Öz
In this study, the free vibration analysis of a functionally graded (FG) micro-beam with tapered cross-section is carried out theoretically. The beam has material distribution according to the power law throughout the thickness. The governing equation is reduced to an ordinary differential equation for the tapered beam with a cross-sectional geometry whose width varies exponentially. Vibrations of functionally graded tapered microbeams were analyzed analytically in the elastic variation range based on the modified stress couple theory. Motion equations and boundary conditions are derived from the Hamilton principle. Analytical results of natural frequencies are calculated for cantilever exponential FG beams. Solutions for natural frequencies were obtained as the ratio of the beam's characteristic size to the material internal length parameter and according to the FGM distribution function characteristics.
Anahtar Kelimeler
Functionally graded materials micro-beam, free vibration analysis tapered cross-section
Kaynakça
- [1] Shariat, Bashir S., et al. "Functionally graded shape memory alloys: Design, fabrication and experimental evaluation." Materials & Design 124 (2017): 225-237.
- [2] Bogaerts, Wim, et al. "MORPHIC: Programmable photonic circuits enabled by silicon photonic MEMS." Silicon Photonics XV. Vol. 11285. International Society for Optics and Photonics, 2020.
- [3] Zarezadeh, Esmail, Vahid Hosseini, and Amin Hadi. "Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory." Mechanics Based Design of Structures and Machines 48.4 (2020): 480-495.
- [4] Granberry, R., Eschen, K., Holschuh, B., & Abel, J. (2019). Functionally Graded Knitted Actuators with NiTi‐Based Shape Memory Alloys for Topographically Self‐Fitting Wearables. Advanced materials technologies, 4(11), 1900548.
- [5] Behrouz, Saman Jabbari, Omid Rahmani, and S. Amirhossein Hosseini. "On nonlinear forced vibration of nano cantilever-based biosensor via couple stress theory." Mechanical Systems and Signal Processing 128 (2019): 19-36.
- [6] Azari, R., H. R. Rezaie, and A. Khavandi. "Investigation of functionally graded HA-TiO2 coating on Ti–6Al–4V substrate fabricated by sol-gel method." Ceramics International 45.14 (2019): 17545-17555.
- [7] Aydogdu M, Taskin V, Free vibration analysis of functionally graded beams with simply supported edges, Material Design, 28, 1651–6, 2007.
- [8] B.V.Sankar, An elasticity solution for functionally graded beams, Composite Science and Technology, 61, 689–696,2001.
- [9] Sina, S.A. et.al, An analytical method for free vibration, Materials and Design, 741–747, 2009.
- [10] Şimşek M., Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nano-rods, Computational Materials Science, 61 257–265, 2012.
- [11] Candan S., Elishakoff I., Apparently first closed-form solution for vibrating inhomogeneous beams, International Journal of Solids Structures, 38, 19 3411–41, 2001.
- [12] Elishakoff I, Becquet R., Closed-form solutions for natural frequency for inhomogeneous beams with one sliding support and the other pinned, Journal of Sound Vibration, 238(3), 529–39, 2000.
- [13] Huang Y, Li XF., A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration, 329(11), 2291–303, 2010.
- [14] Murin J., Aminbaghai M., Kutti V., Exact solution of the bending vibration problem of FGM beams with variation of material properties, Engineering Structures, 32, 1631-1640, 2010.
- [15] R. D. Mindlin, Micro-structure in Linear Elasticity, Rational Mechanics and Analysis, 16, 1, 51-78, 1964
- [16] Fleck N. A., Hutchinson J. W., Strain gradient plasticity, Advances in Applied Mechanics, 33, 1994
- [17] Yang F, Chong ACM, Lam DCC, Tong P., Couple stress based strain gradient theory for elasticity, International Journal of Solids Structures, 39(10), 2731–43, 2002.
- [18] S K Park and X-L Gao., Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 12, 2355–2359, 2006.
- [19] Akgöz B., Civalek Ö., Free vibration analysis of axially functionally graded tapered Bernoulli–Euler, Composite Structures, 2012.
- [20] M. Asghari, M.T. Ahmadian, M.H. Kahrobaiyan, M. Rahaeifard, On the size-dependent behavior of functionally graded micro-beams, Materials and Design, 2324–2329, 2010.
- [21] Y.-A.Kang,X.-F.Li., Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force, International Journal of Non-Linear Mechanics, 696–703, 2009.
- [22] Sulem, J., Vardoulakis, I., & Papamichos, E. (1995). Microstructure and scale effect in granular rocks. Continuum Models in Materials with Microstructure, 201-238.
