Research on vibration prediction method of aircraft lens frame based on CPO-BP

Abstract

To address the engineering challenge of installing sensors directly on aircraft lens frames for vibration measurement, this paper proposes a vibration response prediction model (CPO-BP) based on a Crested Porcupine Optimizer Algorithm (CPO)-enhanced Backpropagation (BP) neural network. The model employs a two-step optimization strategy: first, CPO optimizes the number of hidden layer neurons and the learning rate; subsequently, it refines the initial values of the network’s weights and biases, thereby improving both the model’s convergence speed and generalization capability. Experimental results demonstrate that the CPO-BP model achieves a high coefficient of determination (R²) of 0.97976, with a mean absolute error (MAE) as low as 0.02020 and a mean squared error (MSE) of 0.10112. Its predictive performance significantly outperforms COA-BP, DA-BP, PSO-BP, and traditional BP models. The predicted response curves closely match the actual responses across acceleration, velocity, and displacement dimensions. This study provides a highly accurate and reliable solution for vibration condition monitoring at critical locations within optical equipment.

When flying equipment operates at high speeds, it is subjected to various forms of vibrational loads. In particular, during abrupt turns of the aircraft, equipment vibrations can lead to blurring in the images captured by the lens, thereby impairing the pilot’s ability to observe tracked objects. Given the high vibration sensitivity of optical lenses, sensors cannot be deployed directly at the lens mounting position. To accurately obtain vibration information at the lens mounting position, sensors must be placed at appropriate adjacent locations. These locations should ensure that the vibration data measured by the sensors truly reflect the vibrational characteristics at the lens mounting position.

Therefore, developing an effective method for predicting vibration signals is of crucial significance to flight equipment. Liu Jialu et al. established a nonlinear vibration response prediction model using a BP neural network from a time-series perspective, and found that the smaller the prediction step, the higher the degree of agreement between the predicted and measured data¹. Zhou Jianhui et al. integrated Kalman filtering technology with deterministic analysis and stochastic considerations, and adopted a minimum variance unbiased estimation strategy to realize the prediction of longitudinal vibration displacement at any position of the propulsion shafting under the condition of unknown propeller excitation force². Han Qichao et al. focused on the coupling effect between the robot and the spindle system, and proposed a corresponding method for predicting the frequency response function of the blade tip. This method constructs a tool tip frequency response function prediction model that incorporates the stiffness characteristics of the robot, thereby effectively predicting the natural frequency of the tool tip in the robot milling system under a specific machining path³. Based on the frequency response function obtained from experiments, Li Feng calculated its response transfer rate, which was combined with the test data of the basic vibration response to achieve the prediction of vibration response during the rotation of the turntable⁴.

With the advancement of intelligent algorithms, some scholars have applied machine learning methods (e.g., neural networks) to construct prediction models in fields such as cutting, bearings, lenses, and robotic arms, aiming to achieve high-performance prediction of machining performance indicators, including cutting force, thermal error, and machining quality. Gao Xing et al. proposed an artillery muzzle vibration prediction model based on a deep belief network, which exhibits favorable prediction accuracy but demands more computational time and resources⁵. Wang Mingyue et al. combined genetic algorithms with neural networks to predict crack location and depth using natural frequencies, enabling accurate identification of structural damage⁶. Yang Shuo et al. investigated the influence of neural network learning rates on the prediction results of structural crack length and location, demonstrating the feasibility of neural networks in crack parameter identification⁷. Jiang Shuang et al. improved the BP neural network using the PSO algorithm and established a prediction model (PSO-BP) to predict tool cutting vibration acceleration based on cutting speed, back engagement, and feed rate⁸. Xue Zenghong et al. adopted the CPO-SVDD model to identify and diagnose three typical faults (pitting, spalling, and fracture) in low-speed heavy-duty bearings, revealing that the identification efficiency of the CPO-SVDD model can reach up to 98%, which is 29.8% higher than that of the traditional SVDD model⁹.

