Dynamic analysis of piezoelectric beams under fuzzy initial conditions: A comparative study

Abstract

This study develops a fuzzy-based modelling framework to analyze the uncertain dynamic behaviour of cantilever piezoelectric beams by introducing fuzziness directly into the initial conditions, an aspect not addressed in existing literature. Two widely used fuzzy representations – Triangular Fuzzy Numbers (TFN) and Gaussian Fuzzy Numbers (GFN) – are employed to characterize imprecision in the excitation parameters. The Fuzzy Adomian Decomposition Method (FADM) is applied to derive semi-analytical solutions for the transverse deflection and induced electric potential, followed by an r-cut based uncertainty propagation procedure. Numerical investigations using PZT–5A material reveal that the choice of fuzzy number significantly influences the resulting uncertainty bounds. For mid-span deflection at t = 2 s, the GFN representation produces uncertainty widths approximately 1.8–2.0 times larger than TFN at r = 0.5, and around 1.4–1.6 times larger at r = 0.9. TFN yield compact intervals with lower computational cost, whereas GFN provide smoother and more conservative uncertainty envelopes. Additionally, the uncertainty width monotonically decreases with increasing r and converges to the deterministic response as r → 1. The findings demonstrate that the type of fuzzy number plays a crucial role in the fidelity and robustness of uncertainty modelling in piezoelectric structures. The proposed framework offers a computationally efficient tool for the dynamic analysis, design, and reliability assessment of intelligent piezoelectric devices operating under imprecise or partially known initial conditions.

Keywords

Fuzzy Adomian Decomposition Method (FADM)piezoelectric beam triangular fuzzy number (TFN)Gaussian fuzzy number (GFN)uncertainty quantification

Introduction

Modelling uncertainty has emerged as a crucial element in the analysis of physical and engineering systems, where system parameters, initial conditions, and boundary conditions are often imprecise, variable, or incomplete (Garg and Li, 2025; Gomes and Bendine, 2025; Maji et al., 2025a; Pujari et al., 2024; Tan and Yu, 2023; Zhao and Sheng, 2023). Conventional deterministic models, which presume precise input specifications, often neglect to account for intrinsic uncertainties stemming from measurement inaccuracies, manufacturing tolerances, environmental factors, and operational variances. This constraint may undermine the precision and dependability of engineering forecasts.

To address these shortcomings, researchers have developed probabilistic, interval-based, and fuzzy methodologies. While probabilistic approaches are robust when supported by extensive data, they can be ineffective or misleading when data are scarce or uncertain. Fuzzy logic, introduced by Zadeh (Zadeh, 1965), provides an effective alternative by enabling the representation of vagueness and imprecision using fuzzy sets, without requiring precise statistical information. This foundation has been significantly extended through the generalized Hukuhara derivative (Zadeh et al., 1996), generalized differentiability (Bede and Gal, 2005), and improvements in fuzzy arithmetic and calculus (Stefanini, 2010).

Fuzzy logic continues to play a critical role in uncertainty modelling across diverse engineering domains, with recent studies demonstrating significant advancements. Liu et al. (Liu et al., 2025b,a) introduced enhanced fuzzy interval methods that improved stability, convergence, and robustness in control systems. Jena et al. (Jena et al., 2024) used triangular fuzzy numbers for uncertainty analysis in nanobeam vibration, while Yadao (Yadao, 2024) applied adaptive neuro-fuzzy inference systems for fault detection in rotating machinery, and Sharma et al. (Sharma et al., 2024) validated a fuzzy self-tuned PID-controlled MR damper for vibration isolation in marine engines, demonstrating superior experimental performance over passive systems. Mahmoodabadi and Nejadkourki (Mahmoodabadi and Nejadkourki, 2024) developed a fuzzy adaptive inverse dynamics-based controller for a nonlinear 3DOF RPP manipulator, achieving improved stabilization. Prakash et al. (Prakash et al., 2024) applied fuzzy reasoning to enhance crack detection in structures. Elnozahy et al. (Elnozahy et al., 2023) designed a MATLAB/Simulink-based fuzzy and neuro-fuzzy algorithm for high-precision fault diagnosis in transmission lines with an MSE as low as 0.0042, and Alamin et al. (Alamin et al., 2025) explored a fuzzy logistic fixed effort harvesting model using nonlinear fuzzy difference equations, revealing dynamic stability under uncertainty. Numerous researchers have contributed to this field through diverse approaches and applications (Volchok et al., 2023; Ma et al., 2024; Mishra et al., 2022; Wang et al., 2023; Dhua and Kumar, 2025).

To solve fuzzy differential models analytically, the Adomian Decomposition Method (ADM) pioneered by Adomian (1984, 2013) has become a preferred technique for its ability to efficiently handle nonlinear and fuzzy equations. Its convergence and applicability have been rigorously demonstrated by researchers such as Abbaoui, Cherruault, and Wazwaz Abbaoui and Cherruault (1994, 1995), Cherruault (1989), and Wazwaz (1999). The Fuzzy ADM (FADM) has been successfully applied to fuzzy initial-value problems (Babolian et al., 2004), fuzzy differential inclusions, and fuzzy heat equations (Allahviranloo and Taheri, 2009), with recent studies further validating its effectiveness in modelling uncertainty (Osman et al., 2020; Saeed et al., 2024; Pandit et al., 2021; Al-Hayani and Younis, 2023). Kumar and Dhua (Kumar and Dhua, 2024, 2025) notably explored fuzzy dynamic responses in beam and wave systems using FADM, confirming its adaptability and accuracy.

Despite these developments, limited research has addressed uncertainty in piezoelectric beams, which are fundamental components in high-precision applications such as sensors, actuators, vibration dampers, and energy harvesters (Du et al., 2018; Liu et al., 2013; Maji et al., 2025b; Nath et al., 2025; Ting et al., 2016). These beams convert mechanical energy into electrical energy (and vice versa), and are sensitive to changes in initial and boundary conditions. Conventional models often assume deterministic parameters, overlooking how uncertainty in initial deflection or boundary constraints can influence beam performance, particularly in terms of deflection profiles and induced electric potential. This oversight can lead to substantial discrepancies in applications demanding high accuracy and reliability.

