Dynamic characteristics study of composite base plate: numerical simulation, processing technology research and modal analysis

Abstract

As the “king of artillery”, mortars will continue to play a crucial role in future warfare, which makes the enhancement of their mobility a significant concern. The base plate is a critical component of mortars, and using composite materials is beneficial for achieving a lightweight design. Modal analysis is an effective way to study the dynamic characteristics of structures, and applying it to composite base plates aids in their structural design. This paper investigates an efficient modeling method for composite base plates, designs the composite layup, and establishes a corresponding model. The subspace method serves as the solution algorithm for the eigenvalue problem, which enables the computation of the modes of the composite base plate. To ensure the structural reliability of the prototype, an innovative structural connection scheme and specialized treatment processes are proposed. A modal experiment using the hammering method was conducted to obtain the experimental modes. Comparisons with the computational modes reveal a good consistency. Finally, dynamic response analysis of the composite base plate was performed. This research provides significant guidance for the structural design and optimization of composite base plates and promotes the practical application of composite materials in artillery.

In future wars, fighters are fleeting. Artillery is the main firepower of the army, and the most lethal weapon is probably mortar. Improving the mobility of mortars in all areas is an important problem in mortar design. The traditional metal base plate accounts for about 40% of the total weight and is a key component that affects the mobility of mortars. The application of composite materials can effectively reduce the weight of the base plate. So the dynamic response of the composite base plate is the basis of this research. Modal analysis can be used to determine the vibration characteristics of structural mechanical components and is the basis for dynamic response calculations and other dynamic analysis¹. The degree of agreement between the finite element calculated mode and the experimental mode can reflect the accuracy of dynamic modeling. Therefore, it has great significance to carry out modal analysis and experimental research on composite base plate.

The design of composite material structures is often related to optimization. The research objects in the optimization of composite material structures can be classified into beams, plates, shells, and other structures². The most commonly used design variables are the stacking sequence of laminates, which includes the thickness, angle, and order of each individual layer in the laminate. Optimization objectives or constraints may include fundamental frequency, buckling load, structural mass, strength, and deformation^3,4. In the laminate optimization design, the structure, under different load conditions, not only needs to satisfy constraints such as strength, stiffness, and stability but also has to meet some practical lamination process constraints in engineering. Additionally, considering production costs, single layers usually use the standard lamination angles of ± 45°, 0°, and 90°. For complex laminate structural optimization designs, problems such as an excessive number of variables, large computational loads, and difficulty in convergence often occur. These problems significantly increase the complexity of laminate structure optimization design⁵. Currently, the commonly used optimization methods are mainly divided into two categories⁶: one is the indirect method, which requires not only the calculation of the objective function value but also the gradient information of the function when seeking the optimal solution. The other is the direct method, which does not require any gradient information and only needs to calculate the objective function value to determine the search direction based on certain evaluation criteria. Among these, the genetic algorithm in the direct method is the most frequently used for the optimization design of composite material laminate stacking sequences. Chen, et al.⁷ employed a genetic algorithm to minimize the stress in the laminate by optimizing the fiber angles, achieving the optimal laminate angles and minimum stress at the center of the composite laminate. Chen, et al.⁸ used shape basis vectors to define shape variables, establishing a comprehensive optimization model that included laminate stacking sequence variables and shape variables, thus forming an optimization problem involving continuous and discrete variables. Geng, et al.⁹ based on a genetic optimization algorithm, established a stacking sequence optimization method aiming to enhance the impact damage resistance of composite material cylinders, effectively improving their ability to withstand impact damage. An , et al.¹⁰ proposed a laminate stacking sequence optimization method based on a two—stage approximation algorithm and a genetic algorithm. This method is suitable for structural optimization problems with mixed variables and can directly take the initial stacking angles, sequence, and thickness as design variables. It has a relatively low computational load and requires a small number of structural analyses, demonstrating high computational efficiency in engineering structural design applications¹¹.

