Dynamic Stress Wave Response of Thin-Walled Circular Cylindrical Shell Under Thermal Effects and Axial Harmonic Compression Boundary Condition

Abstract

The interaction between thermal fields and mechanical loads in thin-walled cylindrical shells introduces complex dynamic behaviors relevant to aerospace and mechanical engineering applications. This study investigates the axial stress wave propagation in a circular cylindrical shell subjected to combined thermal gradients and time-dependent harmonic compression. A semi-analytical model based on Donnell–Mushtari–Vlasov (DMV) shells theory is developed to derive the governing equations, incorporating elastic, inertial, and thermal expansion effects. Modal solutions are obtained to evaluate displacement and stress distributions across varying thermal and mechanical excitation conditions. Empirical Mode Decomposition (EMD) and Instantaneous Frequency (IF) analysis are employed to extract time–frequency characteristics of the dynamic response. Complementary Finite Element Analysis (FEA) is conducted to assess modal deformations, stress wave amplification, and the influence of thermal softening on resonance frequencies. Results reveal that increasing thermal gradients leads to significant reductions in natural frequencies and amplifies stress responses at critical excitation frequencies. The combination of analytical and numerical approaches captures the coupled thermomechanical effects on shell dynamics, providing an understanding of resonance amplification, modal energy distribution, and thermal-induced stiffness variation under axial harmonic excitation across thin-walled cylindrical structures.

1. Introduction

Thin-walled circular cylindrical shells are essential structural elements in the domains of aerospace, mechanical, and civil engineering, where their geometric efficiency and load capacity are critical when subjected to dynamic compression loads. In numerous practical scenarios, shells encounter not only mechanical vibrations and impact forces, but also elevated thermal environments and time-dependent axial boundary excitations. The interaction between stress wave propagation and thermal gradients, in conjunction with harmonic compressive loads, generates intricate coupling effects that significantly affect structural integrity, vibrational stability, and failure mechanisms. When subjected to axial harmonic compression, cylindrical shells exhibit dynamic instabilities that may amplify longitudinal and flexural stress wave components. A novel approach for assessing the natural frequencies of thin-walled cylinders using a dynamic matrix derived from Donnell–Mushtari’s shell theory can be effectively simplified to an eigenvalue problem, making it practical for design applications [1,2]. Hart and Semencha explore stress concentration around openings in cylindrical shells, demonstrating that annular inclusions can reduce stress concentration by 13–35%, depending on the inclusion’s properties [3]. Fan et al. investigate the dynamic response of Q235 thin-walled cylindrical shells under lateral shock loading, revealing that deformation patterns are significantly influenced by impact velocity and shell geometry, with distinct deflection behaviors on the impact and rear sides [4]. Sun et al. focus on the dynamic stability of submerged cylindrical shells under combined hydrostatic and shock loading, identifying critical hydrostatic pressure thresholds that determine stability and potential implosion [5]. Zhang et al. introduce a non-classical model incorporating microstructure and surface energy effects, which predicts higher natural frequencies and smaller deflections for very thin shells compared to classical models [6]. Akhmedov et al. address the vibrations of reinforced viscoelastic cylindrical shells, highlighting the role of reinforcement in enhancing rigidity and strength under dynamic influences [7]. The analysis of the stress-strain state of cylindrical shells under harmonic disturbances, particularly in the context of seismic waves, involves complex algebraic modeling to capture the dynamic interactions and structural responses [8,9]. The impact of vibration characteristics during machining operations, particularly the sensitivity of circumferential shell modes to wall thickness variations, affects surface topography [10,11,12]. Amabili and Moghaddasi study the nonlinear dynamics of cantilevered shells containing fluid, noting that fluid presence alters the shell’s dynamic behavior and reduces circumferential contraction during large amplitude vibrations [13]. The stress-deformed state of such shells, particularly when used as storage containers for granular materials, is significantly influenced by temperature and climatic factors, such as solar radiation and ambient temperature changes. These factors induce a temperature field that can be described using Fourier series, affecting the stress-strain state of the shell without significant temperature variation across its thickness due to the high thermal diffusivity of metals [14]. Nonlinear dynamics, including vibrations of polymeric cylindrical shells, are also affected by temperature, which alters instability regions and the magnitude of kinematic responses, leading to more complex dynamics at higher temperatures [15]. Mathematical models, such as those based on Kirchhoff–Love theory, have been developed to analyze thermoelectricity in shells with micro-heterogeneous structures, allowing for the study of effects like microstructure size on global shell behavior [16,17]. Functionally graded materials (FGMs) and coatings further complicate the thermal response, as seen in shells with Functionally Graded Protective Coatings (FGPCs), which enhance degradation resistance under thermal-vibration ageing conditions [18]. The thermal environment also impacts the free vibration characteristics of shells, with temperature-dependent material properties influencing natural frequencies and dynamic stability [19,20,21]. Additionally, thermal loads, such as those from lateral fires, can destabilize thin-walled tanks, with factors like shell thickness distribution and geometrical imperfections significantly affecting critical buckling temperatures [22]. The axial load can influence phenomena such as bulge initiation and propagation in hyperplastic cylindrical shells, where a localized instability can lead to significant deformation [23]. The use of advanced materials and design modifications, such as polygonal shells, can enhance the axial load-carrying capacity and reduce imperfection sensitivity, offering a potential solution to the limitations of traditional circular cylindrical shells [24]. Moreover, the application of stochastic finite element methods allows for a more accurate prediction of buckling loads by accounting for the randomness of geometric imperfections [25]. Donnell–Mushtari–Vlasov (DMV) shell theory has significant limitations regarding its inadequacy accurately predicting the behavior of structures with complex geometries or of those subjected to high aspect ratios and low numbers of longitudinal and circumferential waves [26]. Additionally, the DMV approximation can lead to the emergence of weakly growing disturbances in fluid-structure interaction problems, such as in the stability analysis of Poiseuille flow in compliant pipes [27]. The theory’s reliance on simplified constitutive relationships and its inability to account for anisotropic material properties further constrain its use in accurately predicting the behavior of shells with complex material characteristics [28].