This study aims to fill this gap by developing a comprehensive fuzzy modelling framework to analyze the uncertain dynamic behaviour of piezoelectric beams, where the initial condition is represented by fuzzy numbers. A central contribution of this work is the comparative use of two distinct types of fuzzy numbers: Triangular Fuzzy Number (TFN) and Gaussian Fuzzy Number (GFN) to represent input uncertainty. TFN are simple to define, easy to compute, and widely used in engineering due to their piece-wise linear membership functions. GFN, by contrast, offer a smoother, more continuous, and often more realistic representation of uncertainty, especially when dealing with natural or experimental variability that lacks crisp bounds.

Recent studies have employed finite element methods for analyzing porous and functionally graded beams and plates (Belabed et al., 2024b; Bousmaha et al., 2025; Meftah et al., 2024; Belabed et al., 2024a). These works demonstrate the versatility of FEM, but they rely on deterministic initial and boundary conditions and do not directly accommodate fuzzy uncertainty. Since the present study focuses on epistemic uncertainty in initial conditions, a fuzzy analytical approach is more suitable for representing and propagating such imprecision.

Although fuzzy modelling has been applied in various engineering problems, the incorporation of fuzziness directly into the initial conditions of piezoelectric beams has not been addressed in the existing literature. Most earlier studies treat the initial and boundary conditions as deterministic, even though these quantities are often imprecise and subject to epistemic uncertainty in real applications. Furthermore, the effect of the membership-function shape on uncertainty propagation – particularly the differences between TFN-based and GFN-based representations – has not been investigated for piezoelectric structures. Other uncertainty frameworks, such as robust optimization, probabilistic programming, uncertainty theory, and Z-number theory, were not adopted because they typically require statistical information, reliability indices, or optimization formulations that are not naturally compatible with the partial differential equations governing beam dynamics. These approaches also introduce additional computational complexity without offering clearer interpretability for initial-condition-driven epistemic uncertainty. In contrast, TFN and GFN allow a transparent representation of uncertainty, integrate directly with r-cut analysis, and align well with the Adomian Decomposition framework, enabling semi-analytical solutions while preserving both computational efficiency and physical interpretability.

The purpose of comparing TFN and GFN is twofold: (1) to assess how the shape of the membership function influences the resulting fuzzy dynamic responses (deflection and electric potential), and (2) to identify which type of fuzzy number offers more reliable and interpretable results in modelling real-world uncertainty. To this end, the Fuzzy Adomian Decomposition Method is employed to obtain semi-analytical fuzzy solutions, with uncertainty propagation evaluated through r-cut techniques and parameterized fuzzification.

The results are validated using Lead Zirconate Titanate (PZT-5A), a widely used piezoelectric material. Graphical and numerical comparisons are presented to demonstrate how uncertainty bounds in deflection and electric potential vary with both the α-cut level and the chosen fuzzy number type. The findings emphasize the practical importance of fuzzy number selection in uncertainty modelling and provide guidelines for selecting appropriate representations in piezoelectric beam analysis. Table 1

Table 1.

Compact comparison of related studies and identified research gaps.

Author(s)	Method	Uncertainty	Application	Gap relevant to present work
Jena et al. (Jena et al., 2024)	TFN fuzzy model	Epistemic	Nanobeam vibration	No GFN; no fuzzy initial conditions; not for piezoelectric beams.
Osman et al. (Osman et al., 2020)	Fuzzy differential equations	Fuzzy coefficients	Heat/transport models	Does not address beam dynamics; no TFN–GFN comparison.
Pandit et al. (Pandit et al., 2021)	Fuzzy population model	Growth-rate fuzziness	Population systems	Not PDE-based; no fuzzy beam modelling.
Kumar & Dhua (Kumar and Dhua, 2025)	FLHPA-based fuzzy beam model	Fuzzy loading and fuzzy initial conditions	Fractionally damped viscoelastic beam	Addresses viscoelastic beams; no electromechanical coupling; no piezoelectric dynamic modelling; current work uniquely applies fuzzy ICs to piezoelectric beams with TFN–GFN comparison.
Belabed et al. (Belabed et al., 2024b)	FEM	Deterministic	FG sandwich beams	No uncertainty modelling; assumes crisp parameters.
Bousmaha et al. (Bousmaha et al., 2025)	Advanced FEM	Deterministic	CNT-reinforced plates	Handles complex materials but not fuzzy initial-value uncertainty.
Present study	FADM with TFN & GFN	Fuzzy initial conditions	Piezoelectric beam	First study applying fuzzy ICs to piezoelectric beams and comparing TFN vs GFN for electromechanical responses using semi-analytical solutions.

The rest of this paper is structured as follows. The section Preliminaries of Fuzzy Theory introduces the necessary concepts and definitions. The section Fuzzy Adomian Decomposition Method (FADM) describes the solution methodology. The governing equations of the piezoelectric beam are then presented, followed by the formulation and solution of the fuzzy beam model. Numerical results and comparisons are subsequently discussed. Finally, the section Conclusion and Future Outlook summarizes the main findings and outlines possible future research directions.

Preliminaries of Fuzzy Theory

This section includes several kinds of notations, definitions, and preliminary details that will be used later in this work (Kumar and Dhua, 2024).

On the real line $R$ , a set of ordered pairs $(t, μ_{\tilde{χ}} (t))$ is said to be fuzzy set $\tilde{χ} (t)$ , where $μ_{\tilde{χ}} (t)$ is membership function. The r^th-cut (0 ≤ r ≤ 1) of a fuzzy set is ${[\tilde{χ}]}^{r} = {t \in χ : μ_{\tilde{χ}} (t) \geq r}$ . For r = 1, ${[\tilde{χ}]}^{r}$ is called core of the fuzzy set and $\max {μ_{\tilde{χ}} (t) : t \in χ}$ is called the height of a fuzzy set. A fuzzy set is normal if its height is 1. Otherwise, it is called a subnormal fuzzy set. A fuzzy set is convex, if each r^th-cut is convex, and a normal convex fuzzy set is called a fuzzy number. ${[\tilde{χ}]}^{b} \subseteq {[\tilde{χ}]}^{a} \subseteq {[\tilde{χ}]}^{0}$ , if 0 ≤ a ≤ b For a, b ∈ [0, 1]. Also ${[\tilde{χ}]}^{a} = ⋂_{n = 1}^{\infty} {[\tilde{χ}]}^{a_{n}}$ , If a_n → a.

