DYNAMIC ANALYSIS OF NANOBEAMS UNDER THE EFFECT OF PARTIAL UNIFORM LOAD

Öz

Dynamic analysis of nanobeams under the effect of partial uniform transverse load has been carried out. Governing equation of motion and boundary conditions have been obtained using Eringen’s Nonlocal Elasticity Theory. Partial uniform load effect is modeled with Heaviside function. Present model results have been compared and validated with fragmented model results. Effects of nonlocal parameter, dimensionless uniform load, application point of uniform load to the vibration frequency of nanobeam have been investigated. Effect of various parameters on the amplitude of nanobeam has been shown at different vibration frequencies. Instead of fragmented model which needs extra continuum boundary conditions which leads to increase in size of the matrices, present model needs four boundary conditions. Present study results could be useful at modeling of nano mass sensors like bacteria or virus.

Anahtar Kelimeler

• Partial Uniform Load,Nanobeam,Nonlocal Elasticity,Vibrational Analysis,Heaviside Function

KISMİ YAYILI YÜK ETKİSİNDEKİ NANO KİRİŞLERİN DİNAMİK ANALİZİ

Öz

Kısmi yayılı yük etkisindeki karbon nanotüp kirişlerin dinamik analizi gerçekleştirilmiştir. Nano kiriş için hareketin yönetici denklemi ve sınır şartları Eringen’in Yerel Olmayan Elastisite Teorisi kullanılarak elde edilmiştir. Kısmi yayılı yük etkisi Heaviside fonksiyonu ile modellenmiştir. Oluşturulan model, nano kirişin parçalara bölünmesiyle elde edilen model sonuçlarıyla karşılaştırılarak doğrulanmıştır. Nano kiriş titreşim frekansının yerel olmayan parametre, boyutsuz yayılı yük, yükün başlangıç noktası gibi parametrelerle değişimi incelenmiştir. Farklı titreşim frekanslarında çeşitli parametrelerin nano kirişin genlik değerlerine olan etkisi gösterilmiştir. Nano kirişin parçalara ayrılmasıyla artan süreklilik şartı sayısının oluşturduğu yüksek boyutlu matrislerin çözümüne alternatif olarak oluşturulan modelde dört sınır şartıyla çözüme ulaşılmıştır. Bu çalışmanın sonuçları bakteri veya virüs gibi nano kütle sensörlerinin modellenmesinde kullanılabilir.  

Anahtar Kelimeler

• Kısmi Yayılı Yük,Nano-Kiriş,Yerel Olmayan Elastisite,Titreşim Analizi,Heaviside Fonksiyonu

Kaynakça

1. Adali, S., 2008. Variational Principles for Multi-Walled Carbon Nanotubes Undergoing Buckling Based on Nonlocal Elasticity Theory. Physics Letters A, 372 (35), 5701–5.
2. Akbaş, Ş.D., 2019. Longitudinal Forced Vibration Analysis of Porous a Nanorod. Mühendislik Bilimleri ve Tasarım Dergisi, 7 (4), 736–43.
3. Akgöz, B., Civalek, Ö., Demir, C., 2011. Buckling Analysis of Cantilever Carbon Nanotubes Using the Strain Gradient Elasticity and Modified Couple Stress Theories. Journal of Computational and Theoretical Nanoscience, 8 (9), 1821–27.
4. Akgöz, B, Civalek, Ö., 2012. Investigation of Size Effects on Static Response of Single-Walled Carbon Nanotubes Based on Strain Gradient Elasticity. International Journal of Computational Methods, 09 (02), 1240032.
5. Akgöz, Bekir, 2019. Ritz Yöntemi Ile Değişken Kesitli Kolonların Burkulma Analizi. Mühendislik Bilimleri ve Tasarım Dergisi, 7 (2), 452–58.
6. Akgöz, Bekir, Civalek, Ö., 2016. Bending Analysis of Embedded Carbon Nanotubes Resting on an Elastic Foundation Using Strain Gradient Theory. Acta Astronautica, 119 (February), 1–12.
7. Arda, M., Aydogdu, M., 2016. Bending of CNTs Under The Partial Uniform Load. International Journal Of Engineering & Applied Sciences, 8 (2), 21–21.
8. Avcar, M., 2010. Elastik Zemin Üzerinde Bulunan Her İki Ucu Ankastre Mesnetli Rastgele ve Sürekli Homojen Olmayan Kirişin Serbest Titreşimi. Mühendislik Bilimleri ve Tasarım Dergisi, 1 (1), 33–38.

