A Modeling Approach for Assessing Vibration Immunity in Hydrogen Fuel Cell Stack for Aeronautical Applications

Abstract

Fuel cells offer a promising route to eliminating in-flight emissions from regional aviation, but certification requires proof that stacks can withstand the vibration and shock environment of turboprop aircraft. As part of the EU-funded NEWBORN project, we combined detailed finite element modeling with shaker tests to evaluate the vibration immunity of PowerCell Group’s prototype stack. The numerical model combined an orthotropic, two-zone 3D mesh of the cell package with reduced-order representations of plates, compression bands, disc springs and the mounting cage. The assembled stack was excited between 10 and 300 Hz using pseudo-random and sine-sweep inputs up to 2.0 g, from which 54 frequency response functions were obtained. The tuned model accurately reproduced the first global modes and captured the overall dynamic behavior with good, though not perfect, agreement. The combined numerical–experimental methodology therefore offers a framework for refining test campaigns and delivering early, qualitative evidence of vibration immunity in fuel cell stacks destined for flight.

1. Introduction

Hydrogen fuel cell technology is poised to revolutionize sustainable aviation by delivering zero in-flight emissions and sharply cutting the sector’s environmental footprint [1,2,3]. Reflecting this ambition, the EU-funded NEWBORN project [4] brings together 13 core partners and 5 affiliated entities, for a total of 18 participating organizations from 10 European countries, to develop a qualified, megawatt-class fuel cell propulsion system. A first full-scale demonstration is planned for late 2025, and the scalable architecture targets a propulsion system efficiency of 50% by 2026.

Unlike their automotive counterparts, airborne fuel cell stacks must withstand persistent broadband vibration, high-amplitude mechanical shocks and wide thermal excursions [5]. Although numerous experimental and numerical studies have examined vibration induced degradation, the findings remain contradictory [6]. Some groups report appreciable increases in hydrogen leakage, bolt preload loss and voltage decay after tens to hundreds of hours of excitation [7,8,9], whereas others observe negligible performance drift even after severe loading [10]. This inconsistency underscores the complexity of dynamic load paths within the stack’s multilayer architecture and the difficulty of extrapolating laboratory data to real flight spectra. To clarify these discrepancies, researchers have coupled tests with modeling. Rouss et al. [11,12,13] developed a neural network model capable of detecting local damage under multidirectional excitation, while more conventional finite element and modal analysis techniques, using both detailed and simplified stack representations, have also been applied to map vibration responses [14,15].

Within NEWBORN, CIRA, KU Leuven, Siemens and PowerCell Group adopted an integrated numerical–experimental workflow to characterize the vibration behavior of PowerCell’s prototype stack. An FEM model, with three-dimensional tetrahedral elements for the cell package and mixed two-dimensional/one-dimensional elements for plates, fasteners and the mounting cage, was first prepared. Shaker tests from 10 to 300 Hz (pseudo random loading plus sine sweeps at 0.7 g, 1.0 g and 2.0 g) yielded frequency response functions at 54 sensor locations. Iterative SOL 200 updating in Simcenter 3D (version 2406) tuned material and joint properties until the calculated natural frequencies matched the measurements and the modal assurance criterion (MAC) exceeded 0.8 for the first four global modes. The tuned model was then subjected to the same excitation to replicate test conditions and compare dynamic responses at each sensor. Finally, a representative harmonic base excitation conforming to RTCA/DO-160G [16] was applied.

This paper presents the complete workflow, showing how simulation can reconcile disparate experimental observations, guide optimal sensor placement for future qualification campaigns, and reduce certification risk for fuel cell systems exposed to the unique shock and vibration environment of turboprop aircraft.

2. Materials and Methods

This section details the multidisciplinary workflow used to quantify the structural response of the fuel cell stack. It opens with a brief description of the stack, and then describes the numerical strategy developed to predict its dynamic behavior. A summary of the shaker test campaign follows, along with the correlation procedure employed to calibrate the finite element (FE) model and the subsequent numerical–experimental comparison. The section concludes by defining the load cases selected to demonstrate compliance with the vibration and shock environments specified in RTCA/DO-160G.

2.1. Fuel Cell System Description

The fuel cell stack deployed in this study is an experimental prototype, denoted S3C and illustrated in Figure 1b. It adopts essentially the same mechanical architecture as the NEWBORN demonstrator stack (Figure 1c), but is built around the current-generation S3 stack (Figure 1a), which is one of the leading commercial stacks on the market in terms of power density. This cell technology is used extensively in commercial fuel cell systems that have undergone comprehensive validation, providing a well characterized reference with previous vibration analyses against which the present results can be benchmarked. The S3C prototype was originally conceived as a proof of concept ahead of the NEWBORN development and was selected here because it combines representativeness of the future demonstrator with the availability of detailed test data and sufficient hardware for an extensive vibration campaign. For completeness, the main performance characteristics of the S3, S3C, and NEWBORN stacks are summarized in

Table 1. Comparison of S3, S3C, and NEWBORN stacks.

	S3	S3C	NEWBORN
Power output	125 kW	125 kW	300 kW
Weight (cells)	35 kg	35 kg	45 kg
Springs	Yes	Yes	Yes
Endplates	Aluminum	Thermoset composite	Thermoset composite
Assembly	Variable-Press to force	Constant-Press to height	Constant-Press to height

.

A fuel cell stack operates by compressing membrane electrode assemblies (MEAs) and bipolar plates (BPPs) into a cell package that produces power by combining hydrogen gas and air. Channels for coolant fluid are integrated to maintain the cells within the required operating temperature range. When the cells are compressed, the clamping force is distributed between the channels in the active area and the gaskets that separate the different fluid compartments.

On either side of the cell package, an end-plate seals the stack and applies the compression force to the cells. The end-plates also house the current collectors, which form the electrical interfaces. From the top end-plate, three sets of springs are mounted and compressed between the end-plate and the compression plate. During manufacturing, the compression plate is tied to the bottom end-plate, thereby keeping the stack under a prescribed compression. The springs are required to accommodate variations and movements of the cell package. Small changes in each cell due to thermal expansion, material settlement, pressure induced inflation, and other effects accumulate over hundreds of cells and can substantially affect the overall stack compression.