Definition 1

(Triangular Fuzzy Number)

If a fuzzy number is a triangular fuzzy number (TFN), it has a piece-wise continuing membership function with a singleton core. Arbitrary TFN $\tilde{χ} = (a_{1}, a_{2}, a_{3})$ , membership function is defined by:

μ_{\tilde{χ}} (t) = \{\begin{cases} 0 & if t \leq a_{1} or t \geq a_{3}, \\ \frac{t - a_{1}}{a_{2} - a_{1}} & if a_{1} \leq t \leq a_{2}, \\ \frac{a_{3} - t}{a_{3} - a_{2}} & if a_{2} \leq t \leq a_{3} . \end{cases}

For r ∈ [0, 1], r^th-cut of TFN $\tilde{χ} = (a_{1}, a_{2}, a_{3})$ is $[\underset{̲}{χ (r)}, \bar{χ (r)}] = [(a_{2} - a_{1}) r + a_{1}, - (a_{3} - a_{2}) r + a_{3}]$ .

Definition 2

(Gaussian Fuzzy Number)

The general asymmetrical GFN $\tilde{χ} = (a, σ_{0}, σ_{1})$ is defined by its membership function as follows:

μ_{\tilde{χ}} (t) = \{\begin{cases} e x p [- {(t - a)}^{2} / 2 σ_{0}^{2}] & for t \leq a, \\ e x p [- {(t - a)}^{2} / 2 σ_{1}^{2}] & for t \geq a . \end{cases}

For r ∈ (0, 1], r^th-cut of GFN $\tilde{χ} = (a, σ_{0}, σ_{1})$ is $[\underset{̲}{χ (r)}, \bar{χ (r)}] = [a - \sqrt{- 2 σ_{0}^{2} \log_{e} (r)}, a + \sqrt{- 2 σ_{1}^{2} \log_{e} (r)}]$ .

Note: For any arbitrary fuzzy number $\tilde{χ} (r) = [\underset{̲}{χ} (r), \bar{χ} (r)]$ , ${\tilde{χ}}^{c} (r) = \underset{̲}{χ} (r) + \bar{χ} (r) / 2$ is called Fuzzy Center, $△ \tilde{χ} (r) = \underset{̲}{χ} (r) - \bar{χ} (r) / 2$ is called Fuzzy Radius and $| \underset{̲}{χ} (r) - \bar{χ} (r) |$ is called Fuzzy Width for all 0 ≤ r ≤ 1.

Fuzzy Adomian decomposition method (FADM)

Consider the general form of a nonlinear fuzzy partial differential equation (FPDE) given by:

L \tilde{χ} + R \tilde{χ} + N \tilde{χ} = g,

(1)

where

\tilde{χ}

denotes the fuzzy-valued unknown function. The operator

L

is an invertible linear operator,

R

represents the remaining linear terms,

N

is a nonlinear operator, and

g

is a known fuzzy source function. Applying the inverse of the operator

L

to both sides of equation (1) yields the following equivalent formulation:

\tilde{χ} = h (t) + L^{- 1} (g) - L^{- 1} (R \tilde{χ}) - L^{- 1} (N \tilde{χ}),

where

h (t)

is a function determined by the initial or boundary conditions.

The Adomian Decomposition Method (ADM) is employed to express the fuzzy solution $\tilde{χ}$ as an infinite series:

\tilde{χ} = \sum_{k = 0}^{\infty} {\tilde{χ}}_{k},

(2)

while the nonlinear operator

N \tilde{χ}

in equation (1) is expanded using a series of Adomian polynomials:

N \tilde{χ} = \sum_{k = 0}^{\infty} N_{k} (χ_{0}, χ_{1}, \dots, χ_{n}),

(3)

where each N_k is an Adomian polynomial defined by:

N_{k} = {\frac{1}{k!} \frac{d^{k}}{d λ^{k}} [N (\sum_{i = 1}^{k} λ^{i} {\tilde{χ}}_{i})]|}_{t = 0}, k = 0,1,2, \dots .

(4)

Substituting the series expansions from equations (2) and (3) into equation (1) leads to:

\begin{aligned} \tilde{χ} = \sum_{k = 0}^{\infty} {\tilde{χ}}_{k} = h (t) + L^{- 1} (g) - L^{- 1} (R (\sum_{k = 0}^{\infty} {\tilde{χ}}_{k})) \\ - L^{- 1} (\sum_{k = 0}^{\infty} N_{k}) . \end{aligned}

(5)

The solution components ${\tilde{χ}}_{k}$ are obtained recursively as follows:

\begin{matrix} {\tilde{χ}}_{0} & = h (t) + L^{- 1} (g), \\ {\tilde{χ}}_{1} & = - L^{- 1} (R ({\tilde{χ}}_{0})) - L^{- 1} (N_{0}), \\ {\tilde{χ}}_{2} & = - L^{- 1} (R ({\tilde{χ}}_{1})) - L^{- 1} (N_{1}), \\ ⋮ \\ {\tilde{χ}}_{k} & = - L^{- 1} (R ({\tilde{χ}}_{k - 1})) - L^{- 1} (N_{k - 1}) . \end{matrix}

(6)

The complete fuzzy solution $\tilde{χ} = \sum_{k = 0}^{\infty} {\tilde{χ}}_{k}$ is constructed from the recursively generated terms. The convergence and validity of this series representation are supported by the theoretical results established in the works of Himoun et al. (Himoun et al., 1999), Cherruault (Cherruault, 1989), and Abbaoui and Cherruault (Abbaoui and Cherruault, 1994, 1995).

