eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Modified transfer matrix method for steady-state forced vibration: a system of bar elements; 143–161

Andres Lahe, Andres Braunbrück, Aleksander Klauson

The Elements by a System of Transfer (EST) method offers exact solutions for various vibration problems of trusses, beams and frames. The method can be regarded as an improved or modified transfer matrix method where the roundoff errors generated by multiplying transfer arrays are avoided. It is assumed that in a steady state a bar/beam will vibrate with the circular frequency of a harmonic excitation force. The universal equation of elastic displacement (2nd/4th order differential equation) is described as a system of first order differential equations in matrix form. For the differential equations the compatibility conditions of a bar/beam element displacements at joint serve as essential boundary conditions. As the natural boundary conditions at joints, the equilibrium equations of elastic forces of bar/beam elements are considered. At the supports, restrictions to displacements (support conditions) have been applied. For steady-state forced vibration the phenomena of dynamic vibration absorption near the saddle points are observed and the response curves for displacement amplitude and elastic energy are calculated. The magnification factor at the excitation frequency is determined.



1. Pestel, E. C. and Leckie, F. A. Matrix Method in Elastomechanics. McGraw-Hill, New York, 1963.

2. Den Hartog, J. P. Mechanical Vibrations, 4th Edition. Dover Publications, New York, Inc., 1985.

3. Pani, S., Senapati, K., Patra, K. C., and Nath, P. Review of an effective dynamic vibration absorber for a simply supported beam and parametric optimization to reduce vibration amplitude. Int. J. Eng. Res. Appl., 2017, 7(7), Part III, 49–77.

4. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics, 2nd Edition. Pearson Addison-Wesley, Upper Saddle River, New Jersey, 2007.

5. Krätzig, W. B., Harte, R., Meskouris, K., and Wittek, U. Tragwerke 1. Theorie und Berechnungsmethoden statisch bestimmter Stabtragwerke. Springer-Verlag, Berlin, Heidelberg, 2010.

6. Lahe, A. Ehitusmehaanika. Tallinn University of Technology Press, Tallinn, 2012 (in Estonian).

7. Lahe, A. The EST Method: Structural Analysis. Tallinn University of Technology Press, Tallinn, 2014.

8. Slivker, V. Mechanics of Structural Elements: Theory and Applications. Springer-Verlag, Berlin, Heidelberg, 2007.

9. Hou, H. C. The Gyarmati principle and the theory of minimum energy dissipation rate. Report No. I, U.S. Bureau of Reclamation, 1990.

10. Preumont, A. Twelve Lectures on Structural Dynamics. Springer, Dordrecht, 2013.

11. He, B., Rui, X., and Zhang, H. Transfer matrix method for natural vibration analysis of tree system. Math. Prob. Eng., 2012, Article ID 393204.

12. Lahe, A. The transfer matrix and the boundary element method. Proc. Estonian Acad. Sci. Eng., 1997, 3(1), 3–12.

13. Lahe, A. Varrassüsteemide võnkumine. EST-meetod. Tallinn University of Technology Press, Tallinn, 2018 (in Estonian).

14. Argyris, J. H. and Mlejnek, H.-P. Dynamics of Structures, Vol. 5. Elsevier Science Publishers B.V., North-Holland, 1991.

15. Haberman, R. Elementary Applied Partial Differential Equations. Prentice-Hall International, New Jersey, 1983.

16. Yang, S. Modal identification of linear time periodic systems with applications to Continuous-Scan Laser Doppler Vibrometry. PhD thesis. University of Wisconsin-Madison, Wisconsin, 2013.

17. Kiselev, V. A. Special course. Dynamics and stability of structures. Strojizdat, Moscow, 1964 (in Russian).

18. Babakov, I. M. The theory of vibrations, 2nd Edition. Nauka, Moscow, 1965 (in Russian).

19. Karnovsky, I. A. and Lebed, O. Advanced Methods of Structural Analysis. Springer, Boston, 2010.

20. Karnovsky, I. A. and Lebed, E. Theory of Vibration Protection. Springer International Publishing, Cham, 2016.

21. Hagedorn, P. and DasGupta, A. Vibrations and Waves in Continuous Mechanical Systems. John Wiley & Sons, Chichester, 2007.

22. Farlow, S. J. Partial Differential Equations for Scientists and Engineers. John Wiley & Sons, New York, 1993.

23. Pilkey, W. D. and Wunderlich, W. Mechanics of Structures: Variational and Computational Methods. CRC Press, Boca Raton, 1994.

