Vibration behavior of hybrid sandwich beams with TPU lattice cores and carbon fiber face sheets

Abstract

This study investigates the vibration behavior of hybrid sandwich beams composed of thermoplastic polyurethane (TPU) lattice cores and carbon fiber reinforced polyamide face sheets. Ten different strut-based lattice topologies were designed with approximately constant mass to evaluate the effect of geometry on dynamic response. Experimental modal analysis was conducted under cantilever boundary conditions to determine the fundamental natural frequencies and damping ratios. The results indicate that lattice topology plays a critical role in governing stiffness and damping characteristics. Configurations with higher connectivity exhibit increased stiffness, while structures allowing greater deformation show enhanced damping characteristics. An inverse relationship between stiffness and damping is consistently observed across the investigated designs. These findings highlight the potential of geometry-driven design in tailoring the dynamic performance of lattice-based sandwich structures, and provide insights for the development of lightweight metamaterial-inspired systems with application-specific vibration characteristics.

Introduction

Sandwich composite structures are widely used in engineering applications due to their high stiffness-to-weight ratio and efficient load-carrying capabilities. In such systems, a lightweight core is placed between two stiff face sheets to enhance bending rigidity while minimizing overall mass. Conventional core materials such as honeycombs and foams have been extensively employed; however, their geometric constraints and limited tunability restrict the ability to tailor mechanical and dynamic properties for advanced applications.^1–7 Honeycomb cores have historically been used in aerospace components such as control surfaces, engine nacelles, and landing gear panels. ⁸ Despite their efficiency, their anisotropic behavior and closed-cell geometry impose limitations on design flexibility, particularly under multiaxial loading and vibration conditions. These constraints have motivated the development of alternative core architectures capable of providing improved and more controllable performance. This anisotropic behavior may influence the dynamic response of the structure, as directional variations in stiffness can lead to non-uniform vibration characteristics under time-varying loading conditions. Consequently, a critical need has emerged for novel core architectures capable of transcending the inherent limitations of traditional honeycomb structures.

Recent developments in additive manufacturing have enabled the fabrication of architected lattice cores with highly controllable geometry. By varying unit cell topology, connectivity, and relative density, lattice structures can be designed to achieve targeted stiffness, strength, and energy absorption characteristics. The deformation behavior of lattice structures varies depending on geometry, ranging from bending-dominated to stretch-dominated load transfer mechanisms, which directly influence structural stiffness and damping behavior.^9–15 In this context, architected materials, including metamaterials and lattice structures, have attracted significant attention due to their ability to exhibit tailored mechanical and dynamic properties through geometry rather than composition. These materials enable unprecedented control over wave propagation, stiffness, and energy absorption, addressing the limitations of traditional core designs and offering new opportunities for enhanced performance.^16–22

From a vibration perspective, lattice topology plays a key role in determining modal properties such as natural frequency and damping ratio. Structures with higher connectivity and inclined load paths generally exhibit higher stiffness and natural frequencies, whereas configurations allowing greater local deformation may contribute to increased energy dissipation. In the case of elastomeric materials such as thermoplastic polyurethane (TPU), additional damping mechanisms arise from intrinsic viscoelastic behavior, further influencing the dynamic response. Although the influence of lattice topology on stiffness and damping has been widely reported, most studies focus on either rigid lattice structures or fully polymeric systems. In contrast, the dynamic response of hybrid sandwich structures combining compliant lattice cores with high-stiffness composite face sheets remains insufficiently explored. Moreover, previous studies often consider individual lattice types rather than providing systematic comparisons across multiple geometries under controlled mass conditions.^23–29

The present study aims to address this gap by experimentally investigating the vibration behavior of hybrid sandwich beams composed of TPU lattice cores and carbon fiber reinforced polyamide (PAHT-CF) face sheets. Ten different strut-based lattice topologies are examined under cantilever boundary conditions using modal analysis techniques. The geometries are designed to represent variations in connectivity and deformation mechanisms while maintaining approximately constant mass, enabling a direct comparison of topology-driven effects.

Accordingly, 10 lattice configurations with equalized panel masses were designed to ensure that variations in vibration response arise primarily from differences in structural rigidity rather than mass. In addition to topology, the effects of key geometric parameters, including core presence, strut width, and strut angle, are systematically investigated to better understand their role in determining dynamic behavior. In such configurations, the interaction between geometry-driven deformation mechanisms and the viscoelastic nature of TPU introduces additional complexity into the vibration response and, required more comprehensive analysis. In the scope of this study, machine learning approaches are also employed as a complementary tool to explore the relationship between structural parameters and dynamic performance. Given the relatively limited dataset size (n = 30), the goal is not to establish highly accurate predictive models, but rather to provide a qualitative assessment of parameter influence and to identify underlying trends governing vibrational characteristics. Through this combined experimental and data-driven approach, the study provides new insights into the design of high-performance hybrid lattice sandwich structures for vibration-sensitive applications. Unlike most existing studies that focus on either rigid lattices or purely polymeric systems, this work provides a controlled experimental comparison across multiple topologies under nearly constant mass conditions.

