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The Quarterly Journal of Mechanics and Applied Mathematics 1956 9(3):346-359; doi:10.1093/qjmam/9.3.346
© 1956 by Oxford University Press
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EXPRESSIONS FOR THE DAMPING AND NATURAL FREQUENCY OF LINEAR SYSTEMS

A. T. FULLER and R. H. MACMILLAN

( Department of Engineering, Cambridge University )

Three theorems concerning the values of the roots of polynomial equations are derived by manipulation of the Routh–Hurwitz criteria. When applied to the characteristic equation of a linear system, they yield information on the dynamic constants of the dominant mode. The first theorem gives a test for locating the least negative of the real parts of the roots; the second gives a formula for this real part when its magnitude is small; and the third gives new expressions for the imaginary parts of the corresponding root-pair. Thus the degree of stability of the system may be estimated, and when this is small, the formulae give the natural frequency and the damping of the dominant mode in terms of the coefficients of the characteristic equation; in this condition, it is also shown how to determine the dynamic constants of lesser modes in certain cases.

An incidental discussion is also given of the effects on stability of zero coefficients in the characteristic equation.


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