An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM

Togay Küpeli; Yakup Harun Çavuş; Büşra Uzun; Mustafa Özgür Yaylı

doi:10.17482/uumfd.980105

Research Article

Yay Kütle Sistemi İle Birleştirilmiş Fonksiyonel Olarak Derecelendirilmiş Nanokirişin Özdeğer Problemi İle Çözümü

Year 2021, Volume: 26 Issue: 3, 1097 - 1110, 31.12.2021

Togay Küpeli Yakup Harun Çavuş Büşra Uzun Mustafa Özgür Yaylı

https://doi.org/10.17482/uumfd.980105

Abstract

Nanokirişler günümüzde çok sayıda titreşim frekansı araştırmasında yaygın olarak kullanılmaktadır. Bu çalışmada, nanokirişin ucuna takılı halde bulunan buckyball ve yayın titreşim frekans analizini yapabilmek için bir özdeğer problemi kullanılmıştır. Bu özdeğer probleminde sistemin titreşim frekansları tek bir (2x2) matris kullanılarak hesaplanabilir. Bu çalışmada, nanokirişlere bağlı sensörleri analiz etmek için matematiksel bir yöntem sunulmaktadır. Bu makalede elde edilen sonuçlar literatürde yapılan titreşim frekansı çalışmaları ile uyumlu bir sonuç göstermiştir ve sonuçlar tablo ve grafiklerle sunulmuştur.