For the modal analysis of the composites, Qiu, et al.¹² used MSC Nastran for simulation and the hammering method to conduct the modal analysis of the composite gradient airfoil. Bai, et al.¹³ studied the modal properties of the sandwich structure composed of a carbon fiber braided skin and honeycomb core through experiments and simulations. Fan, et al.¹⁴ proposed an original hybrid numerical experimental identification method and used it to predict the flexural modulus of 3D woven composites. Modal tests were used to obtain the key vibration parameters (natural frequency, damping factor, mode shape) of the composites. Rahul Samyal¹⁵ proposed that the mode shapes were related to the fiber orientation by observing and comparing the different mode shapes of the composite panels under the clamping boundary condition. Eva Kormanikova¹⁶ studied the influence of sandwich panel design parameters such as core thickness and fiber reinforcement angle on the vibration response by considering the shear deformation theory. Wang, et al.¹⁷ considered the influence of thermal stress and material property variations, and they performed a thermal modal analysis of a hypersonic composite wing. Puja Basu Chaudhuri¹⁸ studied the mode frequency analysis of an antisymmetric angle—ply laminated composite stiffened hypar shell with a cutout. It can be seen that for the modal analysis of composite structures, numerical calculation methods and experimental methods are still the main approaches.

To ensure the performance of composite structures under impact loads, it is often necessary to integrate them with metal structures. For the study of composite base plates, the difficulty in numerical calculation lies in the simultaneous presence of composite laminates and alloy frames within the structure. The material nonlinear behavior and anisotropy of composites make the establishment of models and numerical calculations more difficult and complex. Accurately describing the anisotropy of composites and conducting corresponding numerical simulations is a challenge^19,20. The interfacial behavior of composites significantly influences the overall material performance, and the complexity of the computational model may lead to a high demand for computational resources.

On the other hand, the experimental method relies on the fabrication of physical prototypes. The reliable connection performance between composite materials and metals under strong impact loads is essential for ensuring the performance of composite base plates. Therefore, the connection process for composite base plates, which differs from that of traditional base plates, presents unavoidable challenges in the fabrication of physical prototypes. Common connection methods for composite and metal structures include bolting or riveting, adhesive bonding, hybrid connections with screws and adhesives, and welding²¹. Adhesive bonding is a good form of non—destructive connection. Baker, et al.²² analyzed the efficiency and damage of adhesive connections between composites and metals. Although adhesive bonding provides certain advantages for reducing structural weight, it typically requires a large adhesive area to achieve safe and reliable load—bearing capacity, and moreover, the possibility of spontaneous damage to the adhesive layer must also be considered. Srivastava²³ summarized various connection processes for metals, ceramic metals, and composites, as well as factors influencing connections and methods for improving brazed connections²⁴. In contrast, mechanical connections, known for their safety, reliability, simplicity, and reduced sensitivity to environmental factors, are widely used in connection structures. Tan Tiantian, et al.²⁵ explored common connection methods in the assembly of carbon fiber composite structural components, comparing the connection strength of different riveting methods through shear and pull off tests to identify riveting assembly methods that meet product quality standards and are suitable for production site applications, providing guidance for the use of carbon fiber composites in new generation launch vehicles for heavy payloads. Liu²⁶ established several common joint models using finite element software for comparative analysis and compared them with experimental results. Based on existing mesoscopic mechanics analyses, finite element simulations of the novel composite metal comeld connections were conducted, with results indicating that the failure mechanisms and modes of the joints were consistent with the experimental results. Compared to traditional connection methods, comeld connections offer higher load bearing efficiency. Lim, et al.²⁷ experimentally demonstrated the load sharing between adhesive and bolt connections in hybrid joints. From the above literature, it is evident that hybrid connections can integrate several connection methods to leverage their respective advantages in the connection of composite and metal structures. The connection processes between composites and metals present significant challenges, and currently, there is no specific research on the connection processes for composite base plates. Due to the structural characteristics and different load bearing requirements of various regions within composite base plate structures, targeted research should be conducted while minimizing the complexity of the processes as much as possible.

