The Asymptotic Method in the Theory of Random Vibrations of Continuous Systems

Abstract

The method of generalized coordinates, which was considered in detail and illustrated by a number of examples in the preceding chapter, is effective for obtaining numerical results and qualitative conclusions. The application of this method requires summation of the contributions of all the modes of natural vibrations excited by a random loading. If the time spectrum of a loading is sufficiently wide, and the spectrum of the natural frequencies of the system is sufficiently dense, the number of these modes may be very large. As an example, consider the vibration of the covering panels of an aircraft subjected to the acoustic pressures of a jet stream. The spectrum of these pulsations actually occupies the entire acoustic range (from tens to 10,000–20,000 Hertz). On the other hand, the spectrum of frequencies of the natural vibrations of the covering panels is quite dense. The difference between adjacent natural frequencies may be 10 Hertz or less. For instance, in the numerical example of Section 3.5, there are hundreds of vibration modes of the surface (taking account of the large number of panels constituting the surface) and thousands of acoustic modes. Therefore, summation of a large number of terms is necessary when random vibrations of an aircraft surface are analyzed.