Although numerous studies have examined the dynamic behavior of cylindrical shells under mechanical or thermal loads, few have addressed the combined influence of time-dependent axial compression and spatially varying thermal fields. Most existing models assume simplified thermal conditions without fully accounting for the interaction between thermal-induced pre-stress and harmonic excitation in thin-walled geometries. The role of thermal softening in altering natural frequencies, stress amplification, and vibrational mode characteristics remains insufficiently characterized. The present study develops a coupled analytical and numerical model using DMV shell theory, time-frequency signal analysis, and finite element modal analysis to investigate the dynamic stress wave response under simultaneous thermal effects and axial harmonic compression. An advanced approach in shell thermodynamics accounts for the interplay between temperature-dependent material properties, spatial thermal gradients, and structural stiffness, leading to thermal pre-stresses that alter stiffness and mass distribution, influence mode interactions, and shift resonance conditions through weak and strong coupling mechanisms.

2. Proposed Modelling of Thin-Walled Circular Cylindrical Shells

In this section, dynamic axial stress wave propagation in thin-walled cylindrical shells, the governing differential equation for axial deformation under thermomechanical effects, is formulated using the DMV shells theory. This classical theory provides a simplified framework for modelling the coupled mechanical and thermal behavior of cylindrical shells under dynamic loading. By incorporating the assumptions of small strains, moderate rotations, and axisymmetric deformation, the DMV model captures essential structural dynamics while remaining mathematically tractable. The axial motion is governed by a second-order partial differential equation that includes elastic restoring forces, inertial effects, and thermal expansion contributions. In particular, thermal fields introduce time-dependent pre-stresses that alter the stiffness characteristics of the shell, thereby affecting the propagation speed, amplitude, and stability of axial stress waves.

Figure 1 presents a three-dimensional schematic of a thin-walled circular cylindrical shell subjected to axial time-varying compressive load P(t), applied uniformly at both the top and bottom boundaries along the shell axis. The applied load P(t) represents harmonic axial excitations, which include both static and dynamic components. Such loading induces axial stress waves, which interact with the shell’s geometric and material properties, as well as the thermal fields present. The system exhibits axial symmetry, as the loading is uniformly distributed over the circular cross sections, justifying the use of axisymmetric assumptions in the analytical model. The configuration resembles the typical cylindrical skin of a cryogenic tank in a launch vehicle during actual flight conditions. The axial displacement u(x,t) satisfies the following coupled partial differential equation for axial motion with thermal expansion:

$$\rho h{\displaystyle \frac{{\mathit{\partial }}^{2}u(x,t)}{\mathit{\partial }{t}^2}}=Eh{\displaystyle \frac{{\mathit{\partial }}^{2}u(x,t)}{\mathit{\partial }{x}^2}}+Eh\alpha {\displaystyle \frac{\mathit{\partial }T(t)}{\mathit{\partial }x}}$$