Governing equations of the piezoelectric beam

In the absence of body forces and internal heat sources, the governing equations for a homogeneous, transversely isotropic piezoelectric beam are described by the coupled constitutive relations, equations of motion, and Maxwell’s electrostatic equations.

The constitutive equations are given by:

σ_{i j} = c_{ijkl} ε_{k l} - e_{ijk} E_{k},

(7)

ρ S = b_{i j} ε_{i j} + τ_{i} E_{i},

(8)

D_{i} = ζ_{i j} E_{j} + e_{ijk} ε_{j k},

(9)

E_{i} = - ϕ_{, i}, (i, j, k = x, y, z),

(10)

where σ_ij denotes the components of the stress tensor, ɛ_ij = 1/2 (u_i,j + u_j,i) is the strain tensor, and u_i represents the components of the mechanical displacement vector. The electric field components are denoted by E_i, the electric displacement vector by D_i, and ϕ represents the electric potential. The material tensors include the elastic stiffness tensor c_ijkl, the piezoelectric coupling tensor e_ijk, the dielectric permittivity tensor ζ_ij, the thermal modulus tensor b_ij, and the pyroelectric coefficient τ_i. The electric field is derived from the negative gradient of the potential, as shown in equation (10). The notation ‘,’ and ‘.’ indicate partial and time derivatives, respectively.

The equations of motion are given by:

σ_{i j, j} + ρ \ddot{u} i = 0,

(11)

and the electrostatic Gauss law for a charge-free domain is:

D_{i, i} = 0,

(12)

where ρ is the mass density of the material. These coupled field equations form the foundational framework for analyzing the mechanical and electrical responses of piezoelectric beams under various boundary and initial conditions.

Consider a homogeneous piezoelectric beam with length L (0 ≤ x ≤ L), width a (0 ≤ y ≤ a), and thickness h (−h/2 ≤ z ≤ h/2), subjected to small-amplitude flexural deflections. The coordinate system is defined such that the x-axis aligns with the beam’s length, the y-axis with its width, and the z-axis with its thickness, as illustrated in Figure 1.

Figure 1.

Geometric configuration of the piezoelectric beam under flexural deformation.

The beam is assumed to be initially stress-free and strain-free in its equilibrium state. Upon excitation, it undergoes small transverse bending vibrations along the x-axis. The deformation is modelled using linear Euler-Bernoulli beam theory, which assumes that cross-sections initially perpendicular to the beam’s neutral axis remain plane and perpendicular after deformation. Consequently, the transverse deflection w is a function of x and t only, and the displacement field is expressed as:

w (x, y, z, t) = w (x, t) and u = - z \frac{\partial w}{\partial x}, v = 0 .

(13)

Due to the slenderness of the beam, effects such as transverse shear deformation, rotatory inertia, and shear stresses are neglected. Furthermore, the in-plane components of the electric field are considered negligible (E_x = E_y = 0), only the out-of-plane electric field component E_z is retained. Substituting the kinematic expressions from equation (13) into the constitutive equations (7) and (9) yields:

σ_{x} = - z c_{11} \frac{\partial^{2} w}{\partial x^{2}} + e_{13} \frac{\partial ϕ}{\partial z},

(14)

D_{z} = - e_{31} z \frac{\partial^{2} w}{\partial x^{2}} - ζ_{33} \frac{\partial ϕ}{\partial z} .

(15)

Combining Gauss’s law for electrostatics equation (12) with equation (15) gives:

e_{31} \frac{\partial^{2} w}{\partial x^{2}} + ζ_{33} \frac{\partial^{2} ϕ}{\partial z^{2}} = 0 .

(16)

Integration of equation (16) with respect to z results in:

\frac{\partial ϕ}{\partial z} = ϕ_{0} - \frac{e_{31}}{ζ_{33}} z \frac{\partial^{2} w}{\partial x^{2}} .

(17)

where ϕ₀ is an integration constant representing the initial electric field. Integration of equation (17) with respect to z results in:

ϕ (x, z, t) = z ϕ_{0} - \frac{e_{31}}{ζ_{33}} (\frac{z^{2}}{2} - \frac{h^{2}}{8}) \frac{\partial^{2} w}{\partial x^{2}} .

(18)

The flexural moment over the beam’s cross-section is given by:

M (x, t) = - \int_{- \frac{h}{2}}^{\frac{h}{2}} a σ_{x} z d z = \bar{c_{11}} I \frac{\partial^{2} w}{\partial x^{2}},

(19)

where I = ah³/12 denotes the moment of inertia of the cross-section, and

\bar{c_{11}} = c_{11} + e_{31} e_{13} / ζ_{33}

represents the effective flexoelectric stiffness.

The equation governing transverse motion, derived from Newton’s second law and the moment-curvature relationship, is:

\frac{\partial^{2} M}{\partial x^{2}} + ρ A \frac{\partial^{2} w}{\partial t^{2}} = Δ p,

(20)

where A = ah is the cross-sectional area, ρ is the material density, and Δp denotes the pressure difference across the upper and lower surfaces of the beam. Substituting equation (19) into equations (20) yields the governing PDE for transverse vibrations:

\bar{c_{11}} I \frac{\partial^{4} w}{\partial x^{4}} + ρ A \frac{\partial^{2} w}{\partial t^{2}} = Δ p .

(21)

Consequently, equations (17) and (21) constitute the governing equations for the behaviour of a thin piezoelectric beam.