Materials and methodology

In this study, composite sandwich specimens having 10 different lattice geometries were produced by using Fused Deposition Modeling (FDM) printing method. The lattice core material employed was a commercially available thermoplastic polyurethane (TPU), supplied under the trade name CR-TPU (Creality, China). The material exhibits a Shore hardness of 95A and a density of 1.24 g/cm³, indicating a predominantly elastomeric behavior suitable for vibration-sensitive applications. TPU is widely recognized for its combined elastic and viscoelastic response, characterized by pronounced hysteretic energy dissipation under cyclic loading. To maximize flexural rigidity of the overall structure, the face sheets of the sandwich structures were fabricated from carbon fiber reinforced polyamide (PAHT-CF) using FDM. The feedstock material used was Ultrafuse PAHT CF15 supplied by BASF 3D Printing Solutions BV. This composite filament consists of a high-temperature polyamide matrix reinforced with 15% carbon fibers, offering superior dimensional stability and heat resistance up to 130°C. The printed material exhibits a density of 1.23 g/cm³ (1232 kg/m³) and a tensile modulus of approximately 8.4 GPa (8386 MPa) in the XY orientation, ensuring high stiffness-to-weight performance. The face sheets were printed with a symmetric raster orientation of ±45°, providing balanced in-plane shear properties while maintaining high bending resistance. The selection of these high-modulus PAHT-CF face sheets was motivated by their high specific modulus, which enables the core-dominated dynamic response of the sandwich structure to be clearly isolated and analyzed. In the scope of this study, the mechanical properties of tensile and vibration samples printed with 100% infill TPU and PAHT-CF were investigated (Figure 1). Additional FDM process parameters such as printing speed, layer height, and infill strategy were controlled to ensure consistency. The adhesive properties and curing conditions were also standardized to minimize interfacial variability. The printing parameters were controlled to ensure consistency across all specimens. The nozzle temperature was set to 240 and 280°C for TPU and PAHT-CF, respectively, and the printing speed was maintained at 30 mm/s. A layer height of 0.2 mm was used, and the infill strategy was set to 100% controlled lattice infill depending on design. The PAHT-CF face sheets were printed separately using a hardened steel nozzle with a diameter of 0.6 mm to prevent clogging and abrasion. A heated bed temperature of 100°C was maintained to ensure optimal layer adhesion and minimize warping. The specific raster angle of ±45° was applied to align the reinforcement fibers along the principal shear directions. These parameters were selected to achieve stable printing conditions and consistent material properties for all specimens. The bonding between the lattice core and the face sheets was achieved using an epoxy adhesive. The adhesive was applied uniformly at the interface to ensure proper contact and load transfer. The dynamic behavior of multi-material and multilayer systems is significantly influenced by material interfaces. Stiffness distribution and energy dissipation can be greatly impacted by variations in interface performance, including degradation processes like debonding and delamination, which may alter stiffness distribution and energy dissipation, thereby affecting vibration characteristics such as natural frequencies, damping ratios, and mode shapes.^30–32 In the present study, ideal interface bonding is assumed in order to isolate and evaluate the influence of lattice topology on global vibration response. Therefore, interface failure mechanisms are not explicitly considered. This represents a limitation of the current work, and future studies should incorporate interface damage modeling to better capture realistic structural behavior.

Figure 1.

Strain-stress curves for tensile test specimen printed by (a) TPU and (b) PAHT-CF filament.

