To address the insufficient damping and instability tendency of metal coil spring isolators subjected to intense impact loading, a multi-stage vibration isolation configuration integrating polyurethane, springs, and eddy current dampers is proposed. Dynamic models for both single-stage and multi-stage isolation systems are formulated, and a corresponding simulation model is developed in MATLAB R2023b/Simulink to investigate the peak suppression and attenuation characteristics of the multi-stage isolation under impact. To characterize the nonlinear finite deformation and time-dependent response of polyurethane, a hyperelastic-viscoelastic constitutive model is established by coupling the Ogden hyperelastic model with a generalized Maxwell viscoelastic model, with model parameters identified through quasi-static compression and stress relaxation tests. Drop impact experiments are performed to compare the displacement response, top- and bottom-plate peak accelerations, and vibration isolation rate between a polyurethane-spring-eddy-current multi-stage isolator and a spring-spring-eddy-current multi-stage isolator. The results demonstrate that the multi-stage structure enables staged dissipation of the impact energy, substantially reducing both the peak acceleration and the displacement stroke of the isolated mass. Under all drop test conditions, the polyurethane-based multi-stage isolator yields lower top-plate output peak acceleration and higher isolation rate than its all-spring counterpart, confirming its superior isolation performance. Envelope fitting of the simulation-based output acceleration with experimental inputs reveals that the all-spring multi-stage isolator exhibits a higher attenuation rate and equivalent damping ratio, whereas the polyurethane-based isolator achieves more effective suppression of the output peak level under severe impact conditions. 1. Introduction Shock loads are characterized by short durations, high peak accelerations, and concentrated energy release. They typically induce severe structural vibrations or localized damage, and may even lead to structural instability or failure. Spacecraft, naval vessels, and precision equipment are highly vulnerable to such intense shock loads. Without effective protection, these systems often suffer from fatigue damage, fastener loosening, and structural failure [ 1, 2]. Consequently, enhancing structural impact resistance and vibration isolation has become a primary research focus in vibration control and protection engineering. In recent years, extensive research has been conducted on high-damping materials [ 3, 4], composite energy-absorbing structures [ 2, 5], and multi-stage vibration isolation technologies [ 6, 7], providing novel insights and technical means for impact protection design. According to the control strategies, existing vibration isolation technologies are generally classified into three categories: passive [ 8, 9, 10, 11], semi-active [ 12], and active [ 13, 14]. Passive vibration isolation is the most prevalent strategy in engineering fields sensitive to shock loads (e.g., marine, vehicle, and aerospace) because it offers high reliability and structural simplicity without the need for an external power supply. Among passive vibration isolation devices, wire rope isolators were extensively utilized in early impact protection engineering due to their structural simplicity and high equivalent damping ratios [ 15]. Nevertheless, the precise theoretical modeling of their damping and stiffness remains elusive due to the complex interplay between winding configurations, friction mechanisms, and preload. Furthermore, their mechanical performance is highly sensitive to variations in structural parameters, necessitating extensive experimental characterization [ 16]. These inherent limitations have constrained the further application of wire rope isolators in advanced impact protection. Consequently, a solitary metallic spring is seldom sufficient to withstand high-intensity impact loads; it must be integrated with other high-energy-dissipating materials or damping components to construct a multi-stage vibration isolation system. Such a configuration is essential to achieve a synergistic balance between load-bearing capacity and energy dissipation efficiency. In recent years, with the increasing demand for shock protection and isolation under extreme loads, some studies have begun to focus on the application of elastomeric materials and multi-stage hybrid isolation structures in shock environments. For instance, Menga et al. combined rubber-layer roller bearings with nonlinear elastic elements to reduce transient inertial forces in structures subjected to seismic shocks [ 25]. Hajidavalloo et al. proposed a hybrid vibration reduction device integrating an electromagnetic energy harvesting unit and a nonlinear pendulum-type vibration absorber to improve response control under broadband shock and vibration inputs [ 26]. Nevertheless, most existing studies have focused on single energy-dissipating structures or local performance optimization. Systematic comparative investigations of the dynamic