Once all components have been assembled, the stack is compressed and pinned in place using rigid compression bands. These bands lock the bottom end-plate and the compression plate at a fixed distance, while the top end-plate becomes effectively floating, compressed between the springs and the cell package.

2.2. Numerical Modeling Methodology

This subsection discusses the modeling strategy adopted for each stack component, with the objective of balancing computational effort against the fidelity required to capture the fuel cell system’s mechanical response. All simulations were performed in MSC Nastran 2022.4 [17] using linear static (SOL 101), modal (SOL 103), and modal frequency response (SOL 111) analyses. The model combines three-dimensional solid elements for the cell package, shell elements for plates, bands, and cage, beam elements for the reinforcing rods, and scalar spring elements for the disc springs.

Preliminary studies demonstrated that a one-dimensional idealization of the cell package is unsuitable. It cannot be coupled unambiguously to the bottom end-plate or compression plate, and it fails to represent the spatial variations in bending and shear stiffness between the active and the gasket track regions, which demand at least two distinct rigidity values rather than a single equivalent modulus. For these reasons, the cell package was modeled in full three dimensions using linear four-noded tetrahedral solid elements (CTETRA4). By contrast, dimensional reduction is retained for components whose deformation is less critical, thereby concentrating computational effort where geometric precision is indispensable while still maintaining an efficient overall simulation.

The top, bottom, and compression plates were generated from the original 3D CAD (CATIA V5-6R2024) geometry after targeted simplifications designed to yield a regular, high-quality mesh. All fillets were removed to avoid the creation of excessively small elements, and embossed text was deleted because it has a negligible influence on global stiffness while disproportionately increasing meshing effort. The resulting three-dimensional geometries were subsequently modeled as two-dimensional elements, discretized with linear three-noded and four-noded shell elements (CTRIA3 and CQUAD4). This dimensional reduction reduces solution times without compromising fidelity. For the compression plate, for example, MSC Nastran 2022.4 [17], required approximately 10 min to compute the first five natural frequencies in 3D, whereas the 2D representation yielded the same results in approx. 10 s. This dimensional reduction, however, inevitably introduces small departures in inertia and mass properties. Rigorous verification was therefore undertaken to confirm that the 2D model faithfully reproduces the dynamic behavior of the parent 3D system. For these purposes, modal analyses were performed under free-free boundary conditions. Although these constraints do not replicate the in-operation boundary state of the assembled stack, they offer a consistent benchmark for direct comparison and, if necessary, material properties calibration (at this stage, no adjustment was required).

Since the force–displacement relation of the three disc springs is inherently non-linear, their stiffness varies with load. A tangent stiffness corresponding to the stack’s nominal operating compression was therefore evaluated and, in view of the small displacements expected in operation, each spring was assumed linear about this point. In the initial model, the disc springs were represented by spring elements (CELAS2). For compatibility with the SOL 200 model updating procedure in Simcenter 3D (version 2406) [18], these entries were subsequently replaced by equivalent PBUSH elements, while preserving the same linear stiffness values. All remaining degrees of freedom were initially treated as rigidly constrained. This assumption was subsequently relaxed during model calibration to reproduce the behavior observed during the experimental campaign.

The compression bands were modeled with two-dimensional shell elements, an appropriate choice given their geometry, again using CTRIA3 and CQUAD4 formulations. Unlike the end-plates and compression plate, the dynamic response of a band is expected to be strongly modulated by tensile pre-load. To quantify this effect, a preliminary static analysis was performed to recover the in-plane membrane stresses generated by the assembly force. These stresses were then introduced as initial conditions in a subsequent eigenvalue extraction. The results confirmed that tensile pre-stress markedly stiffens bending about the $yz$-plane. Specifically, all compression bands natural frequencies rise, whereas the associated mode shapes remain essentially unchanged. In contrast, bending about the $xz$-plane and torsional modes exhibit only minor frequency shifts, reflecting their higher inertia and lower sensitivity to membrane stress. Ideally, the full-stack model would incorporate these stresses directly via a coupled static–dynamic analysis. That workflow, however, is computationally onerous. A pragmatic alternative is to reproduce the pre-stress effect through “equivalent” material properties. Because the band behaves more like a string than a classical beam when tensile pre-stress is applied, a single Young’s modulus in the unstressed configuration cannot simultaneously reproduce all pre-stressed bending frequencies. To quantify the impact of this simplification, three modal analyses were carried out in which the modulus was separately calibrated to match the first, second, and third bending frequencies of the pre-stressed reference model. The results showed that changing the modulus shifts only compression band dominated modes, whereas the global stack frequencies remain essentially unchanged (Figure 2). For the full-stack simulations, the modulus tuned to reproduce the first bending frequency was therefore adopted. This choice preserves the dynamically most relevant band behavior while keeping the global model tractable.

The remaining stack components were modeled as follows: the spring support plates were discretized with two-dimensional shell elements (CTRIA3/CQUAD4), whereas the rods were represented by one-dimensional beam elements.

After the discrete models of the individual components had been generated, the next step was to prescribe their mutual connectivity. Coincident nodes were created between the compression and bottom end-plates and the active region of the cell package, with additional node matching along the gasket tracks. Further interfaces were required between:

• The rods and both end-plates;
• The top end-plate and the three springs;
• Each spring and its corresponding spring support plate;
• The three spring support plates and the compression plate;
• The rods and the compression bands.

All interfaces were implemented with rigid elements (RBE2) that transfer the full six degrees of freedom between master and slave nodes. No contact non-linearities were introduced at this stage. This strategy preserves kinematic compatibility across components, enforces load transfer without introducing artificial compliance, and yields an assembled FE model that faithfully reproduces the mechanical behavior of the stack.

The final FE representation of the FC stack contains zero-, one-, two-, and three-dimensional elements, with approximately 60,000 shell elements and about 420,000 solid elements. The complete assembly is illustrated in Figure 3.

To reproduce the experimental configuration, the aluminum cage employed during testing was also modeled. Due to its predominantly thin geometry, the cage was meshed with approximately 20,000 shell elements. Individual cage components were joined with rigid elements. The cage model was then coupled to the previously developed FC stack model to create a single, comprehensive FE assembly. Rigid elements were introduced through eight bolt holes at the cage base and four at the cage lid, tying these locations to the corresponding holes in the bottom and top end-plates, respectively.

