Dynamics of spinning axially graded porous nanoscale beams with rectangular cross-section incorporating rotary inertia effects

Abstract

The impacts of axially functionally graded (AFG) materials, porosity distribution patterns, rotary inertia factor, hygro-thermo-magnetic environments, non-uniform elastic substrates, axisymmetric cross-sections, axial and distributed tangential loads on the stability, and dynamics of spinning are analyzed. The governing equations of lateral displacements of the nanobeam are extracted based on the modified nonlocal theory (MNT) and the Rayleigh beam theory assumptions. The vibration frequencies are identified by solving the eigenvalue problem and exploiting the Galerkin discretization scheme. Stability maps and Campbell diagrams are presented to survey divergence and flutter behaviors with the help of numerical and analytical treatments. Comparative studies in various system circumstances are conducted to confirm the accuracy of the model and methodology. Results revealed that contrary to the square cross-section case, a divergence instability region is detected in the stability evolution of the nanobeam with a rectangular cross-section. Also, ascending the porosity factor leads to a more stable structure for low AFG indices. Moreover, the nanobeam can experience flutter instability by considering rotary inertia effects. The present research results could be advantageou for the design of high-tech inhomogeneous spinning systems.

Keywords

axially functionally graded porous materials spinning nanobeam vibration and stability rotary inertia effects non-uniform elastic substrates

1. Introduction

Spinning nanobeams are one of the key components of nanodevices and power transmission nanomachines. The ultra-small-scale structures, as next-generation high-tech nanomachines, lead to intense attainments in various nanoengineering sciences, and a hopeful future is anticipated for them in modern industries (Najmi and Hu, 2023; Najmi et al., 2023). For instance, in a brilliant drug delivery study, Bhirde et al. (Bhirde et al., 2009) applied nanoscale tubes for drug conveyance into the targeted tumors and improved the curing. Tu et al. (Tu et al., 2016) proposed a progressive nanofluidic desalination device utilizing spinning nanotube membrane filters and assessed the effectiveness of their scheme by molecular simulation. Most existing studies in this field are based on the Euler–Bernoulli beam theory, in which the crucial role of rotary inertia effects is neglected (Khorasani et al., 2023; Xia et al., 2023; Katiyar et al., 2022; Zhang et al., 2022). Scientists have devoted considerable empirical and theoretical attempts to perceive the dynamic essence of nanotechnology and nanometer-scale systems. They have acknowledged that implementing size-dependent high-order theories is mandatory to simulate the mechanical behavior of micro/nanosize systems (Hadji et al., 2023; Liu et al., 2022; Faghidian and Tounsi, 2022). The nonlocal Eringen theory (NET) is a well-known nonclassical theory for scrutinizing the scale effects in nanoengineering applications (Eringen, 1983). This theory introduces the nonlocal parameter to justify the softening effects in nanosystems. According to a novel investigation, Lim (Lim, 2010) proved that the governing dynamic equations based on the partial NET are not in equilibrium. He disclosed that the traditional NET cannot be a comprehensive model for the dynamic analysis of nanoscale structures. Also, he established and elaborated upon a modified model with better accuracy in foreseeing the nanostructures’ behavior by considering hardening effects in ultra-small-scale systems. Unfortunately, the published reports on spinning nanobeams’ vibration based on the MNT are sparse.

The mechanical properties of functionally graded materials alter continuously in one or more specific orientations (Ait Atmane et al., 2011; Sofıyev and Kuruoglu, 2015; Isvandzibaei et al., 2016; Singh and Sharma, 2022; Borjalilou et al., 2020). Hence, their material features are more appropriate than those of homogeneous and layered composite materials. Researchers have profoundly distinguished the influences of transverse material gradation on the vibration of spinning nanostructures (Van Vinh et al., 2022; Arshid et al., 2021; Zhang et al., 2023). Nonetheless, inadequate research efforts have been focused on utilizing AFG materials in spinning nanostructures. Due to technical problems in the fabrication of composite materials, nanovoids will probably form in functionally graded structures (Kumar et al., 2021; Van Vinh and Tounsi, 2022b; Khorshidi et al., 2022). In addition to reducing the effective system weight and improving the ability of energy absorption, recent reports have emphasized that these internal pores vary the stability and strength of the structure (Van Vinh and Tounsi, 2022a; Tounsi et al., 2023; Addou, 2023; Abdulmajeed M. Alsubaie, 2023). Consequently, considering different porosity distributions in graded nanosystems is vital and results in a more realistic simulation. Nevertheless, insufficient endeavors have been made to assess the porosity effects on the vibration of spinning graded nanostructures, especially considering the longitudinal material gradient.