response differences, high-frequency shock transmission characteristics, and multi-stage coupled energy dissipation behavior of different isolation systems under severe shock environments remain limited. Moreover, studies combining numerical simulation with shock experiments are still relatively insufficient. In vibration isolation systems, polyurethane serves as a critical energy-dissipating unit. Given that its performance dictates the overall isolation efficacy, a robust constitutive model is essential to characterize its mechanical behavior. Conventional constitutive models for materials such as polyurethane are broadly categorized into two types. The first one is hyper-elastic models, which treat the material as an ideal elastic body and describe the stress–strain relationship via a strain energy density function (e.g., Mooney-Rivlin [ 27, 28], Ogden [ 29], Yeoh [ 30], “trinomial” [ 31], and Exp-Ln models [ 32]). The other one is viscoelastic models, which describe stress relaxation and creep by introducing time-dependent functions. (typically the generalized Maxwell model [ 33], with relaxation functions expressed as Prony series [ 34]). However, employing either approach in isolation to describe the mechanical properties of polyurethane presents inherent limitations. In recent years, hyperelastic–viscoelastic coupling theory and nonlinear viscoelastic modeling have been widely applied in tire modeling, polymer foams, and metamaterials. Esposito [ 35] developed a viscoelastic–hyperelastic coupled model for polymer foams based on a continuous–discrete relaxation spectrum method, which effectively characterized nonlinear energy dissipation behavior and stress relaxation properties. Venturini [ 36] established a digital twin model of a tire–rim coupled system and revealed the nonlinear dynamic response characteristics of elastomeric systems. Shariyat [ 37] investigated the nonlinear damping response of a Mooney–Rivlin hyperelastic plate coupled with a viscoelastic foundation and analyzed the coupled dynamic behavior under complex boundary conditions and dynamic loading. In addition, advanced damping characterization based on elastomeric metamaterials and tire mechanics indicates that a single-mode Prony series with a strain-independent equilibrium modulus is generally insufficient to simultaneously reproduce loading-rate dependence and the residual stress relaxation envelope. Inspired by these studies, the present work adopts a strain-dependent equilibrium modulus provided by the Ogden model and combines it with a second-order Prony series, thereby achieving a balance between representation fidelity and computational tractability in shock-loading simulations. The proposed model can accurately reflect the nonlinear response characteristics of polyurethane materials under complex shock conditions, providing a theoretical basis for the subsequent design and experimental investigation of the multi-stage isolator. To address the limitations of existing studies in the transient constitutive characterization of polyurethane under severe shock loading, the frequency-graded energy dissipation mechanism of multi-stage isolation, and the quantitative comparison of different first-stage isolation units, this study proposes a polyurethane–spring–eddy-current multi-stage isolator. Unlike existing polyurethane-based or conventional multi-stage isolation systems, this study not only embeds an Ogden–generalized Maxwell model with a strain-dependent equilibrium modulus into a lumped-parameter dynamic framework for predicting transient responses under severe shock loading, but also employs polyurethane as the first-stage load-bearing and energy-dissipating unit, which forms a graded energy dissipation structure together with the second-stage spring–eddy-current damper. Furthermore, under the same reference inertia and identical second-stage unit, an equal-input shock comparison between the polyurethane–spring isolator and the all-spring isolator is conducted. The differences between the two isolators in terms of peak shock suppression, vibration attenuation, and high-frequency transmission characteristics are revealed, providing theoretical and experimental support for the design of severe shock protection devices. In summary, to address the deficiencies inherent in conventional wire rope isolators and metallic helical springs, this paper introduces a multi-stage polyurethane-spring-eddy current isolator. Compared to the spring-spring-eddy current configuration, the proposed design offers superior structural stability, enhanced energy dissipation capacity, and high design controllability. The structure of this paper is as follows: Section 2 details the system’s overall design; Section 3 explores the characteristics of polyurethane; Section 4 describes prototype development and drop-test validation. 