Response Analysis and Damping Parameter Identification of Stiffened Plates Under Shock Environment

Abstract

Stiffened plate structures widely used in military aircraft are frequently subjected to severe shock environments, such as those generated by gunfire or explosive blasts, which can significantly compromise the integrity and reliability of onboard equipment and devices. Accurate characterization and prediction of the shock response, especially the damping behavior of such structures, remains a critical yet challenging problem in aeronautical engineering. This study presents an integrated experimental–numerical framework for analyzing the shock response and damping characteristics of representative stiffened plates under shock wave excitation. Controlled shock loading is applied using a shock tube, with real-time acceleration responses measured at multiple locations on both plain and rib-reinforced plates. A high-fidelity finite element model is developed, and three commonly used damping models—Rayleigh Damping, wave attenuation Model, and Maximum Loss Factor Model—are systematically evaluated. Damping parameters are identified through a Particle Swarm Optimization (PSO) algorithm, using the shock response spectrum (SRS) as the performance metric. Experimental results reveal that the incorporation of reinforcing ribs can reduce peak acceleration responses and significantly enhance the damping performance, particularly in the mid-to-high frequency range. The identified damping parameters further show that the maximum loss factor model provides superior agreement with experimental SRS data compared to traditional approaches. The proposed methodology offers a robust method for modeling damping behavior in stiffened plates, providing practical insights for the design of aircraft structures exposed to shock environments.

1. Introduction

Military aircraft structures and their sensitive components—including electronic devices, mechanical assemblies, and valves—frequently encounter shock environments caused by continuous gunfire firing during operational service. This shock environment, commonly referred to as gunfire shock, is characterized by each shot generating a transient mechanical load with high-frequency, high-magnitude impulses. The acceleration amplitudes can reach up to 10,000 g, with durations ranging from several to hundreds of milliseconds and dominant frequencies spanning 100 to 100,000 Hz [1,2]. Given the repeated exposure to these severe shock environments throughout an aircraft’s service life, both the structural and functional integrity of devices are often compromised. Consequently, understanding the dynamic response of aircraft structures and the propagation of shock environments have emerged as a critical research priority in aeronautical engineering [3].

In recent years, researchers have devoted considerable effort to the characterization and theoretical analysis of shock environments, particularly those associated with aerospace pyroshock conditions, which fundamentally share the same physical mechanisms as single gunfire shocks. The severity and frequency characteristics of a shock environment are typically evaluated quantitatively using the Shock Response Spectrum (SRS) [3,4,5] or the Pseudo-Velocity Shock Response Spectrum (PVSRS) [6,7]. These spectra describe the maximum response of a Single-Degree-of-Freedom (SDOF) system subjected to a given shock input, thereby serving as general indicators of potential damage [4]. Supported by testing standards such as MIL-STD-810 [2], NASA-STD-7003A [8], and BV043/85 [9], the aforementioned SRS-based analytical framework has been widely applied to the testing, characterization, and prediction of shock environments across multiple fields, including aerospace separation events [10,11], underwater explosions [12], and other related fields [13]. Building upon this standardized analytical framework, recent theoretical developments have further advanced the physical interpretation of SRS and PVSRS. Hou and Li [7] showed that, contrary to the traditional Hunt relation that links modal stress to modal velocity, modal pseudo-velocity is directly proportional to modal stress. This finding provides rigorous theoretical justification for using PVSRS as a robust severity metric. Additionally, invoking the maximum stress failure criterion, Li et al. [5,6] established two SRS-based damage boundaries governed respectively by pseudo-velocity, absolute acceleration, and relative displacement, thereby bridging empirical environmental test standards with mechanistic damage prediction. Collectively, these studies have established the theoretical foundation for shock environment testing, qualification, and prediction.

In parallel with advances in theoretical research, growing emphasis has been placed on the development of integrated experimental-numerical approaches for the testing and prediction of shock environments [14]. In terms of experimental testing, several types of facilities have been established, such as the floating shock platform [15], the Vertical dual-wave shock tester device [16], and air-gun-based experimental systems [17]. In addition, Babuska [18] and Ott, Richard J [19] investigated the shock response and propagation in mechanical joints and cylindrical structures using practical satellite applications and the Ares I-X primary structure as case studies. Park [20] evaluated the attenuation performance of a novel embedded damping device in a honeycomb sandwich panel equipped with elastic washers under a shock environment. Zhao [21] experimentally studied the damage threshold of a crystal oscillator subjected to shock environments. Lee [22] and Ruan [23] examined the propagation of shock environments in flat plate structures. In terms of shock response signal processing, recent studies have emphasized drift correction, filtering, and decomposition techniques to enhance the fidelity of acceleration data obtained from experiments [24,25]. Advanced signal processing approaches such as Complementary Ensemble Empirical Mode Decomposition (CEEMD) and rigid-body motion correction have significantly improved the fidelity of high-frequency measurement data [16]. In terms of analysis and computational modeling, the Finite Element Method (FEM) and Statistical Energy Analysis (SEA) have been widely applied to the study of shock propagation [26,27]. The semi-analytical models that have been developed are likewise capable of accurately predicting shock propagation behavior and the corresponding response spectra [28]. Based on the comparison of various shock environment simulation methods [14,29], Lee [30,31,32] conducted numerical analyses of the key dynamic characteristics of pyroshocks generated by ridge-cut explosive bolts. Sun et al. [33,34], using an L-shaped representative structure as an example, proposed a Finite Element-Statistical Energy Analysis (FE-SEA) hybrid modeling technique to predict the shock response of complex spacecraft structures.

Although significant progress has been made in the analysis and modeling of shock environments, the damping characteristics under such conditions have historically received little attention. In early or simplified structural dynamic analyses, damping is frequently neglected. It is commonly assumed that due to the extremely short duration of transient shock events, the peak structural response is governed almost entirely by inertial and elastic forces, leaving insufficient time for damping mechanisms to significantly dissipate energy before maximum deformation occurs [1]. However, as engineering demands for structural reliability and precision have increased, ignoring damping is no longer viable. Damping plays a crucial role not only in governing energy dissipation mechanisms and the attenuation of the secondary response under shock loading, but also in structural reliability assessment and the correlation of numerical models with experimental data. Consequently, recent shock response studies have transitioned toward incorporating numerical damping models. For instance, Yan and Li [35] introduced a standard damping ratio into their finite element model. Lee [22] provided comparative validations between several static damping conditions and experimental results. Gao [10] reported the shock responses of onboard equipment under different damping coefficients during spacecraft docking processes. Similarly, Yu [36] investigated the dynamic response of a 3D wedge impacting on water under five prescribed structural damping factors. Despite the explicit inclusion of damping models, these studies still predominantly rely on empirical assumptions, generic constants, and trial-and-error calibration. Such methodologies render the investigation of damping characteristics under shock environments largely passive, lacking a direct mechanistic link to the actual structural dissipation physics. To address the limitations of passive assignment, more recent efforts have shifted toward the active extraction and selection of damping parameters. S. Chandra [37] departed from conventional empirical assumptions by attempting to actively extract the damping characteristics of stiffened laminated composite plates—subjected to uniformly distributed pulse loading in a thermal environment—through free-vibration-based numerical analyses. Nevertheless, the lack of direct experimental validation under actual shock conditions remains a major limitation for these analytically derived parameters. Although white-noise excitation tests were later introduced by Chandra [38] to experimentally validate the damping identification, the investigated scenarios primarily represent narrow-band, small-amplitude vibration. For severe shock environments characterized by strong transients, broadband spectra, and high-frequency energy concentration, the physical representativeness and engineering applicability of such actively extracted parameters have yet to be systematically demonstrated.