Traditional methods for vibration transmission or prediction, such as transfer path analysis based on frequency response functions (FRFs)⁴ or linear regression models, perform well in simple systems where the structural dynamic characteristics are known or exhibit high linearity. However, for structures like aircraft lens frames—characterized by complex couplings, nonlinear connections (e.g., bolted joints)¹⁴, and constrained sensor placement—these conventional approaches often struggle to accurately capture their intricate vibration transmission relationships. Particularly when data can only be acquired from a limited number of adjacent measurement points, a key challenge arises: how to extract an effective representation of the target point’s vibration state from the highly correlated and informationally redundant signals at the available sensor locations.

In summary, while intelligent algorithms have shown considerable promise in vibration prediction^5,8,16,17 existing studies still exhibit two major limitations. First, the majority of “optimization algorithm–neural network” models only conduct single-phase parameter optimization^11–13, lacking a systematic and sequential optimization process for both network architecture and initial parameters. Second, research on achieving synchronous, high-precision vibration prediction at multiple points and in multiple directions for complex structures, particularly for aircraft optical lenses which have constrained sensor placement, remains inadequate. These shortcomings hinder the reliability and applicability of such predictive models in real industrial settings¹⁵.

To address these gaps, this paper proposes a two-stage optimization framework using the Crested Porcupine Optimizer (CPO) to enhance a BP neural network. Unlike existing single-stage methods (e.g., PSO-BP, GA-BP), our approach first optimizes the network’s structural parameters (number of hidden neurons and learning rate) and then refines the initial weights and biases. This sequential optimization ensures a more balanced trade-off between global exploration and local exploitation, leading to improved generalization. Furthermore, we extend the model to multi-point synchronous prediction on both end-face and radial surfaces of an aircraft lens frame, a scenario rarely addressed in previous work.

Among various heuristic algorithms, this paper adopts the Crown Porcupine Optimization Algorithm (CPO)¹⁰ to optimize the BP neural network, mainly due to the following strengths. CPO simulates the four escalating defense behaviors of the crown porcupine—visual intimidation, vocal intimidation, scent attack, and physical attack—thereby establishing a dynamic equilibrium optimization mechanism that shifts from broad global exploration to refined local exploitation. Unlike the particle swarm optimization (PSO) algorithm, which is susceptible to local optima, and genetic algorithms (GA), which often converge slowly^11–13, CPO incorporates a unique cyclic population reduction (CPR) strategy that periodically adjusts the population size. This approach effectively mitigates premature convergence and exhibits stronger robustness and higher convergence accuracy when optimizing parameters for complex nonlinear problems¹⁰. As a result, CPO is particularly suitable for addressing vibration prediction challenges characterized by intricate transmission paths and limited training data, such as those encountered in aircraft lens frame structures.

To address these limitations, this paper proposes a CPO-BP prediction model for aircraft lens frames. The model utilizes the Crested Porcupine Optimizer (CPO)¹⁰ to conduct a two-stage optimization of the Backpropagation neural network, enhancing its performance at both the macro-architectural and micro-parametric levels. This study applies the CPO algorithm to the vibration transmission prediction of aircraft lens frames, establishing a multi-point synchronous prediction model for both end faces and radial cylindrical surfaces. Systematic experiments and comparisons demonstrate the method’s effectiveness and generalization capability in addressing such complex spatial structures under confined sensor placement conditions^15,18.

CPO optimization algorithm

The Crested Porcupine Optimizer (CPO) is a novel meta-heuristic algorithm that simulates the four distinctive defense mechanisms crested porcupines employ against threats in nature. This bio-inspired approach demonstrates a robust balance between global exploration and local exploitation. Unlike traditional unconstrained reinforcement learning methods, the CPO algorithm explicitly incorporates practical constraints—including safety, energy consumption, and resource limitations—making it particularly suitable for complex real-world optimization tasks that require a balance between objective performance and risk control. The population initialization in CPO follows a procedure common to many population-based heuristics, commencing the search from an initial set of individuals as defined by the following equation:

1

Where :N is the number of individuals, is the candidate solution in the search space, L and U are the lower and upper limits of the search range, respectively, and r is a random number between 0 and 1.

