eesti teaduste
akadeemia kirjastus
SINCE 1952
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of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
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Modified transfer matrix method for steady-state forced vibration: a system of beam elements*; pp. 235–256
PDF | 10.3176/proc.2020.3.07

Andres Lahe

The EST (Elements by a System of Transfer equations) 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 beam will vibrate with the circular frequency of an excitation force. The universal equation of elastic displacement (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 beam element displacements at joint serve as essential boundary conditions. As the natural boundary conditions at joints, the equilibrium equations of elastic forces of 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. 

* A sequel to "Modified transfer matrix method for steady-state forced vibration: a system of bar elements [1]


1. Lahe, A., Braunbrück, A., and Klauson, A. Modified transfer matrix method for steady-state forced vibration: a system of bar elements. Proc. Estonian Acad. Sci., 2020, 69(2), 143–161.

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

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

4. 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.

5. He, B., Rui, X., and Zhang, H. Transfer matrix method for natural vibration analysis of tree system. Math. Probl. Eng., 2012, ArticleID393204.

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

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

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

9.Lahe, A. Varrassüsteemide võnkumine. EST-meetod. Tallinn University of Technology Press, Tallinn, 2018 (inEstonian).

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

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

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

13. 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.

14. Kiselev, V. A. Special Course. Dynamics and Stability of Structures. Strojizdat, Moscow, 1964 (in Russian).

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

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

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

18. Kang, B. Exact transfer function analysis of distributed parameter systems by wave propagation techniques. In Recent Advances in Vibrations Analysis (Baddour, N., ed.). IntechOpen, Rijeka, 2011, 1–27.

19. 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, 361–374. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, New York, NY, 2011.

20. Wereley, N. M. Analysis and control of linear periodically time varying systems. PhD thesis. Massachusetts Institute of Technology, Cambridge, 1991.

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. Curtain, R. and Morris, K. Transfer functions of distributed parameter systems: A tutorial. Automatica, 2009, 45(5), 1101–1116.

24. Song, J., Huang, Q.-A. System-Level modeling of packaging effects of MEMS devices. In System-Level Modeling of MEMS (Bechtold, T., Schrag, G., and Feng, L., eds). Wiley-VCH Verlag GmbH & Co. KGaA, 2013, 147–160.

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

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

27. Avcar, M. Free vibration analysis of beams considering different geometric characteristics and boundary conditions. Int. J. Mech. Appl., 2014, 3(3), 94–100.

28. Rosenberg, R. Steady-state forced vibrations. Int. J. Non-Lin. Mech., Elsevier, 1966, 1(2), 95–108.

29. 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.

30. 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 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii, April 23–26, 2012. AIAA, 2012, 11, 9548–9567.

31. Kozień, M. S. Analytical Solutions of Excited Vibrations of a Beam with Application of Distribution. Acta Phys. Pol. A, 2013, 123(6), 1029–1033.

32. Henderson, J. P. Vibration analysis of curved skin-stringer structures having tuned elastomeric dampers. PhD thesis. School of The Ohio State University, 1972.

33. Abu-Hilal, M. Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions. J. Sound Vib., 2003, 267(2), 191–207.

34. Langtangen, H. P. and Linge, S. Vibration ODEs. In Finite Difference Computing with PDEsTexts in Computational Science and Engineering. Springer, Cham, 2017, 1–92.

35. Kartofelev, D. Nonlinear sound generation mechanisms in musical acoustics. PhD thesis. Tallinn University of Technology, Tallinn, 2014.

36. Irofti, D., Boussaada, I., and Niculescu, S.-I. Geometric vs. algebraic approach: A study of double imaginary characteristic roots in time-delay systems. IFAC-PapersOnLine, 2017, 59(1), 1310–1315.

37. Irofti, D., Boussaada, I., and Niculescu, S.-I. Some insights into the migration of double imaginary roots under small deviation of two parameters. Automatica, 2018, 88, 91–97.

38. Sandberg, H., Möllerstedt, E., and Bernhardsson, B. Frequency-domain analysis of linear time-periodic systems. IEEE Trans. Autom. Control50(12), 1971–1983, 2005.

