Abstract <p>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.</p>

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Campbell Diagrams of Size-Dependent Functionally Graded Rotating Shafts via Simple Strain-Velocity Gradient Theory

  • S. Ziaee

摘要

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.