Cyclic Population Reduction (CPR) is a technique within the CPO algorithm that simulates the adaptive behaviors of crested porcupine populations. It balances the exploration and exploitation capabilities of the algorithm by dynamically and cyclically adjusting the population size. Its core mechanisms include a population decay model based on the number of iterations, an alternating individual regeneration strategy, and synergy with other defense phases. This technique not only enhances the convergence speed of the algorithm but also effectively avoids falling into local optima through its cyclic nature. The mathematical model of the cyclic population reduction technique is as follows:

2

Where: N is the population size at the iteration, is the minimum population size, is the initial maximum population size, t is the current number of iterations, T is the number of cycles, is the maximum number of iterations, and % is the remainder symbol. The cyclic reduction forces periodic decreases and increases in population, mimicking natural population fluctuations and preventing stagnation.

There are four stages in the CPO algorithm’s merit seeking process, which are visual intimidation, sound intimidation, odor attack, and physical attack. The specific process is as follows:

1. Visual intimidation stage

This stage simulates the natural behavior of the crowned porcupine, which raises its quills and expands its body size to deter predators upon sensing danger. This behavior corresponds to the algorithm’s exploratory global search, aiming to expand the coverage of the solution space and prevent convergence to local optima. The position update mechanism calculates the new position by generating the predator’s position and integrating the positional relationship between the two with stochastic factors: when the predator is in close proximity, the algorithm explores the region between them to accelerate convergence; when the predator is at a greater distance, it conducts searches across a broader spatial range. This approach enables efficient exploration of diverse regions to identify potential optimal solutions. The mathematical modeling of this behavior is as follows:

3

Where: and are the positions of the individual at the current and next moments, is a random number based on a normal distribution, is a random value in the interval [0,1], is the optimal solution of the evaluation function t, and is the vector generated between the current crown porcupine and the crown porcupine randomly selected from the population. The term encourages exploration by directing movement toward or away from a randomly selected peer.

The mathematical formula for is as follows:

4

Where: r is a random integer between [1, N].

• 2.Voice intimidation stage

This stage simulates the behavior of the crowned porcupine, which emits noise to deter predators and escalates the sound intensity as predators draw near. Within the algorithm, this stage corresponds to the exploratory global search phase, with a focus on integrating high-quality information from the current population to guide the search direction. Its position update mechanism is realized through the dynamic relationship between the “sound intensity coefficient” and the “predator-porcupine distance”: when the predator, representing a potential local optimal region, approaches, the algorithm takes the current optimal solution as a reference and conducts a biased random search within its neighborhood to accelerate convergence toward the optimal solution region; when the predator moves away, the algorithm expands the search coverage via larger-scale random perturbations, balancing randomness and directionality to enhance global exploration efficiency and avoid trapping in local plateaus. This process is simulated by the following equation:

5

Where: r₁ and r₂ are random integers in the range [1, N], and is a random value in the range [0, 1]. The difference between two randomly selected individuals, , generates a random exploration direction. Combined with the step size factor , it introduces perturbations around the current individual. As the population gradually converges to the optimal solution region, this mechanism naturally serves to perform fine searching around the neighborhood of the optimal solution.

• 3.Odor attack stage

This stage simulates the behavior of the crowned porcupine, which secretes a foul odor to deter predators, and corresponds to the exploratory local search phase. Its objective is to conduct precise optimization on potential optimal solution regions. The algorithm dynamically adjusts the search range through an “odor diffusion factor”: when the current solution approaches a high-quality region, it implements high-precision perturbations within the neighborhood of the optimal solution with small step sizes to deepen local exploration; when solutions are dispersed, the search range is moderately expanded to avoid excessive confinement. This stage leverages the characteristic that “odor concentration varies with distance” to balance the precision and scope of local search, focusing on existing high-quality information while preventing entrapment in local optima. It collaborates with the exploration stage to enhance optimization accuracy. This process is simulated by the following equation:

6

Where: r₃ is a random integer in the interval [1, N], is a parameter controlling the search direction, is the position of the individual at iteration t, is the defense factor defined by equation, is a random value in the interval [0, 1], and is the odor diffusion factor defined by Eq. (9). The defense factor and diffusion factor adjust the step size based on the iteration number and fitness, enabling adaptive local search.

7

8

9

Where: is a vector of random numbers generated between [0,1], is a random variable generated between [0,1], is the objective function value of the individual at iteration t, is a small value added to avoid division by zero, N is the population size, t is the current iteration number, and is the maximum iteration number.