(1)

The subsequent formulations are derived under the assumptions of a linear thermoelastic domain, where thermal strains are proportional to temperature changes as governed by the generalized Young–Duhamel relationship. Harmonic thermodynamic loads are represented using Fourier series to characterize periodic thermal variations along the shell’s length.

$$P(t)=P_0+P_1\cos (\omega t)$$

(2)

The axial compressive load at both ends x = 0 and x = L is harmonically varying in time, represented by:

$$T(t)=T_0+\Delta T\cos (\Omega t)$$

(3)

Given axial harmonic compression at both ends, the boundary conditions at x = 0 and x = L are expressed as stress conditions:

$${\sigma }_x(0,t)=-{\displaystyle \frac{P(t)}{2\pi Rh}}$$

(4)

$${\sigma }_x(L,t)=-{\displaystyle \frac{P(t)}{2\pi Rh}}$$

(5)

In linear elasticity applied to a thin shell, small strains are assumed, and higher-order terms in displacement gradients are neglected in accordance with the assumptions of DMV theory. The axial strain ε_x in the axial direction can be defined as:

$${\epsilon }_x(x,t)={\displaystyle \frac{\mathit{\partial }u(x,t)}{\mathit{\partial }x}}$$

(6)

Thermal expansion also contributes to the axial strain. If the temperature changes by an increment ΔT(t), the induced thermal strain can be expressed as:

$${\epsilon }_{thermal}(t)=\alpha \Delta T(t)$$

(7)

The total axial strain combining mechanical and thermal contributions can be expressed as:

$${\epsilon }_x={\displaystyle \frac{\mathit{\partial }u(x,t)}{\mathit{\partial }x}}-\alpha \Delta T(t)$$

(8)

Since the axial stress-strain relationship for a cylindrical shell is under linear elasticity, Hooke’s law establishes a proportional relationship between stress and strain for materials that deform elastically within the limits of small strains.

$${\sigma }_x(x,t)=E{\epsilon }_x(x,t)$$

(9)

From Equation (8) into Equation (9), the axial stress function can be expressed as:

$${\sigma }_x(x,t)=E\left({\displaystyle \frac{\mathit{\partial }u(x,t)}{\mathit{\partial }x}}-\alpha \Delta T(t)\right)$$

(10)

Assuming the axial stress-strain relationship for a cylindrical shell is under linear elastic conditions, and considering that the dominant internal force component is axial, the governing equation simplifies accordingly. By applying the boundary conditions at x = 0 and x = L, the system reduces to the following explicit form:

$$E\left({\displaystyle \frac{\mathit{\partial }u(0,t)}{\mathit{\partial }x}}-\alpha \Delta T(t)\right)=-{\displaystyle \frac{P_0+P_1\cos (\omega t)}{2\pi Rh}}$$

(11)

$$E\left({\displaystyle \frac{\mathit{\partial }u(L,t)}{\mathit{\partial }x}}-\alpha \Delta T(t)\right)=-{\displaystyle \frac{P_0+P_1\cos (\omega t)}{2\pi Rh}}$$

(12)

The resolution of the governing equations presumes classical Lagrangian independence between spatial coordinates and time, allowing the separation of variables technique to decouple the spatial and temporal parts of the solution into distinct functional forms. Assume a separable solution form modal Fourier sine series for axial displacement u(x, t):

$$u(x,t)={\displaystyle \sum _{n=1}^{\infty }{U}_n(t)\sin \left(\frac{n\pi x}{L}\right)}$$

(13)

Equation (9) automatically satisfies geometrically symmetric boundary conditions, and the second spatial derivative of u(x,t) can be expressed as:

$${\displaystyle \frac{{\mathit{\partial }}^{2}u(x,t)}{\mathit{\partial }{x}^2}}=-{\displaystyle \frac{n^2{\pi }^2}{L^2}}{\displaystyle \sum _{n=1}^{\infty }{U}_n(t)\sin \left(\frac{n\pi x}{L}\right)}$$

