Free Vibration Analysis of Rotating Functionally Graded Material Beams on Elastic Foundations Using the Homotopy Perturbation Method

Consider a rotating FGM beam of rectangular cross-section, positioned on an elastic foundation and rotating synchronously with the foundation, as illustrated in the Figure 1. A Cartesian coordinate system is adopted, where the x-axis aligns with the beam’s longitudinal (axial) direction, and the z-axis corresponds to the transverse (thickness) direction. The system rotates about the y-axis at a constant angular velocity Ω. The beam has length L, height h, and width b, while the elastic foundation is characterized by its Winkler elastic foundation K. The beam exhibits a continuous material gradation along its thickness (z-axis), transitioning from a fully ceramic upper surface (z=h/2) to a fully metallic lower surface (z=−h/2). Material properties, including elastic modulus E, Poisson’s ratio ν, and mass density ρ, vary as smooth functions of the thickness coordinate z. These graded properties are governed by a power-law distribution based on the rule of mixtures, expressed as [18-24]:

$P(z)=P_c\left(\frac{z+\frac{h}{2}}{h}\right)^n+P_m\left[1-\left(\frac{z+\frac{h}{2}}{h}\right)^n\right]$             (1)

where, P_c and P_m denote the material properties of the ceramic and metal constituents, respectively, and n represents the volume fraction exponent governing the gradation profile.

Eq. (1) represents the properties of the ceramic and metallic constituents, respectively, and n is the gradient index. The power-law distribution used to describe the material properties of the functionally graded beam guarantees a smooth variation of properties like density and Young's modulus through the thickness. How quickly the material changes through the thickness are determined by the gradient index n. A fully ceramic beam with maximum stiffness and minimum density is produced by n=0, whereas a fully metallic beam with lower stiffness and generally greater ductility is produced by n→∞. Intermediate values of n result in smooth property gradations and enable customized stiffness-to-weight ratios.

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Figure 1. Schematic configuration and coordinate system of a dynamically rotating functionally graded beam resting on a Winkler-type elastic foundation [23]

The governing equations of the rotating FG beam result from an application of Hamilton’s principle, which provides an association of strain energy U, kinetic energy T, and work done by external forces W, within the dynamics of the system. The variational formulation of dynamic equilibrium is obtained by minimizing the action integral from time t₁ to t₂ [24].

$\delta \int_{t_1}^{t_2}(T-U+W) d t=0$             (2)

where, δ is the variational operator, and t₁ and t₂ are the initial and final times of motion, respectively.

Let u(x, t) and w(x, t) denote axial and transverse displacements, respectively, of a point on the neutral axis of the beam. Following Euler-Bernoulli beam theory, displacement at any (x, z) in the cross-section is expressed as:

$u_x(x, t)=u(x, t)-z \frac{\partial w(x, t)}{\partial x}, u_z(x, t)=w(x, t)$                (3)

The strain energy consists of contributions related to bending deformation, centrifugal stiffening, and to the Winkler foundation's elasticity:

$U=\frac{1}{2} \int_0^l\left[A_1\left[\frac{\partial u}{\partial x}\right]^2-2 B_1 \frac{\partial u}{\partial x} * \frac{\partial^2 w}{\partial x^2}+D_1\left[\frac{\partial^2 w}{\partial x^2}\right]^2\right] d x$            (4)

Kinetic energy includes both translational inertia and rotational inertia:

$\begin{array}{r}T=\frac{1}{2} \int_0^l\left[\left[I_1\left[\frac{\partial u}{\partial t}\right]^2+\left[\frac{\partial w}{\partial t}\right]^2 * \frac{\partial^2 w}{\partial x^2}+x^2 \Omega^2\right]\right. \left.-2 I_2 \frac{\partial^2 w}{\partial x \partial t} * \frac{\partial u}{\partial t}+I_3\left[\frac{\partial^2 w}{\partial x \partial t}\right]^2\right] d x\end{array}$              (5)

The external work arises from transverse mechanical loads:

$W=-\frac{1}{2} \int_0^l\left[b K w^2+F(x)\left[\frac{\partial u}{\partial x}\right]^2\right]$               (6)

The terms are defined in the above Eqs. (4)-(6) as follows:

$\begin{gathered}I_1=\rho_m A N_1, I_2=\rho_m h^2 N_2 \\ I_3=\rho_m I N_3, I=\frac{b h^3}{12}, A=b h\end{gathered}$              (7)