• 4.Physical attack phase

This stage simulates the close-range physical stabbing behavior of the crowned porcupine, corresponding to a high-intensity local exploitation phase that focuses on precise intensive searching within the neighborhood of the current optimal solution. Employing a non-elastic collision model, the optimal solution is taken as a reference point, and individual positions are adjusted by integrating dynamically decaying attack intensity with random perturbations: at extremely close distances, small-scale high-frequency searches are performed; at relatively greater distances, the step sizes are moderately expanded. This stage leverages strong interactivity to accelerate convergence, disrupt local equilibrium to avoid falling into optimal traps, and collaborates with the global exploration and mid-range exploitation stages to enhance optimization efficiency and accuracy. This process is simulated by the following equation:

10

Where: is the optimal solution obtained, is the convergence speed factor, and and are random values within the interval [0,1], which are used to control the convergence speed of CP and adjust the average force of CP on the predator, respectively. is the average force affecting the CP of the predator. is calculated by the non-elastic collision law:

11

12

13

14

Where: is a vector of random values in the interval [0,1], is the mass of the individual after t iterations, is the final velocity of the individual at generation t + 1, whose value is assigned by selecting a random solution from the current population, is the initial velocity of the individual at generation t, is the current iteration count, and is the objective function value corresponding to the current position. The mass and velocities simulate a physical collision, driving the solution toward the best-found region with momentum.

BP neural network

The BP neural network is one of the most fundamental and widely adopted models in artificial neural networks. As a multi-layer feedforward neural network trained via the backpropagation algorithm, it emulates the information processing mechanisms of biological neural networks through computer technology. With neurons as the fundamental units, it constructs a network architecture comprising an input layer, hidden layers, and an output layer. The inter-layer connections are defined by weights, and each neuron (except those in the input layer) incorporates a bias term.

The core operational mechanism of neural networks involves forward propagation for output computation. The error between the output results and target values is calculated via the backpropagation algorithm, which utilizes the Sigmoid activation function to compute gradients. This process continuously adjusts the weights and biases of each layer until the error falls within an acceptable range or the iteration limit is reached. This distinctive training method enables neural networks to exert significant efficacy in fields such as machine learning, pattern recognition, and image processing. The Sigmoid function has a domain encompassing all real numbers and is differentiable across its entire domain. Its mathematical expression is presented in Eq. (15), and its derivative is given in Eq. (16). It can be observed that as z approaches positive infinity, the function value converges to 1, resulting in upper saturation; as z approaches negative infinity, the function value converges to 0, leading to lower saturation. In both scenarios, the derivative of the function approaches 0, which may induce the vanishing gradient problem.

15

16

Where: z is the function input (weighted sum of neurons).

The input vector X is passed through the weight matrix and bias vector , and the hidden layer output is calculated by the activation function :

17

The hidden layer output h is transmitted through the weight matrix W and bias vector C, and the predicted value is calculated by the activation function :

18

The learning performance of neural networks can be characterized by the mean square error. The mathematical expression of the mean square error in neural networks is shown in Eq. (19).

19

where is the expected output value of the neural network.

The computational process of a BP neural network can be summarized as follows: (1) Conduct data training by inputting data into the network to enable the network to initially learn data features; (2) Calculate the error of neural network nodes to reflect the deviation between predicted values and actual values; (3) Calculate the mean squared error to quantify the overall error level; (4) Calculate and adjust parameters such as weights and sensitivities of the neural network system; (5) Adjust the weights of the hidden layer based on the sensitivities of the hidden layer. Repeat the above steps until the error is within an acceptable range or the maximum number of iterations is reached.

CPO-BP prediction model

In the prediction process of a BP neural network, parameter configuration is a critical factor influencing prediction accuracy. Among these parameters, the number of hidden layer nodes is closely associated with the network’s predictive performance and inherently correlated with the number of input and output nodes in the system.The number of hidden layer nodes does not display a simple linear positive correlation with the system’s generalization ability: when the number of nodes is excessive, although it may reduce the training error, it can lead to a significant increase in the testing error, markedly prolong the system’s learning duration, and even induce overfitting; conversely, when the number of nodes is insufficient, the system’s fault tolerance is diminished, causing the testing error to approximate the training error and resulting in computational results that are difficult to converge. Thus, both an excessive and an insufficient number of hidden layer nodes will adversely impact the system’s generalization ability.

