In the current study, the higher order Haar wavelet method based formulation is developed for the analysis of the free vibrations of the tapered Timoshenko beam. The clamped-clamped and clamped-pinned boundary conditions are explored and the results with the 4th order and the 6th order of convergence are presented. The results are found to be in good agreement with the corresponding results of the Ritz method. The proposed approach can be considered as the principal improvement of the widely used Haar wavelet method providing the same accuracy with the several magnitudes lower mesh. Thus, the higher order Haar wavelet method has reduced the computational cost in comparison with the widely used Haar wavelet method since the computational complexity of both methods is determined by the mesh used. In the case of the fixed equal mesh used for both methods, the higher order Haar wavelet method results in the several magnitudes lower absolute error without a remarkable increase in computational complexity. The cost needed to pay for higher accuracy is hidden in a certain increase in the implementation complexity compared with the widely used Haar wavelet method.
1. Chen, C. F. and Hsiao, C. H. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl., 1997, 144(1), 87–94.
https://doi.org/10.1049/ip-cta:19970702
2. Hsiao, C. H. State analysis of the linear time delayed systems via Haar wavelets. Math. Comput. Simul., 1997, 44(5), 457–470.
https://doi.org/10.1016/S0378-4754(97)00075-X
3. Lepik, Ü. Buckling of elastic beams by the Haar wavelet method. Estonian J. Eng., 2011, 17(3), 271–284.
https://doi.org/10.3176/eng.2011.3.07
4. Lepik, Ü. Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul., 2005, 68(2), 127–143.
https://doi.org/10.1016/j.matcom.2004.10.005
5. Lepik, Ü. Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput., 2007, 185(1), 695–704.
https://doi.org/10.1016/j.amc.2006.07.077
6. Hein, H. and Feklistova, L. Computationally efficient delamination detection in composite beams using Haar wavelets. Mech. Syst. Signal Process., 2011, 25(6), 2257–2270.
https://doi.org/10.1016/j.ymssp.2011.02.003
7. Hein, H. and Feklistova, L. Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets. Eng. Struct., 2011, 33(12), 3696–3701.
https://doi.org/10.1016/j.engstruct.2011.08.006
8. So, S.-R., Yun, H., Ri, Y., O, R. and Yun, Y.-I. Haar wavelet discretization method for free vibration study of laminated composite beam under generalized boundary conditions. J. Ocean Eng. Sci., 2021, 6(1), 1–11.
https://doi.org/10.1016/j.joes.2020.04.003
9. Aziz, I. and Šarler, B. The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math. Comput. Model., 2010, 52(9–10), 1577–1590.
https://doi.org/10.1016/j.mcm.2010.06.023
10. Aziz, I. and Amin, R. Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl. Math. Model., 2016, 40(23–24), 10286–10299.
https://doi.org/10.1016/j.apm.2016.07.018
11. Pervaiz, N. and Aziz, I. Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations. Physica A, 2020, 545, 123738.
https://doi.org/10.1016/j.physa.2019.123738
12. Majak, J., Shvartsman, B., Karjust, K., Mikola, M., Haavajõe, A. and Pohlak, M. On the accuracy of the Haar wavelet discretization method. Compos. B. Eng., 2015, 80, 321–327.
https://doi.org/10.1016/j.compositesb.2015.06.008
13. Majak J., Pohlak, M., Karjust, K., Eerme, M., Kurnitski, J. and Shvartsman, B. S. New higher order Haar wavelet method: Application to FGM structures.Compos. Struct., 2018, 201, 72–78.
https://doi.org/10.1016/j.compstruct.2018.06.013
14. Majak, J., Pohlak, M., Eerme, M. and Shvartsman, B. Solving ordinary differential equations with higher order Haar wavelet method. AIP Conf. Proc., 2019, 2116, 330002.
https://doi.org/10.1063/1.5114340
15. Majak, J., Shvartsman, B., Ratas, M., Bassir, D., Pohlak, M., Karjust, K. and Eerme, M. Higher-order Haar wavelet method for vibration analysis of nanobeams. Mater. Today Commun., 2020, 25, 101290.