The main innovations of this research are as follows:

- Examination of effects of rotary inertia factor and AFG porous materials on the vibration and stability of gyroscopic nanostructures based on the MNT

- Investigation of the impacts of environmental loads, variable foundations, and axisymmetric cross-sections on the dynamics of spinning nanobeams

Based on the Rayleigh beam assumptions, the vibration equations of spinning nanobeams are derived by considering the scale effects. Then, the nanobeam eigenvalues are achieved, and stability boundaries are determined through analytical and numerical approaches. Finally, the impacts of important system parameters on the vibration behavior and stability of the nanobeam are explored.

2. Problem formulation

Figure 1 depicts a simply supported straight AFG nanobeam with a rectangular cross-section that spins around its longitudinal axis at a speed of Ω. The thickness and width of the nanobeam are indicated by h and b, respectively. In addition, the nanobeam length is denoted by L. According to the figure, the nanobeam is subjected to an axial tensile load P, and a distributed tangential load F. Also, the nanobeam is exposed to external magneto-hygro-thermal fields. The intensity of the magnetic field is B, and the variations in temperature and humidity are expressed by ∆T and ∆H, respectively. Furthermore, it is assumed that an elastic substrate surrounds the nanobeam.

Different porosity distribution patterns, involving uniform and non-uniform patterns, are considered in the longitudinal direction of the nanobeam (Figure 2). It is assumed that there is no void in the perfect nanobeam. For the uniform porosity distribution, the internal pores are supposed to be uniformly spread over the nanobeam length. Also, two types of porosity models are considered for the non-uniform distribution pattern. According to the non-uniform porosity type 1, it is presumed that the porosity volume fraction increases linearly over the longitudinal orientation. While for non-uniform porosity type 2, this trend is reversed. So, the end of the nanobeam has the lowest amount of porosity.

The mechanical characteristics of the nanobeam (G), including Young’s modulus (E), mass density (ρ), thermal expansion factor (𝛼), Poisson’s ratio (λ), magnetic permeability factor (χ), and moisture expansion factor (β), change smoothly in the longitudinal direction. So, the material properties at x = 0 are referred to as pure metal (steel) and at x = L to pure ceramic (alumina). The material properties are characterized according to a power law distribution function in the longitudinal direction, which can be written as below (Akbas, 2021):

Uniform porosity

G (x) = G_{m} g (x) = G_{m} (1 + (η_{G} - 1) {(\frac{x}{L})}^{k} - \frac{μ}{2} (η_{G} + 1))

(1)

Non-uniform porosity (type 1)

G (x) = G_{m} g (x) = G_{m} (1 + (η_{G} - 1) {(\frac{x}{L})}^{k} - \frac{μ}{2} (η_{G} + 1)) (\frac{x}{L})

(2)

Non-uniform porosity (type 2)

G (x) = G_{m} g (x) = G_{m} (1 + (η_{G} - 1) {(\frac{x}{L})}^{k} - \frac{μ}{2} (η_{G} + 1)) (1 - \frac{x}{L})

(3)

where η_G is the material gradient parameter and represents the ratio of the material characteristic of the ceramic phase to that of the metal phase (i.e., η_G = G_c/G_m). In addition, the indices m and c represent the characteristics of metal and ceramic phases, respectively. Also, k is the AFG index, and μ is the porosity factor.

The constituent relationship between axial stress (σ_xx) and axial strain (ε_xx) in nanobeams according to traditional NET can be expressed as below (Eringen, 1983; Mokari et al., 2022, 2023)

(1 - τ^{2} L^{2} \nabla^{2}) σ_{xx} = E (x) ε_{xx}

(4)

in which ∇² = ∂²/∂x² is the Laplacian operator. Also, τ = e₀a/L is the nonlocal parameter, a is the internal characteristic length, and e₀ is the material constant.

The constituent relationship between axial strain and axial stress presented by Lim (Lim, 2010) differs. According to the MNT, the constituent relationship between axial strain and axial stress is stated below (Huang et al., 2022; Wang et al., 2023)

σ_{xx} = E (x) \sum_{n = 1}^{\infty} τ^{2 (n - 1)} \frac{d^{2 (n - 1)} ε_{xx}}{d x^{2 (n - 1)}}

(5)

For spinning systems, the axial strain can be given below (Ebrahimi-Mamaghani et al., 2021)

ε_{xx} = - y \frac{\partial^{2} v}{{\partial x}^{2}} - z \frac{\partial^{2} w}{{\partial x}^{2}}

(6)

where v and w are the lateral displacements in the y- and z-transversal directions, respectively (Akbarzadeh et al., 2020; Khalil et al., 2022).