Boundary conditions were applied through a spider layout; specifically, a central control node was rigidly connected to all perimeter nodes (Figure 4). This arrangement emulates the cage to an adaptation plate bolted joint. Imposing a clamped constraint on the control node distributed a uniform, fully fixed restraint across the interface, faithfully reproducing the experimental setup while keeping the numerical implementation straightforward.

Load Cases for Immunity Assessment

Sinusoidal vibrations are a common phenomenon in rotating or oscillating machinery, including motors, engines, turbines, and other appliances. In the context of hypothetical aircraft, a fuel cell stack is assumed to be mounted on the aircraft airframe. When it is mounted within a chassis housing, and the system is required to operate in a sinusoidal vibration environment, then the chassis will act as the first degree of freedom, because the vibration energy will excite the chassis structure first. The fuel cell stack, being mechanically coupled to the chassis, receives dynamic excitation from it and thereby acts as the secondary degree of freedom. Under such conditions, resonant phenomena may occur in both the chassis and the stack. Due to the mechanical coupling between these components, resonance in one mass induces a corresponding response in the other. If the resonant frequencies of the chassis and the stack are closely aligned, and if both exhibit high transmissibility factors, the vibrational response of the chassis can amplify that of the stack. This amplification can lead to elevated acceleration levels within the stack, significantly increasing the risk of rapid fatigue failure. To mitigate these risks, it is critical to ensure adequate separation between the resonant frequencies of the chassis and the stack.

For qualification simulations, the excitation spectra were derived from the vibration requirements in RTCA/DO-160G, Section 8 (Vibration) [16]. Assuming installation on a fixed-wing aircraft powered by turboprop engines, three harmonic test envelopes are available, each keyed to a specific equipment location. The present stack is intended for fuselage mounting; consequently, the “L” curve, representative of vibration environments on frames, stringers, skin, and other fuselage structures, was adopted for the sinusoidal vibration assessment. In accordance with the standard, the harmonic sweep must be applied sequentially about the equipment’s three orthogonal axes to ensure a comprehensive evaluation.

Because equipment installed on turboprop aircraft must also satisfy the shock criteria in RTCA/DO-160G, Section 7 (Operational Shocks and Crash Safety) [16], a dedicated shock assessment was performed. In this context, shock refers to the rapid injection of mechanical energy into the structure, producing significant increase in stress, acceleration, velocity, or displacement, typically over time scales comparable with one or more natural periods of the system. In complex structures, shock can excite multiple natural frequencies, potentially causing three primary failure types in systems:

• High stresses, leading to fractures or permanent deformations;
• High accelerations, which may lead to loosening of bolted joints;
• Large displacements, resulting in impacts between components.

Among the available evaluation techniques, a time-domain transient analysis with a prescribed pulse load was selected, as it delivers the complete history of displacements, strains, stresses, and reaction forces under arbitrary, time-dependent excitation.

RTCA/DO-160G defines two representative operational shock pulses for turboprop installations:

• Standard pulse: 11 ms pulse;
• Low-frequency pulse: 20 ms pulse.

In accordance with the procedure, the FE model was subjected, along each of its three orthogonal axes, to three terminal saw-tooth shocks having a peak acceleration of 6 g.

Random vibration analysis is not mandated for turboprop applications and was therefore omitted from the qualification simulations.

2.3. Experimental Program

Experimental analysis of the fuel cell stack was carried out with the CUBE (TeamCorporation, Burlington, WA, USA) [19], a six-degree-of-freedom shaker (Figure 5), which operates from 10 to 300 Hz and offers a frequency resolution finer than 0.5 Hz. This bandwidth, set by the shaker capabilities, is consistent with the vibration environment specified for the intended installation of the stack on the fuselage of a turboprop aircraft according to RTCA/DO-160G [16], for which the standard sinusoidal vibration test exhibits significant excitation between approximately 15 and 150 Hz. Moreover, preliminary experimental analyses indicated that the dynamically relevant global modes of the FC stack are located below that frequency. The upper bound of 300 Hz therefore provides a margin above the highest modes of interest, while remaining compatible with the test facility capabilities.

Four excitation schemes were used. These included open-loop pseudo-random excitation with 40 averages to enhance statistical reliability and controlled sine sweep excitation at three distinct amplitudes: 0.7 g, 1.0 g, and 2.0 g. The overall experimental program is summarized in

Table 2. Summary of the experimental test program.

Run	Control Mode	Excitation Signal	Excitation Axis/Averages
1	Open-loop	Pseudo-random	Z/40
2	Open-loop	Pseudo-random	Y/40
3	Open-loop	Pseudo-random	X/40
4	Closed-loop	Sine sweep 0.7 g	Z/1
5	Closed-loop	Sine sweep 1.0 g	Z/1
6	Closed-loop	Sine sweep 2.0 g	Z/1
7	Closed-loop	Sine sweep 0.7 g	Y/1
8	Closed-loop	Sine sweep 1.0 g	Y/1
9	Closed-loop	Sine sweep 2.0 g	Y/1
10	Closed-loop	Sine sweep 0.7 g	X/1
11	Closed-loop	Sine sweep 1.0 g	X/1
12	Closed-loop	Sine sweep 2.0 g	X/1

, which lists all test runs and their main characteristics.

Fifty-four PCB (The Modal Shop, Cincinnati OH, USA) [20] high-sensitivity triaxial accelerometers were distributed across the stack, compression bands, fixture, and shaker, providing a dense sensor network for response measurement. Frequency response functions (FRFs) were referenced to a control accelerometer on the shaker table. A combination of different excitation directions and amplitude runs was performed. Data were acquired with two Siemens Simcenter SCADAS Lab systems (Siemens Digital Industries Software, Leuven, Belgium) [21] and processed in Simcenter Testlab, version 2406 [22], while natural frequencies, mode shapes, and damping ratios were extracted with the PolyMAX algorithm [23]. For model correlation, the natural frequencies plotted in Figure 6 correspond to the open-loop pseudo-random run along the vertical direction z, at mid level. During the campaign, it was observed that changing the excitation direction modified the identified natural frequencies by up to 4%, and increasing the input amplitude from 0.7 g to 2.0 g produced shifts as large as 17%.

