Model-Based Fatigue Life Prediction of Hydraulic Shock Absorbers Equipped with Clamped Shim Stack Valves

Abstract

In modern shock absorber development, the fatigue durability of shim-based clamped valve systems remains a critical factor influencing both performance and operational safety. In this study, the authors extend their previous research achievements by developing a fatigue life prediction methodology that integrates an established finite element framework with a strength-based fatigue model incorporating experimentally derived and validated Wöhler characteristics of the metal alloy used in the valve shims. The focus of this work is the validation of the proposed methodology for hydraulic shock absorbers equipped with shim stack valve systems, supporting the virtual pre-selection of valve configurations during the OEM design process. This approach enables substantial reductions in experimental testing and facilitates cost-effective development under realistic operating conditions. To address random-amplitude loading scenarios, the rainflow-counting algorithm was employed to convert complex load histories into equivalent constant-amplitude cycles, thereby accurately capturing material memory effects associated with stress–strain hysteresis. Experimental validation was conducted using a high-performance servo-hydraulic load frame tester. The validated model demonstrated a prediction uncertainty of 46% for random-amplitude lifetime estimation.

1. Introduction

The continuous evolution of vehicle dynamics and passenger comfort expectations has led to the intensified development of virtual validation methodologies in hydraulic shock absorber design [1,2,3,4,5,6,7,8]. One of the key elements influencing the durability and performance of hydraulic shock absorbers are shim-based valve systems [5], which regulate damping forces and are particularly susceptible to fatigue-induced damage due to cyclic loading. Shim stack valves are fundamental to the damping characteristics of modern hydraulic shock absorbers, providing tunable and highly nonlinear force–velocity relationships [2,9,10]. These valves operate by the controlled deflection of thin, flexible shims under fluid pressure, thereby regulating flow through orifices. The repetitive flexing of these shims, coupled with cyclic stresses on other structural components such as the piston rod, cylinder, and supporting points, makes fatigue a primary failure mode. Accurate prediction of fatigue life is crucial for ensuring both performance longevity and safety [11].

In this context, the present work constitutes the final stage of a long-term research and development project (2009–2024) focused on developing and validating a comprehensive fatigue model of hydraulic shock absorbers equipped with shim stack valves [1,2,12]. The research was initiated due to the limited accuracy and applicability of lumped parameter models, which were traditionally used to estimate stress levels in shim-based valves without appropriate experimental validation or consideration of nonlinear boundary conditions. Initial efforts were directed at the static and fatigue validation of simplified valve system models [2,12], which, despite delivering promising results in axisymmetric configurations, proved insufficient for real-world valve geometries. This motivated the development of advanced finite element (FE) models incorporating the elastic-plastic behavior of shim stacks, as well as their geometrical tolerances and material characteristics. In parallel, significant improvements were introduced to fluid flow modeling (CDF) and fluid–structure interaction (FSI) simulations, enabling the quantification of aeration and cavitation effects at the valve level [13]. These models were validated using experimental techniques such as Particle Image Velocimetry (PIV), allowing for better understanding of fluid–structure coupling and its impact on pressure distribution and damping characteristics [13].

The final stage of this study, presented in this paper, introduces a multi-scale fatigue validation approach [14,15,16] that bridges component-level and system-level lifetime performance for a specific shock absorber configuration, including rod–bore geometry and valve setting. The developed fatigue model of the shock absorber integrates experimentally derived Wöhler curves [17] for shim materials combining FE-based stress analysis and rainflow cycle counting procedures under variable amplitude loading. The fatigue model validation at the shock absorber level was conducted using high-performance servo-hydraulic test benches (Figure 1), employing both the deterministic sine-wave excitation considered in previous work [1] and the stochastic, road-like excitation profiles introduced in the present study [18].

The model-based approach enables pre-selection of valve configurations and the estimation of shock absorber lifetime without the need for extensive full-scale testing, significantly reducing development costs and time. Moreover, the methodology facilitates early detection of configurations prone to aeration, hysteresis, or rapid fatigue damage, which is particularly beneficial in the OEM shock absorber design cycle.

2. Shock Absorber Valve System Design

Numerous types of shim-clamped valves are used in shock absorbers, depending on their specific application requirements. This study focuses on a representative automotive twin-tube shock absorber with a rod-to-bore diameter ratio of 12.4/30 mm. Nevertheless, the methodologies and findings presented in this paper can be readily extended to a broad range of shim-based valve configurations and double-tube shock absorber systems. The shim-clamped stack consists of multiple thin, circular metal plates—referred to as shims—arranged in a layered configuration. By adjusting the number, thickness, and outer diameters of these shims, the stiffness of the stack’s opening response can be finely tuned, thereby controlling the hydraulic flow through the valve cross passages (orifices). The two principal flow control regimes—metered mode and check mode—are activated selectively depending on the direction of piston motion, i.e., compression or rebound, and the internal layout of the valve architecture [19,20,21]. Each valve ensures unidirectional flow of the working fluid (metered mode) while restricting flow in the opposite direction (checked mode), depending on the direction of piston rod movement—compression or rebound, respectively (

Table 1. Flow modes vs. piston rod stroke.