2. The Material Properties of Polyurethane 2.1. Establishment of the Hyperelastic-Viscoelastic Constitutive Model When subjected to shock loading, polyurethane displays significant non-linear deformation and strain-rate sensitivity, manifesting a duality of hyperelastic and viscoelastic traits. Conventional standalone models are inadequate for simultaneously capturing its complex response behaviors under large strains and dynamic shock conditions. Consequently, this study proposes a coupled hyperelastic-viscoelastic constitutive model to more accurately represent the intricate mechanical properties of polyurethane. The hyper-elastic characteristics of the material are characterized by the Ogden model, whose strain energy density function can be expressed as [ 28]: W = ∑ i = 1 n 2 μ i α i 2 λ 1 α i + λ 2 α i + λ 3 α i − 3 (1) where μ i and α i denote the shear modulus and the non-linear parameter of the material, respectively. λ j represents the principal stretch. Under the assumption of volumetric incompressibility, the condition λ 1 · λ 2 · λ 3 = 1 is satisfied. A Prony series is employed to characterize the viscoelastic behavior of the material, and the temporal evolution of its elastic modulus is represented as follows: σ t ε 0 = E t = E ∞ + ∑ i = 1 n E i e − t τ i (2) where E ∞ is the equilibrium modulus, representing the steady-state response of the relaxation modulus as t → ∞ . E i corresponds to the i t h relaxation modulus. τ i is the associated relaxation time constant, which characterizes the rate of stress relaxation. By incorporating the Ogden model as a substitution for the original equilibrium modulus E ∞ within the Prony series, the elastic modulus expression for the coupled hyperelastic-viscoelastic constitutive model is derived as follows: σ t ε 0 = E t = E ∞ ( ε 0 ) + ∑ i = 1 n E i e − t τ i (3) where E ∞ ( ε 0 ) denotes the elastic modulus of the Ogden model at a strain of ε 0 , which is employed to characterise the equilibrium modulus of the material. The exponential decay terms in the Prony series can describe the time-dependent decay of internal stress in the material, essentially reflecting the viscoelastic energy dissipation characteristics of polyurethane under dynamic loading conditions. During shock loading, part of the input mechanical energy is converted into heat through internal viscous damping, resulting in pronounced stress relaxation and hysteretic energy dissipation. Therefore, the Prony series can not only characterize the time-dependent response of the material, but also reflect the damping and energy dissipation capacity of polyurethane in shock environments. By coupling the Ogden model with the Prony series, the established constitutive model can simultaneously describe the large-deformation energy storage characteristics and dynamic energy dissipation behavior of polyurethane. Specifically, the Ogden model is used to describe the nonlinear hyperelastic response of the material under large deformation, while the Prony series is employed to characterize the time-dependent relaxation and energy dissipation behavior. The coupling between these two components is achieved by introducing the strain-dependent equilibrium modulus E ∞ ( ε ) , thereby improving the capability of the model to represent the dynamic response of polyurethane under complex shock loading conditions. 2.2. Material Property Tests and Parameter Fitting To determine the specific parameters of the coupled hyperelastic-viscoelastic model, it is necessary to conduct quasi-static compression tests and stress relaxation tests on the selected polyurethane material. The loading tests were conducted using an electronic universal testing machine with a loading capacity ranging from 0 to 20 t. The testing system was equipped with a force sensor with a measurement range of 0–100 kN and a high-precision displacement sensor, enabling the synchronous acquisition of force, displacement, and time data during the material loading process. Polyurethane specimens (Model 4790-92-15500, Rogers Corporation, Chandler, AZ, USA) measuring 300 mm × 300 mm × 25.4 mm were employed for the study. Quasi-static compression tests were conducted at varying compression rates (20/95 mm/s, 20/750 mm/s, 2 mm/s, and 10 mm/s), designated as Compression Cases 1, 2, 3, and 4, respectively. The stress–strain curves of the polyurethane were subsequently derived by fitting the acquired force-displacement data. To mitigate experimental errors arising from friction, the upper surface of the specimen was covered with a thin steel plate. This serves to reduce frictional effects and thereby ensure unimpeded deformation during the compression process. Three repeated tests were performed under each working condition, and the averaged results were used as the data basis for subsequent parameter fitting and model analysis. Based on the quasi-static compression test results, the third-order Ogden model was employed for constitutive parameter fitting to improve the accuracy in describing the nonlinear stage of the stress–strain curve while avoiding fitting instability caused by the excessive number of parameters in higher-order models. Parameter identification was performed in MATLAB R2023b, where the minimum mean square error between the experimental curve and the theoretical model was defined as the objective function. The model parameters were then iteratively solved using the nonlinear least-squares