Investigation of Crack Effects on the Nonlinear Vibrations of Microbeams with a Tip Mass in a Magnetic Field

Yıl 2025, Cilt: 27 Sayı: 79, 54 - 61, 23.01.2025

Duygu Atcı

https://doi.org/10.21205/deufmd.2025277908

Öz

Microelectromechanical systems (MEMS) are critical members of modern technological devices, due to their applications in various industrial fields. In the physical applications of MEMS, cracks are a common structural problem, affecting the static and dynamic behavior of the system. In this paper, the effects of cracks on microbeams with a tip mass under the influence of a magnetic field have been investigated. The micro-size effect of the beam has been involved into the model by using the modified couple stress theory. The crack has been modeled by using a torsional spring, with the spring coefficient corresponds to the severity of the crack. Thus, the beam has been modeled as consisting of two segments connected by a torsional spring. The equations of motion have been formulated using Hamilton’s principle. The obtained equations have been solved by using the method of multiple scales, a perturbation technique. Frequencies regarding both linear and nonlinear vibrations of the microbeams have been examined. The results obtained in this study have been validated by using available numerical results in the literature. The effects of parameters such as crack severity, crack location, tip mass and the magnetic field force on linear and nonlinear vibrations have been presented. The results indicate a significant decrease in the natural frequencies and nonlinear frequencies of microbeams with increasing crack severity.

Anahtar Kelimeler

Nonlinear Vibrations, Cracked Microbeams, Perturbation Techniques, Modified Couple Stress Theory, Method of Multiple Scales

Etik Beyan

This article does not require ethics committee approval.

Uç Kütlesi Bulunan Mikrokirişlerin Manyetik Alan Altında Lineer Olmayan Titreşimlerinde Çatlakların Etkilerinin İncelenmesi

Öz

Mikroelektromekanik sistemler (MEMS), birçok endüstriyel alana uygulanabilmeleri sayesinde, modern teknolojik cihazların önemli elemanlarından biri haline gelmiştir. MEMS’ in fiziksel uygulamalarında sıklıkla ortaya çıkan çatlaklar, sistemin statik ve dinamik davranışlarını etkilemektedir. Bu makalede, manyetik alan etkisi altındaki, uç kütleye sahip mikrokirişler üzerindeki çatlakların etkileri incelenmiştir. Kirişin mikro boyut etkisi, değiştirilmiş çift gerilme teorisi kullanılarak modele dahil edilmiştir. Çatlak, bir burulma yayı kullanılarak modellenmiştir ve burulma yay katsayısı çatlak şiddetine karşılık gelmektedir. Böylece kiriş, burulma yayı aracılığı ile birbirine bağlı iki kısımdan oluşacak şekilde modellenmiştir. Hareket denklemleri Hamilton prensibi uygulanarak oluşturulmuştur. Elde edilen denklemler, bir perturbasyon yöntemi olan çok ölçekli metot kullanılarak çözülmüştür. Mikrokirişlerin hem lineer hem de nonlineer titreşimlerine ilişkin frekansları incelenmiştir. Bu çalışmada elde edilen sonuçlar, literatürde bulunan mevcut sayısal sonuçlar kullanılarak doğrulanmıştır. Çatlak şiddeti, çatlak konumu, uç kütle ve manyetik alan kuvveti gibi parametrelerin lineer ve nonlineer titreşimler üzerindeki etkileri sunulmuştur. Elde edilen sonuçlar, mikrokirişlerin doğal frekansları ve nonlineer frekanslarında, çatlak şiddetinin artmasıyla birlikte önemli ölçüde düşüş olduğunu göstermektedir.

Anahtar Kelimeler

Nonlineer Titreşimler, Çatlak İçeren Mikrokirişler, Perturbasyon Teknikleri, Değiştirilmiş Çift Gerilme Teorisi, Çok Ölçekli Metot