9. Avcar, M., Mohammed, W.K.M., 2017. Winkler Zemin ve Fonksiyonel Derecelendirilmiş Malzeme Özelliklerinin Kirişin Frekans Parametrelerine Etkilerinin İncelenmesi. Mühendislik Bilimleri ve Tasarım Dergisi, 5 (3), 573–80.
10. Aydogdu, M., 2009. A General Nonlocal Beam Theory: Its Application to Nanobeam Bending, Buckling and Vibration. Physica E: Low-Dimensional Systems and Nanostructures, 41 (9), 1651–55.
11. Aydogdu, M., 2012. Longitudinal Wave Propagation in Nanorods Using a General Nonlocal Unimodal Rod Theory and Calibration of Nonlocal Parameter with Lattice Dynamics. International Journal of Engineering Science, 56 (July), 17–28.
12. Aydogdu, M., Arda, M., 2016. Forced Vibration of Nanorods Using Nonlocal Elasticity. Advances in Nano Research, 4 (4), 265–79.
13. Civalek, Ö., Demir, Ç., Akgöz, B., 2009. Static Analysis of Single Walled Carbon Nanotubes (SWCNT) Based on Eringen’s Nonlocal Elasticity Theory. International Journal of Engineering and Applied Sciences, 1 (2), 47–56.
14. Demir, Ç., Civalek, Ö., 2016. Tek Katmanlı Grafen Tabakaların Eğilme ve Titreşimi. Mühendislik Bilimleri ve Tasarım Dergisi, 4 (3), 173.
15. Dequesnes, M., Tang, Z., Aluru, N.R., 2004. Static and Dynamic Analysis of Carbon Nanotube-Based Switches. Journal of Engineering Materials and Technology, 126 (3), 230.
16. Eftekhari, S.A., 2016. A Differential Quadrature Procedure with Direct Projection of the Heaviside Function for Numerical Solution of Moving Load Problem. Latin American Journal of Solids and Structures, 13 (9), 1763–81.
17. Eftekhari, S.A., Young, 2014. A Differential Quadrature Procedure with Regularization of the Dirac-Delta Function for Numerical Solution of Moving Load Problem. Latin American Journal of Solids and Structures, 121241–65.
18. Elishakoff, I., Pentaras, D., Dujat, K., Versaci, C., Muscolino, G., Storch, J., Bucas, S., et al., 2012. Carbon Nanotubes and Nanosensors. Carbon Nanotubes and Nanosensors: Vibration, Buckling and Ballistic Impact,. ISTE. Hoboken, NJ, USA: John Wiley & Sons, Inc.
19. Eringen, A.C., 1983. On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves. Journal of Applied Physics, 54 (9), 4703–10.
20. Eringen, A.C., 2004. Nonlocal Continuum Field Theories. Edited by A. Cemal Eringen. New York, NY: Springer New York.
21. Eringen, A.C., Edelen, D.G.B., 1972. On Nonlocal Elasticity. International Journal of Engineering Science, 10 (3), 233–48.
22. Feynman, R.P., 2011. There’s Plenty of Room at the Bottom. Resonance, 16 (9), 890–905.
23. Gong, N., Liang, Y.C., Yao, Y.X., Liu, B.G., 2008. Static and Dynamic Analysis of Carbon Nanotube Cantilever Based on Molecular Dynamics Simulation. Key Engineering Materials, 375–376 (August), 631–35.
24. Gul, U., Aydogdu, M., 2017. Wave Propagation in Double Walled Carbon Nanotubes by Using Doublet Mechanics Theory. Physica E: Low-Dimensional Systems and Nanostructures, 93345–57.
25. Gul, U., Aydogdu, M., 2018. Structural Modelling of Nanorods and Nanobeams Using Doublet Mechanics Theory. International Journal of Mechanics and Materials in Design, 14 (2), 195–212.
26. Gul, U., Aydogdu, M., Gaygusuzoglu, G., 2017. Axial Dynamics of a Nanorod Embedded in an Elastic Medium Using Doublet Mechanics. Composite Structures, 1601268–78.