Ten different three-dimensional lattice unit-cell topologies (named as M1–M10) were designed and produced to examine the influence of spatial architecture on the dynamic response of hybrid sandwich beams. The geometries were generated in a CAD environment and constructed within an identical cubic bounding volume of 10 × 10 × 10 mm³. All unit cells were designed to possess identical theoretical relative density based on CAD-calculated solid volumes. Strut diameters were parametrically adjusted according to geometric requirements to maintain equivalent solid volume fractions across configurations while ensuring manufacturability under FDM constraints. The selected topologies exhibit varying degrees of nodal connectivity, diagonal reinforcement and load-path orientation. The characteristic geometric parameters and inclination angles for each topology are presented in Figure 2. It should be noted that the present study focuses exclusively on strut-based lattice geometries. Therefore, TPMS classification is not applicable to the current set of geometries. The investigated lattice structures are characterized based on their unit cell type, connectivity level, and presence of internal diagonal strut arrangements in the cell. These parameters influence load transfer paths and deformation behavior, enabling comparison between configurations with varying structural characteristics. These geometries were selected to represent different connectivity and deformation characteristics rather than standard lattice types. The presented unit cells represent variations in connectivity, strut orientation and load transfer mechanisms, spanning from simple cubic (bending-dominated) to highly triangulated (stretch-dominated) configurations.

Figure 2.

Cell geometries of the lattice structures (dimensions in mm).

The investigated lattice geometries (M1–M10) can be classified according to their connectivity and dominant deformation mechanisms. Low-connectivity structures (M1, M2, and M5) are primarily bending-dominated and exhibit characteristics similar to Kelvin-type lattices. In contrast, highly triangulated configurations (M7–M10) demonstrate stretch-dominated behavior comparable to octet-type structures. Intermediate geometries (M3, M4, and M6) exhibit mixed deformation characteristics, representing transitional configurations between bending- and stretch-dominated mechanisms.

Both the lattice cores and the face sheets were fabricated using the FDM process using a Bambu Lab P1S Combo 3D printer, ensuring manufacturing consistency across the hybrid assembly. The assembled sandwich beams had nominal dimensions of 200 × 10 × 12.2 mm³. A total of 30 specimens (three replicates for each of the 10 lattice topologies) were tested (Figure 3).

Figure 3.

Face sheets and lattice core.

Following the printing process, the PAHT-CF face sheets were bonded to the TPU lattice cores using Vodabond 2K, a commercially available two-component epoxy adhesive. The adhesive was selected for its high gap-filling capability, which accommodates the surface roughness of FDM-printed parts, and its resistance to moisture and chemicals, while providing sufficient stiffness to maintain structural continuity at the face–core interface. Prior to bonding, the contact surfaces were cleaned and degreased. The adhesive was applied to both mating surfaces, and the specimens were placed on a flat, rigid surface. During curing, a uniformly distributed dead weight of 450 g was applied using a flat plate to ensure consistent contact pressure and prevent excessive adhesive squeeze-out. The bonded assemblies were cured at room temperature (20–23°C) under constant pressure for 72 h to ensure full curing and stable interfacial behavior prior to vibration testing.

The dynamic characteristics of the hybrid sandwich structures were investigated using an experimental modal analysis setup designed to simulate cantilever boundary condition. Specimens were clamped over a length of 50 mm, yielding an effective free length (L_eff) of 150 mm. A precision metal spacer (11.5 mm) was inserted between the vice jaws to limit the compression of the 12 mm thick specimen to 0.5 mm, thereby ensuring uniform holding pressure without inducing localized crushing of the lattice core. The vice jaws were lined with 100-grit silicon carbide abrasive paper to enhance friction, effectively preventing slippage during excitation while minimizing artificial boundary stiffening (Figure 4).

Figure 4.

Sandwich composite panels produced by FDM process, (a) design topologies (b) clamped-free panel.

The structures were excited by an impulsive force applied at the mid-span of the specimen (50 mm from the clamped edge) to activate the fundamental bending modes. The dynamic response was measured using a Dytran 3035BG miniature IEPE accelerometer, selected for its low mass (2.5 g) and compact dimensions (diameter 8.3 mm). To minimize mass-loading effects, the accelerometer was mounted 10 mm from the fixed root of the cantilever using a thin layer of mounting wax. This configuration ensured consistent mass-loading condition across all tests and avoided artificial inertia effects that would arise if the sensor were attached near the free end of the approximately 12 g specimen. The acceleration signals were digitized using a Data Physics Quattro dynamic signal analyzer equipped with a 24-bit analog-to-digital converter, providing sufficient dynamic range for accurate vibration measurements. Figure 5 shows the experimental setup used in this study.

Figure 5.

Free vibration test setup.

The acquired acceleration signals were processed using SignalCalc ACE X1 software. Modal parameters were extracted in the frequency domain from time-domain acceleration data. The data acquisition system was configured with a frequency bandwidth of 0–1000 Hz and 3200 spectral lines, resulting in a frequency resolution of 0.3125 Hz. Due to the high intrinsic damping of the TPU lattice core, the transient vibration response decayed completely within the selected 3.2 s observation window; therefore, a rectangular window was employed to preserve the full energy content of the impulse response. To address the low signal-to-noise ratio commonly observed in highly damped lattice structures, a Savitzky-Golay smoothing filter (third-order polynomial, 151-point frame length) was applied exclusively to the magnitude spectrum. The filter parameters were selected to enable reliable identification of the half-power points without shifting the resonance frequency or altering the underlying modal characteristics (Figure 6).