Piston Motion	Primary Flow Path	Active Mode (Damping)	Passive Mode (Check Flow)
Compression (inward)	Compression valve (base valve)	Metered mode (compression)	Rebound valve (check mode)
Rebound (outward)	Rebound valve (piston valve)	Metered mode (rebound)	Compression valve (check mode—bypass)

). The metered mode (Table 1) refers to the condition in which hydraulic oil is compelled to flow through orifices, calibrated flow passages, or deflectable shim stacks that introduce resistance to flow and, in turn, generate a damping force. This resistance is inherently nonlinear and velocity-dependent, allowing the shock absorber to dissipate kinetic energy and deliver a controlled suspension response. The characteristics of metered flow are governed by several factors, including the geometry of the flow passages (e.g., diameter and length), the stiffness and arrangement of the shim stacks, the viscosity of the hydraulic fluid, and the instantaneous velocity of the piston. The check mode (Table 1) is a low-resistance bypass condition, wherein hydraulic oil flows freely through one-way check valves or spring-loaded ports designed to allow rapid pressure equalization without significant damping. This mode is essential for permitting unimpeded oil transfer in the non-damping direction and ensures that the shock absorber can reset effectively between strokes.

During the compression phase, the piston rod assembly moves into the damper body (inwards), reducing the volume in the lower (rod-side) chamber and increasing the pressure in the upper (tube-end) chamber (Figure 2). The associated flow dynamics are as follows: in the metered mode, oil displaced from the rod-side chamber is forced through the tube-end valve (indicated by the orange arrow in Figure 2a), which regulates the flow to generate the compression damping force that resists rapid suspension compression. Simultaneously, in check mode, oil from the tube-end chamber may flow through the piston valve (indicated by the orange arrow in Figure 2b), acting as a check valve, allowing free flow without damping resistance to equalize pressure across the piston. This dual-mode flow ensures effective damping of suspension compression while maintaining hydraulic continuity. During rebound, the piston rod assembly moves outward, increasing the volume in the rod-side chamber and drawing oil from the tube-end chamber. The internal flow paths respond as follows: in the metered mode, oil from the tube-end chamber is forced through the piston valve (indicated by the green arrow in Figure 2a), which is typically integrated within the piston body. This valve includes calibrated flow passages and a rebound shim stack that generates the rebound damping force, effectively opposing the extension velocity of the suspension. Simultaneously, in check mode, oil is allowed to refill the rod-side chamber through the tube-end valve (indicated by the orange arrow in Figure 2b), acting as a check valve, typically via a spring-loaded bypass, which ensures low-resistance oil replenishment during the rebound stroke. The interplay between metered and check flow modes across the compression and rebound phases is central to the shock absorber’s performance. It allows for asymmetrical damping (stronger rebound vs. compression or vice versa), velocity-sensitive response, enhancing ride comfort at low speeds and vehicle control at high speeds, and thermal efficiency, by controlling oil shear and turbulence via flow restriction mechanisms.

In this study, the analyzed shock absorber is equipped with both a piston valve and a tube-end valve, each incorporating dedicated compression and rebound flow circuits (Figure 2). In the automotive industry, valve bodies are typically manufactured from sintered metal alloys, owing to their cost-efficiency and suitability for high-volume production processes. The piston valve assembly includes two shim-based elements: the piston check valve (item 1 in Figure 2a) and the piston main valve (item 2 in Figure 2a).

The tube-end check valve (item 3 in Figure 2b) functions as a sliding intake valve, which is not prone to fatigue failure due to the presence of a supporting coil spring that enables free axial movement of the shim. In contrast, the tube-end main valve (item 4 in Figure 2b) is a conventional clamped shim valve and constitutes the critical component with respect to fatigue damage accumulation. The operation of a twin-tube hydraulic shock absorber [19] can be divided into two main phases as illustrated in Figure 2, compression (a) and rebound (b), respectively. During the compression phase, the piston along with the piston rod moves downward toward the bottom of the working cylinder. Figure 2a illustrates the flow of hydraulic fluid through the valve systems of the piston and the tube-end valve during the compression phase. When the piston moves downward slowly, the pressure in the compression chamber increases gradually. As a result, the force exerted on the shim stack of the tube-end valve remains low and does not lead to the opening of the valve or the transfer of hydraulic fluid to the reservoir chamber. Simultaneously, the hydraulic fluid is displaced through the orifices in the piston into the rebound chamber, which leads to the full opening of the piston check valve (item 1 in Figure 2a) [19]. During the rebound phase, the piston rod moves upward toward the shock absorber head. Figure 2b illustrates the flow of hydraulic fluid through the valve systems of both the piston and the tube-end valve during this phase. At low piston rod velocities, the pressure exerted by the working fluid on the shim stack of the piston’s main valve (item 2 in Figure 2a) remains too low to cause valve opening. As the displacement amplitude increases, the piston starts moving, with higher velocity rising internal pressure in the compression and rebound chambers. This leads to the opening of the piston’s main valve (item 2 in Figure 2b) and the displacement of hydraulic fluid from the rebound chamber to the compression chamber through calibrated orifices in the valve. The fluid then flows through the piston’s main valve as depicted in Figure 2. Shim-clamped valves within hydraulic shock absorbers operate under bi-directional loading conditions, characterized by two distinct functional modes, the metering mode, in which the valve is open and the shim stack experiences maximum tensile stress, and the checking mode, in which the valve is closed and the shim stack is subjected to compressive or reversed loading. During the checking operation, the valve remains closed under the influence of a negative differential pressure, which arises from the shock absorber’s internal flow dynamics. This back pressure acts against the closed valve, imposing a compressive load on the shim stack in the direction opposite to that experienced during metering. Both operational modes are significantly influenced by preload conditions, which stem from the relative axial elevation between the valve seat and the clamping elements of the shim stack. This preload serves multiple purposes. It compensates for manufacturing and assembly tolerances, enhances sealing reliability in the closed position, and influences the dynamic response of the valve—particularly by introducing a delay in the opening process, which affects damping performance. Consequently, the minimum stress level experienced by the shim stack is sensitive to the magnitude and distribution of this preload.

