Semi-Analytical Transverse Vibration Analysis

• The paper presents a rigorous semi-analytical approach that enforces boundary conditions via series expansions, ensuring accurate prediction of frequency spectra and mode shapes.
• It reduces complex three-dimensional PDEs to coupled ODE systems, enabling detailed analysis of displacement and stress fields across varying geometries.
• The model benchmarks classical theories by quantifying higher-order thickness effects and guiding design optimizations in aerospace, civil, and mechanical applications.

A semi-analytical model for transverse vibration analysis refers to solution methodologies that rigorously satisfy governing differential equations and boundary conditions for vibrating continua (such as plates, beams, strings, or shells) by combining analytical representations (series expansions, mode shapes, separation of variables) with auxiliary mathematical or numerical techniques (such as asymptotic expansions, variational principles, or reduced-order discretizations). These models capture essential three-dimensional or nonlinear effects often omitted in classical low-order theories, providing benchmark solutions for frequency spectra, mode shapes, and stress/displacement fields, as well as quantitative predictions for parameter-dependent dynamic responses.

1. Series Representations and Boundary Condition Satisfaction

A defining feature is the representation of solution fields (displacements, stresses) as double or triple trigonometric series (e.g., Fourier or sine/cosine expansions), which guarantee satisfaction of the prescribed boundary conditions on all edges or faces. For simply supported homogeneous rectangular plates, the displacement fields are expanded as

$$\begin{aligned} u(x,y,z,t) &= \sum_{mn} U_{mn}(z) \cos(a_m x) \sin(B_n y) e^{i\omega_{mn}t}, \ v(x,y,z,t) &= \sum_{mn} V_{mn}(z) \sin(a_m x) \cos(B_n y) e^{i\omega_{mn}t}, \ w(x,y,t) &= \sum_{mn} W_{0,mn} \sin(a_m x) \sin(B_n y) e^{i\omega_{mn}t}, \end{aligned}$$

where %%%%1%%%%, $B_n = n\pi/b$ for integers $m,n$ (Batista, 2010). This formulation automatically enforces vanishing displacement and in-plane stresses at the boundaries. For non-classical or elastic boundary conditions, auxiliary sine series are introduced to capture edge behavior and improve convergence, as in recent plate sound radiation models (Deng et al., 16 Sep 2024).

2. Derivation and Structure of Governing Equation Systems

Semi-analytical models are derived by substituting series forms into fully three-dimensional elasticity or beam equations, often with specialized physical restrictions (such as transversely inextensible kinematics). The substitution reduces the initial PDEs to systems of coupled ODEs (for unknown amplitudes or mode functions) in the through-thickness direction or other coordinate(s):

$$\frac{d^2 U_{mn}}{dz^2} - \alpha U_{mn} + \beta V_{mn} = 0, \qquad \frac{d^2 V_{mn}}{dz^2} - \gamma V_{mn} + \delta U_{mn} = 0,$$

with coefficients $\alpha, \beta, \gamma, \delta$ depending on geometry, material, and modal wavenumbers (Batista, 2010). Solutions are expressed in hyperbolic functions of a thickness-dependent frequency parameter, $K_{mn} = (h/2)\sqrt{a_m^2 + B_n^2}$. Boundary conditions on major faces (e.g., $\sigma_z$ vanishing at $z = \pm h/2$) are used to determine integration constants.

For multi-layered spheres and shells, spherical harmonics are used for angular coordinates, with the radial variable discretized by finite elements to produce a decoupled eigenvalue problem for each angular mode (Gallezot et al., 2020).