To compute the strain energy, the below equation can be utilized (Wang, 2011)

E_{s} = \int_{0}^{L} {\frac{1}{2} E (x) ε_{xx}^{2} + \frac{1}{2} E (x) \sum_{n = 1}^{\infty} {(- 1)}^{n + 1} τ^{2 n} {(\frac{d^{n} ε_{xx}}{{d x}^{n}})}^{2} + E (x) \sum_{n = 1}^{\infty} τ^{2 (n + 1)} \sum_{j = 1}^{n} [{(- 1)}^{k + 1} \frac{d^{j} ε_{xx}}{{d x}^{j}} \frac{d^{(2 (n + 1) - j)} ε_{xx}}{{d x}^{(2 (n + 1) - j)}}]} A d x

(7)

in which A is the nanobeam cross-section. The first two nonlocal terms in the series in equation (7) are considered to inspect the scale-dependency of the model, which are the most effectual nonlocal terms signifying the nonlocal effect.

The work done by external environmental fields can be calculated below (Sarparast et al., 2022; Luo et al., 2022)

W_{e} = \frac{1}{2} \int_{0}^{L} ((N_{H} + N_{M} + N_{T}) [{(\frac{\partial v}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}]) d x

(8)

where

N_{H} = - \int_{0}^{L} E (x) β (x) Δ H d A

(9)

N_{T} = - \int_{0}^{L} \frac{E (x) α (x) Δ T}{1 - 2 λ (x)} d A

(10)

N_{M} = \int_{0}^{L} \frac{B^{2}}{χ (x)} d A

(11)

The nanobeam is embedded in non-uniform elastic substrates, including linear, parabolic, and sinusoidal foundations. The external work done by an elastic substrate can be given below (Pradhan and Murmu, 2009)

W_{v} = - \frac{1}{2} \int_{0}^{L} k_{v} (x) [v^{2} + w^{2}] d x

(12)

in which

k_{v} (x) = k_{0} (1 - σ_{L} \frac{x}{L}), 0 \leq σ_{L} \leq 1

(13)

k_{v} (x) = k_{0} (1 - σ_{P} {(\frac{x}{L})}^{2}), 0 \leq σ_{L} \leq 1

(14)

k_{v} (x) = k_{0} (1 - σ_{S} \sin (\frac{π x}{2 L})), 0 \leq σ_{L} \leq 1

(15)

In the above relation, k₀ is the Winkler substrate factor. Also, σ_L, σ_P, and σ_s are constants of linear, parabolic, and sinusoidal substrates, respectively.

The external work done by the axial tensile load is also determined below (Ebrahimi-Mamaghani et al., 2023; Tian et al., 2023)

W_{P} = \frac{1}{2} \int_{0}^{L} P [{(\frac{\partial v}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}] d x

(16)

The conservative and non-conservative work variations done by the distributed tangential compressive load can also be expressed below (Bahaadini et al., 2017; Hao et al., 2022)

δ W_{F}^{c} = - \frac{1}{2} \int_{0}^{L} F (L - x) [\frac{\partial v}{\partial x} \frac{\partial δ v}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial δ w}{\partial x}] d x

(17)

{δ W}_{F}^{nc} = \int_{0}^{L} F [\frac{\partial v}{\partial x} δ v + \frac{\partial w}{\partial x} δ w] d x

(18)

The position vector of an arbitrary point in the nanobeam can be stated according to the below relation (Zhu and Chung, 2019)

r = (x + u - y v^{'} - z w^{'}) i + (y + v) j + (z + w) k

(19)

where the prime represents the spatial derivative.

Since y- and z- coordinates are time-dependent functions. As a result, their time derivatives can be represented by ∂y/∂t = -zΩ and ∂z/∂t = yΩ, respectively (Zhu and Chung, 2019). Thus, according to the material derivative definition, the velocity vector of the spinning nanobeam is calculated below (Lu et al., 2013, 2021)

v = \frac{D}{D t} r + Ω \times r = (Ω z v^{'} - y {\dot{v}}^{'} - Ω y w^{'} - z {\dot{w}}^{'}) i + (\dot{v} - Ω w) j + (\dot{w} + Ω v) k

(20)

in which the dot represents the temporal derivative.