Solution of the Fuzzy Beam Model using FADM

To develop the numerical schemes for uncertain pezoelectric beam, boundary conditions are considered as fuzzy initial conditions (i.e. $\tilde{w} (0, t) = \tilde{K_{1}}, {\tilde{w}}_{x} (0, t) = \tilde{K_{2}} g (t), {\tilde{w}}_{x x} (L, t) = 0, and {\tilde{w}}_{xxx} (L, t) = 0$ ) where $\tilde{K_{1}}$ and $\tilde{K_{2}}$ are fuzzy numbers, also assume Δp = 0. The governing equation and the relation between deflection and electric potential are in fuzzy PDE as follows:

\begin{matrix} \bar{c_{11}} I \frac{\partial^{4} \tilde{w}}{\partial x^{4}} + ρ A \frac{\partial^{2} \tilde{w}}{\partial t^{2}} = 0, \\ \tilde{ϕ} (x, z, t) = z ϕ_{0} - \frac{e_{31}}{ζ_{33}} (\frac{z^{2}}{2} - \frac{h^{2}}{8}) \frac{\partial^{2} \tilde{w}}{\partial x^{2}}, \\ \tilde{w} (0, t) = \tilde{K_{1}}, {\tilde{w}}_{x} (0, t) = \tilde{K_{2}} g (t), \\ {\tilde{w}}_{x x} (L, t) = 0, {\tilde{w}}_{xxx} (L, t) = 0 . \end{matrix}\}

(22)

Let us consider g(t) = sin (πt/L), where L is length of the beam. By using the Max-Min approach, the r^th-cut of the fuzzy PDE of the governing equation in equations (17) and (21) along with boundary conditions can be written into two crisp problems. They are

\begin{matrix} \bar{c_{11}} I \frac{\partial^{4} \bar{w}}{\partial x^{4}} + ρ A \frac{\partial^{2} \bar{w}}{\partial t^{2}} = 0, \\ \bar{ϕ} (x, z, t) = z ϕ_{0} - \frac{e_{31}}{ζ_{33}} (\frac{z^{2}}{2} - \frac{h^{2}}{8}) \frac{\partial^{2} \bar{w}}{\partial x^{2}}, \\ \bar{w} (0, t) = \bar{K_{1}}, {\bar{w}}_{x} (0, t) = \bar{K_{2}} g (t), \\ {\bar{w}}_{x x} (L, t) = 0, {\bar{w}}_{xxx} (L, t) = 0 . \end{matrix}\}

(23)

and

\begin{matrix} \bar{c_{11}} I \frac{\partial^{4} \underset{̲}{w}}{\partial x^{4}} + ρ A \frac{\partial^{2} \underset{̲}{w}}{\partial t^{2}} = 0, \\ \underset{̲}{ϕ} (x, z, t) = z ϕ_{0} - \frac{e_{31}}{ζ_{33}} (\frac{z^{2}}{2} - \frac{h^{2}}{8}) \frac{\partial^{2} \underset{̲}{w}}{\partial x^{2}}, \\ \underset{̲}{w} (0, t) = \underset{̲}{K_{1}}, {\underset{̲}{w}}_{x} (0, t) = \underset{̲}{K_{2}} g (t), \\ {\underset{̲}{w}}_{x x} (L, t) = 0, {\underset{̲}{w}}_{xxx} (L, t) = 0 . \end{matrix}\}

(24)

Equation (23) is solved using the fuzzy Adomian Decomposition Method (ADM). The governing equation for deflection is expressed as follows:

L_{xxxx} (\bar{w}) = - \frac{ρ A}{\bar{c_{11}} I} L_{t t} (\bar{w}) .

(25)

Applying the inverse operator $L_{xxxx}^{- 1}$ to equation (25) yields:

\begin{aligned} \bar{w} = \bar{w} (0, t) + x {\bar{w}}_{x} (0, t) + \frac{x^{2}}{2!} {\bar{w}}_{x x} (0, t) + \frac{x^{3}}{3!} {\bar{w}}_{xxx} (0, t) \\ - \frac{ρ A}{\bar{c_{11}} I} L_{xxxx}^{- 1} (L_{t t} (\bar{w})) . \end{aligned}

Substituting the initial conditions gives:

\begin{aligned} \bar{w} = {\bar{k}}_{1} + {\bar{k}}_{2} x \sin (\frac{π t}{L}) + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!} \\ - \frac{ρ A}{\bar{c_{11}} I} L_{xxxx}^{- 1} (L_{t t} (\bar{w})), \end{aligned}

(26)

Here, $λ_{1} = {\bar{w}}_{x x} (0, t)$ and $λ_{2} = {\bar{w}}_{xxx} (0, t)$ . Assuming $\bar{w} = \sum_{n = 0}^{\infty} {\bar{w}}_{n}$ and substituting it into equation (26) yields:

\begin{aligned} \sum_{n = 0}^{\infty} {\bar{w}}_{n} = {\bar{k}}_{1} + {\bar{k}}_{2} x \sin (\frac{π t}{L}) + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!} \\ - \frac{ρ A}{\bar{c_{11}} I} L_{xxxx}^{- 1} [\sum_{n = 0}^{\infty} L_{t t} ({\bar{w}}_{n})] . \end{aligned}

(27)

from equation (27),

{\bar{w}}_{n}

can be expressed in terms of recursive relations as

\begin{aligned} {\bar{w}}_{0} & = {\bar{k}}_{1} + {\bar{k}}_{2} x \sin (\frac{π t}{L}) + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!}, \\ {\bar{w}}_{1} & = - \frac{ρ A}{\bar{c_{11}} I} L_{xxxx}^{- 1} [L_{t t} ({\bar{w}}_{0})] \\ = {\bar{k}}_{2} \frac{ρ A}{\bar{c_{11}} I} {(\frac{π}{L})}^{2} \sin (\frac{π t}{L}) \frac{x^{5}}{5!}, \\ {\bar{w}}_{2} & = - \frac{ρ A}{\bar{c_{11}} I} L_{xxxx}^{- 1} [L_{t t} ({\bar{w}}_{1})] \\ = {\bar{k}}_{2} {(\frac{ρ A}{\bar{c_{11}} I})}^{2} {(\frac{π}{L})}^{4} \sin (\frac{π t}{L}) \frac{x^{9}}{9!}, \\ ⋮ \\ {\bar{w}}_{n} & = - \frac{ρ A}{\bar{c_{11}} I} L_{xxxx}^{- 1} [L_{t t} ({\bar{w}}_{n - 1})] \\ = {\bar{k}}_{2} {(\frac{ρ A}{\bar{c_{11}} I})}^{n} {(\frac{π}{L})}^{2 n} \sin (\frac{π t}{L}) \frac{x^{4 n + 1}}{(4 n + 1)!} . \end{aligned}\}