(14)

The thermal field T(t) is a function of time alone, implying that temperature gradient ∂T/∂x along the axial direction effectively vanishes. Substituting (13) into (1) and performing algebraic manipulation, the governing equation becomes:

$$\rho h{\displaystyle \sum _{n=1}^{\infty }\frac{d^2{U}_n(t)}{d{t}^2}\sin \left(\frac{n\pi x}{L}\right)}=-Eh{\displaystyle \sum _{n=1}^{\infty }\left(\frac{n^2{\pi }^2}{L^2}\right)}{U}_n(t)\sin \left({\displaystyle \frac{n\pi x}{L}}\right)$$

(15)

Since the sine function is orthogonal for each mode n, the governing Equation (11) can be expressed as:

$$\rho {\displaystyle \frac{d^2{U}_n(t)}{d{t}^2}}+E\left({\displaystyle \frac{n^2{\pi }^2}{L^2}}\right)U_n(t)=0$$

(16)

Thermal loading then directly modifies the stress boundary conditions rather than the internal forcing function. Natural frequencies ω_n of axial vibration modes are given by:

$${\omega }_n={\displaystyle \frac{n\pi }{L}}\sqrt{{\displaystyle \frac{E}{\rho }}}$$

(17)

The full response includes homogeneous free vibration and particular forced vibration solutions induced by harmonic loading, expressed within the framework of d’Alembert–Helmholtz type equations, leading to a wave representation of the system’s dynamic response.

$$U_n(t)=A_n\cos ({\omega }_{n}t)+B_n\sin ({\omega }_{n}t)+C_n\cos ({\omega }_{n}t)+D_n\sin ({\omega }_{n}t)$$

(18)

Coefficients A_n, B_n, C_n and D_n are determined by applying initial conditions and boundary conditions. Axial dynamic stress distribution is given by combining displacement gradients and thermal strains:

$${\sigma }_z(x,t)=E\left[{\displaystyle \sum _{n=1}^{\infty }\frac{n\pi }{L}{U}_n(t)\cos \left(\frac{n\pi x}{L}\right)}-\alpha \Delta T(t)\right]$$

(19)

The elastic strain energy stored in the shell at time t, accounting for both mechanical deformation and thermal strain due to temperature-induced expansion, can be expressed as:

$$U_s(t)={\displaystyle \frac{1}{2}}Eh{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{\partial }u(x,t)}{\mathit{\partial }x}-\alpha \Delta T(t)\right)}^{2}dx}$$

(20)

The kinetic energy per unit length associated with axial motion can be obtained by integrating the square of the particle velocity over the shell’s length.

$$T_k(t)={\displaystyle \frac{1}{2}}\rho h{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\frac{\mathit{\partial }u(x,t)}{\mathit{\partial }x}\right)}^{2}dx}$$

(21)

The total energy is defined as the sum of the strain and kinetic energy components. It reflects the instantaneous mechanical energy content of the shell segment, and is important for evaluating conservation laws, stability, and modal energy exchange, which is consistent with the Rayleigh–Ritz methodology.

$$E_{total}(t)=U_s(t)+T_k(t)$$

(22)

The mechanical power transmitted through the boundaries at x = 0 and x = L is computed as the product of axial stress and axial velocity evaluated at the shell ends, representing the rate of external work through the axial boundaries.

$$P_{flow}(t)={\left.{\sigma }_x(x,t){\displaystyle \frac{\mathit{\partial }u(x,t)}{\mathit{\partial }t}}\right|}_{x=0}^{x=L}$$

(23)

Table 1. Properties of cylindrical shell subjected to axial load.

Parameter	Symbol	Value	Units
Young’s Modulus	E	68.936	GPa
Poisson’s Ratio	ν	0.3	—
Density	ρ	2700	kg/m3
Shell Thickness	h	0.005	m
Radius of Shell	R	0.5	m
Axial Length of Shell	L	2	m
Thermal Expansion Coefficient	α	2.4 × 10−5	/°C
Structural Damping Ratio	ζ	0.02	—

presents the mechanical, geometric, and thermal parameters defining the cylindrical shell model subjected to axial loading. The parameters serve as input for both the modal and thermomechanical simulations.