Given the limitations of existing studies on damping characteristics under shock environments, the accurate prediction of shock responses in stiffened plates commonly used in aircraft structures remains a significant challenge. In this study, a typical aircraft stiffened plate is taken as the research object, and an experiment–numerical integrated iterative analysis framework is established to overcome the conventional damping modeling approaches that rely heavily on empirical assumptions and trial-and-error tuning. The shock response and damping characteristics of the structure under shock environments are systematically investigated. Specifically, this paper moves beyond the simple combination of known methodologies to provide a systematic, experiment-driven investigation of damping behaviors. The main contributions of this study include: (1) We address the crucial but often overlooked role of damping in high-frequency shock environments, bridging the gap between idealized elastic models and real-world structural energy dissipation. (2) Moving away from empirical assumptions and trial-and-error calibration, we establish a rigorous framework to actively acquire physically representative damping parameters directly from experimental shock tube data using PSO. (3) Building upon the identified parameters, we systematically compare various damping formulations. We mechanistically demonstrate that the maximum loss factor model avoids unphysical high-frequency overdamping, providing a highly reliable numerical baseline for the dynamic design of complex stiffened structures.

The structure of this paper is as follows: Section 2 describes the experimental methodology and the procedures for damping parameter identification and optimization. Section 3 reports and analyzes the experimental findings, with a focus on assessing the effects of reinforcing ribs on damping behavior. Section 4 summarizes the research conclusions and outlines the directions for future research.

2. Methodology

This section describes the experimental methodology using shock tube excitation, numerical simulation framework, and PSO-based damping parameter identification procedures. Section 2.1 describes the experimental setup and relevant test configurations. Section 2.2 details the construction procedure of the finite element model. Section 2.3 introduces the method for damping parameter identification of stiffened plates using the PSO method.

2.1. Experimental Setup

Given the physical similarity between shock waves and explosive blast loading, a shock tube was employed to generate ideal blast load excitation on the test plate. The shock tube configuration used in this study is depicted in Figure 1a. The tube features an internal diameter of 100 mm, with the driver section measuring 0.34 m in length and the driven section extending 8.2 m. The shock tube experimental setup is shown in Figure 1b. The experimental pressure measurement system consisted of a piezoelectric pressure sensor (Model 102B, PCB Piezotronics, Depew, NY, USA), a corresponding signal conditioner, and a GEN3i high-speed data acquisition (DAQ) system (HBM, Darmstadt, Germany). The pressure sensor was positioned on the driven section wall at a distance of 100 mm from the shock tube exit to monitor the overpressure profile of the shock wave. The impulse load was applied to the plate center over a circular shock region of 100 mm diameter, positioned at the midpoint of the specimen. The pressure sensor and accelerometers were each connected to a 16-channel DAQ system via their respective signal conditioners. The data acquisition system sampled at a rate of 0.1 MS/s and was triggered by the rising edge of the signals from accelerometers. Prior to each experiment, the plates was tapped to verify that the DAQ system could be properly triggered and record data via the rising edge of the acceleration signal.

Real-time acceleration measurements were conducted using four YK-003PC accelerometers positioned at strategic locations across the plate surface during shock loading events. These accelerometers provided an acceleration measurement range of 100,000 $\mathrm{g}$ with a frequency response spanning 1–15,000 Hz, ensuring adequate performance characteristics for shock loading applications. Accelerometer installation followed manufacturer specifications, with each unit secured using a calibrated torque wrench set to the recommended value of $20\mathrm{kgf}·\mathrm{cm}$ (1.96 N·m). Proper grounding procedures were implemented for the signal conditioner interface to minimize environmental noise interference and prevent data drift artifacts in the measurements. Signal conditioning was implemented with a tenfold amplification factor (×10) to enhance signal resolution and measurement precision.

Maintaining the plates within the elastic deformation range during shock loading is essential for accurate shock environment characterization, as gunfire shock environments are fundamentally elastic wave propagation phenomena. Plastic deformation would not only misrepresent the actual physical scenario but also compromise the repeatability of experimental studies. Through variation in the standoff distance and membrane thickness in a series of preliminary experiments, it was determined that positioning the plate at a distance of 1 cm from the shock tube exit, combined with a membrane thickness of 0.2 mm, provided the acceptable loading conditions. This configuration ensures that the applied shock load remains within the elastic range of the structural material while still generating sufficient excitation for meaningful shock response analysis.

To investigate the shock response of stiffened plate configurations under impulsive excitation, comparative experiments were conducted using various stiffening schemes and positioned measurement points. Two measurement group configurations (A and B) were established on each test plate for data acquisition, as illustrated in Figure 2a. Group A employed four measurement points aligned continuously along the plate’s transverse symmetry axis, creating an axial distribution pattern. Group B utilized a quasi-square configuration with two sensors positioned symmetrically along the transverse axis and two additional sensors located 8 mm from the plate’s bottom edge. Precision 5 mm through-holes were machined in each plate to facilitate accelerometer mounting. The experimental program included 5052 aluminum alloy plain plates (535 mm × 440 mm × 2 mm) and stiffened variants with one or three reinforcing ribs, as shown in Figure 2b–d. Each plate was secured using a steel fixture system with sixteen 10 mm diameter bolts—eight bolts mounted symmetrically on each lateral side.

The experimental scheme comprises six distinct groups, as summarized in

Table 1. Experimental scheme for shock response analysis of structural plates.

Experimental Groups	Shock Tube Loading Conditions		Structural Plate Type	Measurement Points
	Membrane Thickness	Tube-to-Plate Distance
No. 1	0.2 mm	1 cm	Unstiffened plate	Group A
No. 2				Group B
No. 3			Single-stiffened plate	Group A
No. 4				Group B
No. 5			Triple-stiffened plate	Group A
No. 6				Group B

. It systematically covers three plate configurations (unstiffened, single-stiffened, and triple-stiffened) with two measurement group arrangements (Group A and Group B), enabling a comprehensive comparative analysis. Each experimental configuration was subjected to three repeated tests.

2.2. Finite Element Modeling of the Stiffened Plates

To accurately capture the high-frequency transient shock response and systematically evaluate the damping characteristics, a high-fidelity numerical simulation model was established in COMSOL Multiphysics (version 6.2, COMSOL AB, Stockholm, Sweden). This section provides a detailed description of the finite element model construction procedure, including the finite element model formulation, load representation, and boundary conditions.