The governing equation incorporates centrifugal effects and stiffness coupling through the coefficients:

The centrifugal force distribution represents:

$F(x)=\rho(z) A \Omega^2\left(L^2-x^2\right)$            (8)

The tensile stiffness coefficient is:

$A_1=\int_{\frac{-h}{2}}^{\frac{h}{2}} E(z) d z$               (9)

The tensile-bending coupling accounts:

$B_1=\int_{\frac{-h}{2}}^{\frac{h}{2}} E(z) z d z$               (10)

The bending stiffness coefficient defines:

$D_1=\int_{\frac{-h}{2}}^{\frac{h}{2}} E(z) z^2 d z$           (11)

These parameters summarize the coupling of rotational dynamics to the material properties of the functionally graded materials. Dimensionless coefficients result from the nondimensionalization of governing equations, where significant parameters get normalized by quantities (for example, beam length L, reference stiffness E_c, and material density ρ_c). The formulation of these coefficients is expressed as follows:

$N_1=\frac{n+\frac{\rho_c}{\rho_m}}{n+1}$          (12)

$N_2=\frac{-\frac{\rho_c}{\rho_m} n+n}{2\left(n^2+3 n+2\right)}$            (13)

$N_3=1+\frac{3\left(\frac{\rho_c}{\rho_m}\right)\left(n^2+n+2\right)}{\left(n^3+6 n^2+11 n+6\right)}$            (14)

$\emptyset_1=\frac{n+\frac{E_c}{E_m}}{n+1}$             (15)

$\emptyset_2=\frac{-\frac{E_c}{E_m} n+n}{2\left(n^2+3 n+2\right)}$             (16)

$\emptyset_3=1+\frac{3\left(\frac{E_c}{E_m}\right)\left(n^2+n+2\right)}{\left(n^3+6 n^2+11 n+6\right)}$              (17)

By introducing constitutive relations defined in Eqs. (4)-(6) into the formulation as defined in Eq. (2) and not considering longitudinal and rotational components of inertia associated with lateral bending dynamics, the governing differential equation for transverse free vibration of a rotating beam supported by an elastic foundation is expressed as:

$A_1 \frac{\partial^2 u}{\partial x^2}-B_1 \frac{\partial^3 w}{\partial x^3}$             (18)

$B_1 \frac{\partial^3 u}{\partial x^3}-D_1 \frac{\partial^4 w}{\partial x^4}-I_1 \frac{\partial^2 w}{\partial t^2}+\frac{\partial}{\partial x}\left[F(x) \frac{\partial w}{\partial x}\right]-b K w=0$              (19)

By algebraic manipulations of Eqs. (18) and (19), axial displacement component u is eliminated, and an equation governing the partial differential of the transverse motion of w(x, t) is established. This simplification isolates, as it were, the transverse vibrational behavior of the rotating beam to obtain:

$\left[D_1-\frac{B_1}{A_1}\right] \frac{\partial^4 w}{\partial x^4}+I_1 \frac{\partial^2 w}{\partial t^2}-\frac{\partial}{\partial x}\left[F(x) \frac{\partial w}{\partial x}\right]+b K w=0$            (20)

The simple harmonic vibration of the FGM beam can be expressed as:

$w(x, t)=\bar{W}(x) e^{i \omega t}$               (21)

where, $\bar{W}(x)$ is the function of model, $i$ is the imaginary unit, $\omega$ is the natural frequency. the following dimensionless transformation is introduced:

$\left.\begin{array}{c}W=\frac{\bar{W}}{l} \\ \zeta=\frac{x}{l} \\ \eta=\sqrt{\frac{\Omega^2 \rho_m l^4}{E_m h^2}} \\ \lambda=\frac{K l^4}{E_m h^3} \\ \mu=\sqrt{\frac{\omega^2 l^4 \rho_m}{E_m h^2}}\end{array}\right\}$               (22)

where, $\mu$ is the dimensionless of natural frequency, $\lambda$ is the dimensionless elastic foundation modulus, $\eta$ is the dimensionless speed.