3. Fatigue Strength Model—Component and System Level

The overall lifetime of a hydraulic shock absorber is predominantly governed by the fatigue resistance of its most vulnerable component, typically the shim stacks, which are particularly susceptible to cyclic loading-induced damage [22]. The stress state within the shim stack is determined by a combination of factors, including the geometric configuration and material properties of the valve assembly, the operational conditions of the vehicle, the load history, and manufacturing tolerances, as described in [2]. In addition to these structural parameters, fluid dynamic phenomena such as aeration and cavitation substantially reduce the effective pressure acting on the valve components, thereby altering the expected fatigue behavior [13]. The fatigue life of the shim valves is also influenced by mechanical friction [23,24] and the evolving material characteristics—notably elastic modulus degradation—which progressively deteriorate during prolonged operational use. In addition to classical fatigue mechanisms governed by global stress ranges and cycle counts, localized damage phenomena such as fretting fatigue can also significantly contribute to the degradation of hydraulic shock absorber components, particularly in shim stack valves. Fretting fatigue arises from repeated micro-sliding at contact interfaces under cyclic loading and is known to accelerate crack initiation in high-contact-pressure zones such as those found between adjacent shims or between the shim stack and valve seat. Experimental and numerical investigations conducted by Hojjati-Talemi et al. [24] demonstrated that fretting-induced surface damage in automotive shock absorber valves plays a critical role in reducing fatigue life, especially under conditions of limited lubrication and high-frequency excitation. Their work highlights the need for accurate contact modeling and surface treatment considerations when predicting long-term durability of valve systems. Recent studies have shown that the interaction of multiple localized defects significantly influences fatigue strength and structural reliability in metallic components exposed to combined environmental and mechanical loading [25]. Their study combined field testing, numerical simulations, and analytical models to quantify how spatially distributed pitting leads to stress concentration amplification and early crack initiation. The results highlighted that both pit coalescence and the statistical distribution of defects are critical factors governing fatigue life degradation. This approach, although developed for large-scale infrastructure, provides valuable analogies for assessing fatigue reliability in smaller-scale mechanical components, such as shim-based valve systems, where micro-defects and localized stress fields may also interact in a nonlinear manner. Chang introduced a fatigue prediction model for shock absorber cylinders with a correction for surface roughness, enhancing model accuracy for component-scale damage [26]. Zhang developed a multiaxial fatigue life analysis method using deep neural networks for automotive components under variable loads, enabling nuanced time-domain fatigue assessment [27].

The ongoing development of fatigue assessment procedure focuses on improving shock absorber robustness under variable and uncertain road conditions, with the objective of reducing failure rates while maintaining targeted pressure–flow characteristics. These characteristics are critical not only for ensuring passenger comfort but also for sustaining safe long-term operation and preserving fluid performance, particularly the damping force delivered by the shock absorber. Valve fatigue performance is highly dependent on the loading sequence, especially the cumulative amplitude effects experienced over the component’s lifespan [12]. In practice, the complex load history of a valve is typically condensed into an equivalent load representation, which serves as a benchmark for defining fatigue damage thresholds and design specifications. However, it is important to note that both hydraulic and material hysteresis, as well as time-dependent degradation mechanisms such as frictional wear and loss of material elasticity, contribute significantly to the accumulation of fatigue damage [28,29]. These nonlinear and time-dependent effects must be considered in any comprehensive durability evaluation, particularly when simulating real-world loading scenarios and validating digital twin models [12].

The overall procedure for fatigue damage assessment of a shock absorber system is typically carried out on two distinct levels, the component level, focusing specifically on the valve, and the system level, encompassing the complete hydraulic shock absorber.

3.1. Procedure for Valve System Fatigue Strength Damage Assesment

The valve system fatigue damage assessment involves several steps. Initially, pressure–deflection (P-D) and pressure–stress (P-SP-S) shim stack characteristics are generated using the solid 2D/3D finite element model, based on a shim stack configuration from a production bill of materials (BOM). Pressure–flow (P-Q) characteristic is obtained based on P-D shim stack characteristic using the flow model required passage geometries from BOM and flow coefficients. Next, a single shim most susceptible to fatigue—typically experiencing dominant bending loads—is identified as the critical component. A stress–flow (S-QS-Q) characteristic is then derived through the fatigue model for stack of shims using the previously determined P-D and P-Q relationships and based on the known stress–cycles (S-NS-N) characteristic involving shim steel properties. The fatigue life of the valve system is estimated under specific loading conditions (deterministic, random, constant, or variable load). This integrative approach ensures a comprehensive understanding of the valve system’s mechanical, hydraulic, and fatigue behavior, thereby enabling accurate virtual fatigue assessments at the component level.

The valve fatigue damage assessment procedure involves sub-models that are interconnected through the following system of equations:

$$\left\{\begin{array}{l}h={\mathrm{f}}_{PD}\left(\Delta p,h_0, \beta \right)\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\\ \sigma ={\mathrm{f}}_{PS}\left(p,h_0\right)\mathrm{ }\\ \begin{array}{l}q={\mathrm{f}}_{SQ}\left(\Delta p,h,\gamma \right)\\ \begin{array}{c}n={\mathrm{f}}_{SN}\left({\sigma }_{equ}, \beta \right)\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\end{array}\end{array}\mathrm{ }\mathrm{ }\end{array}\right.$$

(1)

where Δp denotes the differential pressure across the shim-based valve in the direction of flow; $\beta$ is shim stack BOM (shim diameter and thickness); h is the valve opening gap between the first sealed shim and the valve seat; h₀ represents the initial negative deflection of the shim (i.e., shim preload); σ_equ is the equivalent alternating stress occurring within the shim stack; q is the volumetric flow rate through the valve body cross-passages and the deflected first sealed shim; γ denotes the flow coefficients; and n is the number of load cycles the shim stack is capable of withstanding. The fatigue damage prediction framework comprises a set of interconnected sub-models, as illustrated in

Table 2. Sub-models constituting the shock absorber lifetime prediction framework.