The stiffened plate was first modeled based on the experiment configuration, assuming perfect bonding with the reinforcing ribs by applying a continuity condition that guarantees identical displacement, stress, and acceleration across the interface. To accurately resolve the high-frequency stress wave propagation through the structure, the stiffened plates were modeled using a 3D solid formulation rather than shell elements. The excitation region in the numerical model was defined as a circular boundary with a diameter of 100 mm at the center of the plate, representing the shock loading area. The shock loading applied to the plate was derived from the actual pressure data measured by the pressure sensor located at the exit of the shock tube in the shock response experiments described in Section 2.1. The material properties of the stiffened plate were defined based on the experimental plate and correspond to the 5052 aluminum alloy available in the COMSOL 6.2 material library, with the corresponding parameters listed in

Table 2. Material properties adopted in the numerical models.

Elastic Modulus E (Pa)	Poisson’s Ratio $\mathit{\nu }$	Density $\mathit{\rho }$ (kg/m3)	Yield Strength ${\mathit{\sigma }}_{\mathit{y}}$ (MPa)	Ultimate Tensile Strength ${\mathit{\sigma }}_{\mathit{u}}$ (MPa)
$7.013\times {10}^{10}$	0.3312	2687.3	193	228

. Since the shock excitation did not induce structural damage or plastic deformation in the structural plates, a homogeneous, isotropic, linear elastic material model was adopted in the numerical simulations. To further substantiate this elastic assumption, the typical yield strength (193 MPa) and ultimate tensile strength (228 MPa) of the 5052 aluminum alloy have been incorporated into Table 2. Throughout the numerical simulations under the prescribed shock loading, the maximum observed von Mises equivalent stress in the structural plates remained significantly below the material’s yield threshold. Under the investigated loading intensities, the structural response remained entirely within the elastic regime, rendering this linear elastic assumption strictly justified and physically valid. Point probes were placed at the center of the respective accelerometer locations to capture the acceleration response. Given the fine mesh size (2 mm) relative to the sensor footprint and the continuous linear elastic deformation of the plate, the single-node extraction provides a highly representative and converged response equivalent to the physical sensor. According to the preliminary experimental setup, eight point constraints were applied on each lateral side of the plate to simulate the bolt-fixed boundary condition. Although local compliance and slip may exist in practical bolted connections, this simplification was adopted to ensure numerical stability and to focus on the global dynamic response under shock loading. The schematic of the finite element model is presented in Figure 3.

In the numerical simulation method adopted in this study, three commonly used damping models suitable for linear elastic materials in COMSOL 6.2 were selected for the structural plate: Rayleigh damping, wave attenuation, and maximum loss factor. In this study, each numerical simulation employed only one specific damping model to ensure independent assessment of its influence. In other words, these three damping models are not applied simultaneously to a single model.

Mesh size and time step are critical parameters influencing both simulation accuracy and computational efficiency. The numerical model was discretized using 8-node linear hexahedral elements. According to the ESA handbook [1], the mesh size should be based on the shortest elastic wavelength, with at least eight elements per wavelength to avoid numerical dispersion or reflection of the incident wave. The governing wavelength depends on wave velocity and the shock frequency [39], as defined by Equations (1)–(3) for expansion, shear, and flexural waves, respectively,

$${\lambda }_e={\displaystyle \frac{\sqrt{\frac{E}{\rho }}}{f}}$$

(1)

$${\lambda }_s={\displaystyle \frac{\sqrt{\frac{G}{\rho }}}{f}}$$

(2)

$${\lambda }_f={\left({\displaystyle \frac{2\pi }{f}}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{E{t}^2}{12\rho }}\right)}^{{\displaystyle \frac{1}{4}}}$$

(3)

where, E and G are the Young’s modulus and shear modulus of the material, respectively; t refers to the thickness of the plate; and f indicates the upper bound of the shock frequency. With a recommended shock frequency of 50,000 Hz, the final mesh size is set to 2 mm so that the wavelength of the shock signal is resolved by at least eight finite element meshes. Within the framework of the explicit finite element integration scheme, the solver time step is constrained by the Courant-Friedrichs-Lewy (CFL) condition [40]. Specifically, the time step must be less than the travel time of the fastest stress wave through the smallest mesh element,

$$\Delta t\le CFL\ast {\displaystyle \frac{\Delta x}{c}}$$

(4)

where, $\Delta x$ corresponds to the smallest spatial discretization size, c represents the propagation speed of the governing wave, CFL is a constant less than or equal to 1, and $\Delta t$ indicates the maximum time step in the simulation. In this study, a time step of $5\times {10}^{-7}\mathrm{s}$ was adopted to satisfy stability and accuracy requirements. To numerically validate the theoretically derived parameters, limited sensitivity analyses were performed during the preliminary modeling stage. Refining the mesh size from 2 mm to 1 mm resulted in negligible changes (less than 2%) in the overall Shock Response Spectrum (SRS) within the dominant frequency range (up to 10 kHz), while substantially increasing the computational cost. Likewise, reducing the time step from 5 × 10⁻⁷ s to 1 × 10⁻⁷ s verified that the selected time increment ensures strict numerical stability without introducing artificial numerical dispersion. Therefore, a mesh size of 2 mm combined with a time step of 5 × 10⁻⁷ s was adopted as the optimal configuration to balance predictive accuracy and computational efficiency.

2.3. Damping Parameter Identification for the Stiffened Plates

In this section, a damping parameter identification methodology based on the commercial finite element software COMSOL 6.2 and MATLAB (version R2023a, The MathWorks, Inc., Natick, MA, USA) is proposed. An adaptive optimization-based damping parameter identification algorithm is developed to achieve inversion and validation of damping parameters from SRS data. The flowchart of the proposed damping parameter identification scheme is shown in Figure 4.

For the Rayleigh damping model, the damping ratio $\zeta$ corresponding to a frequency f is given by

$$\zeta ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\alpha }_{dM}}{2\pi ·f}}+{\beta }_{dK}·2\pi ·f\right)$$

(5)

where, ${\alpha }_{dM}$ and ${\beta }_{dK}$ are referred to as the mass-proportional and stiffness-proportional damping coefficients, respectively. The Rayleigh model adjusts the structural damping value by tuning ${\alpha }_{dM}$ and ${\beta }_{dK}$.