The learning rate of a neural network is typically confined to the range (0, 1). Variations in its value modulate the network weights, thereby influencing the learning speed of the system. Increasing the learning rate can expedite the training process; however, an excessively high learning rate may induce weight oscillations and even result in training stagnation. Conversely, an excessively low learning rate will significantly prolong the training duration and substantially decelerate the convergence speed.

To effectively enhance the system’s generalization ability and convergence speed, this paper adopts the CPO algorithm to optimize the BP neural network through two sequential steps: the first step involves optimizing the number of hidden layer neurons and the learning rate, while the second step focuses on optimizing the initial values of weights and biases. The CPO algorithm’s parameters are chosen based on problem complexity. For the single-point prediction task (end-face), the search space is relatively small, so a population size of 8 and 20 iterations are sufficient to obtain a good solution. For the multi-point synchronous prediction task (radial surfaces), the network has three outputs, requiring optimization over a higher-dimensional parameter space; therefore, the population size is increased to 20 and iterations to 30 to ensure adequate exploration. The algorithmic workflow of the CPO-BP model is primarily divided into three components: determining the structure of the BP neural network, performing iterative optimization via the CPO algorithm, and conducting training and prediction using the BP neural network. The algorithmic workflow is illustrated in Fig. 1.

Fig. 1.

Flow chart of the CPO-BP algorithm.

Experimental study on aircraft lens frames

To simulate the high-speed maneuvering conditions of an aircraft, the lens frame structure was excited in a laboratory setting. Both training and testing data were generated by applying broadband pulse excitation using a force hammer (Model 8206-002) at multiple positions on the frame base, exciting the overall structural modes within the 0–1000 Hz range. Data were acquired while the structure was freely suspended to ensure that vibration responses were not influenced by external constraints. The experiment captured multiple sets of time-history data, including steady-state, transient, and residual vibration responses. The specific training dataset used in this study consisted of 750 time steps (corresponding to approximately 0.18 s at a sampling frequency of 4096 Hz), with 80% allocated for network training and the remaining 20% reserved for validation. Although 750 points may appear limited, the vibration signals contain rich frequency content and are sampled at a high rate, providing sufficient temporal patterns for the neural network to learn the underlying dynamics. Cross-validation during training further confirms the model’s stability.

This paper takes the lens frame of a specific aircraft model as the research object, whose structure and sensor layout are illustrated in Fig. 2. The experiment employs four DeltaTron4533-B accelerometers, characterized by a sampling range of 0.2 Hz to 12,800 Hz, an operating voltage of 13 ± 1 V, an operating temperature range of − 55 °C to 128 °C, and a mass of 8.6 g. These accelerometers are utilized to collect response data from both test points and prediction points, respectively. The excitation device adopted is an 8206-002 force hammer, with a sensitivity of 2.167 mV/N and a frequency range of 10 to 10,000 Hz. Vibration data acquisition is conducted using the LMS system to achieve precise on-site data collection, and the acquired experimental data are employed for model training and comparative analysis of prediction results.

Fig. 2.

Aircraft lens frame.

The experiment selected two sets of data for neural network training and prediction, respectively, with training data accounting for 80% of the total dataset and validation data comprising the remaining 20%. The first dataset uses the responses from sensors 1, 2, and 3 as inputs and the response from sensor 4 as the output for training the CPO-BP neural network. The second dataset employs the responses from sensors 1, 2, and 3 as inputs for model prediction validation, while the response from sensor 4 is used to compare results with the predicted response.

To visually illustrate the composition and characteristics of the input data, Fig. 3 displays segments of the acceleration signals collected by sensors 1, 2, and 3 during training. The signals across all three channels span a wide frequency band, encompassing low to medium‑high frequencies in the excitation response. Their amplitude distributions and temporal evolution are representative of the structural dynamics, supplying the neural network with comprehensive learning samples and ensuring the model’s capacity to capture complex vibration features.

Fig. 3.

Acceleration signals for training.

To objectively evaluate the linear relationships among the input data channels and to reveal the data relationships underlying the prediction task, the Pearson correlation coefficients were first computed between the sensor signals. This statistic measures the linear correlation between two variables, ranging from − 1 to 1, with a larger absolute value indicating a stronger linear relationship. The results of this analysis are presented in Table 1.

Table 1.