The initial FEM prediction (blue markers in Figure 6) exhibits significant scatter about the experimental trend line (black), confirming that further parameter tuning is required to capture the stack’s dynamic behavior accurately. These discrepancies arise from the simplifying assumptions adopted in the initial FE model, such as isotropic material properties for the cell package, incomplete characterization of the spring stiffnesses, the omission of tensile preload effect in the compression bands, which limit the fidelity of the untuned predictions.

Due to confidentiality restrictions associated with the project, the exact measured frequencies presented in this section, as well as the frequencies and amplitudes reported later in the dynamic analysis results, cannot be disclosed. Consequently, the discussion focuses on relative frequency errors and on normalized or dimensionless comparisons between experimental and numerical responses, ensuring that meaningful trends can still be evaluated without revealing proprietary data.

The initial correlation results are summarized in

Table 3. Comparison between experimental and numerical natural frequencies/shapes of the fuel cell stack.

ID Mode	MAC	Frequency Error (%)
1	0.82	$-9.48$
2	0.84	$-18.81$
3	0.57	19.18
4	0.69	$-31.00$
5	0.87	$-49.26$
6	0.60	$-22.07$
7	0.56	$-22.80$

below. The table also reports the MAC, a scalar indicator that quantifies the degree of correlation between the computed and reference mode shapes.

The subsequent subsection exploits the experimental data to refine the numerical model. Specifically, the natural frequencies and mode shapes obtained from the open-loop pseudo-random excitation run in the z-direction (1.0 g) serve as the reference for tuning the model’s elastic properties. The damping ratios derived from the same run are also embedded in the model to enhance the fidelity of the predicted dynamic response.

2.4. Model Tuning and Validation

Parameter tuning of the fuel cell stack was carried out in Simcenter 3D (version 2406) [18], which integrates the SOL 200 model update, an advanced correlation tool designed to update finite element models to match real-life test data or another more complex analysis model as closely as possible. The SOL 200 model update solution process:

• Links and aligns work solution (numerical) and reference solution (experimental) geometries;
• Correlates frequencies and mode shapes between a work solution and a reference solution;
• Allows the definition of design variables;
• Optimizes according to specified criteria;
• Updates the finite element model of the work solution.

The following Figure 7 illustrates the SOL 200 model update workflow.

The main active layer and gasket volumes of the cell package were first modeled as isotropic, limiting the design variables to Young’s modulus and Poisson’s ratio (the shear modulus being derived from these). This simplification failed to achieve convergence with the experimental modal data, indicating that the materials behave anisotropically. The model was therefore upgraded to an orthotropic formulation. Because post-processing was carried out in Simcenter 3D (version 2406) and analysis in MSC Nastran 2022.4, anisotropy was specified via the MAT9 card. Only the diagonal terms of the $6\times 6$ stiffness matrix were treated as independent design variables, while all off-diagonal coupling terms were held at zero.

An important contributor to the overall stiffness of the FC stack is the mechanical response of the three disc springs. Using the available test data, the axial (z-direction) stiffness of each spring was taken to be identical and set to a value obtained from dedicated PowerCell tests. Initially, all remaining spring degrees of freedom (DOF) were assumed rigid, so the compression plate was permitted to translate only along the stack axis, without any lateral motion or rotation. This numerical assumption, however, contradicts experimental observations, which showed significant relative rotations and translations (particularly rotations) between the top-end plate (which follows the cage motion through its eight bolt attachments) and the compression plate (connected to the top-end plate via the three springs). To address this discrepancy, six stiffness parameters, one for each DOF of the springs, were introduced as design variables. To simplify the optimization process and avoid an excessive number of design variables, all three springs were assumed to have identical properties. This assumption is theoretically justified, as no asymmetry is expected in the yz plane. However, the first experimental axial mode exhibited irregular behavior, with two of the springs acting differently and resulting in a non-uniform extension of the FC stack core. These discrepancies were neglected in the numerical modeling process. Specifically, the originally defined CELAS2 spring property could not be used as a design variable within Simcenter3D. To address this limitation, the numerical model was revised by replacing the CELAS2 property with a PBUSH property, which is compatible with the SOL 200.

The final parameter refined was the Young’s modulus assigned to the compression bands. As previously discussed, these bands remain in tension after the stack is clamped, causing their dynamic response to shift from beam-like to string-like behavior. Accordingly, a fictitious modulus was derived so that the first natural frequency of a single prestressed band, obtained via numerical static/modal analysis, matched the corresponding experimental value. This equivalent modulus, which is significantly higher than the nominal Young’s modulus of the bands, should therefore be interpreted as a numerical tuning parameter that reproduces the effective dynamic stiffness of the pre-tensioned bands rather than as a physical material property. Although the simulated frequency was already close to the measurement, the modulus was retained as a tunable variable to secure the best agreement at stack level. Specifically, the three front bands were assigned identical properties, as were the three rear bands, leading to two distinct design variables. Experimental modal tests showed a clear frequency offset between the front and rear groups but negligible variation within each group, an effect attributed to minor differences in boundary conditions. Summarizing, twenty design variables were defined within the correlation tool:

• The diagonal terms of the $6\times 6$ stiffness matrices of the orthotropic materials used for the main active layer and the gasket (MAT9 in MSC Nastran 2022.4), namely $G_{11}^{\mathrm{active}}$, $G_{22}^{\mathrm{active}}$, $G_{33}^{\mathrm{active}}$, $G_{44}^{\mathrm{active}}$, $G_{55}^{\mathrm{active}}$, $G_{66}^{\mathrm{active}}$ for the main active layer, and $G_{11}^{\mathrm{gasket}}$, $G_{22}^{\mathrm{gasket}}$, $G_{33}^{\mathrm{gasket}}$, $G_{44}^{\mathrm{gasket}}$, $G_{55}^{\mathrm{gasket}}$, $G_{66}^{\mathrm{gasket}}$ for the gasket volumes;
• The translational and rotational stiffnesses of the disc springs about the three axes, $k_{tx}$, $k_{ty}$, $k_{tz}$, $k_{rx}$, $k_{ry}$, and $k_{rz}$;
• The Young’s moduli of the front and rear compression bands, $E_{\mathrm{front}}$ and $E_{\mathrm{rear}}$, defined as described above.