Generally, the kinetic energy of the spinning nanobeam can be estimated as below (Shi et al., 2023)

E_{K} = \frac{1}{2} \int_{0}^{L} ρ (x) (v . v) A d x = \frac{1}{2} \int_{0}^{L} ρ (x) A ({\dot{v}}^{2} + {\dot{w}}^{2} + Ω^{2} (v^{2} + w^{2}) - 2 Ω (\dot{v} w - v \dot{w})) d x + \frac{1}{2} \int_{0}^{L} ρ (x) (Ω^{2} (I_{y} {v^{'}}^{2} + I_{z} {w^{'}}^{2}) - 2 Ω (I_{y} v^{'} {\dot{w}}^{'} - I_{z} w^{'} {\dot{v}}^{'}) + I_{z} {\dot{v}}^{'}^{2} + I_{y} {\dot{w}}^{'}^{2}) d x

(21)

where the transversal area moments are determined as below

{I_{y}, I_{z}} = \int_{A} {z^{2}, y^{2}} d A

(22)

The generalized Hamilton’s principle may be utilized to extract the vibration equations of the nanobeam as below (Ebrahimi-Mamaghani et al., 2016; Bai et al., 2022)

\int_{t_{1}}^{t_{2}} (δ E_{K} + δ W_{e} + δ W_{P} + δ W_{v} + δ W_{F}^{c} + δ W_{V}^{nc} - δ E_{s}) d t = 0

(23)

The following dimensionless parameters are given to determine the dimensionless governing motion equations of the nanobeam

(x^{*}, v^{*}, w^{*}) = \frac{(x, v, w)}{L}, t^{*} = \frac{t}{L^{2}} \sqrt{\frac{E_{m} \sqrt{I_{y} I_{z}}}{ρ_{m} A}}, Ω^{*} = Ω L^{2} \sqrt{\frac{ρ_{m} A}{E_{m} \sqrt{I_{y} I_{z}}}}

(P^{*}, F^{*}, k_{F}^{*}) = \frac{(P, F L, k_{F} L^{2}) L^{2}}{E_{m} \sqrt{I_{y} I_{z}}}, (δ_{H}, δ_{T}, δ_{M}) = (N_{H}, N_{T}, N_{M}) \frac{L^{2}}{E_{m} \sqrt{I_{y} I_{z}}}

ε = \frac{I_{z}}{I_{y}}, γ = \frac{\sqrt{I_{y} I_{z}}}{A L^{2}}

(24)

in which γ is the rotary inertia factor.

By using equation (21) and removing the star notation from the above dimensionless parameters, the dynamic equations of the nanobeam can be attained as below

\sqrt{ε} {(g (x) v^{″} - τ^{2} g (x) v^{⁗})}^{″} + (F (1 - x) + δ_{H} + δ_{T} + δ_{M} - P) v^{″} + (δ_{H}^{'} + δ_{T}^{'} + δ_{M}^{'}) w^{'} + g (x) (\ddot{v} - 2 Ω \dot{w} - Ω^{2} v) + γ g (x) (\frac{Ω^{2} v^{″}}{\sqrt{ε}} - Ω (\sqrt{ε} + \frac{1}{\sqrt{ε}}) {\dot{w}}^{″} - \sqrt{ε} {\ddot{v}}^{″}) + γ g^{'} (x) (\frac{Ω^{2} v^{'}}{\sqrt{ε}} - Ω (\sqrt{ε} + \frac{1}{\sqrt{ε}}) {\dot{w}}^{'} - \sqrt{ε} {\ddot{v}}^{'}) + k_{v} (x) v = 0

(25)

\frac{{(g (x) w^{″} - τ^{2} g (x) w^{⁗})}^{″}}{\sqrt{ε}} + (F (1 - x) + δ_{H} + δ_{T} + δ_{M} - P) w^{″} + (δ_{H}^{'} + δ_{T}^{'} + δ_{M}^{'}) w^{'} + g (x) (\ddot{w} + 2 Ω \dot{v} - Ω^{2} w) + γ g (x) (Ω^{2} \sqrt{ε} w^{″} + Ω (\sqrt{ε} + \frac{1}{\sqrt{ε}}) {\dot{w}}^{″} - \frac{{\ddot{w}}^{″}}{\sqrt{ε}}) + γ g^{'} (x) (Ω^{2} \sqrt{ε} w^{'} + Ω (\sqrt{ε} + \frac{1}{\sqrt{ε}}) {\dot{w}}^{'} - \frac{{\ddot{w}}^{'}}{\sqrt{ε}}) + k_{v} (x) w = 0

(26)