method. The identified parameters, μ i , α i ( i = 1,2 , 3 ) , are summarized in Table 1. A comparison between the Ogden model fitting curves and the experimental compression data is presented in Figure 1. As shown in Figure 1, the Ogden model provides a high degree of correlation when fitting Compression Cases 1 and 2, adequately reflecting the hyper-elastic characteristics of the material. However, the fitting performance diminishes for Cases 3 and 4, which is attributed to the fact that the viscoelastic characteristics of the material become significantly more pronounced as the compression rate increases. To further elucidate the material’s time-dependent behavior, stress relaxation experiments were performed. A constant loading was held for 2700 s under distinct initial strains (20%, 30%, and 40%). The resulting temporal stress attenuation profiles are presented in Figure 2. Considering the stability of parameter identification, the computational efficiency of the model, and the fitting accuracy for the stress relaxation behavior of polyurethane, a second-order Prony series was adopted in this study to fit the relaxation test results, yielding the relaxation time constants τ k and the corresponding weighting coefficients E i , as summarized in Table 2. The fitted relaxation results, derived from the Prony series using the identified parameters, are compared with the experimental observations in Figure 3. The figure indicates that the relaxation curves predicted by the second-order Prony series show excellent agreement with the experimental measurements, confirming the validity and applicability of the model. 2.3. Simulation of Constitutive Model A simulation model, shown in Figure 4, was established using MATLAB/Simulink to validate the established coupled constitutive hyperelastic-viscoelastic constitutive model. The model adopts a five-element Maxwell structure, in which the primary spring element k1 is characterized by a nonlinear Ogden function, while the remaining viscoelastic units are composed of linear springs and dashpots in series. The model assumes that the polyurethane material is homogeneous and continuous, without considering the effects of interfacial slip and temperature variation. Uniaxial compression boundary conditions were applied, with the load imposed on the upper end and the lower end fixed. Numerical calculations were performed using the variable-step ode45 solver. The compression rates from the quasi-static tests were implemented in the model, and the resulting simulation data were compared with the experimental measurements, as shown in Figure 5. The results demonstrate that the simulated curves are in excellent agreement with the experimental data, with a maximum deviation of less than 5%. This validates the accuracy of the established Simulink model in characterizing the large deformation and strain-rate effects of the polyurethane material. Contrasting Figure 1 with Figure 5 reveals that the established coupled constitutive model yields accurate representations of the experimental data at different loading rates. This confirms the model’s dual capability in capturing both the hyperelasticity and the viscoelasticity of polyurethane within a wide strain-rate regime. Consequently, it provides a reliable theoretical foundation for the subsequent structural design and performance evaluation of multi-stage vibration isolation systems. 3. Design of Isolation System Vibration isolation systems attenuate impact energy through a combination of elastic and damping elements. The paper establishes mechanical models for both single-stage and multi-stage isolation systems, as illustrated in Figure 6a, upon which the governing equations of equilibrium are derived. Single-stage vibration isolation is typically modelled as a mass-spring-damper system, constituted by the isolated mass m, stiffness k, and damping c. Under the action of external shock loading F ( t ) , the equation of motion is given by: m x ୍ + c x ˙ + k x = F t (4) where x is displacement of mass. Multi-stage isolation systems facilitate the hierarchical dissipation of shock energy. As depicted in the mechanical model in Figure 6b, the system incorporates one stage consisting solely of a spring, and a second stage comprising a spring-damper-inerter assembly. The governing dynamic equations are formulated as follows: m 1 x ୍ 1 + c 1 x ˙ 1 − x ˙ 2 + k 1 x 1 − x 2 = F t m 2 x ୍ 2 + c 2 x ˙ 2 + k 2 x 2 + c 1 x ˙ 2 − x ˙ 1 + k 1 x 2 − x 1 = 0 (5) where x1 is displacement of mass m1, x2 is displacement of mass m2. To investigate the dynamic response characteristics of the proposed multi-stage vibration isolation system under shock loading, a simulation model of the polyurethane-spring-eddy current isolator was developed within the MATLAB/Simulink environment, as illustrated in Figure 7. As shown in Figure 7, the polyurethane component replaces spring k1 in Figure 6b, and k2 is a helical spring. The c2 is identified as an eddy current damper, with m2 representing the inherent inertance of the eddy current system. This model is employed to simulate the force transmission and energy