(28)

Hence, the solution of equation (23) is expressed as:

\begin{aligned} \bar{w} & = \sum_{n = 0}^{\infty} {\bar{w}}_{n} \\ = {\bar{k}}_{1} + {\bar{k}}_{2} x \sin (\frac{π t}{L}) + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!} + \sum_{n = 1}^{\infty} [{\bar{k}}_{2} {(\frac{ρ A}{\bar{c_{11}} I})}^{n} \\ {(\frac{π}{L})}^{2 n} \sin (\frac{π t}{L}) \frac{x^{4 n + 1}}{(4 n + 1)!}] \\ = {\bar{k}}_{1} + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!} + \frac{{\bar{k}}_{2}}{2 ω} \sin (\frac{π t}{L}) (\sinh (ω x) + \sin (ω x)) . \end{aligned}

(29)

Here, $ω = {(ρ A / \bar{c_{11}} I)}^{1 / 4} {(π / L)}^{1 / 2}$ . Substituting the boundary conditions ${\bar{w}}_{x x} (L, t) = 0$ and ${\bar{w}}_{xxx} (L, t) = 0$ into equation (29), the unknown values λ₁ and λ₂ are determined as follows:

\begin{aligned} λ_{2} & = {\bar{k}}_{2} \frac{ω^{2}}{2} \sin (\frac{π t}{L}) (\cos (ω L) - \cosh (ω L)), \\ λ_{1} & = - λ_{2} L + {\bar{k}}_{2} \frac{ω}{2} \sin (\frac{π t}{L}) [(\sin (ω L) - \sinh (ω L))] . \end{aligned}

(30)

Hence, the deflection and electric potential are computed as follows by combining equations (29) and (30), and substituting them into equation (23).

\begin{aligned} \bar{w} & = {\bar{k}}_{1} + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!} + \frac{{\bar{k}}_{2}}{2 ω} \sin (\frac{π t}{L}) (\sinh (ω x) + \sin (ω x)), \\ \bar{ϕ} & = z ϕ_{0} - \frac{e_{31}}{ζ_{33}} (\frac{z^{2}}{2} - \frac{h^{2}}{8}) [λ_{1} + λ_{2} x + {\bar{k}}_{2} \frac{ω}{2} \\ \sin (\frac{π t}{L}) (\sinh (ω x) - \sin (ω x))] . \end{aligned}

(31)

Similarly, the solution of equation (24) is obtained by:

\begin{aligned} \underset{̲}{w} & = {\underset{̲}{k}}_{1} + λ_{1} \frac{x^{2}}{2!} + λ_{2} \frac{x^{3}}{3!} + \frac{{\underset{̲}{k}}_{2}}{2 ω} \sin (\frac{π t}{L}) \\ (\sinh (ω x) + \sin (ω x)), \\ \underset{̲}{ϕ} & = z ϕ_{0} - \frac{e_{31}}{ζ_{33}} (\frac{z^{2}}{2} - \frac{h^{2}}{8}) [λ_{1} + λ_{2} x + {\underset{̲}{k}}_{2} \frac{ω}{2} \\ \sin (\frac{π t}{L}) (\sinh (ω x) - \sin (ω x))] . \end{aligned}\}

(32)

where γ₁, γ₂, and ω are defined as follows:

\begin{aligned} γ_{2} & = {\underset{̲}{k}}_{2} \frac{ω^{2}}{2} \sin (\frac{π t}{L}) (\cos (ω L) - \cosh (ω L)), \\ γ_{1} & = - γ_{2} L + {\underset{̲}{k}}_{2} \frac{ω}{2} \sin (\frac{π t}{L}) [(\sin (ω L) - \sinh (ω L))], \\ ω & = {(\frac{ρ A}{\bar{c_{11}} I})}^{\frac{1}{4}} {(\frac{π}{L})}^{\frac{1}{2}} . \end{aligned}

here,

{\tilde{K}}_{1} = [{\underset{̲}{k}}_{1}, {\bar{k}}_{1}]

and

{\tilde{K}}_{2} = [{\underset{̲}{k}}_{2}, {\bar{k}}_{2}]

represent the rth-cut of the fuzzy functions. Based on the membership function, the following fuzzy functions are obtained:case (1): Consider

{\tilde{K}}_{1} = (- 0.1, 0,0.1)

and

{\tilde{K}}_{2} = (0.9, 1,1.1)

are of TFN, then r^th-cut of each fuzzy function can be written as

{\tilde{K}}_{1} = [{\underset{̲}{k}}_{1}, {\bar{k}}_{1}] = [0.1 r - 0.1, - 0.1 r + 0.1]

{\tilde{K}}_{2} = [{\underset{̲}{k}}_{2}, {\bar{k}}_{2}] = [0.1 r + 0.9, - 0.1 r + 1.1]

case (2): Consider

{\tilde{K}}_{1} = (0,0.1, 0.1)

and

{\tilde{K}}_{2} = (0,0.1, 0.1)

are of GFN, then r^th-cut of each fuzzy function can be written as

{\tilde{K}}_{1} = [{\underset{̲}{k}}_{1}, {\bar{k}}_{1}] = [- \sqrt{- 0.02 \ln (r)}, \sqrt{- 0.02 \ln (r)}],

{\tilde{K}}_{2} = [\underset{̲}{k_{2}}, \bar{k_{2}}] = [1 - \sqrt{- 0.02 \ln (r)}, 1 + \sqrt{- 0.02 \ln (r)}] .

Numerical results and discussion

The present study investigates the numerical behaviour of a piezoelectric beam composed of transversely isotropic material (PZT-5A) under uncertainty. The analysis employs MATLAB to compute the r^th-cut deflection, and electric potential distribution for various values of r (r ∈ (0, 1]), with the uncertainty modelled using Triangular Fuzzy Numbers (TFN) and Gaussian Fuzzy Numbers (GFN).

