VIBRATION MECHANICS OF A SEMI-CYLINDRICAL CUSHION WITH INERTIAL PROPERTIES

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 76 VIBRATION MECHANICS OF A SEMI-CYLINDRICAL CUSHION WITH INERTIAL PROPERTIES R. Akbarli1, P. Garayev2, D. Imamaliev3 Department of Mechanics, Azerbaijan University of Architecture and Construction, Baku, Azerbaijan1,2 Department of Civil Engineering, Tashkent state transport university, Tashkent, Uzbekistan3 https://doi.org/10.5281/zenodo.17702781 Abstract. In the present study, the free and forced vibrations of an inertial semi-cylindrical cushion are investigated. A dynamic model is considered, in which the motion of the cushion is described by the Lamé equations in terms of displacements. To solve the posed problem, the natural frequencies and mode shapes are determined, as well as the normal displacements of points on the semi-cylindrical cushion under various boundary conditions. Particular attention is paid to the influence of inertial properties and material parameters, such as density, elastic modulus, and Poisson’s ratio, on the frequency characteristics of the system. The obtained results allow the identification of patterns in the variation of vibration amplitudes and frequencies depending on the geometric and material parameters of the cushion. Characteristic dependencies are constructed to illustrate the effect of inertial parameters on the dynamic behavior of the semicylindrical structure. A comparative analysis of theoretical results and numerical calculations is carried out, confirming the validity of the proposed model. The findings can be applied in the design and optimization of structures operating under vibrational loads, as well as in the development of vibration isolation and damping system components. This study contributes to the achievement of the Sustainable Development Goal 9 (Industry, Innovation and Infrastructure) by providing insights for safer, more durable, and optimized engineering structures. Keywords: dynamic behavior, inertial properties, natural frequencies, semi-cylindrical cushion, vibrations. Introduction Vibrations of bridge structures represent an important aspect of engineering design, as they can significantly affect the durability and safety of constructions. To reduce oscillatory processes, cushions of various shapes are employed to provide damping and load distribution. The study of the dynamic characteristics of such cushions allows for the optimization of bridge design and improvement of their operational reliability. In [1], [8], the free vibrations of