dissipation processes of the multi-stage device under varying shock conditions. In order to demonstrate the advantages of multi-stage isolation in shock mitigation compared to single-stage isolator, the dynamic behaviors of both systems were analyzed under equivalent excitation. The input shock load is presented in Figure 8. Figure 9 illustrates the acceleration and displacement responses of the isolated mass under both single-stage and multi-stage isolations system. As observed, the mass exhibits a pronounced peak acceleration following the shock with single-stage isolator, accompanied by sustained periodic oscillations and a protracted attenuation process. In contrast, the multi-stage system yields a significantly diminished peak acceleration, with vibrations converging rapidly over time. This indicates that the multi-stage isolator effectively mitigates the transmission of shock loads to the superstructure, thereby demonstrating superior shock suppression capabilities. Displacement data further reveal that while the single-stage system suffers from large-amplitude, persistent reciprocating oscillations, the multi-stage alternative ensures a significantly lower peak displacement and accelerated decay. This highlights the enhanced isolation efficacy achieved by the multi-stage system through its graded energy dissipation and staged transmission mechanisms. Figure 9. Comparison of vibration isolation effects: ( a) Acceleration of mass; ( b) Displacement of mass. Figure 9. Comparison of vibration isolation effects: ( a) Acceleration of mass; ( b) Displacement of mass. 4. Prototype Design and Drop Test 4.1. Design of Multi-Stage Isolator In this study, a spring-spring-eddy current multi-stage isolator was designed as the baseline configuration. The system integrates a metallic helical spring in series with a second-stage assembly. This second stage features a parallel arrangement of a helical spring and an eddy current damper, fixed between the top plate and the middle plate with screws. An isolation mass, representing the protected structure, is mounted on the top plate. To ensure axial stability and preclude lateral instability, a guiding steel tube with a corresponding sleeve is incorporated between the two stages. To enhance the lateral stability and energy dissipation capacity of the isolator, the metallic spring in the first stage of the original isolator was replaced with polyurethane material, resulting in a polyurethane-spring-eddy current multi-stage isolator. In this configuration, the primary stage consists of a multi-layered polyurethane foam stack, which is designed to improve the non-linear deformation capability and energy absorption efficiency. All other components, including the isolation mass, eddy current damper, and the supporting guide mechanism, remain identical to those of the spring-spring-eddy current isolator to ensure the comparability of the performance between the two devices. To preliminary evaluate the isolation performance of the devices, simulation comparisons were conducted across four distinct configurations: single-stage spring-damper, single-stage polyurethane-damper, multi-stage spring-spring-damper, and multi-stage polyurethane- spring-damper. These simulations were based on the same shock load as described in Section 2, with the resulting isolation effects illustrated in Figure 12. As shown in Figure 12a, the single-stage spring isolation scheme exhibits a higher peak acceleration and a prolonged oscillation time, indicating relatively poor isolation performance. In contrast, the multi-stage schemes—particularly the multi-stage polyurethane-spring-damper configuration—demonstrate a marked reduction in peak acceleration, reflecting a robust capacity for shock transmission suppression. In Figure 12b, the single-stage spring scheme shows the largest displacement amplitude with a slow attenuation process. Conversely, the multi-stage schemes effectively diminish the displacement amplitude and accelerate the vibration decay. Overall, the multi-stage polyurethane-spring-damper isolator significantly reduces both peak acceleration and displacement, manifesting superior isolation effectiveness compared to the three other designs. 4.2. Drop Test To explore the dynamic characteristics of the isolation systems under different energy levels, various impact conditions were established by varying the drop height. Three typical scenarios, with peak input accelerations of 230 g, 255 g, and 360 g, were selected for analysis and designated as Test Cases 1, 2, and 3, respectively. To ensure the consistency of input conditions among different isolators, the drop zero position was calibrated before each test. By adjusting the drop test parameters, similar input acceleration amplitudes were achieved for the two isolators, thereby ensuring the feasibility of comparing different vibration isolation schemes. For each shock condition, three independent drop tests were repeated, and one representative dataset was selected for the final analysis. Drop parameters were carefully calibrated to provide nearly identical input acceleration profiles for both