Figure 6.

Impact response for specimen M1-1.

For each specimen, the frequency spectra were obtained from the linear average of 10 valid impacts to reduce random noise effects. To overcome the limitations imposed by discrete frequency resolution, linear interpolation was employed between adjacent spectral bins to accurately determine the 3 dB frequencies ( $f_1$ and $f_2$ ) at the fundamental resonance peak.

The equivalent bending stiffness ( $E{I}_{eq}$ ) of the sandwich beams was estimated from the experimentally measured fundamental natural frequency using Euler-Bernoulli beam theory. For a cantilever beam, the relationship between natural frequency and bending stiffness is given by $f_n=\left(\frac{K_1}{2\mathrm{\pi }}\right)\sqrt{\frac{EI}{m{L}^4}}$ , where for the first bending mode ( $K_1$ ≈ 3.516), L is the beam length, E is the effective Young’s modulus, I is the second moment of area, m is the mass of the structure. Accordingly, the equivalent flexural rigidity is proportional to the square of the natural frequency as $I_{eq}$ $\propto f_n^2$ . Since the specimen mass was approximately constant, variations in natural frequency directly reflect topology-induced differences in structural stiffness. The equivalent stiffness values represent effective structural parameters and may implicitly include contributions from factors such as shear deformation and viscoelastic damping, particularly due to the compliant nature of the TPU core.

The association between lattice design parameters and dynamic response was investigated using machine learning approaches as an auxiliary tool in addition to experimental investigation. Regression-based models were developed using geometric features such as topology type, strut diameter, and orientation as input variables, and natural frequency and damping ratio as target outputs. Given the relatively small dataset, the objective was not to achieve high predictive accuracy, but to identify dominant parameters and reveal underlying trends governing vibration behavior. This approach supports a design-oriented interpretation of the results and provides preliminary insights into geometry-performance relationships.

Results

The physical properties and geometric fidelity of the FDM-fabricated TPU lattice structures were evaluated prior to dynamic characterization. A total of 30 specimens, representing 10 distinct topological configurations with three repetitions per model, were examined to assess manufacturing consistency and experimental repeatability. Figure 7 summarizes the mass measurements and associated statistical deviations for each lattice configuration. Although all lattice cores were designed with a constant target mass, minor experimental variations were observed, with measured values ranging from 11.65 g to 13.28 g.

Figure 7.

Distribution of mass and standard deviations of the specimens.

Despite the compliant nature of TPU and the multi-step hybrid manufacturing approach, the specimens exhibited a high level of mass consistency. The coefficient of variation (CV) for the mass measurements remained below 2.5% for the majority of the lattice groups. For example, Model 7, which observed the highest stiffness in subsequent dynamic tests, showed a mean mass of 12.43 g with a low deviation (CV = 0.35%), indicating repeatability. Similarly, Model 10 exhibited a consistent mass distribution with an average of 12.95 g, confirming reliable fabrication without significant material accumulation or void formation.

The repeatability of the manufacturing process is further corroborated by the dynamic response of the specimens. As illustrated in Figure 8, the error bars corresponding to the standard deviation of both the natural frequency ( $f_n$ ) and damping ratio (ζ) are notably narrow. For instance, the fundamental natural frequencies of Model 2 were measured as 94.69 Hz, 99.06 Hz, and 94.69 Hz across repeated specimen.

Figure 8.

Experimental reliability analysis results for natural frequency and damping of the specimens.

The fundamental natural frequencies obtained from the experimental modal analysis were used to back-calculate the equivalent flexural rigidity ( $E{I}_{eq}$ ) of the hybrid sandwich beams using classical Euler–Bernoulli beam theory (Table 1). This procedure enables the quantitative evaluation of the macroscopic bending stiffness associated with each lattice topology. The measured natural frequencies ranged from 61.15 Hz to 115.83 Hz. Since the mass per unit length remained nearly constant across all specimens, this approximately 90% increase in natural frequency corresponds to a substantial amplification in effective flexural rigidity. Specifically, the equivalent bending stiffness varied from 0.3771 N^.m² (Model 5) to 1.3848 N^.m² (Model 10), representing an approximately 3.6-fold increase primarily governed by internal lattice architecture rather than mass variation. The highest flexural rigidities were recorded for Model 10 (1.3848 N^.m²) and Model 7 (1.3487 N^.m²), indicating the stiffest global structural response among the tested configurations. These topologies incorporate extensive diagonal bracing within the unit cell, which suppresses nodal rotation and promotes stretch-dominated deformation governed predominantly by axial tension and compression. In contrast, Model 5 (0.3771 N^.m²) and Model 1 (0.3985 N^.m²) exhibited the lowest flexural rigidities as seen from the table.