27. Hosseini, M., Gorgani, H.H., Shishesaz, M., Hadi, A., 2017. Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory. International Journal of Applied Mechanics, 9 (6), .
28. Iijima, S., 1991. Helical Microtubules of Graphitic Carbon. Nature,.
29. Janghorban, M., Zare, A., 2012. Harmonic Differential Quadrature Method for Static Analysis of Functionally Graded Single Walled Carbon Nanotubes Based on Euler-Bernoulli Beam Theory. Latin American Journal of Solids and Structures, 9633–41.
30. Leissa, A.W., Qatu, M.S., 2011. Vibrations of Continuous Systems. New York: McGraw-Hill Education.
31. Li, C., Chou, T.-W., 2006. Atomistic Modeling of Carbon Nanotube-Based Mechanical Sensors. Journal of Intelligent Material Systems and Structures, 17 (3), 247–54.
32. Mohammad-Abadi, M., Daneshmehr, A.R., 2014. Size Dependent Buckling Analysis of Microbeams Based on Modified Couple Stress Theory with High Order Theories and General Boundary Conditions. International Journal of Engineering Science, 741–14.
33. Oterkus, E., Diyaroglu, C., Zhu, N., Oterkus, S., Madenci, E., 2015. Utilization of Peridynamic Theory for Modeling at the Nano-Scale. In , 1–16.
34. Reddy, J.N., 2002. Energy Principles and Variational Methods in Applied Mechanics. Wiley.
35. Reddy, J.N., Pang, S.D., 2008. Nonlocal Continuum Theories of Beams for the Analysis of Carbon Nanotubes. Journal of Applied Physics, 103 (2), 023511.
36. Shaban, M., Alibeigloo, A., 2014. Three Dimensional Vibration and Bending Analysis of Carbon Nano- Tubes Embedded in Elastic Medium Based on Theory of Elasticity. Latin American Journal of Solids and Structures, no. 20052122–40.
37. Wang, Q., Shindo, Y., 2006. Nonlocal Continuum Models for Carbon Nanotubes Subjected to Static Loading. Journal of Mechanics of Materials and Structures, 1 (4), 663–80.
38. Wu, Q., Volinsky, A.A., Qiao, L., Su, Y., 2015. Surface Effects on Static Bending of Nanowires Based on Non-Local Elasticity Theory. Progress in Natural Science: Materials International, 25 (5), 520–24.
39. 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 & Applied Sciences, 8 (2), 40–50.
40. Yayli, M.Ö., 2013. Torsion of Nonlocal Bars with Equilateral Triangle Cross Sections. Journal of Computational and Theoretical Nanoscience, 10 (2), 376–79.
41. Yayli, M.Ö., 2014. On the Axial Vibration of Carbon Nanotubes with Different Boundary Conditions. Micro & Nano Letters, 9 (11), 807–11.
42. Yayli, M.Ö., 2016. A Compact Analytical Method for Vibration Analysis of Single-Walled Carbon Nanotubes with Restrained Boundary Conditions. Journal of Vibration and Control, 22 (10), 2542–55.
43. Yayli, M.Ö., 2017. Buckling Analysis of a Cantilever Single-Walled Carbon Nanotube Embedded in an Elastic Medium with an Attached Spring. Micro & Nano Letters, 12 (4), 255–59.
44. Yayli, M.Ö., 2018. Torsional Vibration Analysis of Nanorods with Elastic Torsional Restraints Using Non-Local Elasticity Theory. Micro & Nano Letters, 13 (5), 595–99.
45. Yaylı, M.Ö., 2017. Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. International Journal Of Engineering & Applied Sciences, 9 (2), 103–103.