isolators, ensuring a fair and valid comparison of their respective performance across different isolation configurations. Two different multi-stage vibration isolators for drop tests are shown in Figure 13. To investigate the shock transmission characteristics of the isolation system, accelerometers with measurement ranges of ±500 g and ±100 g were positioned on the bottom plate and top plate to measure the input and output acceleration responses, respectively. Meanwhile, three displacement sensors were deployed at the top plate, middle plate, and bottom plate to record inter-stage relative displacements, thereby assessing the system’s deformation and energy absorption capacity. All signals were synchronously recorded using a high-speed data acquisition system, and the raw data were denoised using a low-pass filtering method. The processed experimental data are summarized in Table 4. Specifically, the input acceleration of bottom plate reflects the intensity of the external shock excitation, while the output acceleration of top plate represents the transmitted response of the shock load through the isolation system. To evaluate the capacity of the isolation system to inhibit the transmission of acceleration under shock loading, an isolation rate T is defined to quantitatively assess the system’s acceleration attenuation capability. The specific calculation formula can be expressed as follows: T = 1 − A output A input ୍ଠ 100 % (6) where A i n p u t and A o u t p u t denote the peak acceleration at the input and output of the isolation system, respectively. A higher value of T indicates a more robust capacity to inhibit shock acceleration, thereby signifying superior isolation effectiveness. Test results confirm that the input acceleration amplitudes for both isolation systems are closely matched, providing a uniform basis for evaluating shock performance. According to Table 4, both multi-stage designs achieve isolation efficiencies exceeding 90% across all cases. Notably, the polyurethane-spring-eddy current isolator outperforms the spring-spring-eddy current isolator in terms of shock attenuation. 4.3. Results of Test To simplify the discussion, the spring-spring-eddy current and polyurethane-spring-eddy current multi-stage isolators are hereafter termed the “all-spring” and “polyurethane” isolators, respectively. Figure 14 presents the displacement-time histories of both isolators under various loading conditions. It is observed that both systems exhibit distinct oscillatory decay under shock loads, though their displacement amplitudes and attenuation histories differ significantly. The all-spring isolator shows generally higher peak displacements for the top and middle plates, with a more pronounced residual vibration and longer-lasting oscillations. This suggests its shock response is dominated by the release of stored elastic energy, leading to a relatively sluggish decay process. In contrast, the polyurethane isolator yields lower displacement peaks and a faster convergence of the vibration envelope, allowing the responses of each stage to reach a stable state. This demonstrates the effective cushioning effect provided by the polyurethane unit. Shock acceleration time-history of 0.1 s was selected as the input signal and applied to the corresponding Simulink dynamic model to obtain the acceleration response of the mass. An exponential decay model was subsequently employed to fit the response envelopes; the fitting outcomes are illustrated in Figure 15 and detailed in Table 5. Figure 15 illustrates that the polyurethane isolator consistently yields lower peak output accelerations, demonstrating its enhanced ability to suppress shock transmission. However, a comparison of the decay characteristics reveals that the all-spring isolator exhibits higher exponential decay constants and greater damping ratios for its envelopes. This reflects that, under the current model assumptions, the all-spring isolator can more rapidly attenuate the vibration response to a low-amplitude range. It should be noted that, whilst the simulation results reflect the general dynamic trends of the two isolators, certain discrepancies remain between the predicted output acceleration amplitudes and decay characteristics compared to the experimental data. In addition to common factors such as material non-linearity and boundary energy dissipation, these deviations are also attributed to the shock propagation mechanisms and loading input methods. The simulation presents an idealized response for the all-spring isolator with swift decay. However, in the physical tests, the shock load contains significant high-frequency components, and stress waves can propagate efficiently through the metallic helical springs and connectors to the upper mass. This excites more intense high-frequency and coupled vibrations, leading to multiple rebounds and sustained oscillations following the drop. Such high-frequency transmission and multiple contact behaviors are typically omitted in the simplified dynamic model, resulting in an overestimation of the isolation effectiveness for the all-spring system. To quantitatively