## Distributed-Parameter Systems

Often in vibration analysis, it is assumed that the inertial (mass), flexibility (spring), and dissipative (damping) characteristics can be ‘‘lumped’’ as a finite number of ‘‘discrete’’ elements. Such models are termed lumped-parameter systems 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, we have line structures, surface or 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 (DoF) and will require an infinite number of coordinates to represent its motion. In other words, extending the concept of a finite-DoF system as analyzed previously (see Chapter 5), an infinite-dimensional vector is needed to represent the general motion of a continuous system. Equivalently, a one-dimensional 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 response will depend on the boundary conditions (BCs) as well as the initial conditions (ICs), and they have to be explicitly accounted for. (Note: BCs are implicitly taken into account in lumped-parameter models.) 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, BCs could be various, individual members may be nonuniform, and the material characteristics may be inhomogeneous and anisotropic. Closedform 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 (see Chapter 5) may be extended from lumped-param-

eter systems to continuous systems. In particular, because the number of principal modes is equal to the number of DoF 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 that particular mode,

is Chapter 5), however, there is no guarantee that such an IC will accurately excite the required mode. A general excitation consisting of a force or an IC will excite more than one mode of motion. But, as in the case of discrete-parameter systems (Chapter 5), the general motion may be analyzed and expressed in terms of modal motions, through modal analysis. In a modal motion, the mass elements of the structure will move at a specific frequency (the natural frequency), bearing a constant proportion in displacement of those elements (i.e., maintaining the corresponding mode shape), and passing the static equilibrium of the structure simultaneously. In view of this behavior, it is possible to separate the time response and the spatial response of a vibrating system in a modal motion. This separability is fundamental to the modal analysis of a continuous system. Furthermore, in practice, all infinite number of natural frequencies and mode shapes are not significant and typically the very high modes may be neglected. Such a modaltruncation procedure, even though carried out by continuous-system analysis, is equivalent to approximating the original infinite-DoF system by a finite DoF one. Vibration analysis of continuous systems may be applied in 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, overhead guideways and walkways, robot arms, and space stations; and transit vehicles, including automobiles, rapid-transit and railway cars, ships, aircraft, and spacecraft.