A set of degrees of freedom corresponding to the experimental sensor placements was selected for modal reduction and exported as USET degrees of freedom. Manual adjustments were necessary since the automatic process, intended to match numerical nodes with experimental sensor locations, occasionally failed due to interferences (for example selecting nodes on the cage rather than on the top-end plate). The adaptation plate was deliberately excluded from the FE model, and its measured responses were consequently removed from the DOF set and from the subsequent optimization. Including those data would have spuriously coupled them to cage nodes, even though the adaptation plate moves relative to the cage, whereas the numerical model treats the cage as rigidly fixed at its bolted interfaces.

More than 50 optimization cycles were executed. At each cycle, the objective function, constructed from selected natural frequencies and mode shapes, was updated on the basis of the preceding results. Weighting factors were varied systematically to emphasize, in turn, the higher-order experimental modes and the fundamental global modes. All design variables were allowed to vary within 50–200% of their nominal values, a range that is effective when the expected value already lies near the physical truth. Given the limited prior information on the spring stiffness (apart from the axial value provided by PowerCell), a series of exploratory SOL 103 modal analyses was first performed to map the plausible stiffness envelope. Visual inspection of those results served to narrow the bounds and to establish consistent nominal values before launching the SOL 200 optimization. Initial guesses for the remaining design variables were taken from the pre-update model, with one important exception: the fuel cell stack core. Whereas the pre-update model treated this region as isotropic, the updating exercise required an orthotropic representation. Accordingly, the diagonal terms of the MAT9 stiffness matrix for the stack core were seeded with values that depart slightly from the original isotropic estimates.

With the initial values and boundaries in place, modal correlation between the numerical and experimental data was enforced by imposing a MAC threshold of greater than 0.55. A least squares optimization scheme iteratively adjusted the design variables. After every SOL 200 cycle, the newly optimized values became the starting point for the next iteration. Once the convergence criteria were met, the FE and simulation files were updated with the finalized material constants and physical properties.

Figure 8 juxtaposes the updated model’s natural frequency predictions with the experimental benchmarks. The correspondence is excellent. The numerically predicted points (magenta) track the experimental trend (black line) almost exactly, with the first global modes (IDs 1, 2, 4, and 5) matching identically and the higher-order modes remaining within an acceptably narrow tolerance.

The MAC values in

Table 4. Comparison between the updated model’s numerical predictions and experimental natural frequencies/shapes of the fuel cell stack.

ID	MAC	Frequency Error (%)
1	0.92	$-0.55$
2	0.89	$-1.99$
3	0.63	0.06
4	0.81	$-0.28$
5	0.88	$-0.09$
6	0.85	13.82
7	0.76	6.64

likewise confirm the close correspondence between the experimental and numerical mode shapes. The comparatively low MAC for mode ID 3 should not be interpreted as a weakness of the FE model. This mode is associated with the compression band, where only a limited number of sensors were installed. The resulting limited instrumentation prevented the reconstruction of the experimental mode shape, consequently reducing the MAC value relative to the numerical prediction.

The updating procedure reduced the discrepancy between the measured and predicted natural frequencies of the first four global modes to a negligible level. The MAC values likewise improved markedly, all exceeding 0.80, which is an excellent result given the complexity of the model. Mode shape comparisons are shown in the following figures, with the experimental shapes displayed on the left and the numerical counterparts on the right for every mode listed in Table 4. As an example, Figure 9 depicts the first bending mode of the stack around the x-direction. Figure 10 presents the first axial (z-direction) mode of the stack. The experimental mode shape displays a slight asymmetry even though no corresponding geometric asymmetry is evident (omitted from the FE model for the sake of simplicity). Future work will investigate the underlying causes of this behavior. Figure 11 presents the first bending mode of a representative compression band. The analogous modes of the remaining five bands, which are qualitatively similar, are omitted for brevity. Figure 12 displays the first bending mode of the stack around the y-axis. The simulation successfully reproduces the spring compliance, manifested in the predicted rotation of the compression plate, and likewise captures the corresponding rotation of the bottom end-plate. Figure 13 depicts the first torsional mode of the stack. The experimental and numerical mode shapes show excellent agreement. Figure 14 shows the second bending mode of the stack around the x-axis. The small residual frequency discrepancy was deliberately left uncorrected, since additional tuning would have compromised the agreement already obtained for the fundamental bending mode. Figure 15 presents the second bending mode of the stack around the y-axis. The corresponding MAC value indicates good correlation, and the remaining frequency discrepancy falls well within acceptable limits.

3. Results

3.1. Numerical–Experimental Comparison of the FC Stack Dynamic Response

Damping values were selected based on experimental results from random vibration tests. Since these damping values correspond to the experimental resonant frequencies, slight frequency shifts were applied to better align the numerical resonant frequencies. The data were then linearly interpolated to cover the entire range from 0 to 300 Hz. For the dynamic analysis, a modal approach was adopted using a modal basis up to 600 Hz, twice the highest frequency investigated. Specifically, the responses were computed over a frequency range using a non-uniform step size, with finer resolution near the resonant frequencies to ensure precise evaluation while balancing accuracy with computational efficiency.

During the experimental campaign, a harmonic base excitation of 0.7 g was applied over the 10–300 Hz bandwidth, sequentially in the x, y and z directions. A control accelerometer mounted on the cube shaker table ensured that the desired input was accurately reproduced in the analyzed direction. The tuning of the FE model natural frequencies relied exclusively on the open-loop pseudo-random excitation run in the z direction, whereas the sine sweep tests in all three directions provide the comparison dataset discussed in the following. This choice allows residual numerical–experimental discrepancies to be more clearly attributed to unmodelled nonlinearities (e.g., bolted joints and contact interfaces) and to limitations in the representation of the test boundary conditions.

In the FE model, the adaptation plate was idealized as rigid and constrained with fully fixed boundary conditions (Figure 16). The bolt locations were kinematically coupled to a single reference node via spider links, and both the clamped constraint and the prescribed base acceleration were applied at this node for the dynamic analyses. This modeling choice effectively assumes that the adaptation plate undergoes a purely translational, spatially uniform base motion, thereby neglecting any rotational motion and spatial variability, even though the experimental data clearly indicate that the plate experiences both rotations and cross-axis motion.