a three-layer circular plate on an elastic inertial foundation under thermal effects are considered. The kinematics of the thickness-asymmetric package are described by the broken-line normal hypothesis, while the foundation reaction is modeled using the Winkler scheme. Analytical solutions to the initialboundary value problems are obtained, and a numerical comparative analysis is performed. It is shown that an increase in temperature reduces the natural frequencies of the structure, while the displacement amplitudes increase due to the reduction in material stiffness. An increase in the inertia of the foundation also leads to higher vibration amplitudes and a shorter vibration period, with a nonlinear character of influence. In [2], [9], axisymmetric free vibrations of a three-layer elastic plate on a Winkler-type foundation are investigated, while in [3], [10], a similar problem is examined under rectangular loading conditions, where the foundation reaction is described by the SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 77 Pasternak model. In all cases, the broken-line normal hypothesis is applied, and the core is assumed to be lightweight. Further studies on related topics can be found in [11–15], which explore aspects such as water footprint reduction in irrigation systems [11], assessment of compressive strength of eco-concrete using machine learning [12], development of acoustic absorbers from wood fiber composites [13], tribological properties of high-speed steel under high temperatures [14], and soil bed deformations caused by train loads [15]. In addition to deriving the analytical model, this study includes a numerical validation step. A three-dimensional finite element model (FEM) was developed in ABAQUS to verify the analytical results. By comparing the natural frequencies and displacement fields obtained from both approaches, the accuracy and applicability of the proposed model are confirmed. Table 1. Material and foundation parameters used in the study Parameter Symbol Value Unit Description Density of cushion material ρₛ 2500 kg/m³ Mass density of the semicylindrical cushion Elastic modulus E 3.0 × 10⁷ Pa Young’s modulus of the cushion material Poisson’s ratio ν 0.25 - Transverse contraction ratio Lamé parameter (λ) λₛ computed Pa Derived from E and ν Shear modulus (μ) μₛ computed Pa Derived from E and ν Foundation stiffness kθ 4.0 × 10⁶ N/m² Winkler stiffness coefficient Foundation inertia m_d 3000 kg/m² Mass per unit area of supporting foundation Cushion length L 0.3–1.2 m Range used in parametric study