Table 1.

Stiffness ( $EI$ ) measurements for 10 lattice configurations, mean $EI$ , standard deviation ( $EI\_STD$ ), and coefficient of variation ( $EI\_CV$ ).

Lattice no	Stiffness, EI (N.m2)	Mean stiffness, $EI$ (N.m2)	$EI\_STD$ (N.m2)	$EI\_CV$ (%)	$f_n^2$ (Hz)2
1-1	0.421	0.3985	0.029676258	7.45	4265.72
1-2	0.409				4064.06
1-3	0.365				3875.06
2-1	0.908	0.9112	0.004076655	0.45	8965.72
2-2	0.916				9813.38
2-3	0.910				8965.72
3-1	0.979	0.9316	0.05666755	6.08	9567.29
3-2	0.869				8730.57
3-3	0.946				9937.60
4-1	0.849	0.9222	0.071881333	7.79	9025.00
4-2	0.926				9204.00
4-3	0.992				9506.25
5-1	0.437	0.3771	0.070449196	18.68	4265.72
5-2	0.395				3828.52
5-3	0.299				3164.06
6-1	0.919	0.8115	0.09333354	11.50	9144.14
6-2	0.765				7601.66
6-3	0.750				7331.64
7-1	1.289	1.3487	0.054658462	4.05	12,797.27
7-2	1.397				13,953.52
7-3	1.360				13,514.06
8-1	1.289	1.2774	0.04377217	3.43	12,797.27
8-2	1.229				12,307.13
8-3	1.314				12,237.89
9-1	1.097	1.1780	0.072945003	6.19	11,025.00
9-2	1.197				11,894.63
9-3	1.239				12,031.35
10-1	1.400	1.3848	0.071318232	5.15	13,297.09
10-2	1.307				12,446.19
10-3	1.447				13,953.52

These specimens, characterized by orthogonal strut arrangements without diagonal reinforcement (Simple cubic type architectures), deform primarily through nodal bending. Increased rotational compliance in such bending-dominated lattices reduces the effective contribution of the core to global bending stiffness, resulting in a more compliant structural response. A near-linear relationship was observed between $E{I}_{eq}$ and $f_n^2$ , consistent with the square-root dependency between natural frequency and flexural rigidity predicted by classical beam theory. For the majority of the configurations, the coefficient of variation (CV) remained below 8%, including the highly rigid triangulated architectures. However, Model 5 and Model 6 exhibited higher variance levels (CV = 18.68% and 11.50%, respectively). This increased deviation suggests that bending-dominated structures lacking extensive diagonal support are more sensitive to minor geometric variations in strut formation and local adhesive thickness introduced during the FDM-based hybrid assembly process. Despite these localized variations, no overlap was observed between the stiffness ranges of bending- and stretch-dominated configurations, and the overall stiffness hierarchy remained distinct and predominantly governed by lattice topology.

The relationship between the equivalent bending rigidity $E{I}_{eq}$ and the square of the natural frequency $f_n^2$ for the 10 lattice configurations is presented in Figure 9. The observed linear relationship between the equivalent bending stiffness ( $E{I}_{eq}$ ) and the square of the natural frequency ( $f_n^2$ ) arises directly from the Euler Bernoulli beam formulation used to estimate stiffness. Therefore, this correlation should not be interpreted as an independent experimental finding, but rather as a consistency check of the applied analytical method. Since the equivalent stiffness is calculated based on the measured natural frequency, the proportional relationship between these parameters reflects the underlying theoretical formulation. Furthermore, as the mass variation among specimens is minimal as shown in Figure 7, the differences in natural frequency primarily reflect variations in structural stiffness associated with lattice topology.

Figure 9.

Correlation between bending stiffness, $EI$ and the square of the natural frequency, $f_n^2$ .

