Document Type : Article

**Authors**

Department of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran

**Abstract**

The optimal position and minimum support stiffness of a vibrating Timoshenko beam are investigated to maximize the fundamental frequency. The Finite element method is employed. The intermediate support's ideal position and minimal stiffness for a wide variety of slenderness proportions were achieved after validating the finite element model with the Euler-Bernoulli and Timoshenko model's analytical solution. It was observed that the ideal position of intermediate support and its minimum stiffness are sensitive to the slenderness ratio. According to the maximum-minimum theorem of Courant, the optimum position is at the zero of the second mode shape function (ZSMS). Also, it was observed that for thick cantilever beams with intermediate support at the optimal location, the minimum support stiffness is less than 266.9, which was reported in the literature for the Euler-Bernoulli beam. The minimum stiffness of familiar end conditions of an optimally located beam is presented for a wide range of slenderness ratios. Since, in many practical applications, it is impossible to locate support at the optimal position, the minimum support stiffness for a beam in which its intermediate support is not located at the optimal position is obtained for various boundary conditions and slenderness ratios.

**Keywords**

**Main Subjects**

References:

1. Courant, R. "Zur Theorie der kleinen Schwingungen", ZAMM - Zeitschriftfur Angewandte Mathematik and Mechanik (1992). DOI: 10.1002/zamm.19220020406.

2. Asgarikia, M. and Kakavand, F. "Minimum diameter of optimally located damping wire to maximize the fundamental frequencies of rotating blade using Timoshenko beam theory", International Journal of Structural Stability and Dynamics, 21(7), 2150090 (2021). DOI: 10.1142/S0219455421500905.

3. Courant, R. and Hilbert, D. "Methods of mathematical physics", Interscience Publishers, New York (1953). DOI: 10.1002/9783527617210.

4. Akesson, B. and Olhoff, N. "Minimum stiffness of optimally located supports for maximum value of beam eigenfrequencies", Journal of Sound and Vibration, 120(3), pp. 457-463 (1988). DOI: 10.1016/S0022-460X(88)80218-9.

5. Wang, C. "Minimum stiffness of an internal elastic support to maximize the fundamental frequency of a vibrating beam", Journal of Sound and Vibration, 259(1), pp. 229-232 (2003). DOI: 10.1006/jsvi.2002.5100.

6. Olhoff, N. and Akesson, B. "Minimum stiffness of optimally located supports for maximum value of column buckling loads", Structural Optimization, 3(3), pp. 163-175 (1991). DOI: 10.1007/BF01743073.

7. Rao, C.K. "Frequency analysis of clamped-clamped uniform beams with intermediate elastic support", Journal of Sound and Vibration, 133(3), pp. 502-509 (1989). DOI: 10.1016/0022-460X(89)90615-9.

8. Won, K.M. and Park, Y.S. "Optimal support positions for a structure to maximize its fundamental natural frequency", Journal of Sound and Vibration, 213, pp. 801-812 (1998). DOI: 10.1006/jsvi.1997.1493.

9. Albaracin, J.M, Zannier, L., and Gross, R.O. "Some observations in the dynamics of beams with intermediate supports", Journal of Sound and Vibration, 271, pp. 475-480 (2004)..DOI: 10.1016/S0022-460X(03)00631-X.

10. Zhu, J. and Zhang, W. "Maximization of structural natural frequency with optimal support layout", Struct Multidiscip Optim, 31, pp. 462-469 (2006). DOI: 10.1007/s00158-005-0593-2.

11. Wang, D. "Optimal design of structural support positions for minimizing maximal bending moment", Finite Elem Anal Des, 43, pp. 95-102 (2006). DOI: 20.1001.1.10275940.1391.12.3.6.5.

12. Wang, D., Friswell, M.I., and Lei, Y. "Maximizing the natural frequency of a beam with an intermediate elastic support", Journal of Sound and Vibration, 291(3-5), pp. 1229-1238 (2006). DOI: 10.1016/j.jsv.2005.06.028.