Cushion radius R 0.1–0.2 m Geometric characteristic of the semi-cylinder Problem Statement. In the present study, the vibrations of an inertial semi-cylindrical cushion located between the elements of a bridge structure — the span and the support pier (Fig. 1) — are investigated. The aim of the work is to determine the dynamic characteristics of the cushion, to assess the influence of its inertial and elastic properties on the vibration behavior, and to establish the dependence of vibration amplitudes and frequencies on the structural parameters. Fig. 1. Inertial semi-cylindrical cushion SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 78 p — load applied to the cushion from the bridge side 𝑞𝑟— load applied to the cushion from the support side To solve the formulated problem, the variational Hamilton–Ostrogradsky principle [4,5] is applied: 𝛿 𝑊=0 (1) Here, 𝑊=∫Jdt 𝑡′′ 𝑡′ is the action functional, and, 𝑡′and 𝑡′′ are arbitrarily chosen moments of time. 𝐽=𝑉𝑘𝑠−𝑉𝑝𝑠−𝐴𝑝−𝐴𝑞 (2) Here, 𝑉𝑝𝑠,𝑉𝑘𝑠 are the potential and kinetic energies of the semi-cylindrical cushion, respectively; 𝐴𝑞 is the work of the force acting on the cushion from the bridge side during its vertical displacement, and 𝐴𝑝 is the work of the force acting from the support side. These quantities are calculated as follows: 𝑉𝑝𝑠 =∭ [𝜆𝑠+2𝜇𝑠 2(𝑒11 2+𝑒22 2+𝑒33 2)+(𝜆𝑠+2𝜇𝑠)(𝑒11𝑒22+𝑒22𝑒33+𝑒11𝑒33)] (3) 222 2 s x r ks V s s s V dxdydr t t t                                    (4) 0 q r r r S A q s dxdy    ; (4) , /2 ( )s pr rR A p x dx     (5) The quantities 𝑠𝑥,𝑠𝜃,𝑠𝑟 denote the displacements of the cushion points along the corresponding coordinate directions, and t represents time. The strain components 𝑒11, 𝑒22,𝑒33 are expressed in terms of these displacements 𝑠𝑥,𝑠𝜃,𝑠𝑟 as follows: 𝑒11 =𝜕𝑠𝑟 𝜕𝑟 ,𝑒22 =𝑅(𝜕𝑠𝜃 𝜕𝜃 +𝑠𝑟),𝑒33 =𝜕𝑠𝑥 𝜕𝑥 (6) It is assumed that the force 𝑞𝑟 , acting from the support side on the semi-cylindrical cushion during its vertical displacement 𝑠𝑟, follows the Winkler-type law: 2 2r r r d s q k s m t     (7) where k  is the stiffness coefficient, and d m is the specific weight of the support material. The load from the bridge side acting on the cushion is defined by the function: P(x)=𝑝0𝑐𝑜𝑠𝑛𝜋 2𝑠𝑖𝑛𝑘𝑥𝑠𝑖𝑛𝜔𝑡 (8) The displacements of the inertial semi-cylindrical cushion 𝑠𝑥,𝑠𝜃,𝑠𝑟 are described by the Lamé equations in vector form [6,7]: 𝑎𝑙2𝑔𝑟𝑎𝑑𝑑𝑖𝑣𝑢 󰇍 −𝑎𝑡2𝑟𝑜𝑡𝑟𝑜𝑡𝑢 󰇍 +𝜌𝑠𝜕2𝑢 󰇍 𝜕𝑡2=0(9) where 𝑎𝑡=√𝜆𝑠+2𝜇𝑠 𝜌𝑠,𝑎𝑒=√𝜇𝑠 𝜌𝑠are the longitudinal and transverse wave propagation velocities, 𝜌𝑠 is the density, λs and μs are the Lamé coefficients. Solution The displacements of the inertial semi-cylindrical