Pearson correlation coefficient matrix for acceleration signals at each measurement point.

Measurement point	Sensor1	Sensor 2	Sensor 3	Sensor 4
Sensor 1	1	− 0.0842	− 0.5774	− 0.6880
Sensor 2	− 0.0842	1	0.4391	0.1194
Sensor 3	− 0.5774	0.4391	1	0.7223
Sensor 4	− 0.6880	0.1194	0.7223	1

Analysis of the Pearson correlation coefficients, as presented in the table, reveals a complex and predominantly weak linear relationship among the sensor signals. Notably, correlations between some input sensors exhibit negative values (e.g., − 1.0842 between Sensor 1 and Sensor 2), while others show positive but modest strengths (e.g., 0.7223 between Sensor 3 and Sensor 4). Similarly, the correlations between input sensors and the target point signals remain limited in magnitude. This pattern of low and variably signed correlations underscores the highly complex nature of vibration transmission paths within the aircraft lens frame. During propagation, vibration energy not only attenuates but may also undergo phase shifts, modal coupling, and nonlinear interactions influenced by joint stiffness, leading to significant alterations in signal waveforms. Consequently, despite spatial proximity, the linear similarity in vibration responses between measurement points is markedly low. This observation strongly justifies the necessity of employing nonlinear models like neural networks, which are capable of learning complex dynamic mappings beyond simple linear relationships. Subsequent optimization results demonstrate that even under these conditions of weak and heterogeneous correlation, the neural network optimized via the two-step CPO process achieves high prediction accuracy, highlighting the robustness and superiority of the proposed method for addressing vibration transmission in complex structures.

Analysis of the Pearson correlation coefficients reveals a mixture of weak and strong linear relationships. For example, Sensor 3 and Sensor 4 exhibit a relatively strong positive correlation (0.7223), indicating that their vibration responses are linearly related to some extent. However, other pairs, such as Sensor 1 and Sensor 2, show very weak negative correlation (− 0.0842), suggesting almost no linear dependence. This heterogeneity in correlation strengths implies that while linear components exist, the overall vibration transmission path through the aircraft lens frame involves complex nonlinear interactions (e.g., phase shifts, modal coupling). Therefore, a purely linear model would be insufficient, and a nonlinear approach like a neural network is justified to capture the underlying dynamics.

The CPO optimization algorithm was configured with a population size of 10 and a maximum iteration count of 20. Figure 4 shows the variation of RMSE during the optimization process. It can be observed that the CPO algorithm rapidly reduces the error in the early iteration stages. Although the error reduction between iterations 1 and 20 is relatively limited, the algorithm stabilizes after approximately the 12th iteration, suggesting that it quickly approaches the optimal solution. This illustrates the efficient convergence of the CPO algorithm within a finite number of iterations. Even when error variation becomes minimal, the algorithm successfully accomplishes the optimization task, demonstrating strong numerical stability.

Fig. 4.

The iteration curve of the CPO algorithm.

Figure 5 presents the regression results of the CPO‑BP model. The model achieved R² values of 0.983 and 0.987 on the training and test sets, respectively, both exceeding 0.98, which indicates excellent fitting performance on both datasets. The slightly higher R² on the test set compared to the training set does not reflect a training deficiency; rather, it suggests that the model avoided overfitting during training and exhibits strong generalization ability on unseen data. All data points cluster closely around the identity line, further confirming the accuracy and consistency of the model’s predictions.

Fig. 5.

Regression results for the CPO-BP model.

The comparison of the prediction results of the CPO-BP algorithm and the traditional BP algorithm with the actual values is shown in Fig. 6. A visual analysis of the predicted results against the actual responses reveals that the prediction curve of the CPO-BP algorithm exhibits a better fit to the test response curve than that of the traditional BP algorithm.As shown in Table 2, a comparison of the prediction error metrics between the two algorithms demonstrates that the CPO-BP algorithm significantly reduces both the mean absolute error (MAE) and the mean square error (MSE). Specifically, the MAE is reduced by 8.31% and the MSE by 5.24% compared to the traditional BP algorithm. This quantitative variation reflects the optimization effect of the CPO-BP algorithm: the reduced MAE indicates a narrower average deviation between the model’s predicted values and the actual values, thereby demonstrating superior overall prediction stability. The decreased MSE signifies enhanced control over extreme errors, which effectively reduces the probability of large deviations occurring. The simultaneous improvement in both metrics further validates the advantage of the CPO-BP algorithm in enhancing prediction accuracy, providing quantitative support for the optimization of model performance.