Keywords

Frekans, Karbon Nanotüp, Buckyball, Özdeğer Problemi

References

1. Akbaş, Ş. D., (2019) Axially Forced Vibration Analysis of Cracked a Nanorod. Journal of Computational Applied Mechanics, 50(1), 63-68. doi:10.22059/jcamech.2019.281285.392
2. Akbaş, Ş. D. (2019) Longitudinal forced vibration analysis of porous a nanorod. Mühendislik Bilimleri ve Tasarım Dergisi,7 (4) , 736-743. DOI: 10.21923/jesd.553328
3. Akbas, S. D. (2020) Modal analysis of viscoelastic nanorods under an axially harmonic load. Advances in Nano Research, 8(4), 277–282. doi: 10.12989/ANR.2020.8.4.277.
4. Alimoradzadeh, M. and Akbaş, Ş. D. (2021) Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory, European Physical Journal Plus, Vol. 136, no. 5. https://doi.org/10.1140/epjp/s13360-021-01539-0
5. Aydogdu, M. (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 41, 1651-1655. DOI:10.1016/j.physe.2009.05.014
6. Civalek, Ö., Uzun, B., & Yaylı, M. Ö. (2020) Stability analysis of nanobeams placed in electromagnetic field using a finite element method. Arabian Journal of Geosciences, 13(21), 1-9. https://doi.org/10.1007/s12517-020-06188-8
7. Civalek, Ö., Demir, Ç. (2011) Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Appl. Math. Model, 35, 2053-2067. https://doi.org/10.1016/j.apm.2010.11.004
8. Civalek, Ö., Akgöz, B. (2010) Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler–Bernoulli beam modeling, Sci. Iranica Trans. B: Mech. Eng., 17, 367-375.
9. Eltaher, M.A., Emam, S.A. (2013) Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88. doi: 10.1016/j.compstruct.2012.09.030.
10. Eringen, A. C. (1972) Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
11. Eringen, A. C. (1983) on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 4703–4710. https://doi.org/10.1063/1.332803
12. Liu, T., Hai, M., Zhao, M. (2008) Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system, Eng. Fract. Mech. 75, 4909-4919. doi:10.1016/j.engfracmech.2008.06.021.
13. Lu, P., Lee, H.P., Lu, C., Zhang, P.Q. (2006) Dynamic properties of flexural beams using a nonlocal elasticity model, J. Appl. Phys., 99, 73510-73518. https://doi.org/10.1063/1.2189213
14. Murmu, T., Adhikari, S., Wang, C.Y. (2011) Torsional vibration of carbon nanotube–buckyball systems based on nonlocal elasticity theory, Physica E Low-dimensional Systems and Nanostructures 43(6):1276-1280. DOI:10.1016/j.physe.2011.02.017
15. Murmu, T., Pradhan, S.C. (2009) Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory, Physica E, 41, 1451-1456. https://doi.org/10.1016/j.physe.2009.04.015.
16. Mustafa Arda & Metin Aydogdu (2020) Vibration analysis of carbon nanotube mass sensors considering both inertia and stiffness of the detected mass, Mechanics Based Design of Structures and Machines. 1-17. doi:10.1080/15397734.2020.1728548.
17. Narendar, S. (2011) Buckling analysis of micro-/nano-scale plates based on two variable refined plate theory incorporating nonlocal scale effects, Compos. Struct, 93, 3093-3103. DOI:10.1016/j.compstruct.2011.06.028
18. Özgür Yaylı, M., & Erdem Çerçevik, A. (2015). Axial vibration analysis of cracked nanorods with arbitrary boundary conditions. Journal of Vibroengineering, 17(6), 2907-2921.
19. Pradhan, S.C., Phadikar, J.K. (2009) Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206-223. DOI:10.1016/j.jsv.2009.03.007.
20. Rahmani, O., Pedram, O. (2014) Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, Int. J. Eng. Sci, 77, 55-70. DOI:10.1016/j.ijengsci.2013.12.003
21. Reddy, J.N. (2007) Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci.45, 288-307. DOI:10.1016/j.ijengsci.2007.04.004
22. Reddy J. N., Pang, S. D. (2008) Nonlocal continuum theories of beam for the analysis of carbon nanotubes,. Journal of Applied Physics, 103, 1-16. DOI:10.1063/1.2833431
23. Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P. (2011) Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737. https://doi.org/10.1016/j.physe.2011.05.032.
24. Shen, L., Shen, H.S., Zhang, C.L. (2010) Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput. Mater. Sci., 48, 680-685. https://doi.org/10.1016/j.commatsci.2010.03.006.
25. Thai, H.T. (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64. DOI: 10.1016/j.ijengsci.2011.11.011
26. Uzun, B., Yaylı, M. Ö., & Deliktaş, B. (2019). Free vibration of FG nanobeam using a finite-element method. Micro & Nano Letters, 15(1), 35-40. doi:10.1049/mnl.2019.0273.
27. Uzun, B., Civalek, Ö., & Yaylı, M. Ö. (2020). Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions. Mechanics Based Design of Structures and Machines, 1-20. DOI: 10.1080/15397734.2020.1846560
28. Uzun, B., Yaylı, M. Ö. (2020) Nonlocal vibration analysis of Ti-6Al-4V/ZrO 2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences 13.4, 1-10. https://doi.org/10.1007/s12517-020-5168-4
29. Yayli, M.Ö. (2015) Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube, Acta Physica Polonica A, 127, 3, 678-683. DOI:10.24107/ijeas.252144
30. Yayli, M.Ö. (2015) Stability analysis of gradient elastic microbeams with arbitrary boundary conditions, Journal of Mechanical Science and Technology, 29, 8, 3373-3380. https://doi.org/10.1007/s12206-015-0735-4
31. Yayli, M.Ö. (2016). Buckling analysis of a rotationally restrained single walled carbon nanotube embedded in an elastic medium using nonlocal elasticity. International Journal of Engineering and Applied Sciences, 8(2), 40-50. DOI: 10.24107/ijeas.252144
32. Yayli, M. Ö. (2017). A compact analytical method for vibration of micro-sized beams with different boundary conditions. Mechanics of Advanced Materials and Structures, 24(6), 496-508. DOI: 10.1080/15376494.2016.1143989.
33. Yaylı, M. Ö. & Yerel Kandemir, S. (2017). Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. International Journal of Engineering and Applied Sciences, Volume: 9 Issue: 2, 103-111. DOI: 10.24107/ijeas.314635
34. Yayli, M. Ö. (2018). Torsional vibration analysis of nanorods with elastic torsional restraints using non-local elasticity theory. Micro & Nano Letters, 13(5), 595-599. doi: 10.1049/mnl.2017.0751
35. Yayli, M. Ö. (2018). Free vibration analysis of a single‐walled carbon nanotube embedded in an elastic matrix under rotational restraints. Micro & Nano Letters, 13(2), 202-206. doi: 10.1049/mnl.2017.0463
36. Yaylı, M. Ö., Uzun, B., & Deliktaş, B. (2021). Buckling analysis of restrained nanobeams using strain gradient elasticity. Waves in Random and Complex Media, 1-20. DOI: 10.1080/17455030.2020.1871112
37. Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M. (2008). Beam bending solutions based on nonlocal Timoshenko beam theory, J. Eng. Mech., 134, 475-481. DOI:10.1061/(ASCE)0733-9399(2008)134:6(475)