The control system operates with a single-input single-output (SISO) strategy, regulating only the acceleration measured by one accelerometer in a single direction. As a result, the actual motion of the stack boundary is not straightforward to reproduce numerically. The four accelerometers mounted on the plate recorded non-zero acceleration components along the axes orthogonal to the intended input, revealing residual cross-axis motion, and their signals were not identical, indicating the presence of a rotational component in the plate motion.

To apply a realistic input to the numerical model and enable a meaningful comparison of dynamic responses, an approximate representation of the plate excitation was therefore constructed from the four accelerometer signals. Two averaging strategies were investigated:

• Mean of the absolute amplitudes, namely the average of the magnitude of each plate sensor measurement (discarding phase information);
• Magnitude of the mean signal, namely the absolute value of the complex-averaged time histories.

The resulting input differs, and in both cases the derived signal must be regarded as an approximation of the true plate excitation, since the use of averaged accelerometer data, although including the x, y, and z components, still neglects the spatial variability of the motion over the plate and the associated rotational degrees of freedom.

The following Figure 17 identifies the sensor locations used to calculate the numerical dynamic responses and to compare them with the experimental measurements.

The curves derived from the arithmetic mean of the absolute plate accelerations reproduce the experimental trend more faithfully than the curve based on the absolute value of the complex-averaged signal. Moreover, since the phase information was purposely discarded in this approach, the comparison is only meaningful in the direction of excitation, where the imposed acceleration is controlled. Accordingly, when the stack is excited along the x-axis, only the x-axis response is analyzed, and when excited along the y-axis, only the y-axis response is examined and the same procedure is applied for z-axis excitation. Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26 compare the predicted acceleration spectra in the x, y, and z directions with the corresponding experimental measurements at each principal instrumentation location, namely, the stack, top end-plate, compression plate, bottom end-plate, cage, and compression bands.

Figure 18 juxtaposes the experimental acceleration spectrum (solid black) with the finite element prediction (red dashed) obtained from accelerometer #1 on the bottom end plate. In the x-direction, two resonant features dominate the response. The first, at low frequency, corresponds to the fundamental bending mode of the stack around the x-direction. Because the imposed excitation is not perfectly uniaxial, small components in the orthogonal directions appear as well. The model reproduces this resonance at the correct frequency but slightly underestimates its amplitude, a discrepancy consistent with the approximations adopted for the input definition. A second resonance, associated with the fundamental bending mode of the stack around the y-direction, appears clearly in the simulation yet is scarcely discernible in the test data, a difference that may stem from the actual manner in which the bottom end plate is attached to the cage. At higher frequency, a sharper peak representing the second y-direction bending mode, in which the cage significantly participates, is located accurately, though its numerical amplitude remains below the measured value. Immediately after the first prominent numerical peak, the experimental trace reveals a deep anti-resonance. The simulation also captures this feature, albeit with a frequency shift. In the y-direction, the experimental spectrum displays a cluster of closely spaced, low-amplitude peaks. The first of these is associated with the fundamental bending mode of the stack around the x-direction. Although this resonance (from pseudo-random analysis) was used for tuning and the numerical model does exhibit a peak at the calibrated frequency, a residual frequency offset remains, attributable to the stack’s nonlinear behavior, which is not characterized in the simulation. The peak magnitude, however, is reproduced accurately. Two further resonances follow, the second y-bending mode and the second x-bending mode. As observed for the previous axis, the largest high-frequency peak evident in the test data is underestimated by the finite element prediction. Along the z-axis, the low-frequency response is governed by the stack’s fundamental axial mode. The numerical analysis pinpoints this resonance with great accuracy, but its amplitude is slightly over-predicted, implying that the real structure is marginally stiffer than assumed. The simulation also shows a secondary peak arising from the fundamental y-bending mode, most likely excited by small out-of-plane components of the shaker input. Coherently, the immunity test results discussed in the next subsection, where only z-axis excitation was applied, exhibit no peak at this frequency. A complementary situation is observed at mid band, where the second y-bending mode of the stack makes a small contribution to the measured z response, yet this feature is absent in the numerical spectra, again suggesting that the idealized input in the model lacks the cross-axis components present in the experiment. At higher frequency, the second axial mode is captured with good fidelity, though both its frequency and amplitude are modestly overestimated.

Figure 19, Figure 20, Figure 21 and Figure 22 present the measured and simulated dynamic responses at sensors #2 to #5, which are positioned progressively from the base to the top of the cell package. Although each subplot refers to a different transducer and one of the three orthogonal directions, the graphs provide similar insights. In every case, the finite element model identifies the fundamental resonances with notable accuracy. The first global mode, namely the stack bending around the x and y directions and stack extension for z, appears at almost the same frequency in both experiment and simulation. The main variation from one sensor to the next lies in amplitude rather than frequency. For the x and y components, the acceleration at the first bending resonance increases from sensor #2 near the base, peaks mid height, and then diminishes toward sensor #5 at the top. The numerical curves reproduce the same distribution, confirming that the model captures the modal shape, yet they slightly overestimate the growth, so the simulated peaks exceed the measurements, most noticeably in the x-response. The z-component behaves differently, specifically, its first resonance amplitude remains almost uniform along the height, a feature that the model replicates, although with a systematic overprediction. At higher frequencies, additional modes govern the response. For the x-direction, the second x-bending mode of the stack is prominent experimentally but is only partially reproduced numerically. In the y-direction, the first y-bending and second y-bending modes appear. The former is matched in amplitude but shows a slight frequency shift, while the latter is poorly represented. This discrepancy can be attributed in part to cross-axis input components that are only approximately prescribed in the numerical model, allowing certain experimentally excited modes to go unpredicted. Furthermore, the model updating procedure calibrated only the first seven experimental modes, those below approximately 300 Hz, leaving modes in the 300-600 Hz range untuned. Although the influence of these higher modes diminishes with frequency, they still affect the spectra beyond 150 Hz, so their exclusion could account for much of the residual discrepancy observed at higher frequencies. In the z-direction, the response is captured reasonably well except at sensor #4, where the model predicts a nodal plane inconsistent with the experiment. Smaller peaks produced by mixed-axis excitation appear in the measurements but are absent from the simulations, reflecting the idealized nature of the imposed input in the numerical analysis.