Characteristic	Reference	Model	Formulation	Parameters	Validation
P-D	[2]	Solid model	First-Principle	Distributed Parameters	Static loading component tests
P-S	[2]	Solid model	First-Principle	Distributed Parameters	Static loading component tests
S-Q	[13]	Fluid model	First-Principle	Lumped Parameters	3D CFD/PIV
S-N	[12]	Fatigue model	Data-Driven	Lumped Parameters	Dynamic loading component tests
Z-DF	[3]	Physical model	First-Principle	Lumped Parameters	Twin-tube shock absorber performance

.

3.2. Procedure for Shock Absorber Fatigue Strength Damage Assesment

The shock absorber fatigue damage assessment procedure involves a twin model of a shock absorber in order to simulate its internal fluid dynamics, accounting for all hydraulic valves and their interactions [3]. The shock absorber model accounts for the test loading signal corresponding to specific road conditions. In the fatigue testing of hydraulic shock absorbers, different categories of loading signals are employed to replicate diverse operating conditions and to assess component fatigue strength under various stress regimes. Based on industry practices and customer specifications, three principal types of load signals are considered for validation testing: constant-amplitude, variable-amplitude, and random-amplitude loading (

Table 3. Common loading types in automotive industry.

Loading Category	Signal Type	Application	Typical Specification
Constant amplitude	Deterministic (sine)	Intercompany validation	200 k cycles
Variable amplitude	Deterministic (sine)	Intercompany and external	0.3 k, 3 k, 300 k cycles
Random amplitude	Stochastic (random)	Final external validation	block repetitions

).

Constant amplitude loading represents the most common approach to testing with the use of sinusoidal excitation signals. The repeatable nature of such tests makes them particularly suitable for inter-company validation, as sine waves can be accurately reproduced by servo-hydraulic testing machines. A typical test scenario includes a fixed number of cycles, e.g., 200 thousand, with a uniform amplitude and frequency. Variable amplitude loading combines short-duration high-amplitude segments to simulate low-cycle fatigue (LCF) and long-duration, low-amplitude segments to address high-cycle fatigue (HCF), thus encompassing a broader fatigue spectrum. While the signals are still typically sinusoidal and therefore reproducible, the controller settings of the servo-hydraulic system must be precisely adjusted to accommodate variations in velocity and displacement. Block loading reflects key points within the expected operational envelope and is often employed in tests for both internal and external customer requirements, e.g., three-stage test with 300, 3000, and 300 thousand cycle blocks. Random amplitude loading emulates road conditions. These loadings are stochastic in nature and derived from actual vehicle usage data. Despite their high relevance to field performance, such signals are characterized by relatively low effective amplitudes, resulting in suboptimal use of the full capacity of servo-hydraulic systems. Moreover, the finite bandwidth of hydraulic testers limits the accurate reproduction of high-frequency components, as signal attenuation typically occurs at elevated frequencies. Nevertheless, wide frequency band signals are considered the most representative for final-stage shock absorber fatigue validation. This study focuses on the most complex and challenging loading scenario—random road-induced excitations. The test signal is applied to a servo-hydraulic test bench (Figure 1) through a PID controller to ensure that the piston rod accurately tracks the rapidly varying displacement amplitude, velocity, and frequency content of the input signal. This approach ensures that the testing conditions closely replicate the dynamic characteristics of real-world vibration loading. The complete framework of the assessment procedure is illustrated in Figure 3.

All sub-models (Table 2) employed in this study have been previously validated within the framework of earlier research and development activities. Data transfer between sub-models is performed in a static, unidirectional manner, following the sequential structure defined in Figure 3. In the initial step, a dynamic model of the shock absorber [3] is applied—either in its classical form (passive suspension system) or in an extended version incorporating an active control valve regulating the damping force (semi-active suspension). This model is driven by displacement z(t) and velocity $\dot{z}$(t) signals corresponding to simulated or experimentally measured road profile, which is transmitted to the shock absorber’s piston rod via the tire and vehicle suspension system. The types and sources of input signals used for this purpose are detailed in Table 3. In the present study, a random amplitude signal was employed—a stochastic input commonly utilized for the final-stage validation of hydraulic shock absorbers in selected test scenarios. This signal is applied in the form of several-minute-long sequences, which are repetitively executed throughout the test until the occurrence of shock absorber failure, identified by a measurable reduction in a shock absorber damping force. The model provides access to all physical variables that can be validated against measurements on an actual twin-tube shock absorber, including internal chamber pressures:

$$[p_{pis,com},p_{pis,reb},p_{base, com},p_{base, reb}]=f(z,\dot{z},\theta ),$$

(2)

Among these, the differential pressures across the valves are of particular importance, as they constitute key parameters governing the fatigue strength of the system. To determine the differential pressures across valve assemblies, the twin-tube shock absorber model integrates the known P-Q relationships with shock absorber parameters—primarily the rod-to-bore diameter ratio—along with other system-specific properties relevant to dynamic simulation of shock absorber [3]. This approach enables accurate simulation-based reconstruction of the worst-case internal pressure distribution across the shim-clamped valves during both metered and check flow operation modes.

As a result of the model implementation, both the maximum pressures attained under a given excitation signal and the functional relationships mapping the piston rod assembly velocity to the valve differential pressure are determined. These outputs are stored as numerical data files within the MATLAB R2024b Simulink environment and subsequently processed using MATLAB scripts (*.m files), which read the data and perform static computations based on the system of Equation (1) for all valves equipped with clamped shims. The sliding tube-end compression valve is excluded from these calculations, as its shim is not subjected to significant cyclic bending or tensile stresses due to its support by a coil spring, allowing it to move freely in the vertical direction. Its operational durability is considered effectively unlimited under typical loading conditions. Subsequently, a high-fidelity finite element simulation model [2] (Table 1) is employed to evaluate the maximum stress levels within the critical regions of the clamped shim stacks according to Equation (1). The model enables estimation of shim stack deflection, which directly affects the valve’s volumetric flow rate and, consequently, the resulting fatigue loading conditions. The finite element model is developed using a leading commercial multi-physics and multi-domain simulation software package.