Using the material parameters of PZT-5A provided by Grover and Sharma (Grover and Sharma, 2012) (Table 2), the study examines the effects of uncertainty on deflection, and electric potential.

Table 2.

Physical data of PZT-5A material.

Parameters	Value of the parameter	Unit
ρ	7.75 × 10³	Kgm ⁻³
c₁₁	13.9 × 10¹⁰	Nm ⁻²
c₁₂	7.78 × 10¹⁰	Nm ⁻²
c₁₃	7.54 × 10¹⁰	Nm ⁻²
c₃₃	11.3 × 10¹⁰	Nm ⁻²
c₄₄	2.56 × 10¹⁰	Nm ⁻²
β₁	1.52 × 10⁶	Nm ⁻² K ⁻¹
β₃	1.53 × 10⁶	Nm ⁻² K ⁻¹
K₁ = K₃	1.5	Wm ⁻¹ K ⁻¹
C_e	420	JKg ⁻¹ K ⁻¹
e₁₃	−6.98	Cm ⁻²
e₃₃	13.8	Cm ⁻²
e₅₁	13.4	Cm ⁻²
ϵ₁₁	60 × 10⁻¹⁰	C ² N ⁻¹ m ⁻²
ϵ₃₃	54.7 × 10⁻¹⁰	C ² N ⁻¹ M ⁻²
p₃	−452 × 10⁻⁶	CK ⁻¹ m ⁻²
T₀	298	°F
a	0.006	m
h	6 × 10⁻⁵	m
L	0.021	m

Deflection

The fuzzy deflection behaviour of the PZT-5A beam was investigated using both TFN and GFN representations of the initial conditions. Figures 2(a)–3(b) present the spatial profiles at t = 2 and the temporal responses at mid-span (x = L/2). In both cases, the deflection intervals narrow as the r-cut level increases, and the solution approaches the deterministic response when r → 1. This trend is reinforced in Figures 4(a)–5(b), where the results for r = {0.2, 0.4, 0.6, 0.8} confirm the progressive reduction in uncertainty width.

Figure 2.

Comparison of fuzziness in beam deflection at t = 2 using TFN and GFN.

Figure 3.

Comparison of fuzziness in deflection at the mid-span position (x = L/2) using TFN and GFN.

Figure 4.

Comparison of fuzziness in beam deflection at t = 2 for multiple r-levels using TFN and GFN.

Figure 5.

Comparison of fuzziness in deflection at the mid-span location (x = L/2) for multiple r-levels using TFN and GFN.

Direct comparisons of TFN and GFN responses are shown in Figure 6(a)–6(b). At both r = 0.5 and r = 0.9, the TFN intervals are strictly contained within those of GFN, demonstrating that Gaussian fuzzy numbers consistently yield wider and smoother uncertainty bounds. Quantitatively, at t = 2 and x = L/2, the uncertainty width under GFN is about 1.8 − −2.0 times larger than TFN at r = 0.5, and approximately 1.5 times larger at r = 0.9. These ratios highlight that TFN produces more compact intervals and is computationally efficient, while GFN captures epistemic variability more conservatively.

Figure 6.

Comparison of fuzziness intervals in mid-span deflection (x = L/2) for different r-levels using TFN and GFN.

The overall trend is summarized in and Table 3, which confirm that increasing r reduces the uncertainty width in both TFN and GFN. Nevertheless, TFN remains bounded within GFN across all r-levels, indicating that the choice of membership function has a significant influence on the magnitude of uncertainty. This behaviour is consistent with earlier reports in fuzzy structural mechanics, where Gaussian-based models were found to produce smoother and more conservative uncertainty bands compared to triangular approximations.

Table 3.

Bounds of the deflection (w (x, t: r)) (for the different r values) for x = L/2 and t = 2 in TFN and GFN.

	TFN		GFN
r	$\underset{̲}{w} (x, t : r)$	$\bar{w} (x, t : r)$	$\underset{̲}{w} (x, t : r)$	$\bar{w} (x, t : r)$
0.1	−0.09334	0.085999	−0.21748	0.21014
0.2	−0.083377	0.076036	−0.18242	0.17508
0.3	−0.073413	0.066073	−0.15828	0.15094
0.4	−0.06345	0.05611	−0.13855	0.13121
0.5	−0.053487	0.046146	−0.12098	0.11364
0.6	−0.043523	0.036183	−0.10438	0.097036
0.7	−0.03356	0.02622	−0.08782	0.08048
0.8	−0.023597	0.016256	−0.07023	0.062889
0.9	−0.013634	0.0062931	−0.049406	0.042066
1	−0.0036702	−0.0036702	−0.0036702	−0.0036702

In practical terms, TFN is advantageous when computational efficiency and narrower bounds are sufficient, while GFN should be preferred in applications such as sensing, vibration control, and energy harvesting, where robustness under uncertain conditions is critical.

Electric potential

The investigation also extends to the fuzziness in the induced electric potential distribution, evaluated at z = h/4 under various scenarios. Figure 7(a)–7(b) present the spatial distribution at t = 2 for both TFN and GFN. In all cases, the uncertainty width decreases monotonically with increasing r, and the solution approaches the crisp deterministic response at r = 1. This trend is confirmed in Figures 8(a)–9(b), where the intervals for r = {0.2, 0.4, 0.6, 0.8} progressively shrink, which show the convergence to crisp values at r = 1.

Figure 7.

Comparison of fuzziness in electric potential distribution at t = 2 and z = h/4 using TFN and GFN.

Figure 8.

Comparison of fuzziness intervals in electric potential distribution at x = L/2 and z = h/4 for multiple r-levels using TFN and GFN.

Figure 9.

Comparison of fuzziness intervals in electric potential distribution at t = 2 and z = h/4 for multiple r-levels using TFN and GFN.

A direct TFN–GFN comparison is shown in Figure 10(a)–10(b). At both r = 0.5 and r = 0.9, the TFN bounds are strictly contained within those of GFN, indicating that Gaussian membership functions capture broader uncertainty. Quantitatively, at x = L/2, z = h/4, t = 2, the GFN uncertainty width is roughly 1.5–1.7 times greater than TFN at r = 0.5, and about 1.3 times at r = 0.9. This behaviour is further substantiated by Table 4, both confirming the progressive narrowing of uncertainty with increasing r and the consistent containment of TFN intervals within GFN.