For the wave attenuation model, four parameters—${\alpha }_p$, ${\alpha }_s$, together with their corresponding $f_{p,\mathrm{ref}}$ and $f_{s,\mathrm{ref}}$—are employed to characterize the bulk and shear viscosity coefficients, thereby revealing the attenuation behavior of elastic and shear waves in space, as expressed in Equations (6) and (7).

$${\eta }_{\nu }={\displaystyle \frac{2G\sqrt{\frac{G}{\rho }}{\alpha }_s·\frac{ln10}{20}}{{(2\pi ·f_{s,ref})}^2}}$$

(6)

$${\eta }_b={\displaystyle \frac{2\left(K+\frac{4}{3}G\right)·\sqrt{\frac{K+\frac{4}{3}G}{\rho }}·{\alpha }_p·\frac{ln10}{20}}{{(2\pi ·f_{p,ref})}^2}}-{\displaystyle \frac{4}{3}}{\eta }_{\nu }$$

(7)

where, ${\alpha }_p$ and ${\alpha }_s$ denote the pressure-wave and shear-wave attenuation coefficients, respectively; K and G are the bulk and shear moduli; ${\eta }_b$ and ${\eta }_{\nu }$ are the bulk and shear viscosity coefficients; and $f_{p,\mathrm{ref}}$ and $f_{s,\mathrm{ref}}$ correspond to the reference frequencies of the pressure- and shear-wave attenuation coefficients, respectively.

For the maximum loss factor model, this approach enables characterization of structural energy dissipation capacity through the loss factor,

$$\eta \left(\omega \right)=2{\eta }_{max}{\displaystyle \frac{{\displaystyle \frac{2\pi f}{2\pi f_{\mathrm{ref}}}}}{1+{\left[{\displaystyle \frac{2\pi f}{2\pi f_{\mathrm{ref}}}}\right]}^2}}$$

(8)

where, ${\eta }_{max}$ and $f_{\mathrm{ref}}$ denote the maximum loss factor and its corresponding reference frequency, respectively. This model characterizes the structural energy dissipation ability under high-frequency shock by adjusting ${\eta }_{max}$ and $f_{\mathrm{ref}}$.

To accurately and effectively identify the damping parameters corresponding to each damping model, the identification process was formulated as an optimization problem. The PSO algorithm was employed to determine the damping parameters that yield the best match between the SRS of the numerical simulation model and that obtained from experimental data. In this study, the damping parameters corresponding to the Rayleigh, wave attenuation, and maximum loss factor models were modeled as particles in the PSO algorithm to facilitate parameter optimization. It should be noted that the parameters for each damping model were optimized independently, with separate optimization processes conducted for each model.

During random initialization, a total of 100 particles are generated to initialize the PSO algorithm. Preliminary tests revealed that smaller populations (e.g., 50 particles) risked premature convergence at local optima, whereas 100 particles provided robust parameter space exploration. The constraints used for identifying the Rayleigh damping parameters are provided in Equation (9),

$$\begin{array}{cc}\hfill \underset{{\mathbf{X}}_{\mathbf{R}}\in {\mathbb{R}}^2}{min}& \mathrm{errorSRS}(\mathit{f},{\mathit{X}}_{\mathit{R}})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {[50,0]}^{\mathrm{T}}\le {\mathit{X}}_{\mathit{R}}\le {[300,0.00001]}^{\mathrm{T}},\hfill \\ \hfill & 0<{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\alpha }_{dM}}{2\pi ·f}}+{\beta }_{dK}·2\pi f\right)<1,\forall f\in [100,100000]\mathrm{Hz}.\hfill \end{array}$$

(9)

where, ${\mathit{X}}_{\mathit{R}}={[{\alpha }_{dM},{\beta }_{dK}]}^{\mathrm{T}}$ represents the damping parameter matrix in the Rayleigh damping model. The constraints define the upper and lower bounds of the particle search space under the Rayleigh damping model setting. The parameter constraints for the wave attenuation model is given in Equation (10).

$$\begin{array}{cc}\hfill \underset{{\mathit{X}}_{\mathit{W}}\in {\mathbb{R}}^4}{min}& \mathrm{errorSRS}(\mathit{f},{\mathit{X}}_{\mathit{W}})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\left[0,0,0,100\right]}^{\mathrm{T}}\le {\mathit{X}}_{\mathit{W}}\le {\left[50,50,1000,10000\right]}^{\mathrm{T}}\hfill \end{array}$$

(10)

where, ${\mathit{X}}_{\mathit{W}}={[{\alpha }_p,{\alpha }_s,f_{p,\mathrm{ref}},f_{s,\mathrm{ref}}]}^{\mathrm{T}}$ represents the damping parameters in the wave attenuation model. Equation (11) provides the constraint conditions for identifying the model parameters corresponding to the maximum loss factor in the algorithm,

$$\begin{array}{cc}\hfill \underset{{\mathit{X}}_{\mathit{M}}\in {\mathbb{R}}^2}{min}& \mathrm{errorSRS}(\mathit{f},{\mathit{X}}_{\mathit{M}})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {[0,0]}^{\mathrm{T}}\le {\mathit{X}}_{\mathit{M}}\le {[1,10000]}^{\mathrm{T}},\hfill \end{array}$$

(11)

where, ${\mathit{X}}_{\mathit{M}}={[{\eta }_{max},f_{\mathrm{ref}}]}^{\mathrm{T}}$ represents the damping ratio corresponding to the maximum loss factor model. Similar to the previous cases, the imposed constraints specify both upper and lower bounds for the damping parameters. These bounds are established by extending commonly accepted parameter intervals from the dynamics literature to ensure the feasibility of the search domain.

The fitness function $\mathrm{errorSRS}(\mathit{f},\mathit{X})$ is jointly defined by,

$$\mathrm{MLAD}=\int \left|{\displaystyle \frac{{log}_{10}SRS(\mathit{f},\mathit{X})-{log}_{10}SR{S}_0\left(f\right)}{{log}_{10}SRS(\mathit{f},\mathit{X})}}\right|df$$

(12)

$$\mathrm{errorSRS}(\mathit{f},\mathit{X})={\displaystyle \frac{{\mathrm{MLAD}}_{\left(A_2\right)}+{\mathrm{MLAD}}_{\left(A_3\right)}}{2}}$$

(13)

where, the Mean Logarithmic Acceleration Difference (MLAD) is introduced to quantitatively evaluate the deviation between the SRS calculated in the numerical simulation model with damping parameter $\mathit{X}$ and the experimentally measured SRS. Here, $SRS\left(f\right)$ denotes the SRS at frequency f obtained from the numerical model, while $SR{S}_0\left(f\right)$ represents the target SRS. In this study, the target SRS curve corresponds to the experimentally measured SRS. According to Equation (12), the MLAD value is always positive, and a smaller value indicates a higher similarity between the simulation result and the reference SRS. As MLAD approaches zero, the simulated response is considered to be in strong agreement with the experimental result.

Since the shock responses at Point $A_2$ (before the reinforcing rib) and Point $A_3$ (after the reinforcing rib) are both critical for analyzing the damping behavior of the stiffened plates, the identification process naturally constitutes a multi-objective optimization problem. Under the above research objective, the fitness function, i.e., Equation (13), accurately and comprehensively reflects the error values between the shock response signals at Point $A_2$ (before the reinforcing rib) and Point $A_3$ (after the reinforcing rib) and the experimentally measured response signals. Ultimately, this forms a multi-objective damping parameter identification system for the stiffened plate based on the PSO algorithm. The termination criteria of the PSO algorithm are jointly defined by two conditions: reaching the maximum number of 20 iterations and achieving an error tolerance of ${10}^{-6}$. Furthermore, the robustness of this stochastic framework was verified through multiple independent runs using varying random seeds for initialization.