Advanced MATLAB-Simulink API functionalities were utilized to develop a custom script integrated with Abaqus (SIMULIA 2017 Dassault Systèmes) finite element (FE) simulation software. MATLAB enables the reading and writing of batch files containing the geometry of the valve and associated shims, represented as nodal meshes in a format compatible with the FE software. The batch file is generated within the MATLAB environment and subsequently used to call the FE solver executable. Upon completion of the FE computation, the resulting output file is automatically re-imported into MATLAB, where further flow-related calculations are performed in accordance with Equation (1), and post-processing routines generate the corresponding visualizations (Figure 4). The implemented features allow for the visualization of shim stack configurations, including maximum deflection and stress distributions (Figure 5). Double-precision floating-point arithmetic combined with interpolation techniques is employed to ensure high accuracy of the final results.

Ultimately, the comprehensive modeling framework enables realistic and predictive fatigue life estimation of hydraulic shock absorbers. The primary output of the fatigue strength assessment (Figure 3) is the predicted number of load cycles to failure for the most critically loaded shim-based valve:

$$n_{min}=\mathit{min}(n_{pis,com},n_{pis, reb},n_{tube-end, reb}),$$

(3)

This modeling approach closely mirrors real-world durability testing conditions, wherein the shock absorber is exposed to continuous cyclic excitation representative of actual operational loading scenarios.

3.3. Rainflow Loading Signal Decomposition

The rainflow-counting algorithm is a widely used method for calculating the fatigue life of components subjected to random-amplitude loading. It transforms complex stress histories into an equivalent set of constant-amplitude stress reversals that induce comparable fatigue damage. The algorithm systematically extracts smaller interruption cycles from the loading sequence, effectively capturing the material memory effects associated with stress–strain hysteresis [30,31,32]. This transformation enables the application of cumulative fatigue damage models, such as Miner’s rule, to estimate the number of cycles to failure for each identified rainflow cycle.

4. Fatigue Strength Prediction and Validation

4.1. Road Endurance Loading

The test was designed to reproduce the operational conditions within the shock absorber; therefore, standard valve settings were applied under non-fully reversed loading conditions (i.e., strain ratio −1 < R < 1). The complete test signal consists of 16,320 repetitions of this fundamental sequence shown in Figure 6.

To derive the velocity signal from the measured displacement in servo-hydraulic automotive testing systems, numerical differentiation techniques are typically employed. The displacement signal, commonly acquired via LVDT sensors or internal hydraulic actuator encoders, is first low-pass filtered to suppress high-frequency noise that would otherwise be amplified during differentiation. A central difference scheme or digital filtering methods, such as the Savitzky-Golay algorithm, are then applied to obtain the velocity profile. This approach ensures an accurate and robust estimation of the piston rod velocity, which is essential for evaluating damping characteristics and energy dissipation in shock absorber testing.

The piston rod velocity range applied in the validation tests was established by identifying peak-to-peak velocity amplitudes derived from the rainflow cycle decomposition of the analyzed loading sequence [31,33] as detailed in

Table 4. Equivalent loading velocities and corresponding cycle counts for the compression stroke.

Velocity (m/s)	−3.6	−3	−2.8	−2.6	−2.4	−2.2	−2	−1.8	−1.6	−1.4	−1.2	−1	−0.8	−0.6	−0.4	−0.2
Cycles	1	3	2	4	9	3	8	12	12	15	15	25	34	29	30	6

and

Table 5. Equivalent loading velocities and corresponding cycle counts for the rebound stroke.

Velocity (m/s)	2.4	2.2	2	1.8	1.6	1.4	1.2	1	0.8	0.6	0.4	0.2
Cycles	2	2	6	16	20	33	29	21	25	28	20	9

.

Ultimately, a package of equivalent sinusoidal signals, derived through the rainflow counting algorithm, was used to substitute the original random-amplitude loading with a variable-amplitude harmonic representation.

4.2. Testing Machinery

The shock absorber fatigue life tests were performed using an IST-50 servo-hydraulic load frame, equipped with two supporting columns, an adjustable traverse, and a custom fixture designed to accommodate up to six shock absorber units simultaneously. The tester is constructed on a rigid frame with a fundamental resonance frequency exceeding 70 Hz, minimizing the influence of structural vibrations. The frame is mechanically isolated from the floor, and its vertical motion does not interfere with load measurement accuracy, thereby ensuring reliable and repeatable test results. The test system is capable of achieving actuator velocities up to 4.0 m/s under a full load of 60 kN, with a total stroke range of 250 mm (±125 mm). The actuator–servo valve configuration allows for peak velocities of up to 4.5 m/s at a reduced load of 30 kN. To maintain a consistent damping force balance during testing, shock absorbers that failed due to fatigue were immediately replaced with new units. This ensured stable dynamic loading conditions on the fixture mechanically coupled to the hydraulic actuator. Certain secondary physical phenomena may manifest during fatigue life testing due to the inherent operating principles of hydraulic shock absorbers. The conversion of mechanical work into heat during damper operation leads to a rise in fluid temperature, which, in turn, affects key oil properties—most notably viscosity—that are highly temperature-dependent and have a direct impact on damping force characteristics. Moreover, high piston velocities induce significant aeration effects, which further alter fluid behaviour and flow resistance within the valve systems. To quantitatively assess the phenomenon of aeration [13], an automated algorithm was developed for the measurement and evaluation of damping force characteristics in hydraulic shock absorbers. To mitigate the adverse effects of fluid temperature increase, custom-designed water-cooling jackets were integrated into the shock absorbers (Figure 7).