Kaynakça

• [1] Fleck, N. A., Hutchinson, J. W. 1993. A Phenomenological Theory for Strain Gradient Effects In Plasticity. Journal of the Mechanics and Physics of Solids, Vol. 41(12), pp. 1825-1857. DOI: https://doi.org/10.1016/0022-5096(93)90072-N
• [2] Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., Tong, P. 2003. Experiments and Theory in Strain Gradient Elasticity. Journal of Mechanics and Physics of Solids, Vol. 51(8), pp. 1477-1508. DOI: 10.1016/S0022-5096(03)00053-X
• [3] Yang, F., Chong, A. C. M., Lam, D. C. C., Tong, P. 2002. Couple Stress Based Strain Gradient Theory for Elasticity. International Journal of Solids and Structures, Vol. 39(10), pp. 2731-2743. DOI: 10.1016/S0020-7683(02)00152-X
• [4] Park, S. K., Gao, X. L. 2006. Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory. Journal of Micromechanics and Microengineering, Vol. 16, pp. 2355-2359. DOI: 0.1088/0960-1317/16/11/015
• [5] Ma, H. M., Gao, X. L., Reddy, J. N. 2008. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory. Journal of Mechanics and Physics of Solids, Vol. 56(12), pp. 3379-3391. DOI: 10.1016/j.jmps.2008.09.007
• [6] Kong, S., Zhou, S., Nie, Z., K. Wang. 2008. The Size-Dependent Natural Frequency of Bernoulli–Euler Micro-Beams. International Journal of Engineering Science, Vol. 46(5), pp. 427-437. DOI: 10.1016/j.ijengsci.2007.10.002
• [7] Reddy, J. N. 2011. Microstructure-Dependent Couple Stress Theories of Functionally Graded Beams. Journal of Mechanics and Physics of Solids, Vol. 59, pp. 2382-2399. DOI: 10.1016/j.jmps.2011.06.008
• [8] Asghari, M., Ahmadian, M. T., Kahrobaiyan, M. H., Rahaeifard, M. 2010. On The Size-Dependent Behavior of Functionally Graded Micro-Beams. Materials and Design, Vol. 31, pp. 2324-2329. DOI: https://doi.org/10.1016/j.matdes.2009.12.006
• [9] Şimşek, M., Aydın, M., Yurtcu, H. H., Reddy, J. N. 2015. Size-Dependent Vibration of a Microplate Under the Action of a Moving Load Based on the Modified Couple Stress Theory. Acta Mechanica, Vol. 226, pp. 3807-3822. DOI: 10.1007/s00707-015-1437-9
• [10] Wang, Y. G., Lin, W. H., Liu, N. 2013. Nonlinear Free Vibration of a Micro Scale Beam Based on Modified Couple Stress Theory. Physica E Low Dimensional Systems and Nanostructures, Vol. 47, pp. 80-85. DOI: 10.1016/j.physe.2012.10.020
• [11] Ghayesh, M. H., Farokhi, H., Amabili, M. 2013. Nonlinear Dynamics Of A Microscale Beam Based on the Modified Couple Stress Theory. Composites Part B, Vol. 50, pp. 318-324. DOI: https://doi.org/10.1016/j.compositesb.2013.02.021
• [12] Hieu, D. V., Hoa, N. T., Duy, L. Q., Thoa, N. T. K. 2021. Nonlinear Vibration of an Electrostatically Actuated Functionally Graded Microbeam under Longitudinal Magnetic Field. Journal of Applied and Computational Mechanics, Vol. 7(3), pp. 1537-1549. DOI: 10.22055/JACM.2021.35504.2670
• [13] Bağdatlı, S. M., Togun, N., Yapanmış, B. E., Akkoca, Ş. 2023. Nonlinear Vibration Of Microbeams Subjected To A Uniform Magnetic Field And Rested On Nonlinear Elastic Foundation. Zeitschrift für Naturforschung A, Vol. 79(1), pp. 17-30. DOI: 10.1515/zna-2023-0225
• [14] Togun, N., Bağdatlı, S. M. 2016. Size Dependent Nonlinear Vibration Of The Tensioned Nanobeam Based On The Modified Couple Stress Theory. Composites Part B-Engineering, Vol. 97, pp. 255-262. DOI: https://doi.org/10.1016/j.compositesb.2016.04.074
• [15] Atcı, D. 2021. Free Vibrations of Nanobeams under Non-Ideal Supports Based on Modified Couple Stress Theory. Zeitschriftt für Naturforschung A, Vol. 76(5), pp. 427-434. DOI: 10.1515/zna-2020-0335
• [16] Loya, J., Lopez-Puente, J., Zaera, R., Fernandez-Saez, J. 2009. Free Transverse Vibrations of Cracked Nanobeams Using a Nonlocal Elasticity Theory. Journal of Applied Physiscs, Vol. 105(4), pp. 044309-9. DOI: 10.1063/1.3068370
• [17] Tadi Beni, Y., Jafari, A., Razavi, H. 2015. Size Effect on Free Transverse Vibration of Cracked Nano-Beams Using Couple Stress Theory. International Journal of Engineering, Vol. 28(2), pp. 296-304. DOI: 10.5829/idosi.ije.2015.28.02b.17
• [18] Hsu, J. C., Lee, H. L., Chang, W. L. 2011. Longitudinal Vibration of Cracked Nanobeams Using Nonlocal Elasticity Theory. Current Applied Physics, Vol. 11(6), pp. 1384-1388. DOI: 10.1016/j.cap.2011.04.026
• [19] Akbaş, Ş. D. 2018. Forced Vibration Analysis of Cracked Nanobeams. Journal of Brazilian Society of Mechanical Science and Engineering, Vol. 40, pp. 392. DOI: 10.1007/s40430-018-1315-1
• [20] Roostai, H., Haghpanahi, M. 2014. Vibration of Nanobeams of Different Boundary Conditions with Multiple Cracks Based on Nonlocal Elasticity Theory. Applied Mathematical Modelling, Vol. 38(3), pp. 1159-1169. DOI: 10.1016/j.apm.2013.08.011
• [21] Larkin, K., Ghommem, M., Hunter, A., Abdelkefi, A. 2020. Nonlinear Modeling and Performance Analysis of Cracked Beam Microgyroscopes. International Journal of Mechanical Science, Vol. 188, pp. 105965. DOI: https://doi.org/10.1016/j.ijmecsci.2020.105965
• [22] Rahi, A., Petoft, H. 2018. Free Vibration Analysis of Multi-Cracked Micro Beams Based on Modified Couple Stress Theory. Journal of Theoretical and Applied Vibration and Acoustics, Vol. 4(2), pp. 205-222. DOI: 10.22064/TAVA.2019.89997.1113
• [23] Esen, İ., Özarpa, C., Eltaher, M. A. 2021. Free Vibration of a Cracked FG Microbeam Embedded in an Elastic Matrix and Exposed to Magnetic Field in a Thermal Environment. Composite Structures, Vol. 261, pp. 113552. DOI: 10.1016/j.compstruct.2021.113552
• [24] Eghbali, M., Hosseini, S. A., Pourseifi, M. 2022. Free Transverse Vibrations Analysis of Size-Dependent Cracked Piezoelectric Nano-Beam Based on the Strain Gradient Theory under Mechanic-Electro Forces. Engineering Analysis with Boundary Elements, Vol. 143, pp. 606-612. DOI: 10.1016/j.enganabound.2022.07.006
• [25] Nayfeh, A. H. 1981 Introduction to Perturbation Techniques. John Wiley, New York, NY, USA. ISBN: 0-471-39917-5.