The dimensionless rotation speed $\eta$, which takes into account centrifugal stiffening from rotation, the dimensionless foundation stiffness $\lambda$, which indicates the Winkler foundation's stiffness in relation to the beam's bending rigidity, and the dimensionless natural frequency $\mu$, which normalizes the frequency regardless of absolute size or material scale, are introduced in order to generalize the governing equations and make parametric analysis easier. These variables enable a methodical investigation of the effects of rotation, foundation stiffness, and material gradation.

Consequently, the dimensionless governing differential equation for the transverse free vibration of a rotating FGM beam resting on an elastic foundation is derived through nondimensionalization, incorporating centrifugal stiffening effects, foundation interaction, and material property gradation. This equation is expressed as:

$L(W)+N(W)=0$              (23)

$L(W)=\left[\frac{\emptyset_3}{12}-\frac{\emptyset_2^2}{\emptyset_1}\right] \frac{d^4 W}{d \zeta^4}-\mu_o N_1 W$             (24)

$\begin{gathered}N(W)=-\eta^2 N_1\left(\frac{1}{2}-\frac{\zeta^2}{2}\right) \frac{d^2 W}{d \zeta^2} +\eta^2 N_1 \zeta \frac{d W}{d \zeta}+\lambda W-\left(\mu^2-\mu_o^2\right) N_1 W\end{gathered}$             (25)

$\begin{aligned} {\left[\frac{\emptyset_3}{12}-\frac{\emptyset_2^2}{\emptyset_1}\right] \frac{d^4 W}{d \zeta^4} } & -\mu_o N_1 W+-\eta^2 N_1\left(\frac{1}{2}-\frac{\zeta^2}{2}\right) \frac{d^2 W}{d \zeta^2}  +\eta^2 N_1 \zeta \frac{d W}{d \zeta}+\lambda W  -\left(\mu^2-\mu_o^2\right) N_1 W=0\end{aligned}$             (26)

The boundary conditions for the rotating FGM beam on an elastic foundation are restricted to the most prevalent configuration in engineering applications:

• C-C boundary conditions
• At both ends $(\zeta=0$ and $\zeta=1)$ :

• ${{W}}(\zeta)=0$, and $\frac{d W}{d \zeta}=0$

• C-S boundary conditions
• At clamped end $\zeta=0$

• ${{W}}(\zeta)=0$, and $\frac{d W}{d \zeta}=0$

• At simply supported end $\zeta$ =1

• ${{W}(\zeta)=0, \text { and } \frac{d^2 W}{d \zeta^2}=0}$

• C-F boundary conditions
• At clamped end $\zeta=0$

• ${{W}}(\zeta)=0$, and $\frac{d W}{d \zeta}=0$

• At free end $\zeta=1$

• $\frac{d^2 W}{d \zeta^2}=0$, and $\frac{d^3 W}{d \zeta^3}=0$

The computation required in the investigation resulted in the development and application of an advanced MATLAB computational portal, which was specifically created to solve the eigenvalue problem using the HPM. Such numerical platform facilitated an extensive investigation of rotating FGM beams resting on an elastic foundation in terms of dimensionless natural frequencies, denoted as μ. The study included variation in gradient indices (n) and investigated three different boundary conditions, namely, C-C, C-S, and C-F. The generated results provided meaningful insights into dynamic response behavior of C-FGM beam modeling, where it showed significant dependencies of natural frequencies on material gradient distributions as well as specific boundary configurations. In addition, an effective demonstration of HPM in exacerbating geometrical and material complexities of eigenvalue problems, as well as quantitative insights into rotational dynamics, foundation elasticity, and graded material properties, were provided by computational outcomes. The findings highlight the efficiency of proposed method in revealing slight variations of frequencies in various mechanical as well as boundary environments, thus promoting further development of analytical methodologies for multibody dynamics in functionally graded systems.

Table 2. First dimensionless natural frequency μ₁ for C-C boundary conditions (ƞ=5, λ=100)

Table 3. First dimensionless natural frequency μ₁ for C-S boundary conditions (ƞ=5, λ=100)

Table 4. First dimensionless natural frequency μ₁ for C-F boundary conditions (ƞ=5, λ=100)

To rigorously validate the numerical precision of the proposed model and evaluate the methodological efficacy, Tables 2-4 present a comparative analysis of the first-order dimensionless natural frequencies (denoted as μ₁) for an FGM beam in a non-rotating configuration, operating in the absence of an elastic foundation under different boundary conditions (C-C, C-S, C-F), across varying gradient indices (n). The results obtained based on this research exhibit significant concordance with standard results obtained based on two well-developed computational methods: Pradhan and Chakraverty's [18] finite element method and Şimşek's [19] Lagrange multiplier method. The fact that these results from multiple analytical and numerical methods converge shows high confidence in the current model's robustness and further verifies the accuracy of the computational methods.