Figure 23 compares the experimental and numerical acceleration spectra obtained from sensor #6 on the compression plate. Along the x-axis, the prediction diverges appreciably from the test data, even though the principal resonance is positioned correctly in frequency. In the y-direction, the response is dominated by the same low-frequency component observed at the bottom end-plate mounted sensors. The simulation, instead, reproduces the sequence of resonance and anti-resonance, but the entire pattern is shifted in frequency, and the computed amplitudes, while of the correct order of magnitude, remain systematically higher. For the z-axis, the model identifies the fundamental and second axial modes within a few hertz of their measured positions yet overestimates the peak amplitudes, most notably at the fundamental.

Figure 24 compares the experimentally acquired and numerically simulated acceleration spectra for sensor #26, which is mounted on the stack’s top end-plate. Along the x-axis, the finite element model reproduces the principal resonant behavior. Specifically, the first resonance is matched in frequency, the anti-resonant trough evident in the test data appears but is less pronounced, the following peak is overestimated, and the model subsequently underpredicts the amplitude of the highest frequency resonance. In the y-direction, the simulation reflects the low-frequency broadband content and the first resonance with comparable accuracy, yet, at higher frequencies, it mirrors the trend seen for the x-axis by underestimating the dominant resonance, notwithstanding its correct frequency prediction. For the z-axis, the response is governed by two marked resonances corresponding to the fundamental and the second axial mode of the stack. Both frequencies are captured with high fidelity, although their peak amplitudes are slightly overpredicted.

Figure 25 compares the measured and simulated acceleration spectra recorded by sensor #36, mounted on the external cage. Along the x-axis, the behavior mirrors that already discussed for sensor #26. For the y-axis, the numerical response traces the gradual broadband rise observed in the test and replicates the first resonance with satisfactory amplitude, though with a slight upward frequency shift. Beyond this, the simulation continues to track the experimentally observed attenuation at higher frequencies. The z-axis again exhibits the strongest agreement. Two dominant resonances govern the response, and both are predicted within a few hertz of the test values. The model slightly overestimates the amplitude of the first peak, matches the second peak almost exactly, and reproduces the intervening anti-resonance with good fidelity.

Finally, Figure 26 presents the acceleration spectra recorded by sensor #52 on the compression band alongside the corresponding FE predictions. Along the x-direction, the model exhibits the same trends already noted for sensors #26 and #36. In the y-direction, the experimental response is dominated by a pronounced low-frequency resonance. The simulation locates this peak slightly higher in frequency and markedly underestimates its amplitude, and it continues to underpredict the response over the rest of the band. The z-direction is again controlled by two principal resonances whose frequencies are reproduced within a few Hertz, while both peak amplitudes are marginally overestimated.

In summary, the FE model captures the fundamental resonant frequencies of the stack with remarkable accuracy across all sensors, correctly identifying bending modes around the x- and y-directions and axial modes in z. Consistency in amplitude is weaker. Responses along z are generally over-predicted, whereas high-frequency peaks in the in-plane directions tend to be underestimated. Even so, the simulation reproduces the overall dynamic behavior of the stack. The remaining gaps are chiefly attributable to the real structure’s nonlinear response, the idealized constraints assumed in the analysis, and the simplified characterization of the base excitation, particularly with regard to the cross-axis input components.

3.2. Preliminary Immunity Analysis

A preliminary immunity assessment of the tuned fuel cell stack was conducted in accordance with RTCA/DO-160G standards. The evaluation considered both harmonic and shock pulse excitations, applied independently along the x, y, and z axes. In each scenario, a single excitation was applied in the positive direction of the considered axis, under the assumption of geometric and boundary condition symmetry, so that the corresponding response to a negative pulse would be identical in magnitude. The dynamic response was measured at the same location used for the experimental–numerical comparison. Figure 27 presents a summary of the harmonic response results, showing the ratio of total displacement amplitudes. This ratio is defined as the measured displacement at a given sensor relative to the maximum displacement observed across all sensors. The corresponding acceleration ratio spectra in the three orthogonal directions are shown in Figure 28.

Figure 27 exhibits three distinct resonance clusters within the frequency range of interest. The lowest frequency peak is governed by bending around the x-axis. At this mode sensor #4, positioned at mid height of the cell package, records the largest normalized displacement for that component. In the compression bands the response is determined mainly by the first bending of the bands themselves rather than by global stack modes. Because of the comparatively large moment of inertia, the x-axis contribution remains almost negligible, whereas the y-axis shows the greatest relative displacement and therefore serves as the reference for the normalization of the displacement ratios. The second intermediate frequency peak is associated with the axial (z-direction) mode and is particularly relevant for the top and bottom end plates. The third and highest frequency peak results from bending around the y-axis. The amplitudes rarely exceed a ratio of 0.2, indicating that this mode is less critical than the others, yet careful control of the y-axis response remains essential because it defines the upper bound of the displacement envelope. In contrast to the displacement results, the acceleration response is dominated by the z-axis. Outside the compression band resonances, it consistently attains the largest normalized amplitudes across the stack. Within the compression bands, however, the dynamics differ. There the y-axis still produces the highest levels, mirroring its behavior in the displacement plots. The remaining two resonant peaks associated with bending around the y- and x-axes remain of comparable magnitude at all sensor locations (except for sensor #52).

For completeness, Figure 29 illustrates contour plots of the total displacement extracted at the resonance frequency corresponding to each of the three principal axis excitations in the standard immunity harmonic analysis.

The shock pulse immunity of the stack was evaluated with the RTCA/DO-160G saw-tooth profile, applying a 6 g peak acceleration for the prescribed 11 ms duration. To isolate the intrinsic response of the stack from that of the supporting cage, the cage motion, recorded at a node adjacent to the central spider where the load was applied, was subtracted from the time-history of every sensor. The resulting relative displacements were then normalized in the same manner adopted for the harmonic immunity study (Figure 30). The accelerations were subsequently derived by finite difference differentiation of the displacement records and were normalized using the same procedure described above (Figure 31).