3. Numerical solution methodology

The transversal displacements of the nanobeam are approximated with the below series (Cui et al., 2023a, 2023b, 2023c)

v (x, t) = \sum_{j = 1}^{\infty} p_{j} (t) ϕ_{j} (x)

(27)

w (x, t) = \sum_{j = 1}^{\infty} q_{j} (t) ϕ_{j} (x)

(28)

where p and q are the time-dependent generalized coordinates of the nanobeam. Also, ϕ is the dimensionless vibration mode shape of the nanobeam, which is stated below (Sun et al., 2023)

ϕ_{j} (x) = \sin (j π x)

(29)

By substituting equation (27) into equations (23), and (24), multiplying the mode shape in the dynamic equations and integrating over the nanobeam length, the equations in matrix form are expressed as below

[\begin{array}{l} M_{11} & 0 \\ 0 & M_{22} \end{array}] {\begin{cases} \ddot{Q} \\ \ddot{P} \end{cases}} + [\begin{array}{l} 0 & C_{12} \\ - C_{12} & 0 \end{array}] {\begin{cases} \dot{Q} \\ \dot{P} \end{cases}} + [\begin{array}{l} K_{11} & 0 \\ 0 & K_{22} \end{array}] {\begin{cases} Q \\ P \end{cases}} = {\begin{cases} 0 \\ 0 \end{cases}}

(30)

in which

{(M_{11})}_{s r} = \int_{0}^{1} ϕ_{s} (x) g (x) ϕ_{r} (x) d x - γ \sqrt{ε} \int_{0}^{1} ϕ_{s} (x) (g (x) ϕ_{r}^{″} (x) + g^{'} (x) ϕ_{r}^{'} (x)) d x

(31)

{(M_{22})}_{s r} = \int_{0}^{1} ϕ_{s} (x) g (x) ϕ_{r} (x) d x - \frac{γ}{\sqrt{ε}} \int_{0}^{1} ϕ_{s} (x) (g (x) ϕ_{r}^{″} (x) + g^{'} (x) ϕ_{r}^{'} (x)) d x

(32)

{(C_{12})}_{s r} = - 2 Ω \int_{0}^{1} ϕ_{s} (x) g (x) ϕ_{r} (x) d x - γ Ω (\sqrt{ε} + \frac{1}{\sqrt{ε}}) \int_{0}^{1} ϕ_{s} (x) (g (x) ϕ_{r}^{″} (x) + g^{'} (x) ϕ_{r}^{'} (x)) d x

(33)

{(K_{11})}_{sr} = \sqrt{ε} \int_{0}^{1} ϕ_{s} (x) (g^{″} (x) ϕ_{r}^{″} (x) + 2 g^{' (x) ϕ_{r}^{‴} (x)} + g (x) ϕ_{r}^{⁗} (x)) d x - \sqrt{ε} τ^{2} \int_{0}^{1} ϕ_{s} (x) (g^{″} (x) ϕ_{r}^{⁗} (x) + 2 g^{'} (x) ϕ_{r}^{⁗} (x) + g (x) ϕ_{r}^{⁗} (x)) d x + (F + δ_{H} + δ_{T} + δ_{M} - P) \int_{0}^{1} ϕ_{s} (x) ϕ_{r}^{″} (x) d x + (δ_{H}^{'} + δ_{T}^{'} + δ_{M}^{'}) \int_{0}^{1} ϕ_{s} (x) ϕ_{r}^{'} (x) d x - F \int_{0}^{1} ϕ_{s} (x) x ϕ_{r}^{'} (x) d x + \frac{γ Ω^{2}}{\sqrt{ε}} \int_{0}^{1} ϕ_{s} (x) (g (x) ϕ_{r}^{″} (x) + g^{'} (x) ϕ_{r}^{'} (x)) d x - Ω^{2} \int_{0}^{1} ϕ_{s} (x) g (x) ϕ_{r} (x) d x + \int_{0}^{1} ϕ_{s} (x) k_{v} (x) ϕ_{r} (x) d x

(34)

