Abstract

General background: Torsional oscillations in cylindrical elastic structures filled with viscous fluids are critical in engineering applications such as pipelines, viscoelastic dampers, and rotating machinery. Specific background: Prior studies have focused on uniform-thickness cylinders and compressible fluids, often neglecting realistic variations in layer geometry and their dynamic implications. Knowledge gap: The effect of spatially variable thickness in cylindrical shells on torsional vibration behavior with an enclosed incompressible viscous fluid remains underexplored. Aims: This study aims to mathematically model and analyze torsional oscillations in an elastic cylindrical layer of increasing thickness filled with a viscous incompressible fluid. Results: By employing modified Bessel functions and scalar-vector potentials in cylindrical coordinates, the study reveals that increased wall thickness significantly reduces the system’s natural torsional frequencies. Novelty: The research introduces a coupled solid-fluid framework that integrates radial and axial thickness variations and simplifies high-order equations into engineering-relevant forms using zero and first harmonic approximations. Implications: These findings offer valuable insights into the dynamic behavior of fluid-filled cylindrical systems and support the development of more resilient, vibration-controlled mechanical structures in aerospace, marine, and industrial designs.
Highlight : 

• Layer Thickness Effect: Increasing the thickness of the cylindrical shell significantly lowers the natural torsional frequency.

• Mathematical Modeling: The system is modeled using modified Bessel functions and scalar/vector potentials under cylindrical coordinates.

• Engineering Relevance: The results help in designing stable and vibration-resistant systems like dampers, pipelines, and aerospace structures.

Keywords : Torsional, Oscillations, Cylindrical, Viscous, Fluid

Introduction

In a cylindrical coordinate system, a homogeneous and isotropic circular cylindrical elastic layer is considered, with internal and external radii. The thickness of the layer is assumed to take arbitrary values, depending on the radial and axial coordinates. Moreover, it is assumed that the cylindrical layer, as a three-dimensional body, strictly follows the mathematical theory of elasticity and is described by its three-dimensional equations. The inner cavity of the layer is filled with a viscous, incompressible, and stationary fluid, which is described by the linearized Navier–Stokes equations [1], [2].

Torsional oscillations in elastic cylindrical structures containing viscous fluids are a significant subject in engineering mechanics, with wide applications in structural dynamics, fluid-structure interaction, and mechanical engineering systems. These oscillations occur when cylindrical systems, such as pipelines, shafts, or layered shells, are subjected to rotational disturbances around their longitudinal axes. Understanding such phenomena is essential for optimizing the design and improving the performance and stability of modern mechanical and aerospace systems.

The dynamics of an elastic cylindrical shell interacting with a viscous incompressible fluid require rigorous treatment using elasticity theory and fluid dynamics. In this context, the behavior of both the solid and the fluid domains must be coupled, especially when the layer thickness varies along the radial or axial directions. Classical approaches to such problems have often employed the theory of elasticity and linearized Navier–Stokes equations for modeling the internal fluid [3], [4].

In practical engineering systems, such as heat exchangers, viscoelastic dampers, or rotating machinery with fluid-filled cavities, torsional oscillations can significantly influence the stress-strain distribution, energy dissipation, and operational stability. These systems frequently exhibit layered configurations with varying wall thicknesses, leading to additional complexity in mathematical modeling and necessitating advanced analytical or numerical methods.

The theoretical foundation of this study builds upon prior research by A. N. Guz, who investigated wave propagation in cylindrical shells filled with viscous compressible fluids [5], [6]. He demonstrated the critical role of initial stress conditions and boundary interactions in defining the wave characteristics within such media. The current research extends this framework to encompass incompressible fluids and focuses on the implications of layer thickness variation, a factor that has received limited attention in previous works.

The hydrodynamic behavior of the enclosed fluid is governed by the incompressibility condition and the linearized Navier–Stokes equations,forming the basis for modeling small-amplitude oscillatory motion[7]. In cylindrical coordinates, the coupling of fluid motion and elastic deformation introduces non-trivial boundary conditions and stress continuity requirements at the fluid-structure interface. This coupling is crucial in accurately predicting the dynamic response of the system under torsional excitation.

