 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 13653 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8223
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| In English (Mech. Solids): | | 5430 |
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| << Previous article | Volume 60, Issue 8 / 2025 | Next article >> |
| S. Ziaee, "Campbell Diagrams of Size-Dependent Functionally Graded Rotating Shafts via Simple Strain-Velocity Gradient Theory," Mech. Solids. 60 (8), 7141-7175 (2025) |
| Year |
2025 |
Volume |
60 |
Number |
8 |
Pages |
7141-7175 |
| DOI |
10.1134/S0025654425606202 |
| Title |
Campbell Diagrams of Size-Dependent Functionally Graded Rotating Shafts via Simple Strain-Velocity Gradient Theory |
| Author(s) |
S. Ziaee (Department of Mechanical Engineering, Yasouj University, Yasouj, 75914-353 Iran, ziaee@yu.ac.ir) |
| Abstract |
A consistent higher-order continuum framework is developed within the context of Euler-Bernoulli and Timoshenko beam theories to analyze the vibratory behavior of spinning
micro/nanoshafts made of functionally graded materials. The pure strain-velocity gradient theory is
employed to capture the size-dependent effects. It is assumed that the material properties are varied
gradually in radial direction, while the potential formation of porosity during manufacturing is considered by incorporating an even-type porosity dispersion model into the property variation equations.
The governing equations of motion are derived by applying Hamilton’s principle in conjunction with
the Ritz method, resulting in a system of linear ordinary differential equations. To facilitate eigenfrequency analysis, the discretized equations are transformed into the state-space form. Following validation of the resulting equations and the proposed solution methodology, several illustrative examples
are provided to investigate the influence of various parameters—including material composition,
length-scale parameter ratio, porosity coefficient, aspect ratio, axial load, and spinning speed—on
both forward and backward natural frequencies. The findings demonstrate that decreasing the aspect
ratio and/or increasing the spinning speed of a simply-supported rotating shaft significantly accentuate the role of shear deformation in accurately predicting natural frequencies, particularly in higher-order modes. Furthermore, compared to simply-supported micro/nanoshafts, shear deformation
exerts a more critical influence on the precise prediction of forward and backward frequencies in fully-fixed, size-dependent spinning shafts. |
| Keywords |
Size-dependent rotating shaft, Campbell diagram, Pure strain-velocity gradient theory, Functionally graded material, Ritz method, Nonclassical Timoshenko beam theory |
| Received |
24 October 2025 | Revised |
08 December 2025 | Accepted |
09 December 2025 |
| Link to Fulltext |
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