https://doi.org/10.1016/j.mtcomm.2020.101290
16. Jena, S. K., Chakraverty, S. and Malikan, M. Implementation of Haar wavelet, higher order Haar wavelet, and differential quadrature methods on buckling response of strain gradient nonlocal beam embedded in an elastic medium. Eng. Comput., 2021, 37(2), 1251–1264.
https://doi.org/10.1007/s00366-019-00883-1
17. Ratas, M., Salupere, S. and Majak, J. Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids. Math. Model. Anal., 2021, 26(1), 147–169.
https://doi.org/10.3846/mma.2021.12920
18. Sorrenti, M., Di Sciuva, M., Majak, J. and Auriemma, F. Static response and buckling loads of multilayered composite beams using the refined zigzag theory and higher-order Haar wavelet method. Mech. Compos. Mater., 2021, 57(1), 1–18.
https://doi.org/10.1007/s11029-021-09929-2
19. Majkut, L. Free and forced vibrations of Timoshenko beams described by single difference equation. J. Theor. Appl. Mech., 2009, 47(1),193–210.
20. Shahba, A., Attarnejad, R., Marvi, M. T. and Hajilar, S. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Compos. B. Eng., 2011, 42(4), 801–808.
https://doi.org/10.1016/j.compositesb.2011.01.017
21. Zhou, D. and Cheung, Y. K. Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions, J. Appl. Mech., 2001, 68(4), 596–602.
https://doi.org/10.1115/1.1357164
22. Pradhan, K. K. and Chakraverty, S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos. B. Eng., 2013, 51,175–184.
https://doi.org/10.1016/j.compositesb.2013.02.027
23. Attarnejad, R., Semnani, S. J. and Shahba, A. Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams. Finite Elem. Anal. Des., 2010, 46(10), 916–929.
https://doi.org/10.1016/j.finel.2010.06.005
24. Sohani, F. and Eipakchi, H. R. Analytical solution for modal analysis of Euler–Bernoulli and Timoshenko beam with an arbitrary varying cross-section. Math. Model. Eng., 2018, 4(3), 164–174.
https://doi.org/10.21595/mme.2018.20116
25. Tang, A.-Y., Wu, J.-X., Li, X.-F. and Lee, K. Y. Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams. Int. J. Mech. Sci., 2014, 89, 1–11.
https://doi.org/10.1016/j.ijmecsci.2014.08.017
26. Huang, Y., Yang, L.-E. and Luo, Q.-Z. Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section. Compos. B. Eng., 2013, 45(1), 1493–1498.
https://doi.org/10.1016/j.compositesb.2012.09.015
27. Guessasma, S. and Bassir, D. Optimization of the mechanical properties of virtual porous slids using a hybrid approach. Acta Mater., 2010, 58(2), 716–725.
https://doi.org/10.1016/j.actamat.2009.09.051
28. Guessasma, S. and Bassir, D. Identification of mechanical properties of biopolymer composites sensitive to interface effect using hybrid approach. Mech. Mater., 2010, 42(3), 344–353.
https://doi.org/10.1016/j.mechmat.2009.12.001
29. Snatkin, A., Eiskop, T., Karjust, K. and Majak, J. Production monitoring system development and modification. Proc. Estonian Acad. Sci., 2015, 64(4S), 567–580.
https://doi.org/10.3176/proc.2015.4S.04
30. Guessasma, S., Bassir, D. and Hedjazi, L. Influence of interphase properties on the effective behaviour of a starch-hemp composite. Mater. Des., 2015, 65, 1053–1063.
https://doi.org/10.1016/j.matdes.2014.10.031
31. Abouzaid, K., Bassir, D., Guessasma, S. and Yue, H. Modelling the process of fused deposition modelling and the effect of temperature on the mechanical, roughness, and porosity properties of resulting composite products. Mech. Compos. Mater., 2021, 56, 805–816.
https://doi.org/10.1007/s11029-021-09925-6