3. Experiment and Simulation Results

In this section, emphasis is placed on the shock response experiments results, along with an analysis and evaluation of the damping parameters. Specifically, in Section 3.1, the response variability at identical locations during repeated experiments is characterized. Subsequently, in Section 3.2, analyses were conducted to investigate the attenuation effect of the ribs on the shock response of the stiffened plate. In Section 3.3, the effects of reinforcing ribs on damping characteristics were examined, and the performance of damping models was evaluated in terms of SRS.

3.1. Repeatability Analysis of Shock Responses in Shock Wave Experiments

Figure 5a presents the pressure loading curves obtained from three independent shock experiments conducted under identical nominal operating conditions. These repeated tests were purposefully performed to verify the stability and repeatability of the shock tube apparatus, thereby ensuring that the input excitation remains consistent and that subsequent structural response analyses are not biased by random loading variations. Quantitatively, the rise front of the shock pulse reaches its peak within approximately 0.09 ms, demonstrating an extremely steep and repeatable pressure increase in all trials. Meanwhile, the effective pulse duration—defined as the time for the pressure to decay from its peak to near zero—remains consistently around 10 ms, further reflecting the stability of the shock-tube energy-release mechanism. The variation in peak pressure values is minimal, and the waveform morphology—including secondary pressure features and low-amplitude oscillations in the later stage—shows negligible deviation, suggesting that various waveform characteristics are well preserved under repeated loading conditions. Subsequently, as illustrated in Figure 5b, the effective portion of the pressure waveform was extracted and applied to the numerical model described in Section 2.3. The high level of waveform consistency observed across repeated experiments reinforces the rationale of selecting a representative load profile and ensures that the input excitation is not influenced by experimental fluctuations.

From the acceleration time-history data obtained in a total of 16 repeated experiments under the four experimental schemes—Experimental Group 1, 2, 5, and 6—in Section 2.1, the datasets with the best and worst acceleration variability were selected and illustrated in Figure 6a,b. As shown in Figure 6a, the acceleration time histories almost coincide in terms of initial peak, rising edge, and decay rate, with their envelopes rapidly converging within the first few milliseconds. After baseline correction, the signals maintain a near-zero mean, and the oscillation amplitudes in the later stage remain similar across repetitions. Figure 6b exhibits similar characteristics, except for slight differences in the time-domain peak. The time-domain results indicate that, under most experimental conditions, the repeated shock experiments demonstrate consistent repeatability and an acceptable level of response variability.

Based on the acceleration time-domain data measured in the experiments, the SRS plots of Experimental Groups 1, 2, 5, and 6 were obtained according to the SRS theory. Similarly, among the sixteen SRS plots, those with the optimal and poorest variability were selected and illustrated in Figure 6c,d. The SRS plots at various points on the plate exhibit highly consistent peak–valley distributions and identical amplification bands across repeated experiments, with dominant resonance peaks appearing in similar natural frequency ranges and comparable peak bandwidths. Even in the worst-case scenario, the variability is confined to a few frequency regions—primarily in the low-frequency range of 100–200 Hz and in some high-frequency bands—while the remaining frequency curves remain highly clustered. The relatively poor SRS repeatability in the 100–200 Hz range mainly results from minor zero-drift phenomena in the time-domain acceleration data. The slight reduction in repeatability observed at several high-frequency points may be attributed to small differences in sensor mounting. Nevertheless, these effects do not alter the peak positions or overall attenuation trends and thus do not affect the overall repeatability conclusion.

To quantitatively evaluate the experimental repeatability across different groups, the mean and standard deviation (SD) of the peak acceleration responses were calculated to characterize the data variability, as summarized in

Table 3. Variability and Difference of Peak Acceleration Response.

Experimental Groups	Measurement Points	Exp. 1 (g)	Exp. 2 (g)	Exp. 3 (g)	Mean (g)	Standard Deviation (g)
Experimental Group 1	$A_1$	2379	2416	2723	2506	188.84
	$A_2$	1102	1234	1126	1154	70.31
	$A_3$	1261	1180	1253	1231	44.64
	$A_4$	1506	1353	1343	1401	91.36
Experimental Group 2	$B_1$	2725	2572	2648	2648	76.50
	$B_2$	1073	1174	950	1066	112.18
	$B_3$	1062	966	928	985	69.06
	$B_4$	1322	1483	1435	1413	82.66
Experimental Group 5	$A_1$	2831	2472	2466	2590	209.02
	$A_2$	1246	1247	1225	1239	12.42
	$A_3$	927	927	1051	968	71.59
	$A_4$	1007	880	980	956	66.91
Experimental Group 6	$B_1$	2369	2733	2724	2609	207.61
	$B_2$	1082	1114	1197	1131	59.35
	$B_3$	1637	1836	1410	1628	213.15
	$B_4$	1348	1068	1094	1170	154.70

. For each measurement point under the respective experimental setups, the statistical dispersion of the responses was rigorously assessed. Across the 16 evaluated configurations, the vast majority exhibited standard deviations that were tightly constrained relative to their means, indicating excellent repeatability and low statistical variance in most cases. It can therefore be inferred that, under the present experimental setup, the measurement uncertainty of the acceleration response remains minimal and well-controlled. However, in a few isolated instances, the acceleration data exhibited slightly larger standard deviations. Such variability is primarily attributed to the intrinsic non-deterministic characteristics of shock response signals in the time domain, as well as localized sensor–structure coupling effects. Despite these inherent sources of uncertainty, the statistical dispersion of peak accelerations under repeated conditions remains finite and well within an acceptable engineering range. Overall, the shock tube apparatus demonstrated high repeatability in the time-domain shock response at various points on the plate. Therefore, the experimental technique employing a shock tube to generate shock wave loads-commonly used to simulate idealized explosive shock wave conditions-exhibits good repeatability and low variability, laying a solid foundation for further investigations into the response behavior of the plate under shock loading.

