A. D. Dimarogonas, "Interval analysis of vibrating systems", Journal of Sound and Vibration, 1995, Vol. 183, No. 4, pp. 739-749.

Rotating turbines can vibrate; it is therefore necessary to to determine the possible vibration frequencies of the turbines and the possible dependencies of vibration on time. In mathematical terms, vibrations are described by a system of ordinary differential equations with constant coefficients. If we know the parameters precisely, then the corresponding computations are easy to perform. In real life, however, there is an uncertainty in the values of these parameters, and the major part of this uncertainty is of interval type.

There are three main sources of interval uncertainty in rotating turbines:

First, the bearing properties depend on temperature, which is controlled in machinery within relatively broad intervals.

Second, manufacturing tolerances introduce another interval parameter.

Third, support is not precisely determined, it is only defined within some interval tolerance.

For all three sources of uncertainty, we rarely know the probabilities of possible values, and therefore, statistical methods are usually not applicable. So, we must use interval methods to determine the possible vibration frequencies of the turbines and the possible dependencies of vibration on time.

The uncertainty intervals $[p^-,p^+]$ in the parameters $p$ of a turbine are reasonably wide (deviations $p-p^-$ can be up to 60\% of $p^-$). The main reason for this width is that the dependence of viscosity on the temperature is described by a double exponential law, so even small temperature changes result in substantial changes of the bearing properties. Since the intervals are large, the intervals of possible frequencies obtained by applying traditional engineering linearized perturbation methods (that are based on neglecting terms quadratic in $p-p^-$) can drastically differ from the actual frequencies. This difference was observed, e.g., for computer hard disk drives.

In view of this difference, we cannot "neglect" anything, but we must use mathematically correct interval methods. Traditional "naive" and more sophisticated interval methods for computing eigenvalues result in an {\it enclosure} for the desired interval that may be much wider than the desired interval of possible values of frequency. To avoid this "overestimation", the author subdivides the domain of possible values of parameters into subboxes on which the frequency is a monotonic function of each parameter: if a function $f$ is monotonic, then we can compute its range precisely by computing its values in two corresponding edges of the subbox.

This subdivision requires, in general, $\ge 2^n$ computation steps, where $n$ is the total number of intervally uncertain parameters. Since $2^n$ grows fast with $n$, this method is usually considered intractable. In the turbine vibration problem, however, $n$ is reasonably small (e.g., $n=5$), and considering, say, $2^5=32$ possible cases is quite feasible.

V. Kreinovich

Reviewer's comment. The author starts the paper with an interesting brief survey of the pre-history of interval computations: first, he mentions Archimedes who used two-sided bounds for $\pi$ Then, "The concept of a function having values which are bounded between limits was discussed by W. H. Young. The concept of operations with a set of multi-valued numbers was introduced by R. C. Young, who developed a formal algebra of multi-valued numbers. The special case of closed intervals was discussed by Dwyer.Interval mathematics was further developed by Sunaga, who introduced the term {\it interval algebra}, and by Moore, who introduced {\it interval analysis}, interval vectors and matrices, and the first non-trivial applications, such as the initial value problem for ordinary differential equations with interval initial conditions".

Archimedes, "On the measurement of the circle", In: Thomas L. heath (ed.), The works of Archimedes, Cambridge University Press, Cambridge, 1897; Dover edition, 1953.

W. H. Young, "Sull due funzioni a piu valori constituite dai limiti d'una funzione di variabile reale a destra ed a sinistra di ciascun punto", Rendiconti Acedemia di Lincei, Classes di Scienza Fiziche, 1908, Vol. 17, No. 5, pp. 582-587.

R. C. Young, "The algebra of multi-valued quantities", Mathematische Annalen, 1931, Vol. 104, pp. 260-290.

P. S. Dwyer, `Linear Computations, J. Wiley, N.Y., 1951.

T. Sunaga, "Theory of interval algebra and its application to numerical analysis", In: RAAG Memoirs II, Ggujutsu Bunken Fukyu-kai, Tokyo, Vol. 2, pp. 547-656.

R. Moore, Interval analysis, Prentice Hall, Englewood Cliffs, NJ, 1966.