To explore the potential applications of composite materials in mortar base plates, the focus is on investigating the mechanical structural properties of composite base plates. This paper first conducts a study on the modeling of the composite base plate, which focus is on the design and optimization of composite material lay—ups and a connection scheme for composite materials and metal frames is designed. Subsequently, the feasibility of the connection scheme is preliminarily verified through theoretical calculations, numerical simulations, and experimental methods. Based on this, a prototype of the composite base plate is manufactured, and the quality of the model design is further examined by comparing the calculated modal analysis results with the experimental modal analysis results. Finally, through dynamic simulations, the strength, fatigue characteristics and stability of the composite base plate are discussed, providing a reference for the lightweight design of mortars. The research outline is shown in Fig. 1.

Fig. 1

Research idea.

Dynamic modeling of composite base plate

Design of composite material layering

The mortar is mainly composed of a barrel, a carriage, and a base plate, as shown in Fig. 2. In past modal analyses of structures, the structural materials were mostly metals. Metal materials generally have a crystal structure, and their molecular structures are repeated arrangements of these crystal structures. However, the structure of a composite material is generally a physical structure, and its different components are connected to each other in various ways. Due to the difference in the main properties of the two materials, as shown in Fig. 3, it is both necessary and interesting to study the modal structure of composite materials.

Fig. 2

Mortar composite base plate for all-territory application.

Fig.3

Depicts the process flow for optimizing the layup sequence in composite materials.

In order to realize the efficient modeling of the composite base plate, it is necessary to consider the physical model, the selection of the element shape, the selection of the element order, and the number and density distribution of the grids. In this paper, the mixed shell element with quadrilateral mesh elements is mainly used. In order to handle the smooth transition between different cell types, the common node method is mainly used to connect each cell part. The linear element is the main element, and the higher—order element is used in the key part²⁸. At the same time, it is necessary to further determine a reasonable number based on the actual research objectives and objective calculation conditions. After research, the total number of model units is determined to be 22,658, and the total number of nodes is 24,116.

The choice of composite material is the key to the design of the composite base plate. The selection of different materials will not only directly affect the performance of the base plate but also influence the molding process and cost of the base plate. For the mortar plate, the structure is required to have high strength, and its structural size should not change with temperature. Fiber—resin matrix composites have excellent mechanical properties, a small specific gravity, a high specific modulus and strength, are easy to process, have rich engineering experience, and feature large stiffness and small weight. Therefore, carbon—fiber resin matrix composites are used for multilayer design to improve the impact resistance of the material. In the process of composite molding, the base material holds the fibers in a fixed position to withstand and transfer the load. Finally, carbon fiber T700 was selected, and epoxy resin 5228 was used as the base material. The material parameters of carbon fiber T700 are listed in Table 1²⁹.

Table 1 Performance parameters of carbon fiber T700.

In addition to the fiber continuity criteria, based on the general principles to be followed for composite material layering³⁰ and the bearing characteristics of the base plate, the layering method of [+ 45°/-45°/0°/90°/0°/-45°/ + 45°] was initially selected. The layering thickness was 0.125 mm, and a total of 56 layers were laid. Among these layers, the 0° layer is mainly subjected to axial load. The ± 45° layers mainly guarantee the shear modulus of the material model to improve the stability and impact resistance of the structure and reduce stress concentration. The 90° layer is mainly subjected to lateral loads to control the Poisson effect. The genetic algorithm was applied, and the population size was selected to be 80. After 100 iterations, the results converged, and the optimal layup sequence was obtained as [+ 45°/-45°/90°/0°/0°/ + 45°/-45°].