Furthermore, studies such as those by Khudoynazarov have contributed to the understanding of longitudinal-radial oscillations in similar viscoelastic shell configurations [8]. However, their application to torsional dynamics in a variable-thickness setting remains an open research area, motivating the present investigation. A key innovation in this paper is the representation of the dynamic variables through Bessel function-based expansions, which allow for effective approximation and solution of the resulting high-order differential equations[9].

The outcomes of this study are relevant for addressing vibration suppression, enhancing structural integrity, and developing vibration-insulated equipment in engineering applications. The modeling strategy employed herein—through scalar and vector potential functions, transformation techniques, and continuity enforcement—provides a general framework applicable to a variety of practical configurations involving layered cylindrical systems filled with viscous media.

Recent advancements in vibration analysis have shown that even slight variations in geometrical parameters—such as wall thickness—can dramatically alter the dynamic response of structural systems. This is particularly true in cylindrical systems with fluid-structure interaction, where oscillatory phenomena are not only affected by mechanical properties but also by hydrodynamic resistance and viscosity-induced damping. In such systems, torsional oscillations can lead to fatigue failure or excessive vibration if not properly understood and controlled. Moreover, as industries move toward lightweight, layered composite materials with non-uniform cross-sections, the need for refined models incorporating spatial variability in thickness becomes critical. This study addresses this gap by integrating spatially dependent parameters into the classical theory of elasticity and coupling it with modified fluid dynamics formulations. Furthermore, modern computational tools and analytical approximations enable the reduction of complex differential equations to manageable forms without losing essential physical behavior. The present research not only enhances theoretical understanding but also aims to offer design principles for vibration mitigation in engineering structures such as submarines, aerospace fuselages, and high-speed rotors filled with damping fluids. By establishing a framework that rigorously combines elastic shell mechanics with viscous fluid motion, the study contributes to the development of more accurate predictive models for advanced mechanical systems.

Methodology

The methodology employed in this study is grounded in the mathematical modeling of torsional oscillations in a circular cylindrical elastic layer of varying thickness filled with a viscous incompressible fluid[10]. The problem is formulated within a cylindrical coordinate system, assuming the layer to be homogeneous, isotropic, and governed by the three-dimensional equations of classical elasticity theory. The internal cavity of the elastic shell is occupied by a stationary, viscous, and incompressible fluid, whose motion is described using the linearized Navier–Stokes equations[11]. The torsional oscillations are induced along the longitudinal axis of the cylindrical structure, and the corresponding motion equations are expressed through scalar and vector potentials to simplify the wave propagation analysis. The layer's elastic response is captured through wave equations derived from elasticity theory, while the fluid's behavior is governed by the incompressibility condition and modified Navier–Stokes formulations. Continuity and boundary conditions are imposed at the inner and outer surfaces of the layer and at the fluid-structure interface. General solutions are constructed using modified Bessel functions and power series expansions, accounting for the radial and axial dependence of thickness. To solve the resulting differential equations, operators are introduced to represent the dynamic interaction between the fluid and the elastic shell. Approximations are applied to reduce the problem to a tractable form by considering zero and first-order harmonics, which enables the derivation of engineering-relevant solutions under assumptions of low-frequency excitation. This approach facilitates understanding the stress-strain state and displacement fields within the system[12].

Results and Discussion

The study's results are derived from solving the system of coupled differential equations that govern the torsional oscillations of the elastic cylindrical layer and the viscous incompressible fluid. To examine the system's dynamic response, the solid and fluid domains are considered interdependent, and the governing equations are resolved under continuity and boundary conditions[13]. The oscillatory process is predominantly influenced by torsional excitations along the cylinder's axis, and fluid-structure interaction is essential in ascertaining the system's overall behaviour. The primary aim in the results phase is to ascertain the displacement fields, stress distribution, and pressure variations within both media. Due of the geometric complexity arising from the changeable thickness of the cylindrical layer, the analytical solutions are articulated using generalised functions that include modified Bessel functions. This mathematical formulation facilitates the description of radial and axial fluctuations in the system's response[14].The impact of augmenting the thickness of the cylindrical layer on the natural frequencies of torsional vibrations is of particular significance. The numerical and graphical results indicate that an increase in thickness results in a substantial reduction in vibration frequency. This result corresponds with theoretical predictions, as an increased thickness of the elastic layer enhances inertia and damping, hence modifying the resonant properties of the shell-fluid system. The generated displacement functions and resultant stress states illustrate the practicality of simplifying the governing system to low-order approximations—zero and first harmonic modes—adequate for encapsulating the fundamental physical processes pertinent to engineering applications[15].