Figure 30 demonstrates that the 6 g/11 ms saw tooth shock predominantly excites the lightly damped fundamental bending mode around the x-axis. This response is most pronounced at the stack mid height, where sensor #4 attains the reference amplitude. Throughout the cell package the y- and z-axis motions remain secondary and of comparable magnitude, becoming significant only at the top and bottom end plates, a behavior already anticipated from the harmonic study. At those end plate sensors the records depart from a purely sinusoidal shape, indicating that additional modes participate in the transient response. Overall, the largest displacements occur within the cell package and the attached compression plates, whereas the end plates and the compression band experience much smaller oscillations. These findings confirm that the cell package, and in particular its y-axis motion, constitutes the most critical element under the prescribed RTCA/DO-160G shock pulse. The acceleration time-histories follow the same spatial trend observed in the displacement plots, although with less pronounced axis to axis difference. Within the cell package the y-direction still exhibits the largest oscillatory content, yet the peak levels on the x- and z-axes now differ by only a few points, indicating that all three directions must withstand nearly comparable inertial loads under the 6 g/11 ms shock. In contrast, the end plates and compression band experience substantially lower acceleration magnitudes, confirming that the cell package remains the critical component from a shock immunity standpoint.

It is important to note that all dynamic simulations in this work are based on a linearized representation of the stack. In addition to the compression bands and disc springs, which are explicitly linearized through a fictitious modulus and a tangent stiffness at the nominal preload, other components and interfaces (e.g., the cell package, gasket contacts and bolted joints) may also exhibit amplitude-dependent behavior. The material and stiffness parameters used in the FE model were tuned using modal data obtained at specific excitation levels, so that the resulting properties are strictly valid in a neighborhood of these operating conditions. For substantially different load amplitudes, both under sinusoidal excitation and under shock pulses, some deviation from purely linear behavior is expected, for instance in the form of modest frequency shifts or changes in effective damping. The preliminary immunity results presented in the following should therefore be interpreted as quasi-linear predictions, which are appropriate for assessing relative trends and critical locations but do not capture all possible non-linear effects at extreme load levels.

4. Conclusions

The numerical–experimental workflow developed within the NEWBORN project demonstrates that a simplified yet balanced finite element model, consisting of a three-dimensional solid representation of the cell package combined with shell and beam elements for the remaining components, can reproduce the global dynamics of a proton exchange membrane fuel cell stack with generally good, though not perfect, agreement.

Before updating, the initial FE model exhibited substantial discrepancies with respect to the experimental modal data: the frequency errors for the considered modes ranged from about $-50\%$ to $+19\%$, and several mode shape correlations yielded MAC values close to 0.6. Iterative SOL 200 updating, based on 54 instrumented points, reduced the frequency discrepancies for the first four global modes to within $2\%$, while the remaining tuned modes were brought within approximately $14\%$ of the measurements. At the same time, the MAC values for these modes increased, reaching 0.92, 0.89, 0.81, and 0.88 for the four global modes and remaining at or above 0.76 for the higher modes. These improvements confirm that orthotropic stiffnesses for the cell package layers, six degree of freedom spring compliances, and equivalent moduli for the prestressed compression bands form a sufficiently rich design variable set to align simulation and experiment in the 10–300 Hz band.

During the shaker campaign, changes in excitation direction were observed to modify the identified natural frequencies by up to 4%, while increasing the sine sweep amplitude from 0.7 g to 2.0 g produced shifts as large as 17%. The tuned, nominally linear model reproduces the modal characteristics around the calibration conditions, but these experimental observations underline the presence of amplitude-dependent effects in the real structure. The remaining discrepancies between numerical and experimental responses, most notably the overprediction of some z-axis accelerations and the underprediction of certain high-frequency in-plane peaks, are therefore attributed to a combination of unmodelled nonlinearities, untuned higher-order modes and idealizations in the boundary conditions and base excitation.

When the tuned model was forced with the experimentally derived adaptation plate spectra, it captured the principal resonances recorded by the distributed accelerometer array and correctly identified the governing mode shapes at all key locations (cell package, end-plates, compression plate, cage, and bands). The fundamental bending around the x-axis and the first axial extension in z were reproduced at nearly identical frequencies to the tests, while the higher y-bending mode was captured with a modest residual frequency offset and generally lower normalized displacement levels. In the harmonic preliminary immunity analysis based on the RTCA/DO-160G “L” curve, three resonance groups were identified: the lowest frequency peak, dominated by x-axis bending with maximum displacement at mid height of the cell package, an intermediate peak associated with the axial mode, particularly relevant for the end-plates, and a higher frequency peak corresponding to y-axis bending, whose normalized displacement ratio rarely exceeds 0.2, indicating a secondary contribution to the overall displacement envelope. Outside the compression band resonances, the z-axis consistently produced the largest normalized acceleration levels across the stack, confirming that axial inertial loading is critical from an acceleration standpoint.

The standard 6 g/11 ms RTCA/DO-160G saw-tooth shock pulse predominantly excited the fundamental x-bending mode of the stack. The largest normalized displacements occurred in the cell package and the attached compression plates, with sensor #4 at mid height defining the reference amplitude, whereas the end-plates and compression bands experienced significantly smaller oscillations. The corresponding acceleration time histories showed that, within the cell package, the y-direction maintained the largest oscillatory content, while the peak acceleration levels on the x- and z-axes differed by only a small margin, implying that all three directions must withstand nearly comparable inertial loads under the prescribed shock pulse. In contrast, the end-plates and compression band exhibited substantially lower normalized acceleration magnitudes, confirming that the cell package remains the mechanically critical component from a shock immunity perspective.

All immunity computations were performed with a linearized model. The calibrated properties are therefore strictly valid in a neighborhood of the modal test conditions, and both harmonic and shock responses at substantially higher amplitudes should be interpreted as quasi-linear predictions. Nevertheless, the combined numerical–experimental methodology provides quantitative evidence that, within the considered operating range and under the RTCA/DO-160G “L” vibration and standard shock environments, the most critical deformations and inertial loads are confined to the central cell package region, while the surrounding structure experiences substantially lower demands.

In summary, the validated model offers a robust physical framework for identifying critical sensors and locations for future qualification campaigns, deriving preliminary margins against harmonic and shock excitation, and guiding the design of isolation and reinforcement measures by highlighting the vibration modes and structural regions most susceptible to operational loading.

Future work will extend this approach to direct simulations on the NEWBORN demonstrator stacks as additional dynamic test data become available.