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<p>This paper presents a comprehensive investigation into the dynamic behavior of rotating Porous Functionally Graded Material (PFGM) shafts using the Timoshenko beam theory, incorporating rotary inertia and gyroscopic effects while undergoing continuous rotation at a constant velocity. Stress–strain relations, along with strain and kinetic energy equations, have been derived to analyze the mechanical behavior under various conditions. A hierarchical beam element formulation discretizes the spinning flexible shaft. The seventh-order shape functions demonstrate stability in the convergence study. Validation against existing literature confirms the accuracy and robustness of the proposed model, using materials such as stainless steel and nickel. A parametric study is performed to illustrate the effects of porosity levels on graded index values, different boundary conditions, and some geometric parameters on natural frequencies. The results reveal that the steady-state natural frequencies decrease with increasing porosity, while for higher FGM indices, they notably increase. Furthermore, boundary conditions significantly affect the vibration behavior. Natural frequencies are analyzed in Campbell’s diagrams to explore the effects of various geometric parameters on PFGM rotating systems such&#xa0;as the thickness ratio <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40997_2025_866_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\((e/D)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">/</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the slenderness ratio <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40997_2025_866_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\((L/D)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">/</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, considering both forward and backward modes. The novelty of this work lies in its investigation of the first critical speed for rotating PFGM shafts using modern simulation methods to achieve more accurate results under various boundary conditions, different geometric parameters, and types of FGMs. These findings contribute significantly to the design and optimization of PFGM rotor systems, emphasizing the interplay between material properties, geometric parameters, and boundary conditions.</p>

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Dynamic Behavior Analysis of a Shear Deformable Rotating Porous FGM Shaft Using \(p\)-FEM

  • 期刊论文

  • 发表日期：

    2025年5月5日

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  • Ahmed Mezrag,
  • Abdelkrim Boukhalfa

摘要

This paper presents a comprehensive investigation into the dynamic behavior of rotating Porous Functionally Graded Material (PFGM) shafts using the Timoshenko beam theory, incorporating rotary inertia and gyroscopic effects while undergoing continuous rotation at a constant velocity. Stress–strain relations, along with strain and kinetic energy equations, have been derived to analyze the mechanical behavior under various conditions. A hierarchical beam element formulation discretizes the spinning flexible shaft. The seventh-order shape functions demonstrate stability in the convergence study. Validation against existing literature confirms the accuracy and robustness of the proposed model, using materials such as stainless steel and nickel. A parametric study is performed to illustrate the effects of porosity levels on graded index values, different boundary conditions, and some geometric parameters on natural frequencies. The results reveal that the steady-state natural frequencies decrease with increasing porosity, while for higher FGM indices, they notably increase. Furthermore, boundary conditions significantly affect the vibration behavior. Natural frequencies are analyzed in Campbell’s diagrams to explore the effects of various geometric parameters on PFGM rotating systems such as the thickness ratio \((e/D)\) ( e / D ) and the slenderness ratio \((L/D)\) ( L / D ) , considering both forward and backward modes. The novelty of this work lies in its investigation of the first critical speed for rotating PFGM shafts using modern simulation methods to achieve more accurate results under various boundary conditions, different geometric parameters, and types of FGMs. These findings contribute significantly to the design and optimization of PFGM rotor systems, emphasizing the interplay between material properties, geometric parameters, and boundary conditions.