Mechanics of Solids

, Volume 43, Issue 2, pp 205–217 | Cite as

Determining the moments of inertia of large bodies from vibrations in elastic suspension

  • A. O. BelyakovEmail author
  • A. P. Seiranyan


To ensure the maneuvering capabilities of aircraft and high-speed sea vessels, designers should know the moments of inertia of their massive parts. But since the structure of some elements such as power units is very complicated, it is impossible to determine their moments of inertia analytically. Thus the problem of measuring the moments of inertia of massive large bodies arises. To this end, a measuring bench was designed in N. E. Zhukovskii Central Institute for Aerohydrodynamics (TsAGI) on the basis of a new method for determining the body moments of inertia from vibrations in the elastic suspension [1]. In this connection, it is necessary to develop the corresponding mathematical algorithms for determining the moments of inertia.

In this paper, we develop mathematical algorithms for determining the body moments of inertia by using methods for identification of linear systems in the state space [2–5]. We present three versions of solving the problem of determining the body moments of inertia depending on the information about the method for exciting the vibrations or about the body parameters and the rigidity of the bench springs. We study the influence of damping on the accuracy of determining the moments of inertia. Numerical results are given for a specific system.


Rigidity Matrix Observation Matrix Body Parameter Pseudoinverse Matrix Vibration Mode Shape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. V. Bogdanov, V. S. Volobuev, A. I. Kudryashov, and V. V. Travin, “A Suite for Measuring Mass, Coordinates of the Center of Mass, and Moments of Inertia of Engineering Components,” Izmer. Tekh., No. 2, 37–39 (2002) [Measurement Techniques (Engl. Transl.) 45 (2), 168–172 (2002)].Google Scholar
  2. 2.
    S. Y. Kung, “A New Identification and Model Reduction Algorithm via Singular Value Decomposition,” in Paper 12th Asilomar Conf. Circuits, Syst. Comput. (Pacific Grove, Calf., 1978).Google Scholar
  3. 3.
    A. O. Belyakov and L. Yu. Blazhennova-Mikulich. “Identification of Inertia Matrix of Conservative Oscillatory System,” Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 3, 25–28 (2005) [Moscow Univ. Math. Bull. (Engl. Transl.)]Google Scholar
  4. 4.
    M. Verhaegen, “Identification of the Deterministic Part of MIMO State Space Models Given in Innovation Forms from Input-Output Data,” Automatica 30(1), 61–74 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Viberg, “Subspace-Based Methods for the Identification of Linear Time-Invariant Systems,” Automatica 31(12), 1835–1851 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. P. Markeev, Theoretical Mechanics (Nauka, Moscow, 1990) [in Russian].zbMATHGoogle Scholar
  7. 7.
    I. V. Novozhilov, Fractional Analysis (Izd-vo MGU, Moscow, 1995) [in Russian].Google Scholar
  8. 8.
    Yu. A. Amenzade, Theory of Elasticity (Vysshaya Shkola, Moscow, 1976) [in Russian].Google Scholar
  9. 9.
    A. Yu. Ishlinskii, Mechanics of Gyroscopic Systems (Izd. AN SSSR, Moscow, 1963) [in Russian].Google Scholar
  10. 10.
    A. O. Belyakov, “Determination of Dynamic Parameters of Massive Bodies by Vibration Modes,” Vestnik Molodykh Uchenykh. Ser. Prikl. Mat. Mekh., St-Petersburg, No. 12, 33–36 (2003).Google Scholar
  11. 11.
    V. F. Zhuravlev and D. M. Klimov, Applied Methods in the Theory of Vibrations (Nauka, Moscow, 1983) [in Russian].Google Scholar
  12. 12.
    M. I. Vishik and L. A. Lyusternik, “The Solution of Some Perturbation Problems for Matrices and Selfadjoint or Non-Selfadjoint Differential Equations I,” Uspekhi Mat. Nauk 15(3), 3–80 (1960) [Russ. Math. Surv. (Engl. Transl.) 15 (3), 1–73 (1960)].Google Scholar
  13. 13.
    A. P. Seyranian and A. A. Mailybaev, Multiparameter Stability Theory with Mechanical Applications (World Scientific, Singapore, 2003).zbMATHGoogle Scholar
  14. 14.
    F. R. Gantmakher, Theory of Matrices (Nauka, Moscow, 1988) [in Russian].zbMATHGoogle Scholar
  15. 15.
    F. R. Gantmakher, Lectures on Analytical Mechanics (Nauka, Moscow, 1966; Chelsea, New York, 1970).Google Scholar
  16. 16.
    A. O. Belyakov, “Numerical Modleing of Measurement Process of Inertia Moments of Large Bodies by Free Vibration Method,” Uchen. Zapiski TsAGI, No. 1–2, 129–136 (2002).Google Scholar

Copyright information

© Allerton Press, Inc. 2008

Authors and Affiliations

  1. 1.Institute of MechanicsLomonosov Moscow State UniversityMoscowRussia

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