evaluate the discrepancies between the simulation and experimental results, three error indices were introduced, namely the peak relative error, the normalized root-mean-square error, and the relative deviation of the attenuation rate. Frequency-domain analysis of the measured top-plate acceleration using fast Fourier transform (FFT) indicates that the dominant energy of the polyurethane isolator is mainly concentrated within the 0–200 Hz frequency band, whereas the all-spring isolator exhibits a pronounced secondary peak in the range of 1000–2000 Hz. This phenomenon is attributed to the propagation of stress waves along the metallic springs and bolted connections. Therefore, the proposed lumped-parameter model is considered quantitatively valid within the 0–100 Hz range, where the errors remain below 11%. Beyond 100 Hz, the high-frequency contact rebound and wave propagation effects neglected under the rigid-contact and single-degree-of-freedom-stage assumptions lead to an underestimation of the sustained oscillations in the all-spring system. These effects may be incorporated in future work by introducing a distributed-parameter spring model or explicit contact-impact elements. Overall, while the simulation captures the general trends within the dominant frequency band, the simplification of high-frequency propagation mechanisms is a primary cause for the discrepancies under severe shock conditions. 5. Conclusions To address the issues of insufficient damping and limited energy dissipation capacity in conventional isolators under severe shock environments, this study proposes a multi-stage isolation configuration incorporating polyurethane materials, eddy current dampers, and metallic springs. Through dynamic modelling, material constitutive parameter identification, numerical simulations, and drop-shock tests, the shock isolation performance of various schemes was systematically investigated. Furthermore, the discrepancies between experimental and simulation results were rationally analyzed. The primary conclusions of this study are as follows: (1) Dynamic models for both single-stage and multi-stage isolation systems were developed and implemented within the MATLAB/Simulink environment. The results demonstrate that multi-stage isolation facilitates the step-by-step transmission and staged dissipation of shock energy. Compared to single-stage systems, the multi-stage configuration effectively reduces the peak acceleration and displacement stroke of the mass, while accelerating response convergence and enhancing shock suppression capabilities. (2) Polyurethane components exhibit non-linear large deformation and time-dependent characteristics during shock loading. A coupled constitutive model coupling Ogden hyper-elastic model with generalized Maxwell viscoelastic model was established. The model parameters were identified based on compression and relaxation tests, providing a robust material modelling foundation for the multi-stage isolation dynamic analysis. (3) Drop tests demonstrate that under the three typical conditions, the peak output acceleration of the polyurethane multi-stage system is consistently lower than that of the all-spring version. This indicates a significantly improved isolation performance and superior effectiveness in shock mitigation. (4) Simulink simulations based on experimental input signals reveal that the all-spring isolator exhibits a higher decay rate and damping ratio, resulting in faster envelope attenuation. Conversely, the advantage of the polyurethane isolator primarily lies in its ability to suppress peak output acceleration. The discrepancies between simulation and experimental results are attributed to the simplification of high-frequency stress wave propagation and the multiple rebound contact processes within the dynamic model. Overall, this research confirms that the proposed multi-stage system effectively mitigates structural responses and suppresses shock transfer across typical shock scenarios, offering valuable theoretical and experimental insights for protective device design. However, current simulations rely on simplified assumptions regarding contact mechanics and the rate-dependent non-linearity of polyurethane. To enhance fidelity and broaden the scope of engineering application, future efforts should focus on high-fidelity contact modelling and more precise parameter recognition techniques specifically tailored for severe shock environments. Author Contributions Conceptualization, H.Z.; Methodology, H.Z.; Software, Y.W. and X.W.; Validation, Y.W.; Investigation, Y.W. and X.W.; Data curation, Y.W.; Writing—original draft, Y.W.; Writing—review & editing, Y.W., Z.C., W.W., Y.C., H.J. and X.W.; Visualization, Y.W.; Project administration, Y.W.; Funding acquisition, H.Z. All authors have read and agreed to the published version of the manuscript. Funding This research was funded by the National Natural Science Foundation of China (Grant No. 52508555) and the Project of Yuelushan Center for Industrial Innovation (Grant No. 2025YCII0116). The APC was funded by the Project of Yuelushan Center for Industrial Innovation (Grant No. 2025YCII0116). Institutional Review Board Statement Not applicable. Informed Consent Statement Not applicable. Data Availability Statement The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Conflicts of Interest The authors declare no conflict of interest. References Figure 1. Ogden fitting results and experimental data curve. Figure 1. Ogden fitting results and experimental data curve. Figure 2. Relaxation test curve. Figure 2. Relaxation test curve. Figure 3. Relaxation fitting result curve. Figure 3. Relaxation fitting result curve. Figure 4. Five-element Maxwell model. Figure 4. Five-element Maxwell model. Figure 5. Simulation and experimental results of polyurethane constitutive model: ( a) Compression Cases 1; ( b) Compression Cases 2; ( c) Compression Cases 3; ( d) Compression Cases 4. Figure 5. Simulation and experimental results of polyurethane constitutive model: ( a) Compression Cases 1; ( b) Compression Cases 2; ( c) Compression Cases 3; ( d) Compression Cases 4. Figure 6. Mechanical models: ( a) Single-stage isolators; ( b) Multi-stage isolators. Figure 6. Mechanical models: ( a) Single-stage isolators; ( b) Multi-stage isolators. Figure 7. Polyurethane-spring-eddy current damper multi-stage vibration isolator simulation model. Figure 7. Polyurethane-spring-eddy current damper multi-stage vibration isolator simulation model. Figure 8. Input acceleration time history for simulation analysis. Figure 8. Input acceleration time history for simulation analysis. Figure 10. Schematic diagram of polyurethane multi-stage vibration isolator. Figure 10. Schematic diagram of polyurethane multi-stage vibration isolator. Figure 11. Schematic diagram of full-spring multi-stage vibration isolator. Figure 11. Schematic diagram of full-spring multi-stage vibration isolator. Figure 12. Vibration isolation effectiveness of four types of vibration isolators: ( a) Acceleration of mass; ( b) Displacement of mass. Figure 12. Vibration isolation effectiveness of four types of vibration isolators: ( a) Acceleration of mass; ( b) Displacement of mass. Figure 13. Multi-stage isolators: ( a) Spring-spring-eddy current multistage vibration isolator; ( b) Polyurethane-spring-eddy current multistage vibration isolator. Figure 13. Multi-stage isolators: ( a) Spring-spring-eddy current multistage vibration isolator; ( b) Polyurethane-spring-eddy current multistage vibration isolator. Figure 14. Displacement of two types of vibration isolators under different test conditions: ( a) Test Case 1; ( b) Test Case 2; ( c) Test Case 3. Figure 14. Displacement of two types of vibration isolators under different test conditions: ( a) Test Case 1; ( b) Test Case 2; ( c) Test Case 3. Figure 15. Simulation of vibration isolator acceleration and decay fitting under different conditions: ( a) Polyurethane isolator (case 1); ( b) All-spring isolator (case 1); ( c) Polyurethane isolator (case 2); ( d) All-spring isolator (case 2); ( e) Polyurethane isolator (case 3); ( f) All-spring isolator (case 3). Figure 15. Simulation of vibration isolator acceleration and decay fitting under different conditions: ( a) Polyurethane isolator (case 1); ( b) All-spring isolator (case 1); ( c) Polyurethane isolator (case 2); ( d) All-spring isolator (case 2); ( e) Polyurethane isolator (case 3); ( f) All-spring isolator (case 3). Table 1. Third-order Ogden model parameters. Table 1. Third-order Ogden model parameters. μ 1 μ 2 μ 3 α 1 α 2 α 3 −0.0337 0.1036 −0.2070 0.9442 −4.7792 9.5610 Table 2. Relaxation parameters table. Table 2. Relaxation parameters table. E 1 τ 1 E 2 τ 2 0.0301 218.1015 0.0713 1724.3399 Table 3. Key mechanical parameters of the vibration isolation device. Table 3. Key mechanical parameters of the vibration isolation device. Polyurethane-Spring-Eddy Current Multistage Vibration Isolator Parameters Spring-Spring-Eddy Current Multistage Vibration Isolator Parameters mass m = 300 kg mass m = 300 kg upper spring k = 200 kN/m upper spring k = 200 kN/m eddy current damper c = 21 kN∙s/m eddy current damper c = 21 kN∙s/m polyurethane h = 304.8 mm bottom spring k = 75 kN/m Table 4. Summary table of drop test acceleration results. Table 4. Summary table of drop test acceleration results. Case Acceleration Spring-Spring-Eddy Current Multistage Vibration Isolator Rate (%) Polyurethane-Spring-Eddy Current Multistage Vibration Isolator Rate (%) 1 Input 224.31 g 92.56 244.60 g 97.30 Output 16.70 g 6.61 g 2 Input 258.89 g 92.70 257.52 g 96.16 Output 18.89 g 9.89 g 3 Input 367.22 g 93.21 356.97 g 94.73 Output 24.93 g 18.81 g Table 5. Summary of simulation acceleration attenuation parameters. Table 5. Summary of simulation acceleration attenuation parameters. Case Isolator Peak Acceleration/g Exponential Decay Rate α/(s −1) Damping Ratio ζ Frequency/Hz 1 Polyurethane isolator 1.7246 0.7427 0.0621 1.9003 All-spring isolator 2.2806 1.2792 0.0854 2.3743 2 Polyurethane isolator 2.4941 0.7913 0.0657 1.9110 All-spring isolator 2.7481 1.2550 0.0838 2.3743 3 Polyurethane isolator 3.0150 0.8266 0.0685 1.9167 All-spring isolator 5.4817 1.3431 0.0897 2.3743 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.
