FFT Plot for Velocity Coefficients 70 60 50 40 30 20 10 180 184 188 192 Frequency, Hz 196 200 204 208 FIGURE 3. A fast Fourier tranform plot for a mid-span point shows high vibration speed Stress distribution 14 12 10 8 6 4 2 180 184 188 192 Frequency, Hz 196 200 204 208 Fig. 5: Stress Distribution. FIGURE 5. The vibration stresses exceed the endurance limit of the piping We have observed the above methods to be conservative and to provide a " cookbook " or a " go/no-go " approach. They tell us only whether or not the vibrations are within acceptable levels. It is not possible to generate a quantitative estimate of the forcing function and of the actual stress levels on the pipes, both of which are essential for a design adequacy check. We studied the problem within the framework of Inverse Theory. We will focus on steady-state vibrations, because they have been found to cause maximum damage. PROPOSED METHOD Theoretically, for a simply supported pipe, the response at any location along the span may be determined by the vibration measurements at two distinct points in the span. The span is a straight portion between two fixed points or supports (Figure 1). A single point measurement near the mid-span is also sufficient. Further mathematical details are included in the second part of this article. The measurements could be realtime displacement, velocity or acceleration with the post-processed fast Fourier transform (FFT) plots. The calculations are straightforward and amenable to simple spreadsheet programming with macros. With errorhttp://WWW.CHE.COM