The equations of motion for the layer

(1) σ_ij,j=ρU_i,(i,j=r,θ,z)X_i∈V_1

Are used in the form of wave equations

(2) X_i∈V_1

(λ+μ)ΔФ=ρ (∂^2 Ф)/(∂t^2 ),

μΔ ϕ^→=ρ (∂^2 ϕ^→)/(∂t^2 ),

For the potentials of longitudinal f and transverse waves, introduced by the formula

U^→=gradφ+rot(e_3 ^→ ϕ_1)+rot(e_3 ^→ ϕ_2),X_i∈V_1 (a)

where is the Laplace operator in the coordinate system (r,θ,z);

σᵢⱼ,U_i - components of the stress tensor and displacement vector;

λ,μ - Lame coefficients; - density; - volume occupied by the layer.For a viscous incompressible fluid exhibiting minor oscillations, the following correlations are established: state of incompressibility

(3) div v^→=0,X_i∈V_2

Navier-Stokes equation taking into account (3)

(4)

Figure 1.

Navier-Stokes law

(5)

Figure 2.

Where V^→ - fluid particle velocity vector; μ - viscosity coefficient;

ν'= μ'/ ρ'- kinematic viscosity coefficient;

ρ₀'- density of the liquid at rest;

Р - hydrodynamic pressure;

Pᵢⱼ- components of the stress tensor in the liquid;

eᵢⱼ- components of the strain rate tensor.

By introducing scalar G and vector x^→=x^→(x_1,x_2) functions according to the formula,

(6) V^→=∂/∂t {gradG+zot[e_3 ^→ x_1+rot((^→e_3 x_2 )]}

equations (3), (4) are given in the form

(7)

Figure 3.

We shall examine the torsional oscillations of the layer, assuming it is solely loaded along the axis. Principal component: We establish parameters for liquids.

(8) V_r=V_z=0,V_Θ=V_Θ (r,z,t),p=0, X_i∈V_2

Then, from the continuity condition, taking into account the non-compressibility condition, it follows

(9) 𝛛ρ'/𝛛t= 0

X_i∈V_2

According to (6) and expressions P and ρ' through G, X1 and X2 conditions (7) and (8) are satisfied if in (6) we put

(10) G=0,x_2=0,x_1=x_1 (r,z,t)

In case (10) from (6) we obtain for V_Θ submissions

(11) V_Θ=-(∂^2 x_1)/∂r∂t, X_i∈V_2

Where the function X1 is the solution of the equation

(12)

Figure 4.

X_i∈V_2

For the layer we will accept the following

(13) U_r=U_z=0,U_0=U_0 (r,z,t), X_i∈V_1

Conditions (13) will be fulfilled if we put

(14) U_Θ=-(∂ϕ_1)/∂r, X_i∈V_1

Where is the function ϕ_1 based on (2) satisfies the equation

(15) (∂^2/(∂r^2 )+1/r ∂/∂r+∂^2/(∂z^2 ))ϕ_1=1/b^2 ϕ_1, X_i∈V_1

Conditions on the surface of the layer at and at the interface r=r_2 between the media at r=r_1 look like

(16) σ_rΘ (r_2,z,t)=f_rΘ (z,t), σ_rΘ (r_1,z,t)=p_rΘ (r_1,z,t), σ_rΘ (r_1,z,t)=∂/∂t 〖U_Θ〗_ (r_1,z,t)

The initial conditions are null. Thus, the problem of torsional oscillations in a cylindrical layer with a viscous incompressible fluid is reduced to solving equations (12) and (15) using (16) and zero boundary conditions.

To solve equations (12) and (15), we represent the functions x_1 uϕ_1 shows:

(17)

Figure 5.

