Free Longitudinal Vibrations Analysis of Nanorods Based on Eringen’s Nonlocal Theory with Considering Surface Energy Using Wave Approach
پذیرفته شده برای پوستر ، صفحه 1-12 (12)
کد مقاله : 1039-ISAV2022 (R3)
نویسندگان
1فارغ التحصیل از دانشگاه آزاد اسلامی واحد تهران شمال
2دپارتمان مهندسی مکانیک، دانشگاه علم و فرهنگ، تهران ، ایران
چکیده
In this paper, analysis of free longitudinal vibrations of continuous nanorod (nanotube) in cylindrical coordinate system, have been done by wave method. therefore, first the axial force equation of nanorod is derived from its motion equation. and then by using of Eringen's nonlocal elasticity theory, nanlocal axial force of nanorod is obtained. after deriving axial force equation which includes surface energy parameters, equations of motion and axial force of nanorod in terms of positive-going and negative-going waves are obtained. by placing the spring at the end of the boundaries of the nanorod, wave propagation matrix is obtained under boundary conditions of clamped-clamped and clamped-free. eventually by calculation the dimensionless frequency of nanorod for both of the mentioned boundary conditions, dimensionless frequency changes vs nonlocal parameter and also vs sureface energy parameters as diagram have been investigated. in each diagram, first three modes of dimensionless natural frequency under difference Poisson ratio are investigated.
کلیدواژه ها
موضوعات
Title
Free Longitudinal Vibrations Analysis of Nanorods Based on Eringen’s Nonlocal Theory with Considering Surface Energy Using Wave Approach
Authors
Mitra Siahnouri, Masih Loghmani
Abstract
In this paper, analysis of free longitudinal vibrations of continuous nanorod (nanotube) in cylindrical coordinate system, have been done by wave method. therefore, first the axial force equation of nanorod is derived from its motion equation. and then by using of Eringen's nonlocal elasticity theory, nanlocal axial force of nanorod is obtained. after deriving axial force equation which includes surface energy parameters, equations of motion and axial force of nanorod in terms of positive-going and negative-going waves are obtained. by placing the spring at the end of the boundaries of the nanorod, wave propagation matrix is obtained under boundary conditions of clamped-clamped and clamped-free. eventually by calculation the dimensionless frequency of nanorod for both of the mentioned boundary conditions, dimensionless frequency changes vs nonlocal parameter and also vs sureface energy parameters as diagram have been investigated. in each diagram, first three modes of dimensionless natural frequency under difference Poisson ratio are investigated.
Keywords
longitudinal vibration, nanorod, Eringens theory, surface energy
مراجع
<p dir="LTR">1. Borowiec, M., G. Litak, M. Friswell, S. Ali, S. Adhikari, A. Lees, and O. Bilgen, "Energy harvesting in piezoelastic systems driven by random excitations", International journal of structural stability and dynamics 13, 1340006, (2013).</p>
<p dir="LTR">2. Joseph, G.V., G. Hao, and V. Pakrashi, "Extreme value estimates using vibration energy harvesting", Journal of Sound and Vibration 437, 29-39, (2018).</p>
<p dir="LTR">3. Liang, H., G. Hao, and O.Z. Olszewski, "A review on vibration-based piezoelectric energy harvesting from the aspect of compliant mechanisms", Sensors and Actuators A: Physical 331, 112743, (2021).</p>
<p dir="LTR">4. Hadas, Z., V. Vetiska, V. Singule, O. Andrs, J. Kovar, and J. Vetiska, "Energy harvesting from mechanical shocks using a sensitive vibration energy harvester". International Journal of Advanced Robotic Systems 9, (2012).</p>
<p dir="LTR">5. Roundy, S., P.K. Wright, and J.M. Rabaey, Energy scavenging for wireless sensor networks, Springer, New York 2003.</p>
<p dir="LTR">6. Blad, T., D.F. Machekposhti, J. Herder, A. Holmes, and N. Tolou. "Vibration energy harvesting from multidirectional motion sources", International Conference on Manipulation, Automation and Robotics at Small Scales, (2018).</p>
<p dir="LTR">7. Daneshjou, K., R. Talebitooti, and M. Kornokar, "Vibroacoustic study on a multilayered functionally graded cylindrical shell with poroelastic core and bonded-unbonded configuration", Journal of Sound and Vibration 393, 157-175, (2017).</p>
<p dir="LTR">8. Sola, A., D. Bellucci, and V. Cannillo, "Functionally graded materials for orthopedic applications–an update on design and manufacturing", Biotechnology Advances 34, 504-531, (2016).</p>