Among the failure criteria commonly used in engineering, such as the maximum stress (strain) criterion, the Tsai—Hill criterion, and the Hoffman criterion, there are different degrees of inconsistency between the strength theory and the experimental results. Tsai and Wu proposed a new strength theory (Tsai—Wu tensor theory) in the form of tensors. The Tsai—Wu failure criterion³¹ is the most comprehensive criterion for describing the failure of composite materials among existing mature criteria, and other criteria can be simplified according to specific loading and stress conditions. In this paper, this failure criterion is used as the failure criterion for each layer in the composite material lay—up. The Tsai—Wu criterion is used to judge the failure of composite laminates, and its criterion expression is as follows:

$$F_{11} \sigma_{1}^{2} + F_{22} \sigma_{2}^{2} + 2F_{12} \sigma_{1} \sigma_{2} + F_{66} \sigma_{6}^{2} + F_{1} \sigma_{1} + F_{2} \sigma_{2} = 1$$

(1)

where,$F_{11} = \frac{1}{{X_{t} X_{c} }}$,$F_{22} = \frac{1}{{Y_{t} Y_{c} }}$,$F_{66} = \frac{1}{{S^{2} }}$,$F_{1} = \frac{1}{{X_{t} }} - \frac{1}{{X_{c} }}$,$F_{2} = \frac{1}{{Y_{t} }} - \frac{1}{{Y_{c} }}$.

By using the bidirectional tensile test of $\sigma_{1} = \sigma_{2} = \sigma_{m}$,$\sigma_{6} = 0$, $F_{12}$ can be obtained:

$$F_{12} = \frac{1}{{2\sigma_{m}^{2} }}\left[ {1 - \left( {\frac{1}{{X_{t} }} - \frac{1}{{X_{c} }} + \frac{1}{{Y_{t} }} - \frac{1}{{Y_{t} }}} \right)\sigma_{m} - \left( {\frac{1}{{X_{t} X_{c} }} + \frac{1}{{Y_{t} Y_{c} }}} \right)\sigma_{m}^{2} } \right]$$

(2)

where,$\sigma_{m}$ stands for bidirectional tensile failure stress. When $F_{12} = - \frac{1}{2}\sqrt {F_{11} F_{22} }$, the theoretical and experimental values can be in good agreement.

Optimization of ply sequence

The stacking sequence of different angles is a discrete variable, and it is relatively difficult to solve it using traditional mathematical programming methods. However, the integer—coding strategy of genetic algorithms has unique advantages in solving optimization problems with discrete variables. Therefore, an integer—coding genetic algorithm is chosen as the optimization algorithm to search for the optimal results.

Using the integer—coding strategy of genetic algorithms, the integers 1, 2, 3, and 4 represent the 0°, + 45°, -45°, and 90° angles, respectively. For the initially designed composite base plate, denoted as [2/3/1/4/1/3/2]_S, which stands for [+ 45°/-45°/0°/90°/0°/-45°/ + 45°]_S, where S indicates the number of layers in the laminate.

In addition to adhering to the general composite material layering principles, the optimization must also meet the performance criteria of the Base Plate structure, primarily its strength indicators. This is because the objective of optimizing the composite material stacking sequence is to achieve the highest possible strength without causing material failure. Therefore, the maximum stress experienced by the composite base plate should not exceed the maximum stress value of the titanium alloy Base Plate, and the stress should be as minimal as possible.

To better compare the changes in composite material strength, the concept of strength ratio is introduced³². The strength ratio is defined as the ratio of the ultimate stress of the material in a specific direction to its actual stress, and it is expressed as:

$$R = \frac{{\sigma_{\max ,i} }}{{\sigma_{i} }}$$

(3)

where: $\sigma_{\max ,i} \left( {i = 1,2,6} \right)$ represents the strength vector;$\sigma_{i}$ represents the applied stress vector. Substituting into Eq. (1), we obtain:

$$\left( {F_{11} \sigma_{1}^{2} + F_{22} \sigma_{2}^{2} + 2F_{12} \sigma_{1} \sigma_{2} + F_{66} \sigma_{6}^{2} } \right)R^{2} + \left( {F_{1} \sigma_{1} + F_{2} \sigma_{2} } \right)R = 1$$

(4)

Equation (4) is a binary linear equation regarding vector $R$.