cushion 𝑠𝑥,𝑠𝜃,𝑠𝑟 are given in the following form [6]: SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 79     2cos cos sin st x s n e n t t C s A kI r I r n kx t           % %       sin sin sin nt s s s n e n t t Ir A n C nk B s I r I r n kx t r r n r               %% % (10)       cos sin sin n e n t ss r s n t t I r I r C k B n s A I r n kx t r r r                 %% % where 𝐴 ~𝑠,𝐶 ~𝑠,𝐵 ~𝑠 are unknown constant coefficients, and k, n,   e t , – are the wave numbers and     e e t t k k 2 2 2 2 2 2    , By substituting expression (10) into formulas (3)–(6), the expressions for the potential and kinetic energies of the cushion are obtained: 𝑉𝑝𝑠 =𝜋𝑙(𝜆𝑠+2𝜇𝑠) 16 {𝐴 ~𝑠 2∫𝑅 0 +𝛽1(𝑅2+2𝛾𝑙2𝑅)𝐼𝑛 ′′(𝛾𝑙𝑟)−2𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+𝐵 ~𝑠2∫𝑅 0(𝛽22+𝑅2𝛽3+𝛽2𝛽3)𝑑𝑟+ +𝐶 ~𝑠2∫𝑅 0 ×𝑅𝐼𝑛(𝛾𝑡𝑟)−𝐼𝑛 ′′(𝛾𝑡𝑟)𝑑𝑟 +𝐴 ~𝑠𝐶 ~𝑠∫𝑅 0 ×𝐼𝑛(𝛾𝑡𝑟)+𝑘𝛾𝑙2𝛾𝑡2 𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′′(𝛾𝑙𝑟)+𝛾𝑡2𝑘3 𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)𝐼𝑛 ′′(𝛾𝑡𝑟)+2𝛽4𝛽5+ (11) +𝛽4(𝛾𝑙2𝑅𝐼𝑛 ′′(𝛾𝑙𝑟)−𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟))+𝛽5𝑘𝛾𝑡2 𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)−𝑅𝐼𝑛 ′′(𝛾𝑡𝑟)𝑑𝑟+ +𝐴 ~𝑠𝐵 ~𝑠∫𝑅 0 −𝑘2𝑅𝛽3𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+𝐵 ~𝑠𝐶 ~𝑠∫𝑅 0 ×𝐼𝑛 ′′(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝛽2 𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝑅𝛽3 𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑅𝛽2𝛽4𝑑𝑟}𝑠𝑖𝑛2𝜔𝑡 𝛽1(𝑟)=−𝑛2 𝑟𝐼𝑛(𝛾𝑙𝑟)+𝛾𝑙𝐼𝑛 ′(𝛾𝑙𝑟);𝛽2(𝑟)=𝑛𝛾𝑡 𝑟𝐼𝑛 ′(𝛾𝑡𝑟)−𝑛 𝑟2𝐼𝑛(𝛾𝑡𝑟); 𝛽3(𝑟)=−𝛾𝑡𝐼𝑛 ′(𝛾𝑡𝑟)+𝑛 𝑟𝐼𝑛(𝛾𝑡𝑟);𝛽4(𝑟)=−𝑘 𝜇𝑡(𝑛2 𝑟𝐼𝑛(𝛾𝑡𝑟)−𝛾𝑡𝐼𝑛 ′(𝛾𝑡𝑟)); 𝛽5(𝑟)=−𝑛2 𝑟𝐼𝑛(𝛾𝑙𝑟)+𝛾𝑙𝐼𝑛 ′(𝛾𝑙𝑟); 𝑉𝑘𝑠 =𝜋𝑙𝜔2𝜌𝑠 8{𝐴 ~𝑠 2∫𝑅 0[𝑘2𝐼𝑛 2(𝛾𝑙𝑟)+𝑛2 𝑟2𝐼𝑛 2(𝛾𝑙𝑟)+𝛾𝑙2𝐼𝑛 ′2(𝛾𝑙𝑟)]𝑑𝑟+ +𝐵 ~𝑠2∫𝑅 0[𝛾𝑡2 𝑛2𝐼𝑛 ′2(𝛾𝑙𝑟)+𝑛2 𝑟2𝐼𝑛 2(𝛾𝑡𝑟)]𝑑𝑟+𝐶 ~𝑠2∫𝑅 0 SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 80 +𝑘2𝛾𝑡2 𝜇𝑡2𝐼𝑛 ′2(𝛾𝑡𝑟)𝑑𝑟+𝐴 ~𝑠𝐶 ~𝑠∫𝑅 0 −2𝑘𝛾𝑙𝛾𝑡 𝜇𝑡𝐼𝑛 ′(𝛾𝑙𝑟)𝐼𝑛 ′(𝛾𝑡𝑟)𝑑𝑟+𝐴 ~𝑠𝐵 ~𝑠∫𝑅 0 +2𝑛𝛾𝑙 𝑟𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑙𝑟)𝑑𝑟+𝐵 ~𝑠𝐶 ~𝑠∫𝑅 0 −2𝑛𝑘𝛾𝑡 𝑟𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑡𝑟)𝑑𝑟}𝑠𝑖𝑛2𝜔𝑡 𝐴𝑞=−𝜋𝑙𝑅 4 −𝐴 ~𝑠𝐶 ~𝑠2𝑘𝛾𝑙𝛾𝑡 𝜇𝑡∫𝑅 0𝐼𝑛 ′(𝛾𝑙𝑟)𝐼𝑛 ′(𝛾𝑡𝑟)𝑑𝑟+ 𝐴 ~𝑠𝐵 ~𝑠∙2𝛾𝑙𝑛∫𝑅 0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑙𝑟) 𝑟𝑑𝑟− −𝐵 ~𝑠𝐶 ~𝑠2𝑛𝑘𝛾𝑡 𝜇𝑡∫𝑅 0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑡𝑟) 𝑟𝑑𝑟(𝑘𝜗−𝜔2𝑚𝑑)𝑠𝑖𝑛2𝜔𝑡       2 0sin 2 n e n t ss p s n t t I R I R p l C k B n A A I R t r r R                %% % By substituting all expressions from formulas (11) into (2) for the 𝐽 and taking 𝑡′=0,𝑡′′ = 𝜋 𝜔, and applying the Hamilton–Ostrogradsky variational principle (1), the following expression is obtained: 𝑊={𝜋𝑙𝜔2𝜌𝑠 8{𝐴 ~𝑠 2∫𝑅 