An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM

Year 2021, Volume: 26 Issue: 3, 1097 - 1110, 31.12.2021

Togay Küpeli Yakup Harun Çavuş Büşra Uzun Mustafa Özgür Yaylı

https://doi.org/10.17482/uumfd.980105

Abstract

Nanobeams are now widely used in numerous vibration frequency research. In this study, an eigenvalue problem has used to determine the vibration frequency analysis of the buckyball and spring attached to the end of the nanobeam. The vibration frequencies of the system may be discovered using a single (2x2) matrix in this eigenvalue problem. A mathematical method for analyzing sensors has attached to nanobeams is presented in this paper. The results, which is obtained in this study, has showed a result that has compatible with the flicker frequency studies conducted in the literature, and the results have presented in tables and graphics.

Keywords

Frequency, Carbon Nanotube, Buckyball, Buckyball, Eigenvalue Problem.

References

1. Akbaş, Ş. D., (2019) Axially Forced Vibration Analysis of Cracked a Nanorod. Journal of Computational Applied Mechanics, 50(1), 63-68. doi:10.22059/jcamech.2019.281285.392
2. Akbaş, Ş. D. (2019) Longitudinal forced vibration analysis of porous a nanorod. Mühendislik Bilimleri ve Tasarım Dergisi,7 (4) , 736-743. DOI: 10.21923/jesd.553328
3. Akbas, S. D. (2020) Modal analysis of viscoelastic nanorods under an axially harmonic load. Advances in Nano Research, 8(4), 277–282. doi: 10.12989/ANR.2020.8.4.277.
4. Alimoradzadeh, M. and Akbaş, Ş. D. (2021) Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory, European Physical Journal Plus, Vol. 136, no. 5. https://doi.org/10.1140/epjp/s13360-021-01539-0
5. Aydogdu, M. (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 41, 1651-1655. DOI:10.1016/j.physe.2009.05.014
6. Civalek, Ö., Uzun, B., & Yaylı, M. Ö. (2020) Stability analysis of nanobeams placed in electromagnetic field using a finite element method. Arabian Journal of Geosciences, 13(21), 1-9. https://doi.org/10.1007/s12517-020-06188-8
7. Civalek, Ö., Demir, Ç. (2011) Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Appl. Math. Model, 35, 2053-2067. https://doi.org/10.1016/j.apm.2010.11.004
8. Civalek, Ö., Akgöz, B. (2010) Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler–Bernoulli beam modeling, Sci. Iranica Trans. B: Mech. Eng., 17, 367-375.
9. Eltaher, M.A., Emam, S.A. (2013) Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88. doi: 10.1016/j.compstruct.2012.09.030.
10. Eringen, A. C. (1972) Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
11. Eringen, A. C. (1983) on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 4703–4710. https://doi.org/10.1063/1.332803
12. Liu, T., Hai, M., Zhao, M. (2008) Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system, Eng. Fract. Mech. 75, 4909-4919. doi:10.1016/j.engfracmech.2008.06.021.
13. Lu, P., Lee, H.P., Lu, C., Zhang, P.Q. (2006) Dynamic properties of flexural beams using a nonlocal elasticity model, J. Appl. Phys., 99, 73510-73518. https://doi.org/10.1063/1.2189213
14. Murmu, T., Adhikari, S., Wang, C.Y. (2011) Torsional vibration of carbon nanotube–buckyball systems based on nonlocal elasticity theory, Physica E Low-dimensional Systems and Nanostructures 43(6):1276-1280. DOI:10.1016/j.physe.2011.02.017
15. Murmu, T., Pradhan, S.C. (2009) Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory, Physica E, 41, 1451-1456. https://doi.org/10.1016/j.physe.2009.04.015.
16. Mustafa Arda & Metin Aydogdu (2020) Vibration analysis of carbon nanotube mass sensors considering both inertia and stiffness of the detected mass, Mechanics Based Design of Structures and Machines. 1-17. doi:10.1080/15397734.2020.1728548.