Fig. 6.

Comparison of test set prediction results.

Table 2.

Comparison of model accuracy parameters.

Model	MAE	MSE
BP	0.022035	0.10671
CPO-BP	0.020204	0.10112

The model obtained from training was employed to predict the second dataset, with the prediction results presented in Fig. 7. As observed in the figure, the predicted responses output by the model exhibit a high degree of consistency with the actual responses, with good agreement in both the overall data trend and the details of local fluctuations. These prediction results further validate the generalization ability of the trained model and confirm that the model possesses reliable predictive performance in practical application scenarios.

Fig. 7.

CPO-BP model prediction results.

To analyze the vibration attitude of the aircraft lens, the measured acceleration signals need to be filtered and integrated to derive the corresponding displacement information. In this study, both the collected acceleration signals and predicted responses were subjected to filtering for the removal of DC components. Figure 8 presents the filtered acceleration curves of the actual and predicted responses, while Fig. 9 illustrates the velocity responses obtained via integration. As indicated in Figs. 8 and 9, the filtered acceleration responses and integrated velocity responses exhibit a high degree of consistency. Figure 10 displays the comparison curve between the displacement response of Sensor 4 and its predicted counterpart, revealing that the predicted response remains highly consistent with the actual response after filtering and integration. Thus, the method proposed in this paper can effectively predict the vibration response of the lens position; combined with the displacement responses of the remaining three sensors, it enables the observation and evaluation of the axial vibration attitude at the lens mounting position.

Fig. 8.

Filtered acceleration response.

Fig. 9.

Speed response comparison.

Fig. 10.

Displacement response comparison.

Since aircraft lenses may be subjected to multidirectional composite vibrations during operation, relying solely on end-face vibration signals is insufficient to fully characterize their dynamic behavior. To more accurately assess the operational status of the lens, this study expands the prediction beyond end-face vibrations to specifically include the radial cylindrical surface vibrations at the lens mounting position. These radial vibrations directly affect the lateral stability and imaging accuracy of the optical system, constituting a critical factor in vibration attitude analysis. Furthermore, to evaluate the model’s performance under complex conditions, a multi-point synchronous prediction model was developed for the radial cylindrical surface. As illustrated in Fig. 11, the responses from Sensors 1, 2, and 3 are used as inputs to synchronously predict the vibrations at Sensor locations 4, 5, and 6. This configuration validates the capability of the CPO-BP model to achieve high-precision, synchronous prediction from a limited set of measurement points to multiple critical locations through complex spatial vibration transmission paths.

Fig. 11.

Sensor layout on aircraft lens frame.

For the aforementioned multi-point prediction task, the training parameters of the prediction model were specifically configured: the population size of the CPO optimization algorithm was set to 20, and the maximum number of iterations was set to 30. This increase, compared to the single-point case, accounts for the expanded search space due to three output neurons and more complex coupling between radial vibration channels. The iteration curve of this optimization process is presented in Fig. 12. Its convergence characteristics clearly show that, in two rounds of optimization, the algorithm reached a stable state with respect to the root mean square error (RMSE) after the 23rd and 18th iterations, respectively. This indicates the efficient completion of the optimization process and the acquisition of a reliable set of network parameters.​The final prediction results are shown in Figs. 13, 14 and 15, where the predicted response curves of Sensors 4, 5, and 6 exhibit high consistency with the actual response curves. In terms of key indicators including the primary vibration trend, peak capture, and phase consistency, the predicted results demonstrate excellent agreement with the actual measurements. This confirms that the BP neural network optimized by the CPO algorithm possesses outstanding synchronous prediction accuracy and generalization ability for the radial multi-point vibration response of the aircraft lens frame.

Fig. 12.

The iteration curve of the CPO algorithm.

Fig. 13.

Comparison of prediction results for Sensor No. 4.

Fig. 14.

Comparison of prediction results for Sensor No. 5.

Fig. 15.

Comparison of prediction results for Sensor No. 6.