{(K_{22})}_{s r} = \frac{1}{\sqrt{ε}} \int_{0}^{1} ϕ_{s} (x) (g^{″} (x) ϕ_{r}^{″} (x) + 2 g^{' (x) ϕ_{r}^{‴} (x)} + g (x) ϕ_{r}^{⁗} (x)) d x - \frac{τ^{2}}{\sqrt{ε}} \int_{0}^{1} ϕ_{s} (x) (g^{″} (x) ϕ_{r}^{⁗} (x) + 2 g^{'} (x) ϕ_{r}^{⁗} (x) + g (x) ϕ_{r}^{⁗} (x)) d x + (F + δ_{H} + δ_{T} + δ_{M} - P) \int_{0}^{1} ϕ_{s} (x) ϕ_{r}^{″} (x) d x + (δ_{H}^{'} + δ_{T}^{'} + δ_{M}^{'}) \int_{0}^{1} ϕ_{s} (x) ϕ_{r}^{'} (x) d x - F \int_{0}^{1} ϕ_{s} (x) x ϕ_{r}^{'} (x) d x + γ Ω^{2} \sqrt{ε} \int_{0}^{1} ϕ_{s} (x) (g (x) ϕ_{r}^{″} (x) + g^{'} (x) ϕ_{r}^{'} (x)) d x - Ω^{2} \int_{0}^{1} ϕ_{s} (x) g (x) ϕ_{r} (x) d x + \int_{0}^{1} ϕ_{s} (x) k_{v} (x) ϕ_{r} (x) d x

(35)

By solving the eigenvalue problem of equation (28), one can determine the vibration frequencies of the nanobeam (the imaginary part of system eigenvalues) in terms of system factors. When one of the vibration frequencies of the nanobeam becomes zero, the nanobeam experiences static instability or divergence. Additionally, if the vibration frequency and the real part of the nanobeam eigenvalue have positive values, the nanobeam undergoes dynamic instability or flutter (Ebrahimi-Mamaghani et al., 2022).

4. Analytical solution methodology

According to the stability theory of linear gyroscopic systems, when the determinant of the stiffness matrix becomes zero for the first mode, the nanobeam eigenvalues become zero, and divergence behavior is observed in the stability evolution (Ebrahimi-Mamaghani et al., 2021). For example, for the spinning homogeneous nanobeam with a square cross-section, one can write

{(K_{11})}_{11} = {(K_{22})}_{11} = π^{4} - τ^{2} π^{6} + (F + δ_{H} + δ_{T} + δ_{M} - P) π^{2} - F / 2 π + Ω^{2} (η π^{2} - 1) = 0

(36)

5. Results and discussion

5.1. Comparative studies

Comparison studies with available scientific reports in the literature are performed in simpler system conditions. Figure 3 shows the impacts of axial and distributed tangential loads on the first two vibration frequencies of a cantilevered non-spinning beam. Based on this figure, the compressive loads induce a stiffness-decrement effect on the beam. Moreover, the outcomes of the current study are in good agreement with those reported in the Refs. (Kazemi-Lari et al., 2012; Karimi-Nobandegani et al., 2018).

Figure 4 depicts the influence of spin speed on the first four vibration frequency branches of a cantilevered beam with a rectangular cross-section. According to the figure, the present research outcomes agree with those calculated in Ref. (Banerjee and Su, 2006) based on the dynamic stiffness matrix method.

Figure 5 reveals the impact of Young’s modulus gradient on the vibration behavior of a simply supported beam. According to the figure, the fundamental vibration frequency of the beam is plotted in terms of the AFG index for different Young’s modulus ratios. As can be seen, the present research results depict a reasonable correlation with those reported by Alshorbagy et al. (Alshorbagy et al., 2011).

The material and physical characteristics of the nanobeam are listed in Table 1 to provide numerical examples. In the continuation of the present research, the results are given for a simply supported nanobeam with a square cross-section unless otherwise stated.

Table 1.

Properties of the nanobeam (Reddy and Chin, 1998; Alshorbagy et al., 2011).

Property	Value
ρ _m	$7800 Kg / m^{3}$
ρ _c	$3960 Kg / m^{3}$
E _m	$210 G P a$
E _c	$390 G P a$
𝛼_m	$12.23 \times 10^{- 6} (1 /^{o} C)$
𝛼_c	$6.82 \times 10^{- 6} (1 /^{o} C)$
β _m	$0.05 {(wt % H_{2} O)}^{- 1}$
β _c	$0.001 {(wt % H_{2} O)}^{- 1}$
λ_m	$0.3262$
λ_c	$0.26$
χ _m	$28.88 \times 10^{- 7} (s / m)$
χ _c	$12.56 \times 10^{- 7} (s / m)$
h	$1 n m$
L	$20 n m$

