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So a spring with mass can be represented by a combination of an ideal spring with the same spring constant and an equivalent mass of one-third of the spring mass attached to the moving end of the spring Similar techniques can be used to obtain equivalent masses for other distributed masses for which the rigid body assumption is not a good approximation Such systems are better analyzed using other techniques, however (Ungar, 1985), so this is not detailed here

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Unlike mass or inertia elements, which store kinetic energy, and springs, which store potential energy, damping elements cannot store energy at all Instead, they remove energy from the system Without dampers, systems modeled as combinations of masses and dampers would always conserve energy Since real systems can, at best, only approximate conservative systems, models without dampers tend to underestimate the force or energy needed to produce a certain system motion The main reason for including dampers in a system model is to capture this energy dissipation effect and more accurately describe the magnitude of input necessary to produce a given output or the ampli cation at resonance As noted before, the addition of damping does not reduce the number of natural frequencies or degrees of freedom of a system As with springs, damping elements describe both components intentionally designed to remove energy from the system (such as dashpots and automotive shock absorbers) and the inherent energy dissipation of other system elements All elements contribute in some way to energy dissipation, either through material damping in springs and structural members or, more importantly, through friction in bearings and other surface contacts In contrast to mass and compliance (or spring) properties, it is generally extremely dif cult to obtain simple analytical expressions for the amount of damping in a system Damping in a gearbox, for instance, will depend upon a number of factors including the type and condition of lubricant, the alignment of the shafts, and the condition and quality of the gears Unless data is available for the element from the manufacturer (as is the case with

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approximated by the length of the curve spanning the pair x j 1 , x j This length was determined above to be about 1 + [ f (x j )]2

shock absorbers) or the system ts a form easily modeled as with the examples in Chen (1982), damping values are usually obtained experimentally Damping results from motion, and as such, experimental determination of damping is complicated by the fact that the system must be maintained in motion in order to generate damping forces As a result, it is rarely worth the effort to xture each component separately to estimate its damping Furthermore, since frictional effects occur at the interface between components, it may even be impossible to determine the amount of damping by looking at components individually Because of these issues, damping is usually determined experimentally by choosing a form or forms of damping and using experimental data from the overall system to t the parameters in those forms Thus damping is often added to a system model after masses and stiffnesses have been determined and the appropriate reductions made Accordingly, the approach of this section is somewhat different than the previous sections Instead of the main focus being on the combination of individual damping elements, this section focuses on describing the broad classes of damping models commonly encountered and appropriate conditions for the use of each 1151 Viscous or Speed-Dependent Damping The most common form of damper applied in linear systems modeling is a damping force that is proportional to the relative velocity of two coordinates in the system A typical schematic for such a damper is illustrated in Fig 1118 The damping force thus takes the form F = bd (1160)

Thus the area contribution of this cylindrical increment of our surface is about 2 f (x j ) 1 + [ f (x j )]2

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