39. Blin, N., Riedinger, P., Daafouz, J., Grimaud, L., and Feyel, P. A comparison of harmonic modeling methods with application to control of switched systems with active filtering. In Proceedings of the 18th European Control Conference (ECC), Naples, Italy, June 25–28, 2019. IEEE, 2019, 4198–4203.

40. Wereley, N. M. and Hall, S. R. Frequency response of linear time periodic systems. In Proceedings of the 29th IEEE Confer- ence on Decision and Control, Honolulu, HI, USA, December 5–7, 1990. IEEE, 1991, 3650–3655.

41. Siddiqi, A. Identification of the Harmonic Transfer Functions of a Helicopter Rotor. MSc thesis. Massachusetts Institute of Technology, Cambridge, 2001.

42. Tcherniak, D., Yang, S., and Allen, M. S. Experimental characterization of operating bladed rotor using harmonic power spectra and stochastic subspace identification. In Proceedings of the 26th International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, September 15–17, 2014 (Sas, P., Moens, D., and Denayer, H., eds), 4421–4436. 

43. Gbur, G. J. Singular Optics. CRC Press, Boca Raton, 2016.

44. Freund, I. and Shvartsman, N. Wave-field phase singularities: The sign principle. Phys. Rev. A, 1994, 50(6), 5164–5172.

45. Dias, C. A. N. General exact harmonic analysis of in-plane timoshenko beam structures. Lat. Am. J. Solids Struct., 2014, 11(12).

46. Yavari, A., Sarkani, S., and Moyer, E. T. Jr. On applications of generalized functions to beam bending problems. Int. J. Solids Struct., 2000, 37(40), 5675–5705.

47. Dong, Y. and Liu, J. Exponential stabilization of uncertain nonlinear time-delay systems. Adv. Difference Equations, 2012, 180.

48. Gu, K., Irofti, D., Boussaada, I., and Niculescu, S. Migration of double imaginary characteristic roots under small deviation of two delay parameters. In Proceedings of the 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, December 15–18, 2015. IEEE, 2016, 6410–6415.

49. Yin, H. and Tao, R. Improved transfer matrix method without numerical instability. Europhys. Lett., 2008, 84(5).

50. Kadisov, G. M. Dynamics and Stability of Structures. Litres, St. Petersburg, 2015.

51. Karnovsky, I. A. Theory of Arched Structures: Strength, Stability, Vibration. Springer-Verlag, NewYork, 2012.

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

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

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

55. 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, May 13–16, 2018. IEEE, 2018, 1–5.

56. Murray, R. M., Li, Z., and Sastry, S. S. A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton– London–NewYork–Washington,D.C.,1994.

57. Smirnov, A. F., Aleksandrov, A. V., Lashchenikov, B. Ya., and Shaposhnikov, N. N. Structural Mechanics. Dynamics and Stability of Structures. Strojizdat, Moscow, 1984 (in Russian).

58. Stepanov, V. V. Course of Differential Equations, 8th Edition. Fizmatgiz, Moscow, 1959 (in Russian).

59. Gajic, Z. Linear Dynamic Systems and Signals. Prentice Hall, Upper Saddle River, 2003.

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

61.Cosşkun, S. B., Atay, M. T., and Öztürk, B. Transverse Vibration Analysis of Euler-Bernoulli Beams Using Analytical Approximate Techniques. In Advances in Vibration Analysis Research (Ebrahimi, F., ed.). IntechOpen, Rijeka, 2011, 1–22.

62. Walker, L. R. Finite element solution: nonlinear flapping beams for use with micro air vehicle design. MSc thesis. Air Univer- sity. Air Force Institute of Technology, Ohio, 2007.

63. Stokey, W. F. Vibration of systems having distributed mass and elasticity. In Harris’ Shock and Vibration Handbook, 5th Edition (Harris, C. M. and Piersol, A. G., eds). McGraw-Hill, New York–Chicago–San Francisco, 2002.

64. Andronov, A. A., Vitt, A. A., and Khaikin, S. E. Theory of Oscillator. Pergamon Press, Oxford–London–Edinburg–New York–Toronto–Paris–Frankfurt, 1966.

65. 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.


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