Structural Engineering and Mechanics

  • 제51권5호
  • /
  • Pages.773-792
  • /
  • 2014
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

  • Sarkar, Korak (Department of Aerospace Engineering, Indian Institute of Science) ;
  • Ganguli, Ranjan (Department of Aerospace Engineering, Indian Institute of Science)
  • 투고 : 2014.04.01
  • 심사 : 2014.06.19
  • 발행 : 2014.09.10
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

키워드

참고문헌

  1. Alshorbagy, A., Eltaher, M. and Mahmoud, F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006
  2. Ananthasuresh, G. (2004), "Inverse mode shape problems for bars and beams with flexible supports", Inverse Problems, Design and Optimization Symposium, Rio de Janeiro, Brazil.
  3. Azizi, N., Saadatpour, M. and Mahzoon, M. (2011), "Using spectral element method for analyzing continuous beams and bridges subjected to a moving load", Appl. Math. Model., 36(8), 3580-3592.
  4. Barcilon, V. (1982), "Inverse problem for the vibrating beam in the free-clamped configuration", Philos. T. Roy. Soc. A, 304(1483), 211-251. https://doi.org/10.1098/rsta.1982.0012
  5. Bayat, M. and Pakar, I. (2012), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., 43(3), 337-347. https://doi.org/10.12989/sem.2012.43.3.337
  6. Burak, S. and Ram, Y. (2001), "The construction of physical parameters from modal data", Mech. Syst. Signal Pr., 15(1), 3-10. https://doi.org/10.1006/mssp.2000.1348
  7. Cha, P. (2002), "Specifying nodes at multiple locations for any normal mode of a linear elastic structure", J. Sound Vib., 250(5), 923-934. https://doi.org/10.1006/jsvi.2001.3964
  8. Cha, P. (2004), "Imposing nodes at arbitrary locations for general elastic structures during harmonic excitations", J. Sound Vib., 272(3), 853-868. https://doi.org/10.1016/S0022-460X(03)00495-4
  9. Cha, P. (2005), "Enforcing nodes at required locations in a harmonically excited structure using simple oscillators", J. Sound Vib., 279(3), 799-816. https://doi.org/10.1016/j.jsv.2003.11.067
  10. Cha, P. and Chen, C. (2011), "Quenching vibration along a harmonically excited linear structure using lumped masses", J. Vib. Control, 17(4), 527-539. https://doi.org/10.1177/1077546310362448
  11. Cha, P. and Pierre, C. (1999), "Imposing nodes to the normal modes of a linear elastic structure", J. Sound Vib., 219(4), 669-687. https://doi.org/10.1006/jsvi.1998.1914
  12. Cha, P. and Rinker, J. (2012), "Enforcing nodes to suppress vibration along a harmonically forced damped Euler-Bernoulli beam", J. Vib. Acoust., 134(5), 051010-051019. https://doi.org/10.1115/1.4006375
  13. Cha, P. and Zhou, X. (2006), "Imposing points of zero displacements and zero slopes along any linear structure during harmonic excitations", J. Sound Vib., 297(1), 55-71. https://doi.org/10.1016/j.jsv.2006.03.032
  14. Elishakoff, I. (2005), Eigenvalues Of Inhomogeneous Structures: Unusual Closed-form Solutions, CRC Press, Boca Raton, Florida, USA.
  15. Eltaher, M., Alshorbagy, A. and Mahmoud, F. (2012), "Vibration analysis of Euler-Bernoulli nanobeams by using finite element method", Appl. Math. Model., 37(7), 4787-4797.
  16. Gafsi, W., Choura, S. and Nayfeh, A. (2009), "Assignment of geometrical and physical parameters for the confinement of vibrations in flexible structures", J. Aerospace Eng., 22(4), 403-414. https://doi.org/10.1061/(ASCE)0893-1321(2009)22:4(403)
  17. Gladwell, G. (1986), "The inverse problem for the Euler-Bernoulli beam", P. Roy. Soc. Lond. A Mat., 407(1832), 199-218. https://doi.org/10.1098/rspa.1986.0093
  18. Gunda, J.B., Singh, A.P., Chhabra, P.S. and Ganguli, R. (2007), "Free vibration analysis of rotating tapered blades using Fourier-p super element", Struct. Eng. Mech., 27(2), 243-257. https://doi.org/10.12989/sem.2007.27.2.243
  19. Hodges, D. Y. and Rutkowski, M. Y. (1981), "Free-vibration analysis of rotating beams by a variable-order finite-element method", AIAA J., 19(11), 1459-1466. https://doi.org/10.2514/3.60082
  20. Kisa, M. (2012), "Vibration and stability of axially loaded cracked beams", Struct. Eng. Mech., 44(3), 305-323. https://doi.org/10.12989/sem.2012.44.3.305