Öz
Bu çalışmada, konik kesitli fonksiyonel derecelendirilmiş (FD) bir mikro kirişin serbest titreşim analizi teorik olarak gerçekleştirilmiştir. Kiriş, kalınlık boyunca güç yasasına göre malzeme dağılımına sahiptir. Yönetim denklemi, genişliği üstel olarak değişen bir enine kesit geometrisine sahip konik kiriş için sıradan bir diferansiyel denkleme indirgenmiştir. Fonksiyonel derecelendirilmiş konik mikro kirişlerin titreşimleri, modifiye edilmiş gerilim çifti teorisine dayalı olarak elastik varyasyon aralığında analitik olarak analiz edilmiştir. Hareket denklemleri ve sınır koşulları Hamilton ilkesinden türetilmiştir. Doğal frekansların analitik sonuçları, konsol üstel FD kiriş için hesaplanmıştır. Doğal frekanslar için çözümler, kirişin karakteristik boyutunun malzeme iç uzunluk parametresine oranı olarak ve FDM dağılım fonksiyonu özelliklerine göre elde edilmiştir.
Anahtar Kelimeler
Fonksiyonel derecelendirilmiş malzeme serbest titreşim analizi konik kesit alanı mikro-kiriş
Kaynakça
- [1] Shariat, Bashir S., et al. "Functionally graded shape memory alloys: Design, fabrication and experimental evaluation." Materials & Design 124 (2017): 225-237.
- [2] Bogaerts, Wim, et al. "MORPHIC: Programmable photonic circuits enabled by silicon photonic MEMS." Silicon Photonics XV. Vol. 11285. International Society for Optics and Photonics, 2020.
- [3] Zarezadeh, Esmail, Vahid Hosseini, and Amin Hadi. "Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory." Mechanics Based Design of Structures and Machines 48.4 (2020): 480-495.
- [4] Granberry, R., Eschen, K., Holschuh, B., & Abel, J. (2019). Functionally Graded Knitted Actuators with NiTi‐Based Shape Memory Alloys for Topographically Self‐Fitting Wearables. Advanced materials technologies, 4(11), 1900548.
- [5] Behrouz, Saman Jabbari, Omid Rahmani, and S. Amirhossein Hosseini. "On nonlinear forced vibration of nano cantilever-based biosensor via couple stress theory." Mechanical Systems and Signal Processing 128 (2019): 19-36.
- [6] Azari, R., H. R. Rezaie, and A. Khavandi. "Investigation of functionally graded HA-TiO2 coating on Ti–6Al–4V substrate fabricated by sol-gel method." Ceramics International 45.14 (2019): 17545-17555.
- [7] Aydogdu M, Taskin V, Free vibration analysis of functionally graded beams with simply supported edges, Material Design, 28, 1651–6, 2007.
- [8] B.V.Sankar, An elasticity solution for functionally graded beams, Composite Science and Technology, 61, 689–696,2001.
- [9] Sina, S.A. et.al, An analytical method for free vibration, Materials and Design, 741–747, 2009.
- [10] Şimşek M., Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nano-rods, Computational Materials Science, 61 257–265, 2012.
- [11] Candan S., Elishakoff I., Apparently first closed-form solution for vibrating inhomogeneous beams, International Journal of Solids Structures, 38, 19 3411–41, 2001.
- [12] Elishakoff I, Becquet R., Closed-form solutions for natural frequency for inhomogeneous beams with one sliding support and the other pinned, Journal of Sound Vibration, 238(3), 529–39, 2000.
- [13] Huang Y, Li XF., A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration, 329(11), 2291–303, 2010.
- [14] Murin J., Aminbaghai M., Kutti V., Exact solution of the bending vibration problem of FGM beams with variation of material properties, Engineering Structures, 32, 1631-1640, 2010.
- [15] R. D. Mindlin, Micro-structure in Linear Elasticity, Rational Mechanics and Analysis, 16, 1, 51-78, 1964
- [16] Fleck N. A., Hutchinson J. W., Strain gradient plasticity, Advances in Applied Mechanics, 33, 1994
- [17] Yang F, Chong ACM, Lam DCC, Tong P., Couple stress based strain gradient theory for elasticity, International Journal of Solids Structures, 39(10), 2731–43, 2002.
- [18] S K Park and X-L Gao., Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 12, 2355–2359, 2006.
- [19] Akgöz B., Civalek Ö., Free vibration analysis of axially functionally graded tapered Bernoulli–Euler, Composite Structures, 2012.
- [20] M. Asghari, M.T. Ahmadian, M.H. Kahrobaiyan, M. Rahaeifard, On the size-dependent behavior of functionally graded micro-beams, Materials and Design, 2324–2329, 2010.
- [21] Y.-A.Kang,X.-F.Li., Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force, International Journal of Non-Linear Mechanics, 696–703, 2009.
- [22] Sulem, J., Vardoulakis, I., & Papamichos, E. (1995). Microstructure and scale effect in granular rocks. Continuum Models in Materials with Microstructure, 201-238.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Mühendislik |
| Bölüm | Tasarım ve Teknoloji |
| Yazarlar | |
| Yayımlanma Tarihi | 27 Haziran 2021 |
| Gönderilme Tarihi | 22 Mart 2021 |
| Yayımlandığı Sayı | Yıl 2021 Cilt: 9 Sayı: 2 |