3.2. Analysis of the Attenuation Effects of Reinforcing Rib on Structural Shock Responses

To comparatively investigate the influence of reinforcing ribs on the shock response of the structural plate—that is, to study the response attenuation effect before and after the ribs—it is necessary to eliminate the influence of small relative distance differences on the response results. Taking Experimental Group 1: the unstiffened plate at Group A points as an example, the effect of propagation distance on the response amplitude along the transverse axis was examined, as illustrated in Figure 7a. At the excitation center (Point $A_1$), the peak acceleration reached about 2415 g, the highest among all measurement points. As the propagation distance increased from 8 mm to Point $A_2$ (102 mm), the peak dropped markedly to about 1225 g; at Point $A_3$ (118 mm), it further decreased slightly to 1190 g; while at the furthest Point $A_4$ (212 mm), it rose to 1350 g, indicating a reversal in the attenuation trend. We speculate that this phenomenon occurs because Point $A_4$ is located near the edge of the plate, where the incident and reflected waves overlap. This superposition of waves results in an increased wave amplitude, which manifests in the acceleration response as a slight rise in amplitude compared with the case of the incident wave acting alone. The variation in acceleration amplitude with propagation distance from Point $A_1$ to Point $A_4$ is illustrated in Figure 7b. The results indicate that the attenuation rate of the shock response in the near field is significantly greater than that in the mid-to-far field, as shown in Figure 7b, where the slope of the line segment from $A_1$ Point to $A_2$ Point in the near-to-mid field is steeper, while the slope from $A_2$ Point to $A_3$ Point in the mid field is noticeably reduced. The nearly flat slope between $A_2$ and $A_3$ Points suggests that the attenuation of the shock response caused by their slight spatial separation is almost negligible, with only about 35 g.

With the propagation distance effect excluded, the attenuation behavior of the reinforcing rib was investigated by comparing the responses measured at $A_2$ Point, located before the rib, and $A_3$ Point, located after the rib, in Experimental Group 5. Figure 8a,b present a comparison of the acceleration time histories and SRS obtained from ensemble-averaged repeated experiments, both showing clear attenuation at Point $A_3$. Specifically, as shown in Figure 8a, the envelope at $A_3$ Point remains consistently lower than that at $A_2$ Point throughout the entire recording period: the early extrema are smaller, the inter-peak differences in the subsequent ringing are reduced, and the late residual vibrations are weaker. Other time-domain characteristics of the responses at the two points—such as rise time and decay trend—exhibit nearly identical behavior. In Figure 8b, the attenuation effect is observed over a broad frequency band, particularly in the mid-to-high frequency range. Within the mid-frequency region, the SRS curve of $A_3$ Point lies almost entirely below that of $A_2$ Point while maintaining a similar peak–valley pattern; the resonance peaks occur at nearly the same positions, indicating that the modal composition remains unchanged, with only amplitude reduction. In the high-frequency range, the separation between the SRS curves before and after the rib becomes more pronounced, confirming the strong attenuation effect of the reinforcing rib. Within the current pair of measurement points, no significant peak shift is observed, whereas the amplitude decreases markedly, demonstrating that the reinforcing rib primarily acts as an energy barrier.

The FRF magnitude curve, shown in Figure 8c, further illustrates the attenuation effect of the reinforcing rib. It is derived by using the time-domain signal at $A_2$ Point as the input and that at $A_3$ Point as the output in the system’s frequency response analysis. The FRF magnitude response in the 100 Hz to 10,000 Hz range clearly demonstrates the damping performance introduced by the reinforcing rib. In particular, the acceleration signal at the natural frequency of 4000 Hz shows a significant amplitude reduction of $-29$ dB, confirming the rib’s effectiveness in suppressing dynamic responses. Despite a resonance peak near 3000 Hz causing local amplification, the overall response is still significantly reduced. Figure 8d illustrates the shock response attenuation performance of the reinforcing rib in the SRS domain, based on the difference in SRS values (in dB) between $A_2$ Point and $A_3$ Point. $-5$ dB differences at 130 Hz and 2100 Hz correspond to 43.8% attenuation, with an average reduction of 1.68 dB across the frequency range. These results are qualitatively consistent with the frequency band spacing between the two curves in Figure 8b, indicating that the reinforcing rib has a significant effect in improving the shock response characteristics, particularly around 130 Hz and in the mid-to-high frequency band.

Table 4. Peak Acceleration Response at Points $A_2$ and $A_3$ in Group A of Triple-stiffened Plate.

Experimental Groups	Experiment IDs	${\mathit{A}}_2$	${\mathit{A}}_3$	Amplitude Attenuation (g)	Attenuation Ratio (%)	Avg. Attenuation (%)
Experimental Group 5	Exp. 1	1246	927	319	25.60	21.82
	Exp. 2	1247	927	320	25.66
	Exp. 3	1225	1051	174	14.20

summarizes the three repeated experiments, where the rib reduced peak accelerations by 319 g, 320 g, and 174 g, with an average attenuation ratio of 21.82%. This confirms that the presence of the rib plays a critical role in reducing the shock response. In most cases, the rib enables a reduction of approximately 25.6% in peak acceleration amplitude. The relatively lower attenuation observed in the third repeated experiment may be attributed to randomness and uncertainty inherent in dynamic responses. Nevertheless, incorporating reinforcing ribs remains an effective approach for significantly reducing shock responses.

3.3. Analysis of the Effect of Reinforcing Rib on Parameters of Damping Models

Using the PSO-based damping parameter identification algorithm proposed in Section 2.3, the damping parameters of three stiffened plates under three different damping models were identified within the framework of the numerical simulation method. The simulation results were validated by comparing the simulated SRS curves with those obtained from the experiments.

Owing to the highly complex and unpredictable nature of shock response signals in multi-objective optimization scenarios, achieving the predefined minimum tolerance during the algorithmic identification process proved to be challenging. In this study, a relatively large number of particles was assigned to each damping parameter to enhance global search capability, resulting in a considerably long computational time per iteration. As a result, all identification processes in this study were concluded upon reaching the maximum number of iterations. To ensure convergence stability during the parameter identification algorithm, the iterative evolution of the objective function was monitored after each iteration. The results showed that the objective function value dropped rapidly during the initial iterations and then converged asymptotically to a stable plateau with minimal oscillation amplitude. This behavior confirms that the proposed algorithm consistently converges to the global optimal solution, indicating that the selected number of iterations fully satisfies the robustness and reliability of the identified parameters. In the subsequent results section, a detailed comparative discussion of the obtained damping parameters will be presented. Figure 9 illustrates the convergence curves obtained by applying the proposed algorithmic framework with three damping models to the three types of structural plates.

Table 5. Damping parameter identified using the PSO algorithm.