5.2. Parametric studies

In Figure 6, the impacts of the rotary inertia factor on the Campbell diagram of the homogeneous nanobeam are illustrated. The backward/forward vibration frequency (lower/higher frequency branch) decreases/increases linearly by ascending the spin speed. When the first background frequency becomes zero, divergence conditions are observed in the stability behavior at a certain spin speed (i.e., the divergence spin speed). In post-divergence conditions, the backward and forward frequency branches have an increasing trend and become parallel. In post/pre-divergence conditions, the backward vibration frequency decreases/increases when rotary inertia effects are considered. While the vibration forward frequency decreases with the increment of the rotary inertia factor. In simple words, the backward and forward frequency branches become closer to each other. Another important point is that by considering rotary inertia effects, the backward and forward frequency branches intersect at spin speeds higher than the divergence spin speed, and in this condition, the nanobeam undergoes flutter instability at higher spin speeds. Note that with the increment in the rotary inertia factor, the spin speed and vibration frequency corresponding to flutter instability diminish. Since flutter instability due to the rotary inertia parameter appears at high spin speeds, considering the rotary inertia effects in modeling and stability evaluation of high-tech high-speed applications is very important.

Figure 7 shows the importance of the cross-section aspect ratio in the evolution of spinning nanobeam dynamics. Unlike the nanobeam with a square cross-section (i.e., h = b), the variations in the vibration frequency of the nanobeam with a rectangular cross-section are nonlinear. Because the transversal area moments differ for rectangular cross-sections, the forward and backward frequency branches do not bifurcate from the same point at Ω = 0. In addition, the backward/forward vibration frequency decreases/increases by ascending the cross-section aspect ratio. In other words, the distance between the lower and upper frequency branches increases as the aspect ratio increases. Another important feature of rectangular cross-section nanobeams is that divergence instability happens in a range of spin speeds, unlike nanobeams with a square cross-section. In addition, the range of the divergence spin speed becomes wider with an increase in the aspect ratio. Thus, the stable regions of a nanobeam with a rectangular cross-section are smaller than those of a nanobeam with a square cross-section. Ignoring rotary inertia effects, nanobeams with square and rectangular cross-sections do not undergo flutter instability in post-divergence conditions, and upper and lower frequency branches intersect at infinity. The cross-section geometry significantly affects the spinning nanobeam dynamics and fine-adjusting the vibration characteristics by precisely determining the cross-section aspect ratio is feasible.

Figure 8 presents the coupled effect of the longitudinal material gradation and rotary inertia factor on the Campbell diagram of nanobeams with a rectangular cross-section. By ascending the AFG index, the volume fraction of the metal phase is enhanced. As a result, the nanobeam’s effective mass density and elastic modulus increase and decrease, respectively. The vibration frequencies will reduce with the increment of the AFG index. Divergence and flutter spin speed ranges are observed in the Campbell diagram of the AFG nanobeam with rectangular cross-sections by considering the inertia effects. By comparing Figures 6–8, one can realize that the Campbell diagram significantly depends on the cross-section geometry, and considering rotary inertia effects in the vibration equations is very important. Nanostructures with rectangular, elliptical, or triangular sections produce an attractive appearance and better performance. Non-circular cross-sections have superior strength against impact loading than the typical circular cross-sections due to their torsional rigidity (Ebrahimi-Mamaghani et al., 2023). Furthermore, rectangular cross-sections have a better bending capacity in diverse loading conditions than circular cross-sections due to unequal flexural rigidity about their principal axes. AFG materials can be exploited in numerous high-tech engineering devices, such as thin films, pressure vessels, biomedical implants, atomic force microscopies, and biomass sensors (Ghayesh and Farajpour, 2019). It can be inferred that the instability of high-tech spinning engineering systems, such as nanomotors, nanoturbines, nanomolecular bearings, and nanoscale gears, can be considerably hindered by utilizing AFG materials.

Figure 9 reveals the impacts of environmental conditions and axial tensile load on the first backward frequency. Since the effective stiffness improves by applying the axial tensile load and magnetic field, the vibration frequency is enhanced in the presence of the magnetic field and axial tensile load. On the other hand, since the hygro-thermal fields have destructive effects on the equivalent rigidity, the vibration frequency is weakened by the temperature and humidity rise in the environment. It can be comprehended that in the design of spinning nanostructures, great attention should be paid to the impacts of environmental conditions, and the performance of high-tech spinning structures can be enhanced by fine-tuning of system parameters in complex environments.

Figure 10 depicts the importance of rotary inertia effects on the stability map in the F-Ω plane. Without considering rotary inertia effects, for low values of the distributed tangential compressive load, as the spin speed ascends, the nanobeam undergoes divergence instability only at a certain spin speed and does not experience flutter instability. While at high distributed tangential compressive loads, the nanobeam undergoes flutter instability for all spin speed values. The divergence spin speed declines by considering rotary inertia effects, and the flutter instability region enlarges. So, flutter conditions are provided for the spinning nanobeam at low values of the distributed tangential compressive loads.