These jackets serve to dissipate excess heat generated during cyclic excitation, thereby stabilizing internal fluid properties throughout the test duration. Additionally, an off-line error correction technique was employed to enhance the accuracy of displacement and velocity tracking. This method involves iterative replay of the applied road loading signal, allowing the servo-hydraulic controller path to be refined through successive adjustments. As a result, the target kinematic parameters—namely peak displacements and velocities—could be reproduced with nearly 100% fidelity.

4.3. Fatigue Strength Prediction Based on Shim-Clamped Valve Model

The shock absorbers considered for fatigue tests are standard production units (pressure tube inner diameter = 30 mm; rod diameter 12.4 mm) equipped with customized valve settings for the piston metering (compression) and check (rebound) modes and the base metering valve, as detailed in

Table 6. Layering and geometric arrangement of shims in the piston rebound valve.

No.	Part Name	ID	OD	Thickness
1	A3	9600	13,500	0.200
2	B2	9600	24,525	0.150
3	C2	9600	27,000	0.150
4	C2	9600	27,000	0.150
5	C2	9600	27,000	0.150
6	C1	9600	27,000	0.100
7	D2	9600	27,005	0.150

. The same shock absorber configurations were used as in previous studies, with harmonic loading conditions applied as described in [1].

The considered piston valve is a symmetrical construction, which provides damping in both damper movement directions, namely rebound and compression side. The back pressure in case of the rebound valve is the metered pressure at the compression valve and vice versa. The stress range ($\Delta \mathrm{\sigma })$ is defined as the algebraic difference between the maximum and minimum stress within a single loading cycle:

$$\Delta \mathrm{\sigma }={\mathrm{\sigma }}_{max}-\mathrm{ }{\mathrm{\sigma }}_{min},$$

(4)

The stress amplitude is defined as one-half the stress range:

$${\mathrm{\sigma }}_{alt}={\displaystyle \frac{\Delta \sigma }{2}}={\displaystyle \frac{{\sigma }_{max} - {\sigma }_{min}}{2}},$$

(5)

In the context of shim-based clamped valve systems, σ_max corresponds to the metered mode of the valve (i.e., the valve is open), while σ_min corresponds to the checked mode (i.e., the valve is closed). The mean stress ($\Delta {\mathrm{\sigma }}_{mean})$ is the arithmetic average of the maximum and minimum stress values within a cycle:

$${\mathrm{\sigma }}_{mean}={\displaystyle \frac{{\mathrm{\sigma }}_{max}+{\mathrm{\sigma }}_{min}}{2}},$$

(6)

To quantitatively describe the influence of mean stress on fatigue behavior, two dimensionless parameters are commonly employed, the stress ratio R, also referred to as the asymmetry coefficient, and the amplitude ratio A, also known as the stability coefficient [31]:

$$\mathrm{R}={\displaystyle \frac{{\mathrm{\sigma }}_{min}}{{\mathrm{\sigma }}_{max}}}, \mathrm{A}={\displaystyle \frac{{\mathrm{\sigma }}_{alt}}{{\mathrm{\sigma }}_{mean}}}={\displaystyle \frac{1-R}{1+R}},$$

(7)

These parameters are widely used in fatigue analysis to characterize the nature of cyclic loading. Specifically, a fully reversed loading condition [31] is represented by R = −1, while a static (non-cyclic) loading condition corresponds to R = 1. A loading case with R = 0 implies that the mean stress is equal in magnitude to the alternating stress and is entirely tensile. In fatigue testing, a common practice involves employing cycles with R = 0, indicative of tension–tension loading, where the minimum stress is 10% of the maximum stress. Shim-based clamped valve systems typically operate under stress ratios within the range R ∈ (−1,0), depending on the specific geometry of the clamping components and the applied service loading conditions.

Several empirical models have been developed to account for the influence of mean stress on fatigue life prediction [31,34]. Among the most widely used are the Goodman, Gerber, Soderberg, Morrow, Smith–Watson–Topper, and Walker relations [17]. These models establish relationships between the alternating stress (σ_alt) and the mean stress (σ_mean), based on material properties such as ultimate tensile strength and endurance limit. The Goodman equation represents one of the earliest and most commonly employed mean stress correction models. It assumes a linear relationship between fatigue strength and mean stress and is expressed as:

$${\displaystyle \frac{{\mathrm{\sigma }}_{alt}}{{\mathrm{\sigma }}_{equ}}}+{\displaystyle \frac{{\sigma }_{mean}}{{\sigma }_u}}=1,$$

(8)

where

${\mathrm{\sigma }}_{alt}$ is the alternating stress;

${\mathrm{\sigma }}_{mean}$ is the mean stress;

${\mathrm{\sigma }}_{ult}$ is the ultimate stress;

${\mathrm{\sigma }}_{equ}$ is the effective alternating stress (i.e., endurance limit).