Model Type	Parameter	Structural Plate Type
		Triple-Stiffened Plate	Single-Stiffened Plate	Unstiffened Plate
Rayleigh damping	${\alpha }_{dM}$	200	127.914	108.623
	${\beta }_{dK}$	$2.225\times {10}^{-8}$	$2.845\times {10}^{-8}$	$2.317\times {10}^{-8}$
wave attenuation	${\alpha }_s$	0.1	6.653	0.00223
	${\alpha }_p$	15.098	33.572	20
	$f_s$ (Hz)	5000	4786.963	6270.776
	$f_p$ (Hz)	104.363	283.576	493.021
	${\eta }_v$ (Pa·s)	$1.92\times {10}^3$	$1.39\times {10}^5$	$2.71\times {10}^1$
	${\eta }_b$ (Pa·s)	$5.25\times {10}^9$	$1.58\times {10}^9$	$3.11\times {10}^8$
maximum loss factor	${\eta }_{\max }$	0.509	0.252	0.319
	$f_{\mathrm{ref}}$ (Hz)	29.017	26.871	12.256

presents the damping parameters of the three structural plates identified using the PSO algorithm under the Rayleigh damping, wave attenuation, and maximum loss factor models. The effective shear viscosity (${\eta }_v$) and bulk viscosity (${\eta }_b$) for a linear elastic isotropic material are determined based on the spatial attenuation behavior of elastic shear and compressional waves, as governed by Equations (6) and (7). The incorporation of reinforcing ribs leads to a substantial increase in the bulk viscosity ${\eta }_b$, reaching values on the order of ${10}^9$, indicating that ribs effectively suppress the transmission of pressure waves and promote significant energy attenuation. However, their influence on the shear viscosity ${\eta }_v$ is relatively minor. A comparison among the triple-stiffened, single-stiffened, and unstiffened plates shows that ${\eta }_v$ varies only slightly, unlike the considerable changes observed in ${\eta }_b$.

Furthermore, it is crucial to acknowledge the implications of modeling idealizations—specifically regarding the boundary conditions and the stiffener-plate interfaces—on the identified damping parameters. In the physical experimental setup, energy is dissipated not only through intrinsic material friction but also through micro-slip and local compliance at the bolted boundary connections, as well as wave scattering and interfacial micro-slip at the stiffener-plate joints. However, to maintain computational efficiency and numerical stability during the optimization process, the finite element model simplifies the boundaries as idealized rigid point constraints and enforces a perfect bonding (continuity) condition at the stiffener-plate interfaces. These simplifications artificially eliminate explicit joint-related and interfacial energy dissipation mechanisms. Consequently, during the parameter identification process, the optimization algorithm inevitably compensates for these unmodeled energy losses by augmenting the global structural damping values. Therefore, the damping parameters reported in Table 5 should not be interpreted purely as intrinsic material properties. Rather, they function as effective, lumped calibration parameters that implicitly account for both the inherent structural damping and the complex, unmodeled dissipation mechanisms at the boundaries and internal interfaces.

For the Rayleigh damping model and the maximum loss factor model, the frequency-dependent characteristics can be clearly illustrated using the plots of damping ratio or loss factor versus frequency, based on the damping parameter values obtained from Table 5. This graphical representation facilitates the interpretation and comparison of the results. Figure 10a shows the frequency-dependent damping ratio curve corresponding to the Rayleigh damping parameters. According to the formulation of Rayleigh damping, the Rayleigh damping model exhibits frequency-dependent behavior: at lower frequencies, the mass-related damping term ${\alpha }_{dM}$ is predominant, whereas at higher frequencies, the stiffness-related damping term ${\beta }_{dK}$ governs the system response. The presence of reinforcing ribs significantly increases both the mass and stiffness of the structure, leading to notable increases in the damping parameters ${\alpha }_{dM}$ and ${\beta }_{dK}$. The increase in ${\alpha }_{dM}$ enhances the damping ratio in the low- to mid-frequency range, while the increase in ${\beta }_{dK}$ contributes to improved damping in the high-frequency range. As a result, the triple-stiffened plate exhibits a much higher damping ratio across the frequency spectrum compared to the single-stiffened and unstiffened plates, with the single-stiffened plate also outperforming the unstiffened one. Therefore, the presence of reinforcing ribs significantly improves the damping characteristics of the stiffened plates, which is reflected in the increased damping parameter values. It should be noted that the ${\beta }_{dK}$ value obtained for the triple-stiffened plate appears unusually low. This behavior mainly arises from the coupling effects between different damping terms during the optimization process of damping parameter identification. In the inverse identification procedure, a necessary trade-off was introduced for the ${\beta }_{dK}$ term to compensate for the relatively large ${\alpha }_{dM}$ value, which could otherwise lead to insufficient fitting accuracy in the high-frequency regime. This engineering-oriented approximation of ${\beta }_{dK}$ is primarily intended to achieve a reliable characterization and prediction of the overall structural shock response across the entire frequency range.

Figure 10b illustrates the relationship curve of the loss factor varying with frequency. The presence of reinforcing ribs leads to an increase in the parameter ${\eta }_{\max }$, as evidenced by the significantly higher value observed in the triple-stiffened plate compared to the single-stiffened and unstiffened plates. However, the influence of ribs on the loss factor $\eta$ is not limited to simply increasing ${\eta }_{\max }$; it can also affect the structural damping behavior by altering the reference frequency $f_{\mathrm{ref}}$. For instance, the comparison between the single-stiffened and unstiffened plates provides a clear example: although the single-stiffened plate does not exhibit a higher ${\eta }_{\max }$ than the unstiffened plate, its $f_{\mathrm{ref}}$ is noticeably elevated. As a result, the loss factor of the single-stiffened plate is higher than that of the unstiffened plate across nearly the entire frequency range, as illustrated by the yellow and red curves in Figure 10b. In addition, by examining the Rayleigh damping ratio in Figure 10a and the loss factor in Figure 10b, it can be observed that the indicative parameters representing energy dissipation take relatively higher values at low- and mid-order modal frequencies. As the frequency increases toward the high-frequency range, both the damping ratio and the loss factor gradually decay and converge to a stable level. The consistent variation pattern observed in both damping models further demonstrates the energy dissipation mechanism of the reinforcing rib.

It is essential to interpret the superior performance of the maximum loss factor model not merely as a consequence of improved numerical fitting, but as a result of its more physically consistent representation of the underlying damping mechanisms. The standard Rayleigh damping formulation imposes a restrictive mathematical structure that may be inadequate for broadband shock environments. In particular, its stiffness-proportional term increases linearly with frequency, which tends to introduce excessive damping of high-frequency components (up to 10,000 Hz). As a result, during the calibration process, the optimization procedure is driven to significantly reduce the stiffness-related coefficient, down to the order of ${10}^{-8}$ (as shown in Table 5), in order to preserve high-frequency shock content. This adjustment leads to a distortion of the damping distribution, particularly in the mid-to-high frequency range, where the response may become overly weak and, in some cases, approach negligible levels, which is not physically representative of realistic structural behavior. In contrast, the maximum loss factor model does not impose such a frequency-dependent artificial penalty. Instead, it yields smoother peak responses and more gradual frequency-band attenuation, which is more consistent with the distribution of hysteretic energy dissipation in broadband structural systems. Therefore, its superior performance is primarily attributed to a more physically grounded representation of energy dissipation mechanisms under broadband shock loading conditions.