3. Dynamic Response of the Analytical Thin-Walled Circular Cylindrical Shell System

The governing differential equations account for axial displacement, thermal expansion effects, and inertial forces under linear elastic assumptions. Modal analysis is employed to identify the natural frequencies and corresponding mode shapes, while frequency response analysis is conducted to assess the system’s sensitivity to excitation parameters.

Figure 2a shows the axial displacement distributions associated with the first six vibration modes. Mode 1 exhibits a single smooth half-wave across the shell length, characteristic of the fundamental axial mode. As the mode number increases, the modal shapes develop additional oscillations, indicating higher spatial frequencies and the presence of more nodal points along the axis. In Figure 2b, the stress response of the shell is shown as a function of axial position under different levels of thermal loading, represented by temperature increments ranging from ΔT = 20 °C to ΔT = 100 °C. As temperature increases, the axial stress distribution becomes more pronounced, with compressive stress intensifying near the fixed end and tensile stress emerging toward the opposite boundary. The stress curves reveal a nonlinear progression, where higher thermal gradients lead to steeper stress gradients and larger deviations from the neutral stress state.

In Figure 3a, the variation of axial stress σ_x with excitation frequency Ω is shown for a series of increasing dynamic load amplitudes P₁ ranging from 10 kN to 50 kN. As the excitation frequency approaches the shell’s natural frequency, sharp peaks emerge in the stress response, indicating resonance. The inset highlights how the magnitude of axial stress increases progressively with higher load amplitudes, while the resonance frequency remains nearly constant. The system exhibits linear frequency behaviour under the given conditions, though the amplitude response is load-sensitive. The stress peaks become more pronounced as the load increases, indicating the shell’s increasing susceptibility to dynamic amplification at resonant conditions. Figure 3b displays the axial displacement response as a function of excitation frequency for various temperature increments ΔT, ranging from 20 °C to 100 °C. The displacement curves show that temperature significantly affects the system’s resonant behaviour. The coupling between mechanical dynamics and thermal behavior manifests implicitly through thermal softening effects, where temperature changes modify the shell’s effective stiffness and natural frequencies. No explicit geometric coupling is introduced in the frequency-domain harmonic response representation. As the thermal gradient increases, the location of the resonance peaks shifts toward lower excitation frequencies, indicating a reduction in structural stiffness due to thermal softening. The predicted downward shift in natural frequencies with increasing thermal gradients is consistent with experimental trends documented in [29].

4. Power Spectral Density Model

The Power Spectral Density (PSD) characterization in such systems is derived from the temporal stress response, often computed through semi-analytical solutions of the shell’s governing equations. The equations are based on DMV shells theory, extended to incorporate thermal pre-stresses and dynamic axial excitation. The resulting stress field in the axial direction exhibits a complex interplay between thermal softening and modal amplification at resonant frequencies.

$$S_{x_i}(f)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{R}_{x_i}(\tau )e^{-j2\pi f\tau }d\tau }$$

(24)

$$P_i(f)={\displaystyle \frac{1}{N{f}_{s}U}}{\left|{\displaystyle \sum _{n=0}^{N-1}{x}_i[n]w[n]e^{-j2\pi fn/f_s}}\right|}^2$$

(25)

$$U={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=0}^{N-1}{w}^2[n]}$$

(26)

$$PS{D}_{bB}^i(f)=10{\log }_{10}\left({\displaystyle \frac{1}{N{f}_{s}U}}{\left|{\displaystyle \sum _{n=0}^{N-1}{x}_i[n]w[n]e^{-j2\pi fn/f_s}}\right|}^2\right)i\in \left\{1,2,\dots n\right\}$$

(27)

Modal resonances, stress wave propagation, and energy exchanges manifest as peaks and patterns in the PSD representation. Peaks in the PSD indicate resonance phenomena, which are influenced by temperature-induced stiffness reduction, and dynamic axial loading. To quantify this behaviour, the stress signal σ_x(t) is transformed into the frequency domain using the Fourier transform, and their energy content is estimated through the PSD.

In Figure 4a, the time-domain waveform of the axial stress σ_x(t) is shown over a time interval of 0.1 s. The waveform exhibits amplitude-modulated structure, indicating the presence of excitation components applied by the harmonic load. The periodic nature and modulation of the signal indicates constructive interference between vibration modes. The amplitude reaches approximately 2.6 MPa, which indicates significant stress magnification. In Figure 4b, the PSD of the axial stress response is presented as a function of frequency for five dominant vibration modes. Distinct resonance peaks are observed at characteristic modal frequencies, each corresponding to one of the axial modes. The peaks indicate the frequencies at which the shell exhibits maximum energy amplification due to resonance. However, the magnitude of each peak in dB/Hz reflects the relative contribution of each mode to the overall stress response, with mode 2 and 4 in particular exhibiting pronounced energy content.

