Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model
Dynamic stability is an important mechanical behavior of nanobeams, which has been studied extensively using the Euler–Bernoulli and Timoshenko beam theories, while the Levinson-beam-theory-based dynamic instability analysis of nanobeams has not been investigated yet. Shear deformation is not or is...
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MDPI AG
2023-04-01
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| Series: | Materials |
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| Online Access: | https://www.mdpi.com/1996-1944/16/9/3404 |
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| author | Youqin Huang Richeng Huang Yonghui Huang |
| author_facet | Youqin Huang Richeng Huang Yonghui Huang |
| author_sort | Youqin Huang |
| collection | DOAJ |
| description | Dynamic stability is an important mechanical behavior of nanobeams, which has been studied extensively using the Euler–Bernoulli and Timoshenko beam theories, while the Levinson-beam-theory-based dynamic instability analysis of nanobeams has not been investigated yet. Shear deformation is not or is not suitably considered in the Euler–Bernoulli and Timoshenko theories, so it is very important to introduce the Levinson beam theory in the dynamic stability analysis of nanobeams, which correctly models the combined action of bending and shear in nanobeams with smaller length/height ratios. In this work, the equation of the transverse vibration of a Levinson beam embedded in an elastic foundation is firstly formulated based on the displacement field of Levinson beam theory, and the nonlocal theory is further applied to the Levinson nanobeam. Then, the governing equation of the dynamic stability of the Levinson nanobeam is derived using Bolotin’s method to achieve a generalized eigenvalue problem corresponding to the boundaries of regions of dynamic instability. The principal instability region (PIR) is the most important among all regions, so the boundary of the PIR is focused on in this work to investigate the dynamic stability of the Levinson nanobeam. When the width, length/height ratio, density, Young’s modulus, Poisson’s ratio, size scale parameter, and medium stiffness increase by about 1.5 times, the width of the PIR changes by about 19%, −57%, −20%, 65%, 0, −9%, and −11%, respectively. If a smaller critical excitation frequency and narrower width of the PIR correspond to the better performance of dynamic stability, the study shows that the dynamic stability of the Levinson nanobeam embedded in an elastic medium improves under a larger length and density and a smaller width, height, and Young’s modulus, since these factors are related to the natural frequency of the nanobeam which controls the width of the PIR. Additionally, the local model would overestimate the dynamic stability behavior of the Levinson nanobeam. |
| first_indexed | 2024-03-11T04:13:51Z |
| format | Article |
| id | doaj.art-1d8748b763e9459dbcf56ecd9798863c |
| institution | Directory Open Access Journal |
| issn | 1996-1944 |
| language | English |
| last_indexed | 2024-03-11T04:13:51Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Materials |
| spelling | doaj.art-1d8748b763e9459dbcf56ecd9798863c2023-11-17T23:15:37ZengMDPI AGMaterials1996-19442023-04-01169340410.3390/ma16093404Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam ModelYouqin Huang0Richeng Huang1Yonghui Huang2Research Centre for Wind Engineering and Engineering Vibration, Guangzhou University, Guangzhou 510006, ChinaResearch Centre for Wind Engineering and Engineering Vibration, Guangzhou University, Guangzhou 510006, ChinaResearch Centre for Wind Engineering and Engineering Vibration, Guangzhou University, Guangzhou 510006, ChinaDynamic stability is an important mechanical behavior of nanobeams, which has been studied extensively using the Euler–Bernoulli and Timoshenko beam theories, while the Levinson-beam-theory-based dynamic instability analysis of nanobeams has not been investigated yet. Shear deformation is not or is not suitably considered in the Euler–Bernoulli and Timoshenko theories, so it is very important to introduce the Levinson beam theory in the dynamic stability analysis of nanobeams, which correctly models the combined action of bending and shear in nanobeams with smaller length/height ratios. In this work, the equation of the transverse vibration of a Levinson beam embedded in an elastic foundation is firstly formulated based on the displacement field of Levinson beam theory, and the nonlocal theory is further applied to the Levinson nanobeam. Then, the governing equation of the dynamic stability of the Levinson nanobeam is derived using Bolotin’s method to achieve a generalized eigenvalue problem corresponding to the boundaries of regions of dynamic instability. The principal instability region (PIR) is the most important among all regions, so the boundary of the PIR is focused on in this work to investigate the dynamic stability of the Levinson nanobeam. When the width, length/height ratio, density, Young’s modulus, Poisson’s ratio, size scale parameter, and medium stiffness increase by about 1.5 times, the width of the PIR changes by about 19%, −57%, −20%, 65%, 0, −9%, and −11%, respectively. If a smaller critical excitation frequency and narrower width of the PIR correspond to the better performance of dynamic stability, the study shows that the dynamic stability of the Levinson nanobeam embedded in an elastic medium improves under a larger length and density and a smaller width, height, and Young’s modulus, since these factors are related to the natural frequency of the nanobeam which controls the width of the PIR. Additionally, the local model would overestimate the dynamic stability behavior of the Levinson nanobeam.https://www.mdpi.com/1996-1944/16/9/3404dynamic instabilitynanobeamsLevinson beamelastic mediumBolotin’s methodparametric analysis |
| spellingShingle | Youqin Huang Richeng Huang Yonghui Huang Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model Materials dynamic instability nanobeams Levinson beam elastic medium Bolotin’s method parametric analysis |
| title | Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model |
| title_full | Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model |
| title_fullStr | Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model |
| title_full_unstemmed | Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model |
| title_short | Investigation into the Dynamic Stability of Nanobeams by Using the Levinson Beam Model |
| title_sort | investigation into the dynamic stability of nanobeams by using the levinson beam model |
| topic | dynamic instability nanobeams Levinson beam elastic medium Bolotin’s method parametric analysis |
| url | https://www.mdpi.com/1996-1944/16/9/3404 |
| work_keys_str_mv | AT youqinhuang investigationintothedynamicstabilityofnanobeamsbyusingthelevinsonbeammodel AT richenghuang investigationintothedynamicstabilityofnanobeamsbyusingthelevinsonbeammodel AT yonghuihuang investigationintothedynamicstabilityofnanobeamsbyusingthelevinsonbeammodel |