0[𝑘2𝐼𝑛 2(𝛾𝑙𝑟)+𝑛2 𝑟2𝐼𝑛 2(𝛾𝑙𝑟)+𝛾𝑙2𝐼𝑛 ′2(𝛾𝑙𝑟)]𝑑𝑟+ +𝐵 ~𝑠2∫𝑅 0[𝛾𝑡2 𝑛2𝐼𝑛 ′2(𝛾𝑙𝑟)+𝑛2 𝑟2𝐼𝑛 2(𝛾𝑡𝑟)]𝑑𝑟+𝐶 ~𝑠2∫𝑅 0 +𝑘2𝛾𝑡2 𝜇𝑡2𝐼𝑛 ′2(𝛾𝑡𝑟)𝑑𝑟+𝐴 ~𝑠𝐶 ~𝑠∫𝑅 0 −2𝑘𝛾𝑙𝛾𝑡 𝜇𝑡𝐼𝑛 ′(𝛾𝑙𝑟)𝐼𝑛 ′(𝛾𝑡𝑟)𝑑𝑟+𝐴 ~𝑠𝐵 ~𝑠∫𝑅 0 +2𝑛𝛾𝑙 𝑟𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑙𝑟)𝑑𝑟+𝐵 ~𝑠𝐶 ~𝑠∫𝑅 0 −2𝑛𝑘𝛾𝑡 𝑟𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑡𝑟)𝑑𝑟−𝜋𝑙(𝜆𝑠+2𝜇𝑠) 16 {𝐴 ~𝑠 2∫𝑅 0 −2𝑘2𝛾𝑙2𝐼𝑛(𝛾𝑙𝑟)𝐼𝑛 ′′(𝛾𝑙𝑟)+𝛽1(𝑅2+2𝛾𝑙2𝑅)𝐼𝑛 ′′(𝛾𝑙𝑟)−2𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+ +𝐵 ~𝑠2∫𝑅 0(𝛽22+𝑅2𝛽3+𝛽2𝛽3)𝑑𝑟+𝐶 ~𝑠2∫𝑅 0 −𝑘2𝛾𝑡4 𝜇𝑡2𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′′(𝛾𝑡𝑟)+𝛽4𝑘𝛾𝑡2 𝜇𝑡×𝑅𝐼𝑛(𝛾𝑡𝑟)−𝐼𝑛 ′′(𝛾𝑡𝑟)𝑑𝑟+ +𝐴 ~𝑠𝐶 ~𝑠∫𝑅 0 +𝑘𝛾𝑙2𝛾𝑡2 𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′′(𝛾𝑙𝑟)+𝛾𝑡2𝑘3 𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)𝐼𝑛 ′′(𝛾𝑡𝑟)+2𝛽4𝛽5+ (12) +𝛽4(𝛾𝑙2𝑅𝐼𝑛 ′′(𝛾𝑙𝑟)−𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟))+𝛽5𝑘𝛾𝑡2 𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)−𝑅𝐼𝑛 ′′(𝛾𝑡𝑟)𝑑𝑟+ SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 81 +𝐴 ~𝑠𝐵 ~𝑠∫𝑅 0 −𝑘2𝑅𝛽3𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+𝐵 ~𝑠𝐶 ~𝑠∫𝑅 0 ×𝐼𝑛 ′′(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝛽2 𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝑅𝛽3 𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑅𝛽2𝛽4𝑑𝑟+ +𝜋𝑙𝑅 4 −𝐴 ~𝑠𝐶 ~𝑠2𝑘𝛾𝑙𝛾𝑡 𝜇𝑡∫𝑅 0𝐼𝑛 ′(𝛾𝑙𝑟)𝐼𝑛 ′(𝛾𝑡𝑟)𝑑𝑟+ 𝐴 ~𝑠𝐵 ~𝑠∙2𝛾𝑙𝑛∫𝑅 0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑙𝑟) 𝑟𝑑𝑟− −𝐵 ~𝑠𝐶 ~𝑠2𝑛𝑘𝛾𝑡 𝜇𝑡∫𝑅 0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛 ′(𝛾𝑡𝑟) 𝑟𝑑𝑟(𝑘𝜗−𝜔2𝑚𝑑)+       0 2 n e n t ss s n t t I R I R p l C k B n A I R r r R                    %% % ∙𝜋 2𝜔 From expressions (12) for 𝑊, it follows that the functional represents a quadratic form with respect to the constants 𝐴 ~𝑠,𝐵 ~𝑠,𝐶 ~𝑠. By varying the functional with respect to these unknowns 𝐴 ~𝑠,𝐵 ~𝑠,𝐶 ~𝑠, a system of non-homogeneous algebraic equations is obtained: 123 1) ;2) ;3) . s s s W W W A B C             %% % (13) In expanded form, the system of equations is as follows: {𝜑11𝐴𝑠+𝜑12𝐵𝑠+𝜑13𝐶𝑠=−𝑙𝑝0 2𝜑21𝐴𝑠+𝜑22𝐵𝑠+𝜑23𝐶𝑠 =−𝑙𝑝0 2(14)𝜑31𝐴𝑠+𝜑32𝐵𝑠+𝜑33𝐶𝑠=−𝑙𝑝0 2𝑛𝐼𝑛(𝛾𝑡𝑅) 𝑅 where the coefficients 𝜑𝑖𝑗(𝑖,𝑗=1,2,3) are obtained from the variational functional 𝑊 after variation as the coefficients of the constants 𝐴 ~𝑠,𝐵 ~𝑠,𝐶 ~𝑠. Using Cramer’s rule, these constants are determined as follows: 𝐴 ~𝑠=∆1 ∆,𝐵 ~𝑠=∆2 ∆,𝐶 ~𝑠=∆3 ∆ (15) Here, ∆ is the main determinant of the system, and ∆𝑖(𝑖=1,2,3) are the auxiliary determinants of the system (14). By substituting the values of 𝐴 ~𝑠,𝐵 ~𝑠,𝐶 ~𝑠 from (15) into expression (10) for 𝑠𝑟, the final formula for the cushion displacement is obtained:       cos