13. Wang, D. and Friswell, M.I. "Support position optimization with minimum stiffness for plate structures including support mass", Journal of Sound and Vibration, 499, 116003 (2021)..DOI: 10.1016/j.jsv.2021.116003.

14. Kong, J. "Vibration of isotropic and composite plates using computed shape function and its application to elastic support optimization", Journal of Sound and Vibration, 326(3-5), pp. 671-686 (2009). DOI: 10.1016/j.jsv.2009.05.022.

15. Wang, D., Yang, Z., and Yu, Z. "Minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by Rayleigh-Ritz method", Journal of Sound and Vibration, 329(14), pp. 2792-2808 (2010). DOI: 10.1016/j.jsv.2010.01.034.

16. Aydin, E. "Minimum dynamic response of cantilever beams supported by optimal elastic springs", Structural Engineering and Mechanics, 51(3), pp. 377-402 (2014). DOI: 10.12989/sem.2014.51.3.377.

17. Aydin, E., Dutkiewicz, M., Ozturk, B., et al. "Optimization of elastic spring supports for cantilever beams", Struct Multidiscip Optim, 62, pp. 55-81 (2020). DOI: 10.1007/s00158-019-02469-3.

18. Roncevic, G.S., Roncevic, B., Skoblar, A., et al. "Closed form solutions for frequency equation and mode shapes of elastically supported Euler-Bernoulli beams", Journal of Sound and Vibration, 457, pp. 118-138 (2019). DOI: 10.1016/j.jsv.2019.04.036.

19. Abdullatif, M. and Mukherjee, R. "Effect of intermediate support on critical stability of a cantilever with non-conservative loading, some new results", Journal of Sound and Vibration, 485, 115564 (2020). DOI: 10.1016/j.jsv.2020.115564.

20. Kukla, S. "Free vibrations of axially loaded beams with concentrated masses and intermediate elastic supports", Journal of Sound and Vibration, 172(4), pp. 449-458 (1994). DOI: 10.1006/jsvi.1994.1188.

21. Lin, H.P. and Chang, S. "Free vibration analysis of multi-span beams with intermediate flexible constraints", Journal of Sound and Vibration, 281(2), pp. 155-169 (2005). DOI: 10.1016/j.jsv.2004.01.010.

22. Lin, H.Y. "Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements", Journal of Sound and Vibration, 309(1), pp. 262-275 (2008). DOI: 10.1016/j.jsv.2007.07.015.

23. Magrab, E.B. "Natural frequencies and mode shapes of Timoshenko beams with attachments", Journal of Vibration and Control, 13, pp. 905-934 (2007). DOI:10.1177/1077546307078828.

24. Han, F., Dan, D., and Deng, Z. "A dynamic stiffnessbased modal analysis method for a double-beam system with elastic supports", Mechanical System and Signal Process, 146, 106978 (2021). DOI: 10.1016/j.ymssp.2020.106978.

25. Lei, Y., Gao, K., Wang, X., et al. "Dynamic behaviors of single- and multi-span functionally graded porous beams with exible boundary constraints", Applied Mathematical Modelling, 83, pp. 754-76 (2020). DOI: 10.1016/j.apm.2020.03.017.

26. Kukla, S. "The Green function method in frequency analysis of a beam with intermediate elastic supports", Journal of Sound and Vibration, 149, pp. 154-9 (1991). DOI: 10.1016/0022-460X(91)90920-F.

27. Roncevic, G.S., Roncevic, B., Skoblar, A., et al. "A comparative evaluation of some solution methods in free vibration analysis of elastically supported beams", Journal of Polytech Rijeka, 6, pp. 285-98 (2018). DOI: 10.31784/zvr.6.1.5.