The close alignment of all data points with the regression line (R² = 0.9973) indicates both the high repeatability of the measurements and the dominant influence of lattice topology on global stiffness. Topologies such as M7, M8, M9, and M10 cluster at higher $f_n^2$ values, reflecting their stretch-dominated behavior and enhanced rigidity, whereas bending-dominated configurations like M1 and M5 appear at the lower end of the spectrum these findings indicate that lattice architecture functions as a primary structural tuning parameter for controlling macroscopic bending stiffness in hybrid elastomer-composite sandwich systems. This relationship arises from the Euler-Bernoulli formulation used to estimate equivalent stiffness from the natural frequency and therefore represents a consistency check rather than an independent finding. It should be also considered that the equivalent bending stiffness (EI) used in this study is obtained inversely from experimental data. Therefore, it represents an effective structural parameter that inherently may implicitly include contributions from shear deformation and damping effects of the flexible TPU core. As such, the use of Euler-Bernoulli beam theory here should be interpreted as an effective engineering approximation rather than a rigorous structural model.

Figures 10 and 11 summarize the energy dissipation performance of the lattice samples by ordering the topologies according to their experimentally measured damping ratios. The modal damping ratios (ζ) are calculated using the half-power bandwidth method in this study. The measured values ranged from 0.1199 to 0.1771, corresponding to an approximately 48% variation in damping performance across the lattice topologies investigated. Model 1 exhibited the highest damping ratio (ζ = 0.1771), followed by Model 6 (ζ = 0.1636) and Model 5 (ζ = 0.1560). It is observed that Model 7 has the lowest damping ratio (ζ = 0.1199), while Model 10 (ζ = 0.1393) and Model 4 (ζ = 0.1270) have low damping capacities. The highly triangulated, stretch-dominated architectures of these rigid models restrict nodal rotation and minimize local strut curvature. Consequently, while this mechanism maximizes macroscopic flexural rigidity, it limits the local deformation required to fully activate the intrinsic damping potential of the elastomeric core. The standard deviations of the damping ratios remained consistently small (exemplary it is 0.0037 for Model 1), indicating highly repeatable measurement results across replicated specimens.

Figure 10.

Damping performance of the lattice topologies, ranked according to their measured damping ratios.

Figure 11.

Distribution of effect of topological performance on the damping ratio versus natural frequency.

Figure 11 indicates an inverse trend between stiffness and damping within the investigated lattice configurations. The results divide the lattice configurations into two dominant mechanical behaviors. Models M1 and M5 fall within the damping-dominated region, characterized by low natural frequencies and high damping ratios, indicating bending-dominated deformation. In contrast, models such as M7, M8, M9, and M10 cluster in the stiffness-dominated region, showing high natural frequencies and relatively low damping. Intermediate configurations (M2, M3, M4, M6) occupy the transition region between the two extremes, reflecting mixed deformation mechanisms. When compared with the dynamic stiffness trends presented in the previous section, a clear inverse trend between structural rigidity and damping capacity is observed. Within the investigated frequency range (60–115 Hz), this behavior suggests that topology-driven deformation mechanisms dominate over frequency-dependent material effects in governing vibration attenuation. As illustrated in the figure plotting the mean fundamental natural frequency against the mean damping ratio (ζ) reveals a stiffness-damping trade-off across the investigated lattice topologies. This trend is observed within the investigated configurations and is attributed to topology-dependent deformation behavior.

Machine learning analysis is employed as an exploratory tool to investigate the relationship between structural parameters and dynamic performance. The data are utilized to confirm and explain trends in the experimental results in this study rather than to draw separate conclusions. No independent test dataset was available; therefore, model validation is limited to internal consistency. Consequently, the predictive capability of the models for unseen configurations is inherently limited. Within the scope of this study, three different regression approaches were utilized Artificial Neural Networks (ANN) based Support Vector Regression (SVR), and Gaussian Process Regression (GPR). SVR and GPR were selected due to their suitability for small datasets. These models were selected to provide a comparative framework for evaluating parameter influence across different modeling methodologies. In particular, SVR and GPR are known to perform relatively well with small datasets due to their inherent regularization and ability to capture nonlinear relationships with limited training samples. However, despite these advantages, the dataset size may remain insufficient to establish robust predictive models. The input parameters used in the models include the number of struts, strut width, strut angle, and presence of internal diagonal strut in each cell associated with the lattice core. Because the dataset size is limited, multiple regressors were employed to ensure robustness of parameter ranking. The use of multiple models enables cross-comparison of parameter sensitivity trends rather than reliance on a single modeling framework. MATLAB codes were developed to analyze four key design parameters such as the number of struts within the lattice, strut width, strut angle, and the presence or absence of an inner diagonal struts in each cell. ³³ By comparing the results among these methods, the consistency and robustness of the input importance rankings were evaluated. The models are not intended for extrapolation beyond the configurations investigated. Model performance was evaluated by comparing predicted and experimentally measured values of equivalent stiffness and damping ratio.