<p dir="LTR">9. Kang, Y.-A. and 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 44, 696-703, (2009).</p>
<p dir="LTR">10. Shi, Y., H. Yao, and Y.-w. Gao, "A functionally graded composite cantilever to harvest energy from magnetic field", Journal of Alloys and Compounds 693, 989-999, (2017).</p>
<p dir="LTR">11. Van Suchtelen, J., "Product properties: a new application of composite materials", Philips Research Reports 27, 28- 37, (1972).</p>
<p dir="LTR">12. Vaezi, M., M.M. Shirbani, and A. Hajnayeb, "Free vibration analysis of magneto-electro-elastic microbeams subjected to magneto-electric loads", Physica E: Low-dimensional Systems and Nanostructures 75, 280-286, (2016).</p>
<p dir="LTR">13. Lee, J., J.G. Boyd IV, and D.C. Lagoudas, "Effective properties of three-phase electro-magneto-elastic composites", International Journal of Engineering Science 43, 790-825, (2005).</p>
<p dir="LTR">14. Shishesaz, M., M.M. Shirbani, H.M. Sedighi, and A. Hajnayeb, "Design and analytical modeling of magneto-electromechanical characteristics of a novel magneto-electro-elastic vibration-based energy harvesting system", Journal of Sound and Vibration 425, 149-169, (2018).</p>
<p dir="LTR">15. Ylli, K., D. Hoffmann, A. Willmann, P. Becker, B. Folkmer, and Y. Manoli, "Energy harvesting from human motion: exploiting swing and shock excitations", Smart Materials and Structures 24, 025029, (2015).</p>
<p dir="LTR">16. Younis, M.I., D. Jordy, and J.M. Pitarresi, "Computationally efficient approaches to characterize the dynamic response of microstructures under mechanical shock", Journal of Microelectromechanical Systems 16, 628-638, (2007).</p>
<p dir="LTR">17. Askari, A.R. and S. Lenci, "Size-dependent response of electrically pre-deformed micro-plates under mechanical shock incorporating the effect of packaging a frequency-domain analysis", Journal of the Brazilian Society of Mechanical Sciences and Engineering 43, 1-21, (2021).</p>
<p dir="LTR">18. Rekik, M., S. El-Borgi, and Z. Ounaies, "An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium", Applied Mathematical Modeling 38, 1193-1210, (2014).</p>
<p dir="LTR">19. Ma, J., L.-L. Ke, and Y.-S. Wang, "Sliding frictional contact of functionally graded magneto-electro-elastic materials under a conducting flat punch", Journal of Applied Mechanics 82, (2015).</p>
<p dir="LTR">20. Li, X., H. Ding, and W. Chen, "Three-dimensional analytical solution for functionally graded magneto–electro-elastic circular plates subjected to uniform load", Composite Structures 83, 381-390, (2008).</p>
<p dir="LTR">21. Singh, S. and I. Singh, "Extended isogeometric analysis for fracture in functionally graded magneto-electro-elastic material", Engineering Fracture Mechanics 247, 107640, (2021).</p>
<p dir="LTR">22. Suresh, S. and A. Mortensen, Fundamentals of functionally graded materials. The Institut of Materials, 1998.</p>
<p dir="LTR">23. Hashemi, R., "Magneto-electro-elastic properties of multiferroic composites containing periodic distribution of general multi-coated inhomogeneities", International Journal of Engineering Science 103, 59-76, (2016).</p>
<p dir="LTR">24. Reddy, J., "Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates" International Journal of Engineering Science 48, 1507-1518, (2010).</p>
<p dir="LTR">25. JESD22, J.S., B111: "Board level drop test method of components for handheld electronic products", JEDEC Solid State Technology Association, (2003).</p>
<p dir="LTR">26. Meirovitch, L. and R. Parker, "Fundamentals of vibrations", Applied Mechanics Reviews 54, 100-101, (2001).</p>
<p dir="LTR">27. Reddy, J.N., Energy principles and variational methods in applied mechanics. John Wiley & Sons, New York, 2017.</p>
<p dir="LTR">28. Derayatifar, M., M. Tahani, and H. Moeenfard, "Nonlinear analysis of functionally graded piezoelectric energy harvesters", Composite Structures 182, 199-208, (2017).</p>
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<p dir="LTR">ارتعاشات طولی؛ نانومیله؛ تئوری ارینگن؛ انرژی سطحی</p>
