Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-23T19:17:30.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of the Effect of a Parameter Change on the Roots of Stability Equations

Published online by Cambridge University Press:  07 June 2016

K. Mitchell
K. Mitchell*
Affiliation:
King's College, Newcastle upon Tyne
Get access

Summary

A method is given for calculating approximately the changes in the roots of a stability secular equation caused by a change in any of the parameters involved. General formulas are given applicable to any quartic equation, and special formulae are also given applicable to the stability of an aeroplane: lateral stability in the text, and longitudinal stability in an appendix.

The method of using the formulae is illustrated by applying them to a particular calculation of the lateral stability of an aeroplane, and a check of the results is made by comparing the predicted approximate changes with those calculated by solution of the modified period equations. It is shown that the formulae are reliable, for this typical case, for any reasonable changes in any parameter other than nv.

If the changes in the derivatives are made equal to the probable error with which they can be measured, the formulae enable us to evaluate the probable errors of the roots. These are found to be considerable, and to arise mainly from uncertainties in yv, nv and nr: if these could be reduced to 0.03 in yv and 0.006 in the others, the uncertainties in the roots would be reduced to some ten per cent, of their values, except for a larger uncertainty in the root corresponding to the slow spiral motion.

Type
Research Article
Information
Aeronautical Quarterly , Volume 1 , Issue 1 , May 1949 , pp. 39 - 58
Copyright
Copyright © Royal Aeronautical Society. 1949

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bryant, L. W., and Hopwood, M. L. (1942). Calculation of the Damping of the Lateral Oscillation of an Aeroplane. R. & M. 1956, 1942.Google Scholar
2. Gandy, R. W. (1943). The Response of an Aeroplane to the Application of Ailerons and Rudders. Part I—Response in Roll. R. & M. 1915, 1943.Google Scholar
3. Mitchell, K., Thorpe, A. W., and Frayn, E. M. (1945). The Theoretical Response of a High-Speed Aeroplane to a Side Gust. Part of R. & M. 2294, 1945.Google Scholar
4. Cowley, W. L., and Skan, S. W. (1930). A Study of Polynomial Equations. R. & M. 1325, 1930.Google Scholar
5. Mitchell, K., Thorpe, A. W., and Frayn, E. M. (1945). The Theoretical Response of a High-Speed Aeroplane to the Application of Ailerons and Rudder. Part of R. & M. 2294, 1945.Google Scholar
6. Neumark, S. (1943). The Disturbed Longitudinal Motion of an Uncontrolled Aeroplane and of an Aeroplane with Automatic Control. R. & M. 2078, 1943.Google Scholar
7. Bryant, L. W., and Gates, S. B. (1937). Nomenclature for Stability Coefficients. R. & M. 1801, 1937.Google Scholar
8. Mitchell, K. (1943). A Supplementary Notation for Theoretical Lateral Stability Investigations. R.A.E. Tech. Note No. Aero. 1183, 1943.Google Scholar
9. Buxton, D. H. L., and Minhinnick, I. T. (1945). Expressions for the Rates of Change of Critical Flutter Speeds and Frequencies with Inertial Aerodynamic and Elastic Coefficients. R. & M. 2444, 1945. 58Google Scholar