Figure 12 illustrates the relative influence of strut width, number of struts, strut angle, and internal diagonal strut. However, due to the limited dataset size, the calculated importance values should be considered as indicative rather than definitive. When the dataset was trained using an ANN, the relative importance of the input variables was evaluated through Garson’s algorithm and the connection weights approach. Both methods yielded very similar trends, which can be attributed to the limited dataset size. Results indicate that internal diagonal strut in the cell and its configuration play a dominant role in determining structural response.

Figure 12.

Relative importance of structural parameters for (a) stiffness and (b) damping values obtained from ANN-based analysis.

Within the scope of this study, both Garson’s Algorithm and Olden’s Algorithm were applied to an Artificial Neural Network architecture with two hidden layers. Several ANN analyses using different number of hidden layers, and dual or triple combinations of functions were carried out to obtain the best architecture. Initial analyses were conducted using a more complex ANN architecture with two hidden layers to explore model capability. The first hidden layer contains 10 neurons and the second hidden layer includes 8 neurons, each applying weighted summation followed by a nonlinear activation function. The output layer comprises two neurons responsible for predicting the target variables, corresponding to the stiffness and damping responses of the lattice structures. Weight matrices (W), bias terms (b), and activation operations are shown explicitly in the diagram, highlighting the flow of information from the inputs through the hidden representations to the final outputs. This architecture enables the model to capture nonlinear relationships between the structural design parameters and their resulting mechanical performance. Figure 13 illustrates the architecture of the artificial neural network used in this study. The predictive performance of the models was evaluated using regression plots comparing predicted and measured values. While high R-values were obtained, these results should be interpreted cautiously, as they may reflect overfitting due to the small dataset rather than true generalization capability.

Figure 13.

Regression results of the ANN model showing predicted versus experimental values for equivalent stiffness and damping ratio.

To mitigate overfitting, model complexity was kept low and the analysis focused on identifying consistent trends rather than achieving high predictive accuracy. However, given the limited number of samples, overfitting cannot be fully avoided, and this constitutes an inherent limitation of the present analysis. The relative importance of the input variables was visualized through a heatmap representation given in Figure 14. As shown in the figure, certain predictors exhibit noticeable differences between the two methods. Specifically, Garson’s approach tends to distribute importance values more evenly across inputs, resulting in relatively similar contributions, whereas Olden’s method highlights sharper contrasts among variables, revealing stronger and weaker influences more distinctly. One of the main reasons for this difference is that the Garson algorithm proportionally redistributes latent-output weights among input-latent connections, which naturally smooths contributions and produces more balanced significance scores. The Olden algorithm, on the other hand, directly multiplies input-latent and latent-output weights, preserving both the magnitude and sign of the connections. This allows it to capture subtler variations in how individual inputs affect the output. Therefore, while Garson prioritizes overall balance and interpretability, Olden is more sensitive to the true weight dynamics of the network. The observed differences in the heatmap thus reflect the opposing philosophies of the two algorithms: one smooths contributions, while the other emphasizes differences. When the results of the Olden algorithm were examined, the analysis revealed that the presence of an internal core within the lattice structure exerts a significant influence on the overall resistance of the structure. In contrast, the damping ratio was found to be more closely associated with the number of struts in the lattice. The heatmap visualization highlights these relationships, showing that structural features related to the internal diagonal strut primarily govern load-bearing capacity, while geometric repetition through strut count modulates energy dissipation mechanisms.

Figure 14.

Comparative heatmap of strut parameters using Garson and Olden approaches.

Then, due to the limited dataset size, simplified neural network architecture was also employed to reduce model complexity, to investigate the sensitivity of the results obtained from the previous model and mitigate overfitting risk in the scope of this study. A single hidden-layer feedforward ANN with five neurons was used to model the relationship between geometric parameters and dynamic response. The model was evaluated using 5-fold cross-validation to reduce overfitting and assess generalization performance. The dataset was randomly partitioned into five subsets, and the model was trained and tested iteratively to evaluate generalization performance. The performance of the ANN model was evaluated using mean squared error (MSE). The training MSE was found to be 0.0212, while the average cross-validation MSE was 0.0659. The ratio between validation and training error was approximately 3.10, indicating a moderate level of overfitting. This behavior is consistent with the limited dataset size and reflects the sensitivity of the model to data partitioning. However, the overall consistency between training and validation results suggests that the model captures general trends rather than memorizing the data.