In the case of shim-based valve systems, the valve preload introduces a non-zero mean stress component, resulting in a deviation from the fully reversed loading line. Consequently, the equivalent alternating stress σ_equ, accounting for the influence of mean stress, can be calculated using a modified form of the Goodman relation:

$${\mathrm{\sigma }}_{equ}={\mathrm{\sigma }}_{alt}+\left({\displaystyle \frac{{\mathrm{\sigma }}_{equ}}{{\mathrm{\sigma }}_{ult}}}\right)·\left|{\sigma }_{mean}\right|,$$

(9)

Assuming σ_equ = 0.5·σ_ult for shims made of alloy steel [34] and substituting the expressions for ${\mathrm{\sigma }}_{max}$ and ${\mathrm{\sigma }}_{min}$ into Equation (10), we obtain:

$${\mathrm{\sigma }}_{equ}={\displaystyle \frac{{\mathrm{\sigma }}_{max}-{\mathrm{\sigma }}_{min}}{2}}+{\displaystyle \frac{\left|{\sigma }_{max}+{\sigma }_{min}\right|}{4}},$$

(10)

The use of the absolute value of σ_mean ensures a conservative estimate by accounting for the more severe stress side of the shim—whether it occurs in the metering or check mode. The equivalent stress values were determined, as illustrated in

Table 7. Partial equivalent stress (${\mathrm{\sigma }}_{equ}$) in MPa obtained for piston metered valve calculated for equivalent load velocities of the test signal (stress values were rounded to integers).

		Velocity of Compression Stroke (m/s)
		−3.6	−3.4	−3.2	−3.0	−2.8	−2.6	−2.4	−2.2	−2.0	−1.8	−1.6	−1.4	−1.2	−1.0	−0.8	−0.6	−0.4	−0.2	0.0
Velocity of rebound stroke (m/s)	2.4	2627	2328	2050	1791	1553	1335	1137	1041	988	941	899	861	794	2627	2328	2050	1791	1553	1335
	2.2	2617	2318	2040	1781	1543	1325	1127	1011	958	911	869	831	764	2617	2318	2040	1781	1543	1325
	2	2607	2308	2030	1771	1533	1315	1116	980	927	880	838	800	733	2607	2308	2030	1771	1533	1315
	1.8	2597	2298	2019	1761	1522	1304	1106	949	896	849	807	769	702	2597	2298	2019	1761	1522	1304
	1.6	2586	2287	2009	1750	1512	1294	1095	917	864	817	775	737	670	2586	2287	2009	1750	1512	1294
	1.4	2575	2276	1998	1739	1501	1283	1085	906	832	785	743	704	637	2575	2276	1998	1739	1501	1283
	1.2	2564	2265	1987	1728	1490	1272	1073	895	798	751	710	671	604	2564	2265	1987	1728	1490	1272
	1	2553	2254	1975	1717	1478	1260	1062	884	764	717	675	637	570	2553	2254	1975	1717	1478	1260
	0.8	2541	2242	1964	1705	1467	1249	1050	872	729	682	640	602	535	2541	2242	1964	1705	1467	1249
	0.6	2529	2230	1951	1693	1455	1237	1038	860	701	646	604	566	499	2529	2230	1951	1693	1455	1237
	0.4	2516	2218	1939	1681	1442	1224	1026	847	688	608	567	528	461	2516	2218	1939	1681	1442	1224
	0.2	2503	2205	1926	1668	1429	1211	1013	834	675	569	528	489	422	2503	2205	1926	1668	1429	1211
	0	2490	2191	1912	1654	1416	1197	999	821	661	529	487	448	382	2490	2191	1912	1654	1416	1197

, for equivalent load velocities considered during the signal decomposition stage using the rainflow counting method. To prevent material failure, the equivalent stress must not exceed the tensile strength of the material:

$${\mathrm{\sigma }}_{eq} \le R_m,$$

(11)

where R_m denotes the shim material’s ultimate tensile strength introduced in [1,2,12]. Fatigue strength damage is evaluated according to the Palmgren–Miner rule as the cumulative sum of partial damage contributions associated with equivalent load velocities during metered ($v_{metered}$) and check ($v_{checked}$) mode operation, as illustrated in

Table 8. Partial strength fatigue damage (D_ij) obtained for piston metered valve calculated for equivalent velocities of the test signal (damage values were rounded to four decimal places).

		Velocity of Compression Stroke (m/s)
		−3.6	−3.4	−3.2	−3.0	−2.8	−2.6	−2.4	−2.2	−2.0	−1.8	−1.6	−1.4	−1.2	−1.0	−0.8	−0.6	−0.4	−0.2	0.0
Velocity of rebound stroke (m/s)	2.4	0.336	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	2.2	0	0	0	0	0.120	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	2	0	0	0	0.095	0	0	0.127	0	0	0	0	0	0	0	0	0	0	0	0
	1.8	0	0	0	0.017	0	0.047	0.045	0	0.054	0	0.024	0	0	0.010	0.009	0	0	0	0
	1.6	0	0	0	0	0	0.011	0.010	0.009	0.004	0.021	0.020	0.015	0.003	0	0	0.003	0	0	0
	1.4	0	0	0	0	0	0	0.002	0.001	0.004	0.005	0.003	0.004	0.004	0.005	0.002	0.001	0.001	0	0
	1.2	0	0	0	0	0	0	0	0	0	0.001	0	0.001	0.001	0.001	0.002	0.001	0	0	0
	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	0.8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	0.6	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	0.4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	0.2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

, which summarizes the corresponding partial fatigue strength damage values across the velocity intervals. The cumulative fatigue damage is obtained for equivalent load velocities (Table 5) according to the following equation:

$$D=\sum _{v_{metered}=1}^i \sum _{v_{check}=1}^j{\displaystyle \frac{n_{ij}}{N_{ij}}}$$

(12)

where n denotes the number of applied load cycles and N denotes the number of cycles to failure (i.e., fatigue life or lifetime). The Palmgren–Miner rule (12) was applied to all valve systems equipped with shim-based clamped valves, including piston compression, piston rebound, and base compression valve configurations. For illustrative purposes, detailed partial damage results are presented for the piston valve, while synthetic summary data for the remaining valves are provided in

Table 9. Damage and lifetime factor for evaluated shim-clamped valve systems.

