Model updating of gun recoil mechanism based on time domain response

For gun launching dynamics modeling, the accuracy of the recoil and counterrecoil model is important. In order to improve the accuracy of gun recoil mechanism modeling, the rigid-flexible coupling firing dynamics model of a small-caliber gun was established, and the recoil displacement of the gun was tested by live firing test. Aiming at the error between the simulation curve and the test curve of the recoil displacement changing with time, taking the correlation coefficient of the two curves as the objective function, the dynamics model of the recoil mechanism was modified based on the time-domain response using the optimization algorithm. The modified results show that the difference of the maximum recoil displacement decreases from 9% to 0.1%, and the similarity between the simulation curve and the test curve reaches 0.99. The modified results based on the time-domain response effectively improve the accuracy of the recoil model, so that the established dynamics model can more truly reflect the dynamic characteristics of the recoil mechanisms, and provide the basic conditions for ensuring the simulation accuracy of the launch dynamics model.

The 2020 Spring International Conference on Defence Technology Journal of Physics: Conference Series 1507 (2020) 082004 IOP Publishing doi: 10.1088/1742-6596/1507/8/082004 2 methods, the time-domain signals carry more information and can reflect the dynamic response of the test structure more truly and effectively. Moreover, the time-domain response signal can avoid frequency truncation, and can cause little error due to re-analysis of the data, which makes the model updating accuracy highly [4]. The comparison method of time-domain signals has been studied, and the methods of residual errors between time-domain signals have been analyzed, which provided a theoretical basis for model modification based on time-domain responses [5]. The residuals of feature quantities were defined by using the experiment and finite element simulation signals to obtain the mathematical expressions, and multiple optimization algorithms were used to obtain the optimal solution [6]. For the updating of the rigid-flexible coupled launch dynamics model of the infantry fighting vehicles, the similarity between the simulated and measured data was used as the objective function, and the support vector machine response surface method was used [7]. Up to now, the research on the parameters of the gun recoil mechanism is mainly focused on the optimization design of recoil mechanism, while the research on the model updating of recoil mechanism parameters is little [8][9][10].
In this paper, the rigid-flexible coupling firing dynamics model of a small-caliber gun has been established. The parameters of the recoil mechanism were studied based on the time-domain response correction method and the genetic algorithm. It has been shown that the time-domain response correction method effectively improved the accuracy of the recoil mechanism model and provided the basis for the muzzle vibration simulation.

Basic assumptions
The basic assumptions for establishing the rigid-flexible coupling firing dynamics model of the smallcaliber gun are as follows.  The inertia of the spring is ignored.  The gun barrel is assumed to be an elastic body, and other components are rigid bodies.  The gun recoil mechanism connects the recoil part and the cradle. The recoil part moves back and forth relative to the cradle.

Barrel Finite Element Model
Figure1 shows the the finite element meshes and constraints of the barrel. Figure 2 shows the calculated first six order bending mode shapes of the flexible barrel. By calculation, the frequencies of the first six bending modes are: 71Hz, 71Hz, 437Hz, 437Hz, 1136Hz, and 1136Hz. Due to the axisymmetric characteristic of the barrel, each frequency in the first six modes of the barrel corresponds to two mode shapes, and the bending directions of the two mode shapes are perpendicular to each other.

Rigid-flexible coupling launch dynamics model
A rigid-flexible coupling launch dynamics model including turret, cradle, gun box, barrel and recoil mechanism was established [11]. The topological graph of the connection relationship of the components of the gun is shown in figure 2, where H1 is a fixed pair, H2 is a rotating pair, H3 is a contact pair, H4 is a contact pair, and H5 is a contact pair. The rigid-flexible coupling model has 1 flexible body and 9 rigid bodies as shown in figure 3.

Differential equation of recoil motion
The differential equation of the gun recoil motion is: where, is the mass of the recoil part of the gun, is the recoil displacement, 0 is the preload of the bumper spring, is the stiffness of the bumper spring, is the bottom pressure, is the cross section area, is the friction of the guide rail between the recoil part and the cradle, is the hydraulic damping coefficient, is the shooting angle, is the forward force generated by the powder gas ejecting through the muzzle brake. After solving the equation (1), the recoil resistance can be calculated as follows, = ( + 0 ) + +̇2 (2) The recoil displacement result of one shot with a 0-degree shooting angle calculated according to the design values is shown in figure 5.

Recoil Displacement Measurement
By pasting the markers on the gun barrel and the cradle respectively, and by using the high-speed camera measurement system, through optical calibration and image processing, the relative positions of the two markers in each frame were extracted. The layout of the high-speed camera and the markers is shown in figure 6.
The high-speed camera measurement system captures the movement of the marker and performs digital image processing to obtain the marker displacement. The measurement system mainly includes high-resolution high-speed cameras and installation devices such as brackets, heads, and tripods. In order to meet the requirements for high-precision measurement of marker displacement, the camera should have sufficiently high resolution and frame rate. The high-speed camera used in this test has a resolution of 1280 × 800 pixels. The shooting frame rate at full resolution can reach 16000fps. Then, the grayscale processing was performed on the captured image, and then other related processes such as smoothing, sharpening, enhancing, filtering, and denoising are performed, and the grayscale images were converted to binarized images in oredr to reduce the amount of the data processing. Finally, the measured curve of the recoil displacement changing with time is shown in figure 7.  By comparing with the test curve, as shown in figure 8, there was about 9% difference in the maximum value of the recoil displacement for the simulation curve. It is to say, there is a large error between the measurement result and the simulation result of the rigid-flexible coupling launch dynamics model by using the design value of the recoil mechanism parameters.