Figure 15 illustrates the results obtained from simplified neural network models. As seen from the figure, the results show more balanced contributions while preserving consistent trends observed in the complex model. Specifically, this model also identifies the presence of internal cross braces (core configuration) as the dominant parameter governing stiffness behavior. Similarly, the parameters influencing damping behavior differ from those controlling stiffness, indicating that energy distribution is governed by different geometric mechanisms. While the simplified model provides a more balanced distribution of parameter importance, the more complex model clearer emphasizes dominant characteristics. Despite these differences, the similarity of the trends identified in both architectures demonstrates fundamental physical relationships in lattice structures.

Figure 15.

Input importance obtained using a simplified ANN model with a single hidden layer.

In addition to the ANN-based input importance analysis, the study further investigated predictive performance using Support Vector Regression (SVR) and Gaussian Process Regression (GPR) models. A total of six models were constructed by employing different kernel functions, including radial basis function (RBF), linear, polynomial, rational quadratic (RQ), Matern (MAT), and squared exponential (RSE). These models were evaluated with respect to their ability to predict both structural stiffness and damping ratio. Figure 16 presents a comparison of predicted and experimental values across these models. While differences are observed between model types, the overall trends remain consistent, supporting experimental observations. Overall, the machine learning analysis provides supportive insight into the relative importance of geometric parameters such as strut number, angle, and width. However, these findings should be considered exploratory and complementary to the experimental results rather than definitive conclusions. Among the tested configurations, the most pronounced differences were observed between the SVR-linear and SVR-RBF models.

Figure 16.

Comparison of predicted and experimental values for stiffness and damping obtained using different regression models (SVR and GPR).

Conclusions

This study experimentally investigated the topology-dependent vibration behavior of hybrid sandwich beams composed of TPU lattice cores and carbon fiber reinforced polyamide face sheets.

The results demonstrate that lattice topology governs the dynamic performance of the structures. Despite minimal mass variation, the fundamental natural frequency varied by approximately 90%, corresponding to a 3.6-fold increase in equivalent flexural rigidity. This confirms that variations in stiffness and natural frequency are primarily driven by geometric architecture rather than density effects. Stretch-dominated topologies with pronounced diagonal bracing promote axial load transfer and exhibit higher stiffness and natural frequencies, whereas bending-dominated structures allow greater local deformation and result in lower stiffness. It should be considered that the classification of lattice structures as bending- or stretch-dominated is based on qualitative geometric observations. A rigorous classification based on Maxwell’s criterion or energy-based analysis was not performed within the scope of this study because the geometries are non-standard and parametrically varied and direct application of Maxwell’s criterion would not yield reliable classification.

An inverse relationship between stiffness and damping was observed. Bending-dominated configurations exhibited higher damping due to enhanced local deformation and viscoelastic strain, whereas stretch-dominated configurations provided higher stiffness but reduced energy dissipation.

In contrast, stretch-dominated lattices limited flexural deformation and reduced hysteretic losses. These results indicate that damping in hybrid lattice sandwich systems arises from the coupled interaction between topology-induced deformation modes and intrinsic viscoelastic material behavior. The findings are in line with established deformation mechanisms in lattice materials, reinforcing the validity of the observed stiffness-damping trends. However, significant tuning of structural performance can be achieved solely through geometric design, without modifying material composition or overall mass.

Machine learning analysis provided additional qualitative insight into the relative influence of geometric parameters. While high correlation values were obtained, the limited dataset size restricts predictive capability, and the results should be interpreted as indicative trends rather than generalizable models. These methods include inherent regularization mechanisms and can model nonlinear relationships with fewer training samples. Results from both simplified and more complex artificial neural network models used for comparison show a similar trend when the importance of input parameters is considered. High correlation coefficients indicate agreement between predicted and experimental values, supporting the validity of the simplified ANN model. Although moderate overfitting is observed due to limited data, it remains within acceptable limits and does not affect the qualitative interpretation of the results.

The analysis was limited to cantilever boundary conditions and the first bending mode, with stiffness estimated using Euler-Bernoulli beam theory, neglecting shear deformation effects. Furthermore, only a single TPU material and fixed face-sheet configuration were considered. Future work should incorporate finite element modeling, alternative material systems, and expanded datasets to improve generality and predictive capability.

Overall, the results demonstrate that architected elastomeric lattice cores can serve as effective design parameters for controlling the vibration response of hybrid sandwich structures. By tailoring lattice topology, structures can be designed for either enhanced vibration attenuation or increased stiffness, depending on application requirements. However, the observed stiffness-damping relationship is specific to the investigated configurations and should not be generalized without further validation.