17. Narendar, S. (2011) Buckling analysis of micro-/nano-scale plates based on two variable refined plate theory incorporating nonlocal scale effects, Compos. Struct, 93, 3093-3103. DOI:10.1016/j.compstruct.2011.06.028
18. Özgür Yaylı, M., & Erdem Çerçevik, A. (2015). Axial vibration analysis of cracked nanorods with arbitrary boundary conditions. Journal of Vibroengineering, 17(6), 2907-2921.
19. Pradhan, S.C., Phadikar, J.K. (2009) Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206-223. DOI:10.1016/j.jsv.2009.03.007.
20. Rahmani, O., Pedram, O. (2014) Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, Int. J. Eng. Sci, 77, 55-70. DOI:10.1016/j.ijengsci.2013.12.003
21. Reddy, J.N. (2007) Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci.45, 288-307. DOI:10.1016/j.ijengsci.2007.04.004
22. Reddy J. N., Pang, S. D. (2008) Nonlocal continuum theories of beam for the analysis of carbon nanotubes,. Journal of Applied Physics, 103, 1-16. DOI:10.1063/1.2833431
23. Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P. (2011) Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737. https://doi.org/10.1016/j.physe.2011.05.032.
24. Shen, L., Shen, H.S., Zhang, C.L. (2010) Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput. Mater. Sci., 48, 680-685. https://doi.org/10.1016/j.commatsci.2010.03.006.
25. Thai, H.T. (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64. DOI: 10.1016/j.ijengsci.2011.11.011
26. Uzun, B., Yaylı, M. Ö., & Deliktaş, B. (2019). Free vibration of FG nanobeam using a finite-element method. Micro & Nano Letters, 15(1), 35-40. doi:10.1049/mnl.2019.0273.
27. Uzun, B., Civalek, Ö., & Yaylı, M. Ö. (2020). Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions. Mechanics Based Design of Structures and Machines, 1-20. DOI: 10.1080/15397734.2020.1846560
28. Uzun, B., Yaylı, M. Ö. (2020) Nonlocal vibration analysis of Ti-6Al-4V/ZrO 2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences 13.4, 1-10. https://doi.org/10.1007/s12517-020-5168-4
29. Yayli, M.Ö. (2015) Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube, Acta Physica Polonica A, 127, 3, 678-683. DOI:10.24107/ijeas.252144
30. Yayli, M.Ö. (2015) Stability analysis of gradient elastic microbeams with arbitrary boundary conditions, Journal of Mechanical Science and Technology, 29, 8, 3373-3380. https://doi.org/10.1007/s12206-015-0735-4
31. Yayli, M.Ö. (2016). Buckling analysis of a rotationally restrained single walled carbon nanotube embedded in an elastic medium using nonlocal elasticity. International Journal of Engineering and Applied Sciences, 8(2), 40-50. DOI: 10.24107/ijeas.252144
32. Yayli, M. Ö. (2017). A compact analytical method for vibration of micro-sized beams with different boundary conditions. Mechanics of Advanced Materials and Structures, 24(6), 496-508. DOI: 10.1080/15376494.2016.1143989.
33. Yaylı, M. Ö. & Yerel Kandemir, S. (2017). Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. International Journal of Engineering and Applied Sciences, Volume: 9 Issue: 2, 103-111. DOI: 10.24107/ijeas.314635
34. Yayli, M. Ö. (2018). Torsional vibration analysis of nanorods with elastic torsional restraints using non-local elasticity theory. Micro & Nano Letters, 13(5), 595-599. doi: 10.1049/mnl.2017.0751
35. Yayli, M. Ö. (2018). Free vibration analysis of a single‐walled carbon nanotube embedded in an elastic matrix under rotational restraints. Micro & Nano Letters, 13(2), 202-206. doi: 10.1049/mnl.2017.0463
36. Yaylı, M. Ö., Uzun, B., & Deliktaş, B. (2021). Buckling analysis of restrained nanobeams using strain gradient elasticity. Waves in Random and Complex Media, 1-20. DOI: 10.1080/17455030.2020.1871112
37. Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M. (2008). Beam bending solutions based on nonlocal Timoshenko beam theory, J. Eng. Mech., 134, 475-481. DOI:10.1061/(ASCE)0733-9399(2008)134:6(475)