5. Empirical Mode Decomposition and Instantaneous Frequency of the Time Domain Response

The analysis of the time-domain response of a structural system under dynamic loading often reveals complex oscillatory behaviour arising from multiple interacting modes and external excitations. Although the governing equations are linear, EMD and PSD are employed as signal processing techniques to extract modal contributions and to visualize the frequency content of axial stress response arising from the superposition of linear modes. To gain a more detailed understanding of the underlying dynamic content, the signal can be decomposed into its intrinsic oscillatory components using Intrinsic Mode Functions (IMFs). The decomposition can be achieved using Empirical Mode Decomposition (EMD) by isolating narrowband components that retain physical meaning and are suitable for further time–frequency analysis. The EMD decomposes the time-domain of dynamic stress containing multiple oscillatory components resulting from combined thermal and mechanical effects into a finite set of IMFs. Each IMF captures a specific oscillatory mode embedded in the original signal, corresponding to a particular frequency range, and its IF can be computed using the Hilbert Transform. Each IMF satisfies two conditions: (i) the number of extrema and the number of zero crossings in the signal must either be equal or differ at most by one, and (ii) the mean value of the upper and lower envelopes, defined by local maxima and minima respectively, must be zero at every point.

Considering a measured axial stress signal X(t) defined over a time interval t ∈ [0, T], the EMD seeks a representation of X(t) as a finite sum of IMFs, and the residual term can be expressed as:

$$X(t)={\displaystyle \sum _{j=1}^n{C}_j(t)+R_n(t)}$$

(28)

$$F_{j,k}(t)={\displaystyle \frac{E_{\max }(t)+E_{\min }(t)}{2}}$$

(29)

$$H_{j,k}(t)=X(t)-F_{j,k}(t)$$

(30)

$$SD={\displaystyle \frac{{\displaystyle \underset{0}{\overset{T}{\int }}{\left|X(t)-H_{j,k}(t)\right|}^{2}dt}}{{\displaystyle \underset{0}{\overset{T}{\int }}{\left(X(t)\right)}^{2}dt}}}<ϵ$$

(31)

$$R_j(t)=X(t)-C_j(t)$$

(32)

Each extracted IMF C_j(t) is transformed into an analytic signal via the Hilbert Transform, and the instantaneous amplitude a_j(t) and instantaneous phase θ_j(t) can be expressed as:

$$z_j(t)=C_j(t)+i{\widehat{C}}_j(t)$$

(33)

$$a_j(t)=\sqrt{C_j^2(t)+{\widehat{C}}_j^2}$$

(34)

$${\theta }_j(t)={\tan }^{-1}\left({\displaystyle \frac{{\widehat{C}}_j(t)}{C_j(t)}}\right)$$

(35)

The instantaneous frequency profiles f_j(t) obtained from all IMFs constitute the Hilbert spectrum, offering a time–frequency representation of the dynamic content of X(t).

$$f_j(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d{\theta }_j(t)}{dt}}$$

(36)

The decomposition is achieved through a sifting process, in which the following steps are performed iteratively as summarised in Figure 5.

Figure 5 conceptually illustrates the underlying logic of the EMD method, which is designed to adaptively extract meaningful oscillatory modes from complex time-domain signals. The structure of the flowchart emphasizes the recursive nature of the algorithm, where the signal is progressively stripped of its dominant frequency components through a sifting process based on local extrema and envelope construction.