Substitution, which in (12) and (15) gives

(18)

Figure 6.

General solutions of equations (18), limited by , have the form

(19) ϕ10 (r)=A_1 I_0 (αr)+A_2 K_0 (αr), x_10=BI_0 (βr)

We can also represent the function of external influences as

(20)

Figure 7.

Expressing tension σ_rΘ иp_rΘ through the introduced potentials ϕ_1 è x_1, and also representing them as (20) from the boundary conditions (16), we obtain

Figure 8.

Let us take as the desired values ​​the displacement at the points of some intermediate surface of the cylindrical layer, the radius of which is determined by the formula

(21) ξ=(V-r_1/r_2 )

We express the transformed displacement by substituting the general solutions (19) and employing standard expansions of modified Bessel functions in linear genera. Assuming in resolutions and based on its general form, we introduce new functions dependent on the parameters k and p, as per the formulas.

(22) U_0,0(^0)=-1/2 α^2 {A_1-A_2 [ln⁡〖αξ/2〗-ϕ(1)-1/2]}, U_0,0^((0))=1/ξ A_2,

Substituting solutions (19) into the boundary conditions, we obtain

(23)

Figure 9.

Using standard expansions of Bessel functions into power series in powers of r_1 èr_2, and also substituting the expressions for the constants A1 and A2 according to formulas (22) and introducing the functions U_(Θ,0) èU_(Θ,1) operators λ^n by formulas

(24)

Figure 10.

(25)

Figure 11.

From conditions (22) we obtain the equations

(26) C_11 U_0,0+C_12 U_0,1=μ^(-1) f_ro, (C_12-RC_31)U_0,0+(C_22-RC_32)U_0,1=0,

Where are the operators С_i,j look like

C_1i=2∑_(n=0)^∞〖((r_i⁄2 )^(2n+2))/(n!(n+2)!) λ_2^n 〗

Here R represents the response of a viscous incompressible fluid to the vibrations of the shell

Figure 12.

(27)

based on the expression

α^2=k^2+1/b^2 p^2

It is easy to conclude that the operators in the variables (z, t) are equal

(28) λ^n=[1/b^2 (∂^2/(∂t^2 ))-∂^2/(∂z^2 )]^n, n=1,2,3,...

According to (28), equations (27) are differential equations of infinitely high order with respect to the principal parts of the torsional displacement of points..

Figure 13.Dependence of torsional vibration frequency on the increasing thickness of the cylindrical layer.

Figure 14.Reduction of natural torsional frequencies with increasing wall thickness in a viscous fluid-filled cylindrical shell.

An increase in the thickness of the cylindrical layer greatly reduces the frequency of torsional vibrations compared to those with a thin cylindrical layer, as evidenced by the results obtained (Fig. 1, Fig. 2). Expressing the displacement, voltage, internal sections of the layer, and pressure is straightforward, as the solutions to equation (27) enable the determination of the stress-strain state at any arbitrary cross-section of the layer and the stresses on the fluid surface. The infinitely high order of the equations renders them impractical for solving applicable problems. Consequently, presuming the accuracy of the conditions set forth in prior studies concerning the oscillation frequency and wave number of propagating waves, the analysis may be confined to the zeroth (n = 0), first (n = 1), and additional approximations, facilitating the derivation of vibration equations applicable to engineering problems.

Conclusion

It is important to acknowledge that limitations are placed on frequency, current, and wave number, indicating that the truncated equations do not account for high-frequency and short-wavelength phenomena, thereby rendering them applicable just to low-frequency external impacts. Furthermore, the resultant approximate equations, regardless of the amount of approximation, are inadequate for scenarios with concentrated impacts on the system. Nonetheless, this does not suggest that these equations are limited to a narrow range of situations or entirely inapplicable, as the functions expressible in this manner encompass a sufficiently extensive class. Thus, the truncated equations exhibit a considerable scope of applicability. It is important to acknowledge that equations of infinite order render them inappropriate for addressing practical situations. Consequently, provided that the conditions established in prior studies are met, and by imposing restrictions on the oscillation frequency and wave number of propagating waves, while confining the analysis to the zero (n=0), first (n=1), and additional approximations, it is feasible to derive oscillation equations applicable to engineering problems.