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Details

Primary Language	English
Subjects	Civil Engineering
Journal Section	Research Articles
Authors	Togay Küpeli 0000-0002-5921-8667 Yakup Harun Çavuş 0000-0002-6607-9650 Büşra Uzun 0000-0002-7636-7170 Mustafa Özgür Yaylı 0000-0003-2231-170X
Publication Date	December 31, 2021
Submission Date	August 7, 2021
Acceptance Date	October 8, 2021
Published in Issue	Year 2021 Volume: 26 Issue: 3

Cite

APA	Küpeli, T., Çavuş, Y. H., Uzun, B., Yaylı, M. Ö. (2021). An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi, 26(3), 1097-1110. https://doi.org/10.17482/uumfd.980105
AMA	Küpeli T, Çavuş YH, Uzun B, Yaylı MÖ. An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM. UUJFE. December 2021;26(3):1097-1110. doi:10.17482/uumfd.980105
Chicago	Küpeli, Togay, Yakup Harun Çavuş, Büşra Uzun, and Mustafa Özgür Yaylı. “An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM With an ATTACHED SPRING MASS SYSTEM”. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi 26, no. 3 (December 2021): 1097-1110. https://doi.org/10.17482/uumfd.980105.
EndNote	Küpeli T, Çavuş YH, Uzun B, Yaylı MÖ (December 1, 2021) An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi 26 3 1097–1110.
IEEE	T. Küpeli, Y. H. Çavuş, B. Uzun, and M. Ö. Yaylı, “An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM”, UUJFE, vol. 26, no. 3, pp. 1097–1110, 2021, doi: 10.17482/uumfd.980105.
ISNAD	Küpeli, Togay et al. “An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM With an ATTACHED SPRING MASS SYSTEM”. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi 26/3 (December 2021), 1097-1110. https://doi.org/10.17482/uumfd.980105.
JAMA	Küpeli T, Çavuş YH, Uzun B, Yaylı MÖ. An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM. UUJFE. 2021;26:1097–1110.
MLA	Küpeli, Togay et al. “An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM With an ATTACHED SPRING MASS SYSTEM”. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi, vol. 26, no. 3, 2021, pp. 1097-10, doi:10.17482/uumfd.980105.
Vancouver	Küpeli T, Çavuş YH, Uzun B, Yaylı MÖ. An EIGENVALUE SOLUTION OF FUNCTIONALLY GRADED NANOBEAM with an ATTACHED SPRING MASS SYSTEM. UUJFE. 2021;26(3):1097-110.

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