Figure 6 presents the time-domain IMFs and their corresponding IF profiles obtained via Hilbert spectrum analysis of the axial stress signal at the mid-span location x = L/2. The left column shows the temporal evolution of each IMF, while the right column displays the instantaneous frequency content as a function of time. IMF 1 contains high-frequency oscillations with significant amplitude, capturing the dominant high-frequency content embedded in the original signal. Its corresponding IF profile indicates a nearly constant frequency near 4500 Hz, indicating a narrowband oscillation mode with minimal frequency modulation. IMF 2 reveals a moderately high-frequency component with more visible amplitude modulation. The IF for IMF 2 indicates a main oscillation in the range of approximately 1000 Hz, with minor variations over time, pointing to a physically meaningful resonant mode. IMF 3 exhibits a lower frequency signal with smoothly varying amplitude and phase characteristics. The corresponding IF plot shows two dominant spectral ridges below 1000 Hz, clearly separated and temporally localized, indicating modal interference or closely spaced natural frequencies activated in the system’s low-frequency response. IMF 4 presents the lowest-frequency component with significant amplitude variation and fewer oscillations across the time window. The IF spectrum confirms dominant content around 200–600 Hz, and it likely represents the global structural motion with thermal-induced baseline shifts in the axial stress field.

6. Finite Element Modal Analysis

The finite element modal analysis of a thin-walled circular cylindrical shell under axial compression serves as a basis for examining the interaction between thermal and mechanical effects in dynamic conditions. Through this approach, the study explores how finite element model analysis accounts for geometrical and material parameters and characteristics to compute the natural frequencies and associated mode shapes in axial, circumferential, and radial directions. The model captures the full geometric curvature of the shell and accommodates realistic boundary constraints that reflect practical implementation. The finite element analysis explores not only axial displacements but also the coupling between axial, circumferential, and radial deformations by revealing detailed vibrational modes and stress patterns across the shell surface. Particular attention is given to the role of axial and thermal pre-stresses. Thermal effects introduce non-uniform pre-strains arising from temperature gradients that modify the local stiffness distribution across the shell. The variation in stiffness influences the vibrational characteristics of the system, resulting in modifications to the dynamic response that require consideration in the prediction of accurate models.

Figure 7 conceptually illustrates the sequential workflow used in finite element modal analysis for cylindrical shell structures. The process begins with the creation of the geometry, where a computational representation of the thin-walled cylindrical shell is defined according to its physical dimensions and topology. The next stage is meshing, in which the geometry is discretized into finite elements that enable numerical approximation of field variables. Once the shell elements mesh is established, appropriate boundary conditions are imposed to replicate realistic constraints that exclude torsional and bending effects. Loading scenarios are then defined, incorporating axial compression forces and thermal fields to simulate the mechanical and thermal environment experienced by the thin-walled circular cylindrical shells. The engineering problem is then solved using a numerical FFEPlus solver to compute eigenvalues and mode shapes, capturing the shell’s dynamic characteristics under the specified conditions. Finally, post-processing is conducted to extract meaningful results, such as natural frequencies and mode shapes.

The finite element mesh applied in the analysis was configured as a solid mesh using a blended curvature-based meshing algorithm. The mesh quality was set to high, with 16 Jacobian points per element to ensure accurate element distortion assessment. Both the maximum and minimum element sizes were maintained at 100.264 mm, indicating uniform mesh sizing across the domain. The mesh consisted of a total of 18,026 nodes and 8920 elements. The maximum aspect ratio recorded within the mesh was 26.741, although no elements exhibited an aspect ratio below three. A small proportion of elements (5.27%) had an aspect ratio greater than 10. Importantly, no distorted elements were present, confirming that the mesh met acceptable quality standards for structural modal analysis.

Table 2. Material properties of cylindrical shell.

Property	Value
Material Name	1060 Alloy
Model Type	Linear Elastic Isotropic
Failure Criterion	Max von Mises Stress
Yield Strength	27.574 GPa
Tensile Strength	68.936 GPa
Elastic Modulus (E)	68.936 GPa
Poisson’s Ratio (ν)	0.33
Mass Density (ρ)	2700 kg/m3
Shear Modulus (G)	27 GPa
Thermal Expansion Coefficient (α)	2.4 × 10−5/°C

summarizes the essential material properties assigned to the cylindrical shell model. The data define the mechanical and thermal behavior of the structure and serve as the foundation for conducting the finite element analysis. The parameters ensure that the simulation reflects the physical response of the shell under the prescribed loading and thermal conditions.

Table 3. Natural frequencies and mode shapes of cylindrical shell structure.