28. Luo, J., Zhu, S., and Zhai, W. "Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports", International Journal of Mechanical Sciences, 213, 106842 (2022). DOI: 10.1016/j.ijmecsci.2021.106842.

29. Han, S.M., Benaroya, H., and Wei, T. "Dynamics of transversely vibrating beams using four engineering theories", Journal of Sound and Vibration, 225(5), pp. 935-988 (1999). DOI: 10.1006/jsvi.1999.2257.

30. Zohoor, H. and Kakavand, F. "Vibration of Euler- Bernoulli, and beams in large overall motion on flying support using finite element method", Scientia Iranica, 19(4), pp. 1105-1116 (2012). DOI: 10.1016/j.scient.2012.06.019.

2. Asgarikia, M. and Kakavand, F. "Minimum diameter of optimally located damping wire to maximize the fundamental frequencies of rotating blade using Timoshenko beam theory", International Journal of Structural Stability and Dynamics, 21(7), 2150090 (2021). DOI: 10.1142/S0219455421500905.

3. Courant, R. and Hilbert, D. "Methods of mathematical physics", Interscience Publishers, New York (1953). DOI: 10.1002/9783527617210.

4. Akesson, B. and Olhoff, N. "Minimum stiffness of optimally located supports for maximum value of beam eigenfrequencies", Journal of Sound and Vibration, 120(3), pp. 457-463 (1988). DOI: 10.1016/S0022-460X(88)80218-9.

5. Wang, C. "Minimum stiffness of an internal elastic support to maximize the fundamental frequency of a vibrating beam", Journal of Sound and Vibration, 259(1), pp. 229-232 (2003). DOI: 10.1006/jsvi.2002.5100.

6. Olhoff, N. and Akesson, B. "Minimum stiffness of optimally located supports for maximum value of column buckling loads", Structural Optimization, 3(3), pp. 163-175 (1991). DOI: 10.1007/BF01743073.

7. Rao, C.K. "Frequency analysis of clamped-clamped uniform beams with intermediate elastic support", Journal of Sound and Vibration, 133(3), pp. 502-509 (1989). DOI: 10.1016/0022-460X(89)90615-9.

8. Won, K.M. and Park, Y.S. "Optimal support positions for a structure to maximize its fundamental natural frequency", Journal of Sound and Vibration, 213, pp. 801-812 (1998). DOI: 10.1006/jsvi.1997.1493.

9. Albaracin, J.M, Zannier, L., and Gross, R.O. "Some observations in the dynamics of beams with intermediate supports", Journal of Sound and Vibration, 271, pp. 475-480 (2004)..DOI: 10.1016/S0022-460X(03)00631-X.

10. Zhu, J. and Zhang, W. "Maximization of structural natural frequency with optimal support layout", Struct Multidiscip Optim, 31, pp. 462-469 (2006). DOI: 10.1007/s00158-005-0593-2.

11. Wang, D. "Optimal design of structural support positions for minimizing maximal bending moment", Finite Elem Anal Des, 43, pp. 95-102 (2006). DOI: 20.1001.1.10275940.1391.12.3.6.5.

12. Wang, D., Friswell, M.I., and Lei, Y. "Maximizing the natural frequency of a beam with an intermediate elastic support", Journal of Sound and Vibration, 291(3-5), pp. 1229-1238 (2006). DOI: 10.1016/j.jsv.2005.06.028.

13. Wang, D. and Friswell, M.I. "Support position optimization with minimum stiffness for plate structures including support mass", Journal of Sound and Vibration, 499, 116003 (2021)..DOI: 10.1016/j.jsv.2021.116003.

14. Kong, J. "Vibration of isotropic and composite plates using computed shape function and its application to elastic support optimization", Journal of Sound and Vibration, 326(3-5), pp. 671-686 (2009). DOI: 10.1016/j.jsv.2009.05.022.

15. Wang, D., Yang, Z., and Yu, Z. "Minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by Rayleigh-Ritz method", Journal of Sound and Vibration, 329(14), pp. 2792-2808 (2010). DOI: 10.1016/j.jsv.2010.01.034.