  21. Kural, S. and Ozkaya, E. (2012), "Vibrations of an axially accelerating, multiple supported flexible beam", Struct. Eng. Mech., 44(4), 521-538. https://doi.org/10.12989/sem.2012.44.4.521
  22. Lai, E. and Ananthasuresh, G. (2002), "On the design of bars and beams for desired mode shapes", J. Sound Vib., 254(2), 393-406. https://doi.org/10.1006/jsvi.2001.4101
  23. Liu, Z., Yin, Y., Wang, F., Zhao, Y. and Cai, L. (2013), "Study on modified differential transform method for free vibration analysis of uniform Euler-Bernoulli beam", Struct. Eng. Mech., 48(5), 697-709. https://doi.org/10.12989/sem.2013.48.5.697
  24. Maeda, Y., Nishiwaki, S., Izui, K., Yoshimura, M., Matsui, K. and Terada, K. (2006), "Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes", Int. J. Numer. Meth. Eng., 67(5), 597-628. https://doi.org/10.1002/nme.1626
  25. Meirovitch, L. (1986), Elements Of Vibration Analysis, Vol. 2, McGraw-Hill, New York.
  26. Mottershead, J., Mares, C. and Friswell, M. (2001), "An inverse method for the assignment of vibration nodes", Mech. Syst. Signal Pr., 15(1), 87-100. https://doi.org/10.1006/mssp.2000.1353
  27. Neuringer, J. and Elishakoff, I. (2001), "Inhomogeneous beams that may possess a prescribed polynomial second mode", Chaos Soliton. Fract., 12(5), 881-896. https://doi.org/10.1016/S0960-0779(00)00043-6
  28. Niels, L. (2000), "Design of cantilever probes for atomic force microscopy (AFM)", Eng. Optimiz., 32(3), 373-392. https://doi.org/10.1080/03052150008941305
  29. Ram, Y. and Elishakoff, I. (2004), "Reconstructing the cross-sectional area of an axially vibrating nonuniform rod from one of its mode shapes", P. Roy. Soc. Lond. A Mat., 460(2046), 1583-1596. https://doi.org/10.1098/rspa.2003.1214
  30. Rubio, W., Paulino, G. and Silva, E. (2011), "Tailoring vibration mode shapes using topology optimization and functionally graded material concepts", Smart Mater. Struct., 20(2), 025009. https://doi.org/10.1088/0964-1726/20/2/025009
  31. Saffari, H., Mohammadnejad, M. and Bagheripour, M. (2012), "Free vibration analysis of non-prismatic beams under variable axial forces", Struct. Eng. Mech., 43(5), 561-582. https://doi.org/10.12989/sem.2012.43.5.561
  32. Sarkar, K. and Ganguli, R. (2013), "Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition", Meccanica, 48(7), 1661-1676. https://doi.org/10.1007/s11012-013-9695-x
  33. Shahba, A. and Rajasekaran, S. (2011), "Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials", Appl. Math. Model., 36(7), 3094-3111.
  34. Song, Z., Li, W. and Liu, G. (2012), "Stability and non-stationary vibration analysis of beams subjected to periodic axial forces using discrete singular convolution", Struct. Eng. Mech., 44(4), 487-499. https://doi.org/10.12989/sem.2012.44.4.487
  35. Spletzer, M., Raman, A., Wu, A., Xu, X. and Reifenberger, R. (2006), "Ultrasensitive mass sensing using mode localization in coupled microcantilevers", Appl. Phys. Lett., 88(25), 254102-254102. https://doi.org/10.1063/1.2216889
  36. Sundaram, M.M. and Ananthasuresh, G. (2013), "A note on the inverse mode shape problem for bars, beams, and plates", Inverse Prob. Sci. Eng., 21(1), 1-16. https://doi.org/10.1080/17415977.2012.665905
  37. Takezawa, A. and Kitamura, M. (2013), "Sensitivity analysis and optimization of vibration modes in continuum systems", J. Sound Vib., 332(6), 1553-1566. https://doi.org/10.1016/j.jsv.2012.11.015
  38. Tufekci, E. and Yigit, O.O. (2012), "Effects of geometric parameters on in-plane vibrations of two-stepped circular beams", Struct. Eng. Mech., 42(2), 131-152. https://doi.org/10.12989/sem.2012.42.2.131
  39. Udupa, K.M. and Varadan, T.K. (1990). Hierarchical finite element method for rotating beams", J. Sound Vib., 138(3), 447-456. https://doi.org/10.1016/0022-460X(90)90598-T
  40. Vinod, K.G., Gopalakrishnan, S. and Ganguli, R. (2007), "Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements", Int. J. Solids Struct., 44(18), 5875-5893. https://doi.org/10.1016/j.ijsolstr.2007.02.002

피인용 문헌

  1. Closed-form solutions for non-uniform axially loaded Rayleigh cantilever beams vol.60, pp.3, 2016, https://doi.org/10.12989/sem.2016.60.3.455