Figures 2-4 induced the first five dimensionless natural frequencies (μ) of rotating FGM beams on elastic foundations under C-C, C-S, and C-F boundaries. The frequencies decrease as the gradient index (n) increases due to the transition from stiff ceramic (n=0) to flexible metal (n=∞). The higher modes (e.g., μ5) feature more sensitivity to n compared to the lower modes (μ1), as reducing the stiffness increases curvature-dependent energy dissipation. The stiffness is determined by boundary conditions: C-C > C-S > C-F, where frequencies for C-C achieve levels 4 to 5 times larger than those for C-F (e.g., μ1=12.43 for C-C vs. μ1=3.52 for C-F when n=0). The results confirm those of FEM/DTM, thus establishing the accuracy of HPM for material gradients and complex boundaries.

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Figure 2. First five dimensionless natural frequencies μ vs. n (C-C boundary, η=5, λ=100)

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Figure 3. First five dimensionless natural frequencies μ vs. n (C-S boundary, η=5, λ=100)

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Figure 4. First five dimensionless natural frequencies μ vs. n (C-S boundary, η=5, λ=100)

The dimensionless natural frequencies (μ) of rotating FGM beams exhibit a pronounced dependence on the material gradient index (n) and kinematic boundary constraints. A systematic reduction in n, corresponding to ceramic-rich compositions with elevated structural rigidity (E_c $\gg$ E_m), where E_c and E_m denote Young’s moduli of ceramic and metal phases, respectively, induces a marked increase in natural frequencies (μ) due to enhanced elastic stiffness). Conversely, high values of n (representing metal-dominated gradations) lead to a decrease in rigidity, thereby decreasing μ. For instance, for C-C boundary conditions (see Figure 2), the first-mode natural frequency (μ₁) decreases from 12.43 (when n = 0, corresponding to purely ceramic) to 6.46 (as n → ∞, corresponding to purely metallic), indicating a 48% drop in stiffness. This trend is followed for different modes and is tabulated systematically in Table 5. The behavior is also affected by the particular boundary condition: C-C setups, which provide full rotational and translational constraints, increase the structural rigidity, leading to the highest frequencies (e.g., μ₁ = 12.43 at n = 0). In contrast, C-F boundaries, being compliant at the free end, provide the lowest frequencies (μ₁ = 3.52 at n = 0), while C-S setups give intermediate frequencies due to the partial rotational freedom at the supported end.

The higher modes, e.g., μ₅, show increased sensitivity to the parameter n due to an inverse relationship between their frequencies and localized curvature distributions, which are strongly dependent on stiffness gradients in beams. The results validate the classical beam theory principles, where natural frequencies are proportional to √(E/ρ), where E is the effective elastic modulus and ρ is density. The computational robustness of the HPM is rigorously validated against FEM, with discrepancies confined to less than 2% (e.g., μ₁ = 9.57 via HPM vs. 9.55 via FEM for C-C, n = 1).

The vibrational response of FGM beams is thus governed by the synergistic interplay of material gradation (n) and boundary constraints. Compositions with mostly ceramics (n → 0) tend to increase both stiffness and natural frequencies when coupled with fully clamped boundaries, while metal-dominant gradations (n → ∞) with free-end conditions minimize them. These findings underscore the crucial role of tailor-made material designs and constraint engineering in optimizing the dynamic behavior of FGMs, offering useful guidelines for advanced mechanical design in applications demanding precise frequency tuning.

Table 5. Statistical comparison of frequency reductions (%)

The sub-2% deviation between HPM and FEM or Lagrange benchmarks not only corroborates the mathematical fidelity of the proposed formulation but also highlights HPM’s computational efficiency in resolving eigenvalue problems for complex, inhomogeneous structures, a pivotal advantage for iterative design workflows. These results collectively establish a foundational framework for predicting and manipulating the dynamic response of FGMs, bridging theoretical beam dynamics with application-driven material engineering.

