analytical solutions can be obtained for the vibration of these members. A

general structure may consist of more than one such member, and furthermore, boundary conditions

(BCs) could be various, individual members may be nonuniform, and the material characteristics may be

inhomogeneous and anisotropic. Closed-form analytical solutions would not be generally possible in such

cases. Nevertheless, the insight gained by analyzing the vibration of standard members will be quite

beneficial in studying the vibration behavior of more complex structures.

The concepts of modal analysis may be extended from lumped-parameter systems to continuous

systems. In particular, since the number of principal modes is equal to the number of DoFs of the

system, a distributed-parameter system will have an infinite number of natural modes of vibration. A

particular mode may be excited by deflecting the member so that its elastic curve assumes the shape

of the particular mode, and then releasing from this initial condition. When damping is significant

and nonproportional, however, there is no guarantee that such an initial condition could accurately

excite the required mode. A general excitation consisting of a force or an initial condition will excite

more than one mode of motion. However, as in the case of discrete-parameter systems, the general

motion may be analyzed and expressed in terms of modal motions, through modal analysis. In a

modal motion, the mass elements will move at a specific frequency (the natural frequency), and

bearing a constant proportion in displacement (i.e., maintaining the mode shape), and passing the

static equilibrium of the system simultaneously. In view of this behavior, it is possible to separate the

time response and spatial response of a vibrating system in a modal motion. This separability is

fundamental to modal analysis of a continuous system. Furthermore, in practice an infinite number

of natural frequencies and mode shapes are not significant and typically the very high modes may be

neglected. Such a modal-truncation procedure, even though carried out by continuous-system

analysis, is equivalent to approximating the original infinite-degree-of-freedom system by a finitedegree-

of-freedom one. Vibration analysis of continuous systems may be applied in the modeling,

analysis, design, and evaluation of such practical systems as cables; musical instruments; transmission

belts and chains; containers of fluid; animals; structures including buildings, bridges, guideways, and

space stations; and transit vehicles, including automobiles, ships, aircraft, and spacecraft.

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