Modal spectrum analysis for an imperfect beam model with random vibration

The cross-Spectral Density (SD) is a method of analyzing the relationship between randomly-input forces and their responses in the frequency domain in spectrum analysis. It is essential for analyzing antiresonant eigenfrequencies that are difficult to understand using only direct-SD. Additionally, the mass in a beam is often analyzed in modern engineering to control the vibration characteristics of a simple beam or as a major model for error. Especially, the scalability can be expected in various fields such as nanosensors as well as macroscopic scale by studying imperfect mass. Firstly, the correlation function and SD are defined for an arbitrary vibration input in a beam. Then, the response function is established according to a harmonic input of some unit force amplitude. Moreover, the direct- and cross-SDs for the response are calculated and mathematically compared with both results. Further analysis is performed by comparing the variation of the SD with response position, the standard deviation (StDev) of the mean of maximum responses, and the bandwidth. The developed cross-SD of the imperfect beam extends the previously developed SD models of the perfect beam and can provide ideas for new types of eigenfrequency analysis, vibration isolation, etc.