<p dir="LTR">در این مقاله به تحلیل ارتعاشات طولی آزاد نانومیله (نانولوله) یکنواخت در دستگاه مختصات استوانه ای، به روش موج پرداخته شده است. به این ترتیب، ابتدا معادله نیروی محوری نانومیله از معادله حرکت آن استخراج شده و سپس با استفاده از تئوری الاستیسیته غیر محلی ارینگن، معادله نیروی محوری غیر محلی نانومیله به دست آمده است. پس از استخراج معادله نیروی محوری که شامل پارامترهای انرژی سطحی است، معادلات حرکت و نیروی محوری نانومیله را بر حسب امواج مثبت-رونده و منفی-رونده به دست آورده ایم. با به دست آوردن عدد موج و با قرار دادن فنر در مرزهای نانومیله، ماتریس انتشار موج تحت شرایط مرزی گیردار--گیردار و گیردار-آزاد به دست آمده است. در نهایت با محاسبه فرکانس بدون بعد نانومیله برای هر دو شرط مرزی ذکر شده ، تغییرات فرکانس بدون بعد بر حسب پارامترهای غیر محلی و همچنین بر حسب پارامترهای انرژی سطحی به صورت نمودار بررسی شده است. در هر یک از نمودارها سه مود اول فرکانس طبیعی بدون بعد نانومیله تحت ضرایب پواسون متفاوت بررسی شده است. در این مقاله به تحلیل ارتعاشات طولی آزاد نانومیله (نانولوله) یکنواخت در دستگاه مختصات استوانه ای، به روش موج پرداخته شده است. به این ترتیب، ابتدا معادله نیروی محوری نانومیله از معادله حرکت آن استخراج شده و سپس با استفاده از تئوری الاستیسیته غیر محلی ارینگن، معادله نیروی محوری غیر محلی نانومیله به دست آمده است. پس از استخراج معادله نیروی محوری که شامل پارامترهای انرژی سطحی است، معادلات حرکت و نیروی محوری نانومیله را بر حسب امواج مثبت-رونده و منفی-رونده به دست آورده ایم. با به دست آوردن عدد موج و با قرار دادن فنر در مرزهای نانومیله، ماتریس انتشار موج تحت شرایط مرزی گیردار--گیردار و گیردار-آزاد به دست آمده است. در نهایت با محاسبه فرکانس بدون بعد نانومیله برای هر دو شرط مرزی ذکر شده ، تغییرات فرکانس بدون بعد بر حسب پارامترهای غیر محلی و همچنین بر حسب پارامترهای انرژی سطحی به صورت نمودار بررسی شده است. در هر یک از نمودارها سه مود اول فرکانس طبیعی بدون بعد نانومیله تحت ضرایب پواسون متفاوت بررسی شده است.</p>
<p dir="LTR"> </p>
<p dir="LTR">تحلیل ارتعاشات طولی آزاد نانومیله بر اساس نظریه غیر محلی ارینگن با در نظر گرفتن انرژی سطحی به روش موج</p>
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<p dir="LTR"> </p>
<p dir="LTR"> </p>
<p dir="LTR"> </p>
<p dir="LTR"> </p>
<p dir="LTR"> </p>
<p dir="LTR"> </p>
<p dir="LTR"> </p>
<p dir="LTR"> </p>
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<p dir="LTR">1. S. Ijima , Helical microtubules of graphitic carbon, nature, Vol. 345, No.6348, pp. 56-58, 1991.</p>
<p dir="LTR">2. P. J. Harris, P. J. F. Harris, carbon nanotubes and related structures : new materials for the twenty-first century : Cambridge university press, 2001.</p>
<p dir="LTR">3. Yan, K., Choi, J., Kim, S.-K., Song O, Flow-indused vibration and stability analysis of multi-wall carbon nanotubes, Jornal of Mechanical Science and Technology, 26, 3911-20, 2012. doi: 10.1007/s12206-012-0888-3.</p>
<p dir="LTR">4. Aydogdu , M., Longitudinal wave propagation in multiwalled carbon nanotubes, Composite structures, 107 , 578-84, 2014. doi : 10.1016/j.compstruct.2013.08.031</p>
<p dir="LTR">5. Loghmani, M., Hairi Yazdi, M., R., Nikkhah Bahrami, M., Longitudinal vibration analysis of nanorods with multiple discontinuities based on nonlocal elasticity theory using wave approach, Technical paper. doi: 10.107/s00542-017-3619.</p>
<p dir="LTR">6. Arash, B., & Ansari, R. (2010). Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain. Physica E:Low dimensional system and nanostructures, 42, 2058-2064.</p>
<p dir="LTR">7. K. Kiani, Free longitudinal vibration of tapered nanowires in the context of nonlocal Continuum theory via a perturbation technique, Physica E: Low –dimensionless System and Nanostructures, 43 (2010) 387-397.</p>
<p dir="LTR">8. Danesh, M., Farajpour, A., and mohammadi, M. (2012) Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mechanics Research Communications. 39 (1) , 23-27.</p>
<p dir="LTR">9. T.Murmu, S.Adhikari, Nonlocal effects in the longitudinal vibration of double-nan- orod systems, Physica E: Low-dimensionless systems and nanostructures,43 (2010) 415-422.</p>
<p dir="LTR">10. Chang, W., J., Lee, H .-L., Vibration analysis of viscoelastic carbon nanotubes, Micro&Nano Letters, 7, 1308- 12, 2102. doi: 10.1049/mnl.2012.0612.</p>
<p dir="LTR">11. Reddy, J. N. (2007). Nonlocal theories for buckling bending and vibration of beams. International Jornal of Engineering Science, 45, 288-307.</p>
<p dir="LTR">12. Loya, J., Lopez-Puente, J., Zaera, R., and Fernandez-Saez, J. (2009) Free transverse Vibrations of cracked nanobeams using a nonlocal elasticity model. Jornal of Applied Physics. 105 (4), 044309.</p>
<p dir="LTR">13. Aydogdu, M. (2012) Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics. International Jornal of Engineering Science. 56 17-28.</p>
<p dir="LTR">14. Eringen, A. C. (1983). on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Jornal of Applied Physica, 54(9), 4703-4710.</p>