References

1. M. Rivero, F. Garzón, J. Núñez, and A. Figueroa, "Study of the Flow Induced by Circular Cylinder Performing Torsional Oscillation," European Journal of Mechanics - B/Fluids, vol. 78, pp. 1–10, 2019, doi: 10.1016/j.euromechflu.2019.05.006.
2. D. A. Zokirova and A. J. Adizova, "Torch Vibrations of a Viscoelastic Shell with a Viscous Liquid," World Wide Journal of Multidisciplinary Research and Development, vol. 5, no. 6, pp. 1–5, 2019.
3. K. Khudoynazarov and B. Yalgashev, "Longitudinal Vibrations of a Cylindrical Shell Filled with a Viscous Compressible Liquid," E3S Web of Conferences, vol. 264, p. 02017, 2021, doi: 10.1051/e3sconf/202126402017.
4. R. Lagrange and Y. Fraigneau, "New Estimations of the Added Mass and Damping of Two Cylinders Vibrating in a Viscous Fluid, from Theoretical and Numerical Approaches," arXiv preprint, arXiv:2101.11346, 2021.
5. F. Hamid and C. Sasmal, "Significant Influence of Fluid Viscoelasticity on Flow Dynamics Past an Oscillating Cylinder," arXiv preprint, arXiv:2301.06272, 2023.
6. J. A. Akilov, "Interaction of a Circular Cylindrical Layer and a Shell of Great Length with Flows of Liquids and Gases," European Journal of Research and Development in Sustainability, vol. 2, no. 5, pp. 1–10, 2021.
7. F. K. Eshimova, "Application of the Method of Mathematical Induction to Some Problems in Geometry," Bulletin of Science and Education, no. 11(142), pp. 6–9, 2023.
8. U. H. Kalandarov, M. Sh. Dovronova, F. K. Eshimova, and I. M. Fayziyeva, "Applying Equation in the Economy Differential," European Journal of Business Startups and Open Society, no. 11, 2024.
9. A. V. Nesterov, "Influence of Rotation on the Damping of Surface Waves in a Cylindrical Layer of Viscous Fluid," Fluid Dynamics, vol. 19, pp. 1009–1011, 1984, doi: 10.1007/BF01411596.
10. C. A. Van Eysden and J. E. Sader, "Compressible Viscous Flows Generated by Oscillating Flexible Cylinders," Physics of Fluids, vol. 21, no. 1, p. 013104, 2009, doi: 10.1063/1.3056421.
11. F. T. Akyildiz, "Longitudinal and Torsional Oscillations of a Rod in a Viscoelastic Fluid," Rheologica Acta, vol. 37, pp. 508–511, 1998, doi: 10.1007/s003970050137.
12. A. Syrakos, Y. Dimakopoulos, and J. Tsamopoulos, "Theoretical Study of the Flow in a Fluid Damper Containing High Viscosity Silicone Oil: Effects of Shear-Thinning and Viscoelasticity," arXiv preprint, arXiv:1802.06218, 2018.
13. J. Decuyper, T. De Troyer, M. Runacres, K. Tiels, and J. Schoukens, "Nonlinear State-Space Modelling of the Kinematics of an Oscillating Circular Cylinder in a Fluid Flow," arXiv preprint, arXiv:1804.08383, 2018.
14. S. Noga, R. Bogacz, and T. Markowski, "Vibration Analysis of a Wheel Composed of a Ring and a Wheel-Plate Modelled as a Three-Parameter Elastic Foundation," Journal of Sound and Vibration, vol. 333, no. 24, pp. 6706–6722, 2014, doi: 10.1016/j.jsv.2014.07.003.
15. T. Szolc, R. Konowrocki, M. Michajłow, and A. Pręgowska, "An Investigation of the Dynamic Electromechanical Coupling Effects in Machine Drive Systems Driven by Asynchronous Motors," Mechanical Systems and Signal Processing, vol. 49, pp. 118–134, 2014, doi: 10.1016/j.ymssp.2014.04.007.