16. Aydin, E. "Minimum dynamic response of cantilever beams supported by optimal elastic springs", Structural Engineering and Mechanics, 51(3), pp. 377-402 (2014). DOI: 10.12989/sem.2014.51.3.377.

17. Aydin, E., Dutkiewicz, M., Ozturk, B., et al. "Optimization of elastic spring supports for cantilever beams", Struct Multidiscip Optim, 62, pp. 55-81 (2020). DOI: 10.1007/s00158-019-02469-3.

18. Roncevic, G.S., Roncevic, B., Skoblar, A., et al. "Closed form solutions for frequency equation and mode shapes of elastically supported Euler-Bernoulli beams", Journal of Sound and Vibration, 457, pp. 118-138 (2019). DOI: 10.1016/j.jsv.2019.04.036.

19. Abdullatif, M. and Mukherjee, R. "Effect of intermediate support on critical stability of a cantilever with non-conservative loading, some new results", Journal of Sound and Vibration, 485, 115564 (2020). DOI: 10.1016/j.jsv.2020.115564.

20. Kukla, S. "Free vibrations of axially loaded beams with concentrated masses and intermediate elastic supports", Journal of Sound and Vibration, 172(4), pp. 449-458 (1994). DOI: 10.1006/jsvi.1994.1188.

21. Lin, H.P. and Chang, S. "Free vibration analysis of multi-span beams with intermediate flexible constraints", Journal of Sound and Vibration, 281(2), pp. 155-169 (2005). DOI: 10.1016/j.jsv.2004.01.010.

22. Lin, H.Y. "Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements", Journal of Sound and Vibration, 309(1), pp. 262-275 (2008). DOI: 10.1016/j.jsv.2007.07.015.

23. Magrab, E.B. "Natural frequencies and mode shapes of Timoshenko beams with attachments", Journal of Vibration and Control, 13, pp. 905-934 (2007). DOI:10.1177/1077546307078828.

24. Han, F., Dan, D., and Deng, Z. "A dynamic stiffnessbased modal analysis method for a double-beam system with elastic supports", Mechanical System and Signal Process, 146, 106978 (2021). DOI: 10.1016/j.ymssp.2020.106978.

25. Lei, Y., Gao, K., Wang, X., et al. "Dynamic behaviors of single- and multi-span functionally graded porous beams with exible boundary constraints", Applied Mathematical Modelling, 83, pp. 754-76 (2020). DOI: 10.1016/j.apm.2020.03.017.

26. Kukla, S. "The Green function method in frequency analysis of a beam with intermediate elastic supports", Journal of Sound and Vibration, 149, pp. 154-9 (1991). DOI: 10.1016/0022-460X(91)90920-F.

27. Roncevic, G.S., Roncevic, B., Skoblar, A., et al. "A comparative evaluation of some solution methods in free vibration analysis of elastically supported beams", Journal of Polytech Rijeka, 6, pp. 285-98 (2018). DOI: 10.31784/zvr.6.1.5.

28. Luo, J., Zhu, S., and Zhai, W. "Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports", International Journal of Mechanical Sciences, 213, 106842 (2022). DOI: 10.1016/j.ijmecsci.2021.106842.

29. Han, S.M., Benaroya, H., and Wei, T. "Dynamics of transversely vibrating beams using four engineering theories", Journal of Sound and Vibration, 225(5), pp. 935-988 (1999). DOI: 10.1006/jsvi.1999.2257.

30. Zohoor, H. and Kakavand, F. "Vibration of Euler- Bernoulli, and beams in large overall motion on flying support using finite element method", Scientia Iranica, 19(4), pp. 1105-1116 (2012). DOI: 10.1016/j.scient.2012.06.019.

Transactions on Mechanical Engineering (B)

July and August 2024Pages 967-979