Figure 10.

Comparison of fuzziness intervals in electric potential distribution at the mid-span location (x = L/2) for different r-levels using TFN and GFN.

Table 4.

Bounds of the distribution of electric potential (ϕ(x, t, z: r)) (for the different r values) for x = L/2, z = h/4 and t = 2 in TFN and GFN.

	TFN		GFN
r	$\underset{̲}{ϕ} (x, t, z : r)$	$\bar{ϕ} (x, t, z : r)$	$\underset{̲}{ϕ} (x, t, z : r)$	$\bar{ϕ} (x, t, z : r)$
0.1	2.0417	2.4455	1.7621	2.7251
0.2	2.0641	2.4231	1.8411	2.6461
0.3	2.0866	2.4007	1.8955	2.5918
0.4	2.109	2.3782	1.9399	2.5473
0.5	2.1314	2.3558	1.9794	2.5078
0.6	2.1539	2.3334	2.0168	2.4704
0.7	2.1763	2.3109	2.0541	2.4331
0.8	2.1987	2.2885	2.0937	2.3935
0.9	2.2212	2.266	2.1406	2.3466
1	2.2436	2.2436	2.2436	2.2436

The influence of through-thickness position is illustrated in Figures 11(a)–11(b). On the beam surfaces (z = ±h/2), the electric potential vanishes, while finite non-zero values are observed inside the beam (z = h/4). Importantly, fuzziness is clearly manifested in the interior region, reflecting how uncertainty in initial conditions propagates into the induced electric field. This observation is particularly relevant for piezoelectric sensing and energy harvesting applications, where internal fields govern device efficiency and robustness under uncertainty.

Figure 11.

Comparison of fuzziness intervals in electric potential distribution at x = L/2 for r = 0.5 using TFN and GFN.

Overall, the numerical results clearly demonstrate that the choice of fuzzy number has a significant impact on uncertainty quantification. TFN provides computationally efficient solutions with compact intervals, whereas GFN yields smoother and broader uncertainty bands, making it more suitable for conservative and robustness-oriented designs. These findings are consistent with recent fuzzy dynamic analyses (Tan and Yu, 2023; Wang et al., 2023) and highlight the importance of selecting appropriate fuzzy representations for intelligent piezoelectric devices such as sensors, actuators, and energy harvesters.

Conclusion and future outlook

In this study, the Fuzzy Adomian Decomposition Method (FADM) was employed to investigate the uncertain dynamic behaviour of a transversely isotropic piezoelectric beam. The initial conditions were represented by Triangular and Gaussian fuzzy numbers (TFN and GFN), and uncertainty was incorporated analytically using generalized Hukuhara differentiability and r-cut representations. This approach enabled the computation of fuzzy deflection and electric potential responses in a semi-analytical framework, demonstrating the efficiency of FADM for solving fuzzy partial differential equations.

The comparative analysis revealed that TFN produces compact and computationally efficient intervals, whereas GFN consistently yields 1.5–2.0 times wider and smoother uncertainty bands, providing a more conservative and realistic representation of epistemic uncertainty in smart materials. These results underscore the critical influence of fuzzy number selection in uncertainty quantification and confirm the capability of FADM to model real-world engineering systems with inherent imprecision. The findings have direct implications for the robust design of intelligent piezoelectric devices used in vibration control, sensing, and energy harvesting. Future work may extend this framework to multilayered or nonlinear piezoelectric systems and integrate experimental data to further validate the fuzzy models.

The behaviour identified in this analysis also highlights a modelling aspect that has not been examined in earlier studies on piezoelectric structures: the quantitative impact of membership-function shape on the evolution of uncertainty. This distinction between TFN- and GFN-based representations establishes that fuzzy modelling choices directly influence the interpretation and reliability of dynamic responses, thus providing a new analytical perspective on uncertainty treatment in smart materials.

While the fuzzy modelling framework demonstrates strong interpretability and computational efficiency, several limitations must be acknowledged. The current formulation is restricted to the Euler–Bernoulli beam theory, which neglects shear deformation and rotary inertia, and therefore may not capture the behaviour of short or thick beams. Only the initial conditions were treated as fuzzy variables, although material constants, boundary constraints, and external actions may also contain uncertainty in practical systems. The focus on TFN and GFN excludes other possible membership functions that may exhibit different uncertainty propagation characteristics. Experimental validation is not included, and the comparison with finite element methods is conceptual rather than numerical. These limitations constrain the generalizability of the results largely to slender piezoelectric beams operating under small deformations and scenarios dominated by initial-condition uncertainty.

Despite these constraints, the modelling strategy remains adaptable and may be extended to higher-order beam theories, multilayered or graded piezoelectric structures, fuzzy boundary conditions, and hybrid probabilistic–fuzzy approaches. Incorporating experimental studies in future work would further strengthen the practical relevance and validation of the analytical predictions.

Footnotes

Acknowledgement

The authors sincerely thank the NIT Andhra Pradesh, India, for facilitating a good research environment. Also, convey sincere thanks to RGUKT Nuzvid, Andhra Pradesh, India, for encouraging and providing support in doing research.

ORCID iDs

Anapagaddi Sudhir Kumar

Sudarshan Dhua

Ethical considerations

The authors confirm that no research ethics approval was needed for this study.

Author contributions

The contributions of each author to the manuscript are as follows:

Anapagaddi Sudhir Kumar conceptualization, formal analysis, investigation, methodology, validation, software, visualization, original draft, review, and editing.

Sudarshan Dhua conceptualization, project administration, supervision, writing, review, and editing.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data Availability Statement

This article has no additional data.*

Human participants and/or animals

The study did not include any human or animal subjects.

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(2023) Uncertain support vector machine based on uncertain set theory. Journal of Intelligent and Fuzzy Systems 45(2): 2133–2144. https://doi.org/10.3233/jifs-230935