	Piston Rebound (Metered Mode)	Piston Compression (Check Mode)	Tube-End (Metered Mode)
Damage D = nij/Nij	0.000273	1.023731	0.163352
Lifetime L = 1/D	3663.529	0.976819	6.121756

.

4.4. Fatigue Strength Test Results

The test results conducted for six tested units until damage occurred are presented in

Table 10. Reported number of cycles withstood by each unit tested until failure occurred.

Units Applied in the Test	Loading Sequence Repetition
Unit 1	5840
Unit 2	16,160
Unit 3	22,880
Unit 4	25,600
Unit 5	17,520
Unit 6	9920
Mean result of 6 units	16,320
Standard deviation result of 6 units	7499
Statistical scatter	46%

. The results exhibit a moderate level of sample scatter, which may be attributed to material inhomogeneity and manufacturing tolerances. The observed variability within the sample indicates sensitivity to slight changes in loading conditions.

The mean number of cycles sustained by all six tested shock absorbers was approximately 16,320 repetitions of the signal realization, with a standard deviation of 7499 repetitions. This corresponds to a spread of nearly 46% indicating a relatively high degree of scatter within the test sample [35]. Nevertheless, in the context of early-phase virtual design validation, an uncertainty of ±46% is acceptable and typical in the automotive sector for fatigue assessments involving random road loads. According to industry benchmarks (e.g., SAE AE-22), acceptable prediction deviations in fatigue lifetime can range from ±50% to ±100% depending on component criticality, data availability, and loading complexity [22]. The experimental results were compared with the predictions obtained from the fatigue model, as summarized in

Table 11. Validation of fatigue strength prediction for a batch of shock absorbers.

	Predicted Test Repetitions	Measured Test Repetitions	Test Results
Piston compression valve	0.97	1 ± 46%	failed
Piston rebound valve	3663.52	1 ± 46%	passed
Tube-end compression valve	6.12	1 ± 46%	passed

.

The “Predicted” column in Tabel 11 shows model-estimated lifetime in units of test repetitions until failure; the “Measured” column shows the normalized average of the six tested units. For the piston compression valve, failure occurred within predicted margins, validating the damage accumulation model, while, for the piston rebound valve and tube-end valve, the predicted lifetime exceeded test duration, indicating conservative fatigue durability design.

5. Discussion

During the development of a comprehensive fatigue life prediction and assessment procedure for automotive hydraulic shock absorbers, the research team encountered numerous scientific and engineering challenges.

From a scientific perspective, a key task involved the formulation of an accurate nonlinear first-principle distributed-parameter model [2,12], incorporating both symmetric and asymmetric plate deflection theories to account for valve systems commonly used in automotive shock absorbers. The primary purpose of this model was to determine the deflection of clamped shim stacks under the influence of hydraulic pressure within the damper working chambers. These results were subsequently used to derive pressure–flow characteristics through a calibrated hydraulic model, which was tuned using experimental data obtained from a servo-hydraulic test bench and a dedicated flow coefficient identification methodology. In the area of fluid mechanics, additional studies were undertaken to quantify the effects of fluid aeration. This involved the use of a distributed-parameter Computational Fluid Dynamics (CFD) model [13], validated through advanced flow visualization techniques such as Particle Image Velocimetry (PIV) [13]. To estimate the internal pressure distribution within individual chambers of the shock absorber, a lumped-parameter system-level model was also developed [3].

From an engineering standpoint, advanced analytical tools were created within the MATLAB-Simulink environment, enabling direct integration with FEM/CFD simulation results via system-level API interfaces. A range of engineering challenges were addressed in the experimental testing phase, resulting in patented solutions. LabVIEW 2017 software was employed to enhance the precision of excitation tracking, and several auxiliary testing tools were designed to support the experimental campaign.

The results presented in this study can be regarded as promising; however, the experimental findings indicate that achieving high predictive accuracy through the use of a digital twin and a fully virtual simulation protocol is associated with considerable variability in physical test outcomes. This observation highlights the inherent challenges in validating numerical models against real-world data when applied to mass-produced automotive components.

It is important to emphasize that components used in the automotive industry are typically manufactured under cost-driven industrial conditions, where low unit production cost is prioritized over tight manufacturing tolerances. In the present study, production-grade shock absorber units were tested—assembled using components with industrial tolerances rather than laboratory precision. These include variations in alloy composition and sheet rolling processes at the metallurgical level, leading to anisotropy in mechanical properties and deviations in shim thickness/diameter. Furthermore, random axial orientation applied during assembly, limited control over hydraulic fluid properties, and variations in test bench performance (e.g., fluid thermal energy dissipation and ability to reach target velocities under load) all contribute to the observed scatter in results.

Given the wide range of influencing factors, a detailed sensitivity analysis is warranted to assess the impact of individual variables on fatigue life predictions. Additionally, increasing the sample size of tested shock absorbers would improve statistical significance and enhance model validation. These considerations form the basis for future research aimed at refining the accuracy and reliability of digital-twin-based fatigue prediction methodologies.

6. Conclusions

This paper presents the final stage in the development of a predictive framework for evaluating the fatigue life of hydraulic shock absorbers, incorporating complex stochastic road loading inputs decomposed using the rainflow counting method. Previous research efforts [1] enabled fatigue life predictions only under simplified harmonic loading conditions. The results of the current study confirm the practical applicability of the proposed fatigue model in an engineering context, with prediction uncertainties not exceeding ±46% under road loading scenarios. The comprehensive methodology introduced herein facilitates the design and implementation of a fully functional virtual test environment—effectively serving as a digital twin of an industrial-grade shock absorber testing system. This virtual test environment offers a reliable and computationally efficient analytical tool to support digital-twin-based product development and to ensure compliance with fatigue strength requirements under realistic operational loading conditions.