0 Hz	98 Hz	110 Hz	112 Hz
139 Hz	165 Hz	179 Hz	202 Hz

presents the natural frequencies and associated mode shapes of a thin-walled cylindrical shell structure, obtained through finite element modal analysis. Each column illustrates a vibrational mode at a specific frequency, accompanied by a 3D deformation plot and a top-view projection that reveals the circumferential wave pattern of each mode. The first column, 0 Hz, corresponds to the rigid-body mode, where no deformation occurs, representing the shell’s undeformed equilibrium configuration. The subsequent modes, beginning from 98 Hz, depict the onset of dynamic behavior with increasing complexity in shape and frequency. As the frequency increases, the number of circumferential lobes visible in both the 3D and top-views increases, indicating higher-order vibrational modes. For instance, the mode at 98 Hz shows a moderate number of lobes, while the mode at 202 Hz exhibits a much more complex deformation pattern with significantly more lobes and a higher degree of curvature distortion along the shell circumference. The shape and nature of the modal displacements field exhibit the same principal modal shapes and sequences of phase transitions presented in [30,31], where dynamic divergence was shown to manifest through amplitude oscillations involving similar geometric configurations and mode interactions.

Table 4. Modal displacements and stress deformation of cylindrical shell structure.

presents the modal displacement fields and corresponding stress distributions of a thin-walled cylindrical shell structure under modal excitation. The results are separated into two categories: displacements in the global Cartesian directions UX, UY, UZ, and normal stress components SX, SY, SZ, each visualized over the shell surface. In the top row, the displacement fields are shown in mm. The UX axial direction exhibits the highest magnitude of displacement, peaking at approximately 0.131 mm, with distinct alternating bands of high and low amplitude regions along the axial length. The UY component of circumferential direction displays much smaller magnitudes, with a maximum of roughly 0.0066 mm, indicating that circumferential motion contributes less to the overall dynamic deformation. The UZ radial direction reveals a displacement pattern consistent with radial expansion and contraction modes, with peak values reaching 0.118 mm. The spatial patterns in all three components are characteristic of modal deformation, where different parts of the shell oscillate out of phase depending on the mode shape. In the bottom row, the corresponding stress components are shown in N/m². The SX field of axial stress reaches a maximum value of approximately 2.38 × 10⁶ N/m², indicating that the axial direction experiences significant dynamic loading, particularly in the regions with high displacement gradients. The SY stress of circumferential direction peaks around 1.63 × 10⁶ N/m² and follows a similar banded distribution, aligned with the modal waveforms observed in the displacement plots. The SZ stress of radial direction demonstrates the highest stress concentration, reaching values up to 2.76 × 10⁶ N/m², especially in areas where radial displacement amplitudes are greatest.

7. Conclusions

The dynamic stress wave response of thin-walled circular cylindrical shell subjected to simultaneous axial harmonic compression and thermal loading was investigated using a combined analytical and numerical approach. A semi-analytical model based on DMV shells theory was formulated to characterize the propagation of purely axial stress waves under thermomechanical coupling. The present study addresses weak implicit coupling between thermal and mechanical behaviors, where thermal effects adjust stiffness and natural frequencies without inducing geometric non-linearities. The model accounted for elastic stiffness, inertial effects, and temperature-dependent pre-stress, and its predictions were evaluated using finite element modal analysis and time–frequency signal decomposition techniques. Results show that thermal gradients cause a measurable reduction in natural frequencies. The first axial mode shifted from 112 Hz to below 98 Hz as the temperature increment increased from 20 °C to 100 °C, indicating thermal softening of the shell structure. Finite element modal analysis revealed displacement amplitudes up to 0.131 mm in the axial direction, with corresponding axial stress magnitudes reaching 2.38 × 10⁶ N/m². Power spectral density analysis confirmed pronounced energy concentration at resonant frequencies, with mode 2 and mode 4 exhibiting dominant peaks in the frequency range of 1000–4500 Hz. Empirical Mode Decomposition of the axial stress signal at mid-span identified four dominant intrinsic modes, with IMF1 showing high-frequency content centered near 4500 Hz and IMF4 capturing the low-frequency global motion in the 200–600 Hz range. The axial displacement response also demonstrated a downward shift in resonance frequency as temperature increased, confirming the influence of thermal pre-strain on dynamic stiffness. Future research could focus on designing experimental campaigns to capture dynamic stress wave responses under combined thermal and harmonic loading for direct comparison with the analytical and numerical predictions presented in this study. Explicit modeling of strong coupling effects, including geometric non-linearities and iterative updates of the deformation field of nonlinear material could also be explored.