The stability map of the AFG nanobeam with a rectangular cross-section in the F-Ω plane is portrayed in Figure 11 by considering rotary inertia effects. Compared with the square cross-section case (viz., Figure 10), a divergence instability region is observed in the stability map instead of a divergence instability border. At low and high spin speeds, the nanobeam experiences either divergence or flutter instability. While at moderate spin speeds, both divergence and flutter behaviors are observed in the stability evolution. Consequently, it is vital to consider rotary inertia effects for spinning nanostructures. Due to the stiffness-decrement effect of the distributed tangential compressive load, the divergence and flutter instability areas expand by ascending the distributed tangential compressive load.

Figure 12 highlights the impacts of nonlocal parameter and different elastic substrates on stability. Contrary to the softening effects of nonlocality in NET, increasing the nonlocal parameter enhances stability in the MNT. Furthermore, the overall flexural rigidity is strengthened by increasing the Winkler substrate factor; hence, the divergence spin speed is enhanced. Compared with non-uniform elastic substrates, the toughening effect of uniform Winkler substrate is more noticeable. The parabolic/sinusoidal substrate induces the most/least stiffening effects among the non-uniform substrates. Since the spinning nanobeam stability can be considerably enhanced by exploiting a suitable medium, one of the most applicable approaches to enhancing the performance and stability of high-tech applications is fine-tuning the foundation parameters. Also, the outcomes of the numerical approach are validated by the analytical method.

Figures 13 and 14 indicate the critical spin speeds regarding the cross-section aspect ratio. The divergence regions condense as the aspect ratio approaches unity (viz., symmetric cross-section). In other words, a nanobeam with a square cross-section (i.e., h = b) undergoes divergence instability only at a particular spin speed. Otherwise, increasing or decreasing the aspect ratio amplifies the divergence spin speed range. Moreover, considering rotary inertia effects, the divergence thresholds displace to lower spin speeds, and the flutter instability zone appears in the stability map (Figure 14). These destabilizing effects can be attributed to the mass-addition effect of the rotary inertia factor. The flutter spin speed decreases first and then increases by ascending the aspect ratio. Compared with the flutter instability region, the divergence instability areas are more sensitive to the geometric features of the cross-section.

Figure 15 demonstrates the porosity effects on the AFG nanobeam stability. For all the porosity distributions, the divergence spin speed of the nanobeam is reduced by incrementing the AFG index (viz., decrementing the volume fraction of the ceramic phase). The variation in divergence spin speed is notable for low AFG indices. For high values of the AFG index, the divergence spin speed converges to that of homogeneous nanobeams. For low AFG indices, the increment in the porosity factor enhances the divergence spin speed. While for high AFG indices, this trend is reversed. In simple words, the perfect system is more stable/unstable than the imperfect system at low/high AFG indices. Mass density and Young’s modulus decline with the porosity factor increment; hence, for low/high AFG indices, the decrement rate of the mass density is higher/lower than that of Young’s modulus. The divergence speed of the AFG nanobeam with the uniform porosity model has the highest sensitivity to porosity variations. Generally, in addition to the axial distribution of materials, porosity provides an additional degree of freedom to adjust the stability characteristics of high-tech applications.

6. Conclusions

This work examines the vibration and stability of spinning AFG porous nanobeams with various cross-sections. Dynamic equations are extracted, and vibration frequencies and critical spin speeds are computed. The numerical technique’s accuracy is confirmed by an analytical method. The results showed that the flutter stability region emerges in stability maps by considering rotary inertia effects. The divergence spin speed decreases, and the flutter instability region enlarges by ascending the rotary inertia factor. Also, the divergence instability region shrinks as the cross-section aspect ratio approaches unity. Variations in the cross-section aspect ratio can lead to the extension or shrinkage of the flutter region. Moreover, vibration frequencies and critical speeds are declined by ascending the AFG index. Besides, the undesirable effects of hygro-thermal fields on vibration frequencies can be alleviated by amplifying the magnetic field and axial tensile load. The nanobeam porosity can be increased/decreased to enhance stability at low/high AFG indices. The uniform porosity model also has the maximum effect on the nanobeam dynamics among the porosity distribution patterns. Also, stability can be strengthened/weakened by ascending the nonlocal parameter/distributed tangential compressive load. Furthermore, non-uniform substrates have less stiffening effects than the uniform Winkler substrate. The linear elastic substrate has less/more stabilizing effects than the parabolic/sinusoidal elastic substrate.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Forat Falih Hasan

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