To further validate the superiority of the proposed CPO-BP algorithm, this study conducted comparative experiments to evaluate its performance against the COA-BP, DA-BP, and PSO-BP algorithms. All experiments were performed using the same dataset to ensure consistent comparison conditions and reliable results.​.

Performance was assessed using the coefficient of determination (R²), mean squared error (MSE), and mean absolute error (MAE). The results in Table 3 indicate that the CPO-BP model achieved higher R² values and lower MAE and MSE compared to other algorithms. This enhancement stems directly from the intrinsic properties of the CPO optimizer. While conventional algorithms like PSO often converge to suboptimal solutions, CPO strikes a superior balance between exploration and exploitation through its bio-inspired defensive mechanisms. Specifically, the ‘visual intimidation’ behavior expands the global search scope, whereas the ‘physical attack’ behavior intensifies the local search around current best candidates. This coordinated strategy effectively mitigates premature convergence, leading to the identification of superior parameter configurations for the network. Consequently, the observed performance fully validates the predictive efficacy of the CPO-BP algorithm.

Table 3.

Comparison of evaluation indicators for relevant models.

Model	R2	MAE	MSE
CPO-BP	0.97976	0.02020	0.10112
COA-BP	0.97433	0.02392	0.11781
DA-BP	0.9727	0.02427	0.10890
PSO-BP	0.97387	0.02288	0.10939

The comparative results in Table 3 show that the CPO‑BP model achieves higher prediction accuracy than the other benchmark models. Notably, all meta‑heuristic‑optimized BP models (CPO‑BP, PSO‑BP, DA‑BP, COA‑BP) perform significantly better than the unoptimized traditional BP model (see Table 2). This first confirms the general effectiveness of using meta‑heuristic algorithms to optimize BP neural networks. The superior performance of CPO‑BP among these models reflects the advantages of the CPO algorithm in balancing exploration and exploitation, enabling it to identify better network structures and initial parameters for this specific prediction task. The main engineering significance of this study lies in its systematic experimental validation that the CPO‑BP combination offers a more reliable and accurate solution for this type of vibration prediction problem.

Conclusion

This study has addressed the challenge of vibration measurement at aircraft lens mounting positions by introducing a prediction model based on a CPO-optimized BP neural network. The principal conclusions are as follows:

1. The proposed two‑step CPO optimization strategy effectively improves the performance of the BP neural network. Compared with the conventional BP network, the CPO‑BP model reduces the Mean Absolute Error (MAE) by 8.31% (from 0.022035 to 0.020204) and the Mean Squared Error (MSE) by 5.24% (from 0.10671 to 0.10112).

2. The CPO-BP model exhibits comprehensive superiority over other heuristic-optimized models. As shown in Table 3, it achieves the highest R² value while maintaining the lowest MAE and MSE among all compared models (including PSO-BP, DA-BP, and COA-BP). This leading performance across all key metrics is primarily attributed to the inherent characteristics of the CPO algorithm. Unlike PSO and GA algorithms, which are prone to local optima or slow convergence, CPO establishes a more effective dynamic balance between global exploration and local exploitation by simulating the crested porcupine’s four-phase defense behaviors (visual intimidation, vocal intimidation, odor attack, and physical attack). Furthermore, its Cyclic Population Reduction (CPR) strategy helps suppress premature convergence. Consequently, CPO can identify more robust network architecture parameters (including the number of hidden neurons and learning rate) as well as more suitable initial weights and biases for the BP neural network, thereby significantly enhancing the final network’s prediction accuracy, stability, and generalization capability.

3. The developed model successfully accomplishes synchronous vibration prediction at multiple points on both the end-face and radial surfaces. This demonstrates its efficacy in handling complex spatial vibration transmission problems in structures, offering a novel and efficient technical pathway for engineering condition monitoring where direct measurement of critical states is infeasible.

Future work will focus on testing the model’s performance under broader operational conditions and integrating it into real-time health monitoring systems.

Author contributions

X.Z. and T.G. wrote the main manuscript text. H.C. prepared Figs. 1, 2, 3, 4, 5, 6, 7, 8 and 9, and S.G. prepared Figures10–17. All authors reviewed the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51875009 and 52475502) and the Key Project of the 2025 Educational Science Planning Project of Xinjiang Uygur Autonomous Region (Grant No. HEK2025013).

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.