Objective function
The correlation coefficient was utilized to characterize the similarity between the test curve and the simulation curve of the recoil displacement changing with time. The correlation coefficient between the two time-domain signals can be expressed as: where, and represent test values and simulation values, , represents the test value and simulation value corresponding to the ith time point. The correlation coefficient value ranges from -1 to +1, and +1 means that the two time-domain signals are exactly the same, and −1 means that the two

Design variables
According to the design principle of the recoil mechanism and the nonlinear differential motion equation of the recoil mechanism, it can be concluded that the parameters of the dynamics model of the recoil mechanism are: recoil hydraulic resistance coefficient 1 , the counter recoil hydraulic resistance coefficient 2 , spring stiffness , and friction coefficient μ of the guide rail between the recoil part and the cradle。

Optimization Algorithm
Because the model updating can actually be changed to optimization of the objective function, the choice of the optimization algorithm is very important. The rigid-flexible coupled dynamics model of this example has nonlinear factors such as friction and contact, which is highly nonlinear. It makes traditional optimization search methods such as gradient method and Newton iteration method difficult to get the global optimal solution. So the genetic algorithm which has great advantages for the global optimization was used as the optimization algorithm for the model updating.
Genetic algorithm is a global probability search algorithm based on biological mechanisms such as natural selection and genetic mutation [12]. For the genetic algorithm, the chromosomes represented by one-dimensional string structure data correspond to data or arrays. Each position of the string corresponds to the gene, and the value at each position corresponds to the value of the gene. The string composed of genes is the chromosome. A certain number of individuals make up a group, and the degree of adaptation of each individual to the environment is called fitness.
The relevant parameters and selection of the genetic algorithm in the application process are as follows:  Population size. The population size indicates the number of individuals in each generation of the population. If the size of the population is large, the efficiency of the calculation is low. Oppositely, if the size of the population is too small, the efficiency of the calculation will increase. However, the diversity of the population will also be greatly reduced, which can increase the possibility of premature genetic calculation. In this calculation model, the population size is selected to be 20.  Selection probability. The selection probability determines the number of copies of the individual parents to the offspring. If the selection probability value is too large, it will easily lead to duplication of groups, which is not conducive to the evolution of individuals. In some genetic calculations, the selection probability is zero, that is, the progeny population is composed of individuals produced by crossover or mutation. In order to save the optimal individual, the selection probability selected during the genetic calculation is 0.1.  Crossover probability. It is the proportion of chromosomes participating in the crossover operation to the total number of chromosomes. Crossover is the main method of generating new individuals in genetic algorithms, so the value is generally large. However, if the value is too large, it is easy to destroy the excellent genes contained in the existing individuals. If the value is too small, the generation rate of new individuals will be very slow. The value range is generally selected between 0.4 and 0.99.  Probability of mutation. Probability of mutation refers to the proportion of the number of mutated genes in the total number of chromosomal genes. Too large value can easily lead to the loss of outstanding individual genes, and makes genetic algorithms deviate from directed search similarly to random search. Generally, The value ranges from 0.0001 to 0.1.  Maximum iteration algebra. The maximum iteration algebra determines the end of genetic calculation. Its selection is closely related to the content of the actual genetic calculations. If some evolutions are slow, the genetic algebra must be larger, or the genetic calculations have not yet reached stability and will be terminated, which is not in line with the original intention of genetic calculations. For the fast evolution process, the genetic algebra should be small. Otherwise it is a waste of computing time, and genetic computing may not continue to evolve. The basic genetic algorithm generates a new generation of population by copying, transforming and mutating the biological genes of a certain generation of population, and then repeats this process until the performance of the population or the most advantageous reaches a satisfactory level. The genetic algorithm process is shown in figure 9.

Optimization results
The genetic algorithm was used to update the parameters of the recoil mechanism model based on the test time-domain curve of the recoil displacement. The simulated recoil displacement curve and the measurement curve are shown in figure 10, the correlation coefficient reaches 0.99, and the maximum value of the recoil displacement differs by 0.1%.  Figure 10. The comparison of the test curve and the updated simulation curve of the recoil displacement.

Conclusion
Aiming at the parameters of a recoil mechanism of a small caliber gun, combing with the measurement and genetic algorithm, the parameters of the recoil mechanism for the rigid-flexible coupling launch dynamics model were corrected based on the time-domain response method. The difference of the maximum value of the recoil displacement between the simulation and measurement was reduced from 9% to 0.1%, and the similarity of the recoil displacement curves of the simulation and the measurement reached 0.99. The updated result based on the time-domain response effectively increased the accuracy of the dynamics model of the recoil mechanism, and made the dynamics model can more accurately reflect the dynamic characteristics of the recoil mechanism. Thus the updating result is useful for ensuring the simulation accuracy of the launch dynamics model.