Figures 5-7 confirm that rotating functionally graded material (FGM) beams' dimensionless natural frequencies (μ) experience substantial increases as a function of foundation modulus (λ) that apply to all three boundary classes (C-C, C-S, C-F). This increase is due to added transverse stiffness of the foundation, which combats bending of the beam and hence increases its natural frequencies of structure. The lowest modes (μ1, μ2) experience the most significant percentage increases; e.g., μ1 for C-F increases by 172.9% as λ varies from 0 to 500 due to global bending dependency that directly includes distributed stiffness of foundation, as explored in Table 6. In contrast, higher modes (μ3-μ5) experience smaller but still significant increases based on local deformation, while μ5 for C-F increases by 101.2%. Boundary conditions play an important role in specifying the scope of these: the C-C boundary, which features fully fixed ends, features highest absolute frequencies (e.g., μ1 =23.45 for λ =500) but relatively lesser percentage increases (+145.0% for μ1) compared to the C-F boundary (+172.9%), where baseline stiffness of free end enhances magnitude of foundation's influence. The C-S boundary, which shows in-between stiffness, enjoys moderate increases (+148.4% for μ1). They affirm results carried out using FEM and HPM while keeping errors within 4%. They highlight basic significance of foundation to enhance dynamics, especially for flexible systems like cantilevers, and illustrate the interactive sensitivity of material gradation, boundary restraints, and flexural support in design considerations, which is of interest to practicing engineers.

The observed trends in natural frequencies with respect to the gradient index $n$ and elastic foundation modulus $\lambda$ can be explained by examining the governing dimensionless differential equation, Eq. (23), which includes the stiffness term $E I(x)$ and the foundation contribution $\lambda_w$. The material distribution moves towards the metallic phase, which has a lower elastic modulus $E_m$, as the gradient index $n$ increases. As a result, the effective stiffness $E I$ decreases, resulting in lower natural frequencies since Eq. (39); on the other hand, an increase in the elastic foundation modulus $\lambda$ introduces an additional restoring force term $\lambda_w$, effectively stiffening the system and raising the natural frequencies, especially for lower-order modes, which have higher modal displacements and therefore greater interaction with the foundation; additionally, centrifugal stiffening due to rotation (parameter $\eta$) increases axial tension, resulting in an increase in bending stiffness and a subsequent increase in frequencies. These relationships demonstrate the coupled nature of material gradation, boundary conditions, and foundation stiffness in determining dynamic performance.

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Figure 5. First five dimensionless natural frequencies (μ) vs. elastic foundation modulus (λ) for C-C boundary (n=1, η=5)

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Figure 6. First five dimensionless natural frequencies (μ) vs. elastic foundation modulus (λ) for C-S boundary (n=1, η=5)

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Figure 7. First five dimensionless natural frequencies (μ) vs. elastic foundation modulus (λ) for C-F boundary (n=1, η=5)

Table 6. Percentage increase (%) in natural frequencies (μ) from λ=0 to λ=500 for C-C, C-S, and C-F boundaries

The design and optimization of rotating FGM beams for engineering applications are directly impacted by the study's findings. For instance, in order to minimize vibrational strain and prevent resonance, turbine blades used in gas turbines or aviation engines frequently need to strike a balance between low weight and high stiffness. Designers can regulate the stiffness-to-weight ratio and permit frequency tuning without sacrificing structural integrity by choosing an appropriate gradient index $n$. Elastic foundations can also be used by robotic arms and precision positioning systems to improve positional accuracy and reduce vibration. By anticipating how foundation stiffness and material gradation would affect modal properties, engineers can design systems that are quieter, more dependable, and require less maintenance because resonance-induced wear is decreased.

It is important to recognize a number of limitations even though the current analysis offers comprehensive insight into the dynamic behavior of rotating FGM beams. The formulation is less accurate for thick beams or high-frequency modes because it is based on Euler-Bernoulli beam theory, which ignores rotary inertia effects and transverse shear deformation. The elastic foundation was modelled using a straightforward linear Winkler representation, which ignores shear interaction and non-linear foundation behavior, and the beam's cross-section was taken to be uniform. Furthermore, damping, geometric non-linearities, and thermal effects were not taken into account. Thus, the analysis should be expanded in future work to include Pasternak or non-linear foundations, non-uniform cross-sectional geometries, coupled thermo-mechanical or transient dynamic effects, and Timoshenko or higher-order shear deformation theories.