Figure 11, Figure 12 and Figure 13 present the time-domain response signals and SRS at Point $A_2$ and Point $A_3$ for the three structural plates under the Rayleigh damping model, the wave attenuation model, and the maximum loss factor model, respectively. These simulated results are directly compared with the corresponding experimental data to validate the accuracy of the numerical models. In the time domain, the numerical simulation successfully reproduces the typical transient response characteristics observed in the experiments, including the initial steep shock, the subsequent broadband oscillatory decay process, and the intermittent high-frequency components. The simulated peak amplitudes generally match well with the experimental results, particularly within the first 1–2 ms, where the response energy is highly concentrated. Notably, the maximum loss factor model exhibits excellent performance, whereas the simulation of the single-stiffened plate using the widely adopted Rayleigh damping model somewhat limits the overall effectiveness of the Rayleigh model in reproducing the response behavior. The SRS curves of the response signals calso incorporate $\pm 6\mathrm{dB}$ and $\pm 3\mathrm{dB}$ tolerance curves derived from the experimental SRS data. These tolerance margins serve to highlight the accuracy of the numerical simulations by indicating the acceptable deviation range from the measured results. The SRS curves generated for all three types of structural plates, across all three damping models, fall largely within the $\pm 6\mathrm{dB}$ experimental tolerance bounds. This confirms that the numerical simulation framework provides reliable predictions of shock response characteristics. For the maximum loss factor model, the simulated curves almost entirely fall within the ±3 dB or ±6 dB tolerance bands, indicating that this model effectively characterizes the energy dissipation behavior during both the shock signal propagation and response processes. Even when using the least effective wave attenuation model, which results in a deficiency of high-frequency energy components, the overall SRS curves of the three structural plates under all three damping models remain largely within the ±6 dB experimental tolerance range. The above results verify the reliability of the proposed numerical simulation framework in predicting shock response characteristics, as well as its robustness and high-accuracy features in identifying the damping parameters of structural plates.

Table 6. Response results of three types of structural plates under three damping model schemes.

	Damping Types	Rayleigh Damping		Wave Attenuation		Maximum Loss Factor		Mean of Various Plate
	Point	${\mathit{A}}_{\mathbf{2}}$	${\mathit{A}}_{\mathbf{3}}$	${\mathit{A}}_{\mathbf{2}}$	${\mathit{A}}_{\mathbf{3}}$	${\mathit{A}}_{\mathbf{2}}$	${\mathit{A}}_{\mathbf{3}}$
Triple- stiffened plate	MLAD	0.0611	0.0712	0.0713	0.0849	0.0549	0.0589	0.0671
	errorSRS	0.0662		0.0781		0.0569		0.0671
	Proportion (±3 dB) (%)	48.68	44.74	36.84	32.89	56.58	48.68	44.74
	Proportion (±6 dB) (%)	84.21	73.68	77.63	69.74	82.89	90.79	79.82
Single- stiffened plate	MLAD	0.0464	0.0565	0.0651	0.0726	0.0417	0.0411	0.0539
	errorSRS	0.0515		0.0689		0.0414		0.0539
	Proportion (±3 dB) (%)	59.21	56.58	40.79	35.53	67.11	63.16	53.73
	Proportion (±6 dB) (%)	94.74	90.79	84.21	76.32	100	97.37	90.57
Unstiffened plate	MLAD	0.0705	0.0796	0.0892	0.0917	0.0502	0.0497	0.0718
	errorSRS	0.0751		0.0905		0.0500		0.0718
	Proportion (±3 dB) (%)	38.16	25.00	17.11	19.74	51.32	52.63	33.99
	Proportion (±6 dB) (%)	78.95	72.37	69.74	59.21	96.05	92.11	78.07
Mean of Various Damping Types	MLAD	0.0593	0.0691	0.0752	0.0831	0.0489	0.0499	0.0643
	errorSRS	0.0642		0.0791		0.0494		0.0643
	Proportion (±3 dB) (%)	48.68	42.11	31.58	29.39	58.33	54.82	44.15
	Proportion (±6 dB) (%)	85.96	78.95	77.19	68.42	92.98	93.42	82.82

presents the quantitative analysis results of response errors for different structural plates under various damping model configurations. It includes the fitness function values obtained from the PSO algorithm, as well as the proportions of the simulated SRS results falling within $\pm 3\mathrm{dB}$ and $\pm 6\mathrm{dB}$ of the experimental SRS data. The case of single-stiffened plate demonstrated the best predictive performance among all three structural types, with its simulated SRS results covering 90.57% within the $\pm 6\mathrm{dB}$ range and 53.7% within the $\pm 3\mathrm{dB}$ range. These results reflect the superior reliability of the numerical simulation for the single-stiffened configuration. Furthermore, the performance of the three damping models in the numerical simulation was evaluated. Compared with the widely used Rayleigh damping model, the maximum loss factor model offers superior simulation accuracy. The maximum loss factor model achieved the lowest fitness function value (errorSRS) at 0.0494, in contrast to 0.0642 for the Rayleigh model and 0.0791 for the wave attenuation model. Furthermore, it demonstrated exceptional predictive agreement in the SRS analysis: both Point $A_2$ and Point $A_3$ recorded $\pm 6\mathrm{dB}$ coverage ratios of 92.98% and 93.42%, respectively. At the $\pm 3\mathrm{dB}$ level, the model also maintained strong performance, with coverage ratios of 58.33% at Point $A_2$ and 54.82% at Point $A_3$.

4. Conclusions

This study introduces an innovative experimental methodology coupled with a numerical simulation framework. The shock response attenuation effects of reinforcing ribs on stiffened plates were systematically investigated. A parameter identification method based on PSO was employed to quantitatively assess the influence of ribs on the damping parameters. The main conclusions are summarized as follows:

• The damping characteristics of the structure are strongly affected by the presence of the reinforcing rib. The attenuation in time-domain response amplitude was consistently around 25.0%, while the most significant SRS reduction reached $-5$ dB.
• Among the damping models considered, the maximum loss factor model provides the best damping prediction performance, significantly outperforming the engineering-standard Rayleigh damping model. In contrast, the wave attenuation model yields the lowest prediction accuracy, reaching only 78.11% of that achieved by the maximum loss factor model.
• Among the parameters affecting the accuracy of shock environment prediction, damping is the most critical modeling parameter. By adjusting only the damping model parameters, an average of 82.82% of the two-point prediction results can be brought within the 6 dB tolerance range. However, it should be noted that these results primarily demonstrate the localized calibration fidelity of the model across the reinforcing rib, rather than its global predictive capability.
• The present study is limited to linear damping formulations and shock environment prediction for plate-type structures operating within the elastic regime. It should be noted that under more realistic and severe shock environments, energy dissipation is often governed by complex nonlinear mechanisms that are not explicitly captured in the current model, including localized plastic deformation, structural yielding, and mechanical joint slip. In this context, the damping parameters identified in this study should be regarded as effective calibration parameters that phenomenologically characterize macroscopic energy dissipation, rather than intrinsic physical material properties. Therefore, the proposed methodology is primarily applicable to the structural dynamic design and environmental qualification of airborne equipment subjected to moderate shock conditions where elastic assumptions remain valid. Future work will focus on nonlinear damping behavior, joint friction effects, and more complex structural configurations involving fluid–structure interaction, with the aim of developing data-driven or hybrid identification approaches to improve predictive robustness under severe nonlinear shock environments.