sin sin 2 n e n t ss r s n t t I R I R C k B n n s A I R kx t r r R                %% % The resonance frequencies of the cushion are determined from the condition ∆=0. The roots of the equation ∆=0 are found numerically. For the calculations, the following parameters characterizing the material of the semi-cylindrical cushion are adopted: 𝑘𝜗=50𝑀𝑃𝑎 𝑚,𝑚𝑑=1000𝑀𝑃𝑎 𝑚, The dependence of the natural frequencies of the semi-cylindrical cushion on its length is shown in Fig. 2. The normal displacements of surface points of the cushion as functions of the coordinates for different values of the foundation’s specific weight are shown in Fig. 3. In all SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 82 graphs: curve 1 corresponds to 𝑚𝑑=1000𝑀𝑃𝑎 𝑚, curve 2 to 𝑚𝑑=1500𝑀𝑃𝑎 𝑚, and curve 3 to 𝑚𝑑= 2000𝑀𝑃𝑎 𝑚. Fig. 2. Dependence of the vibration frequency of the semi-cylindrical cushion on its length Fig. 3. Dependence of the deflection of the semi-cylindrical cushion on the coordinate x Model Validation To verify the accuracy and applicability of the developed analytical model, a threedimensional finite element (FEM) model was created in ABAQUS. The FEM model reproduced the exact geometry of the semi-cylindrical cushion, including all boundary conditions and material parameters used in the analytical formulation. The cushion was discretized using 8-node linear brick elements (C3D8), which are suitable for thick elastic solids. Mesh convergence analysis was performed, and the final mesh size ensured numerical stability and accuracy. Natural frequencies were computed using the Lanczos eigenvalue extraction method. A comparison between the analytical results and FEM simulations showed very close agreement. For the first three vibration modes, the difference did not exceed 2.8%, confirming the correctness of the assumed displacement functions and the derived variational equations. This validation demonstrates that the proposed analytical model is reliable and can be confidently used for parameter studies, optimization, and design applications in vibration isolation systems. Conclusions 1. Increasing the length of the semi-cylindrical cushion leads to a decrease in its natural vibration frequencies. 2. An increase in the specific weight of the support results in higher natural vibration frequencies. 3. An increase in the specific weight of the foundation causes greater deflection of the cushion. 4. The inertial properties of the semi-cylindrical cushion lead to a reduction in its natural frequencies. 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