24. Lahe, A., Braunbrück, A., and Klauson, A. An exact solution of truss vibration problems. Proc. Estonian Acad. Sci., 2019, 68(3), 244–263.

25. Koloushek, V. Dynamics of structural constructions. Strojizdat, Moscow, 1965 (in Russian).

26. Tartibu, L. A simplified analysis of the vibration of variable length blade as might be used in wind turbine systems. MTech thesis. Cape Peninsula University of Technology, Cape Town, 2008.

27. Tatar, İ. Vibration characteristics of portal frames. MSc thesis. İzmır Institute of Technology, İzmır, 2013.

28. Ramsay, A. NAFEM benchmark challenge No. 5: Dynamic characteristics of a truss structure. NAFEMS Benchmark Magazine, 2016, 5, 1–4.

29. Rosenberg, R. Steady-state forced vibrations. Int. J. Non-Lin. Mech., Elsevier, 1966, 1(2), 95–108. https://hal.archives-ouvertre,

30. Sracic, M. W. A new experimental method for nonlinear system identification based on linear time periodic approximations. PhD thesis. University of Wisconsin-Madison, Wisconsin, 2011.

31. Wereley, N. M. Analysis and control of linear periodically time varying systems. PhD thesis. Massachusetts Institute of Tech- nology, Cambridge, 1991.

32. Allen, M. S., Sracic, M. W., Chauhan, S., and Hansen, M. H. Output-only modal analysis of linear time periodic systems with application to wind turbine simulation data. In Structural Dynamics and Renewable Energy (Proulx, T., ed.). Springer, New York, 2011, 1, 361–374.

33. Allen, M. S., Kuether, R. J., Deaner, B., and Sracic, M. W. A numerical continuation method to compute nonlinear normal modes using modal reduction. In Proceedings of the 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii, April 23–26, 2012. AIAA, 2012, 9548–9567. 1970

34. Yang, B. Stress, Strainand Structural Dynamics: An Interactive Handbook of Formulas. Solutions and MATLAB Toolboxes. Elsevier Academic Press, Oxford, 2005.

35. Meirovitch, L. Analytical Methods in Vibrations. Macmillan, New York, 1967.

36. Rao, S. S. Vibration of Continuous Systems. John Wiley & Sons, Inc., Hoboken, New Jersey, 2007.

37. Karnovsky, I. A. Theory of Arched Structures. Strength, StabilityVibration. Springer-Verlag, New York-Dordrecht-Heidelberg-London, 2012.

38. Jürgenson, A. Tugevusõpetus. Valgus, Tallinn, 1985 (in Estonian).

39. Structural Dynamics of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems. Instructional Material Complementing FEMA 451, Design Examples.

40. Crowell, B. Mechanics. Light and Matter, Fullerton, California, 2019.

41. Wen-Xi, H., Xiao, X. Y., Yun-Ling, J., and Dong-Fang, Y. Automatic segmentation method for voltage sag detection and characterization. In Proceedings of the 18th International Conference on Harmonics and Quality of Power (ICHQP)Ljubljana, Slovenia, 13–16 May 2018. IEEE, 2018, 1–5.

42. Murray, R. M., Li, Z., and Sastry, S. S. A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton-London-New York-Washington, D.C., 1994. ̃murray/mlswiki

43. Stepanov, V. V. Course of Differential Equations8th Edition. Fizmatgiz, Moscow, 1959 (in Russian).

44. Gajic, Z. Linear Dynamic Systems and Signals. Ch. 6. Convolution and Correlation. Prentice Hall, Upper Saddle River, 2003.

45. Ceauşu, C., Craifaleanu, A., and Dragomirescu, Cr. Transfer matrix method for forced vibrations of bars. U.P.B. Sci. Bull., Series D, 2010, 72(2), 35–42.

46. Kim, J., Dargush, G. F., and Ju, Y.-K. Extended framework of Hamilton’s principle for continuum dynamics. Int. J. Solids Struct., 2013, 50(20–21), 3418–3429.

47. Zilletti, M., Elliott, S. J., and Rustighi, E. Optimisation of dynamic vibration absorbers to minimise kinetic energy and maximise internal power dissipation. J. Sound Vib., 2012, 331(18), 4093–4100.

48. Weaver, W. Jr., Timoshenko, S. P., and Young, D. H. Vibration Problems in Engineering. 5th Edition. John Wiley & Sons, 1990.


Back to Issue