3. Displacement and Stress Field Prediction

Once amplitudes are determined, the full displacement and stress fields throughout the structure are recoverable by back-substitution, ensuring correct three-dimensional distribution of strains and stresses. For transversally inextensible plates, the out-of-plane displacement $w$ is independent of $z$; in-plane displacements $u$, $v$ vary across thickness, typically via terms like

$$U_{mn}(z) = \frac{h W_{0,mn}}{2K_{mn}} \frac{\sinh(K_{mn} z^2)}{\cosh(K_{mn}/2)}.$$

Stress components employ standard constitutive relations:

$$\sigma_x(x, y, z, t) = \frac{E}{1 - \nu^2} \left[ \frac{\partial U_{mn}}{\partial x} + \nu \frac{\partial V_{mn}}{\partial y} \right] \sin(a_m x) \sin(B_n y) e^{i\omega_{mn} t},$$

with analogous expressions for $\sigma_y$, $\tau_{xy}$, etc. (Batista, 2010).

4. Asymptotic Expansions and Approximation of Thin Plate Limits

For thin plates ($h/a \ll 1$), solutions in terms of hyperbolic functions are expanded into power series of thickness ratio $K_{mn}$:

$$\sinh(K_{mn}/2) \approx K_{mn}/2 + (K_{mn}/2)^3/6, \qquad \cosh(K_{mn}/2) \approx 1 + (K_{mn}/2)^2/2,$$

yielding asymptotic corrections to amplitudes and frequencies:

$$W_{0,mn} \simeq W_{0,mn} \left[ 1 - \frac{1}{6} K_{mn}^2 + \mathcal{O}(K_{mn}^4) \right],$$

which recover two-dimensional (e.g., Reissner) plate theory in leading order and quantify higher-order thickness effects. This also enables direct comparison with first and higher-order plate theories in benchmark studies (Batista, 2010).

5. Frequency Factor Determination and Parameter Dependence

Characteristic vibration frequencies are obtained by imposing face boundary conditions, leading to transcendental equations for modal frequencies. The normalized frequency factor,

$$\lambda_{mn} = \omega_{mn} \left( \frac{a^2}{2} \right) \left( \frac{h}{P} \right)^{1/2},$$

is tabulated for various ratios of thickness to width ($h/a=0.01, 0.1, 0.2, 0.4$) and length to width ($b/a$). As thickness increases, frequency factors rise, reflecting increased shear deformation contribution; aspect ratio also strongly affects the natural frequencies. These factors serve as benchmarks for plate design and allow error quantification in approximate (2D) theories (Batista, 2010).

6. Comparative and Benchmark Role of Semi-Analytical Solutions

Direct comparison between semi-analytical three-dimensional solutions and lower-order theories is a haLLMark of these models. For example, Table 5 in (Batista, 2010) shows that three-dimensional results provide more accurate predictions for higher ($h/a$) ratios than the first-order plate theory (FOPT) and higher-order plate theory (HOPT). The explicit expressions facilitate code-based verification, parametric studies, and guide the development of reduced-order models for engineering analyses.

7. Extension, Limitations, and Applications

Semi-analytical models have been extended to accommodate various complexities: nonlocal effects (for nanostructures) (Mustapha, 2015), arbitrary boundary and interface conditions (auxiliary sine/cosine series) (Deng et al., 16 Sep 2024), geometric and material nonhomogeneities (Kumar, 2018), nonlinearities (Dalir, 2019Varghaei et al., 2019), viscoelastic or piezo-active elements (Varghaei et al., 2019Zuyev et al., 21 Jun 2024), crack-induced disturbances (Rezaee et al., 2023), and soil-structure interaction (Banushi, 30 Jul 2025Banushi et al., 21 Sep 2025).

Limitations include assumptions regarding boundary conditions, geometric linearity (except explicit nonlinear models (Dalir, 2019)), and material symmetry, as well as computational scalability for high modal densities or complex geometries. Nonetheless, semi-analytical models serve as gold standards for validation, design optimization, and benchmarking of more approximate or numerical methods in vibration and dynamic response analysis across aerospace, civil, mechanical, and materials engineering.

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The semi-analytical model for transverse vibration analysis combines mathematically rigorous representations of vibrating continua with analytical and computational techniques to accurately characterize displacement and stress fields, modal frequency spectra, and the influence of system parameters, providing robust and reliable benchmarks for both theoretical understanding and practical engineering applications.