Let’s denote: $F_{11} \sigma_{1}^{2} + F_{22} \sigma_{2}^{2} + 2F_{12} \sigma_{1} \sigma_{2} + F_{66} \sigma_{6}^{2} = a$, $F_{1} \sigma_{1} + F_{2} \sigma_{2} { = }b$, solve the equation to get $R$ as:

$$R = - \left( \frac{b}{2a} \right) \pm \sqrt {\left( \frac{b}{2a} \right)^{2} + \frac{1}{a}}$$

(5)

In the above equation, the square root term generally takes the positive sign. However, when all applied stress components are reversed, the negative square root conjugate root is taken, and its absolute value is considered as the strength ratio. From the definition of the strength ratio, when this value is greater than or equal to 1, the laminate does not fail. For the laminate to fail, the stress must increase by a corresponding factor. Therefore, the optimization objective for the composite material stacking sequence is to find the maximum value of the minimum ply strength ratio of the laminate. This objective is defined as the fitness function for the genetic algorithm, which is expressed as:

$$F = \min \left( {R\left( i \right) - 1} \right)\begin{array}{*{20}c} {} & {\left( {i = 1,2, \cdots ,n} \right)} \\ \end{array}$$

(6)

In the equation, $i$ represents the total number of layers in the laminate, and $R\left( i \right)$ represents the strength ratio of each individual layer. As a result, the optimization process of the genetic algorithm to maximize laminate strength is essentially the process of finding the maximum value of the minimum ply strength ratio within the population.

The area for laying composite materials is divided into three main components: the main plate, the conical basin, and the package rib, with each component having a specified thickness of 7 mm. Typically, the thickness of a polymer fiber composite layer is approximately 0.125 mm. Choosing each unidirectional layer to be of equal thickness, the total number of layers is set to 56, represented as [+ 45°/-45°/0°/90°/0°/-45°/ + 45°]8, which corresponds to [2/3/1/4/1/3/2]8. The genetic algorithm is applied with a population size of 80, 100 generations, a crossover probability of 0.85, and a mutation rate of 0.05.

For assessing the fitting accuracy of the regression model, the coefficient of determination $R^{2}$ test can be employed, expressed as:

$$R^{2} { = }1{ - }\frac{{\sum\limits_{i = 1}^{n} {\left( {y_{i} - \hat{y}_{i} } \right)^{2} } }}{{\sum\limits_{i = 1}^{n} {\left( {y_{i} - \overline{y}} \right)^{2} } }}$$

(7)

where, $y_{i}$ represents the true value of the dynamic model for the base plate,$\hat{y}_{i}$ represents the calculated value of the approximate model,$\overline{y}$ is the mean of the true values,$n$ is the number of test samples.$R^{2} \in \left[ {0,1} \right]$, the closer $R^{2}$ is to 1, the higher the fitting accuracy of the polynomial regression model.

Equation (7) was used to verify the correctness of the surrogate model, achieving a coefficient of determination $R^{2}$ greater than 0.9, indicating that the surrogate model has good accuracy. The optimization process for the composite material stacking sequence is illustrated in Fig. 3.

The optimization involves six main steps executed in Matlab, employing genetic algorithm principles and surrogate modeling for efficiency.

Step 1: Generate an initial population of chromosomes based on specified initial parameters. These chromosomes represent potential solutions for the layup sequence.

Step 2: Decompose and transform the chromosomes from the initial population to create new offspring chromosomes. This likely involves genetic operators such as crossover and mutation.

Step 3: Decode the selected chromosomes into specific ply angle values, translating the genetic representations into actual angular orientations for the composite layers.

Step 4: Utilize a surrogate model to compute the objective function values for the decoded layup sequences. The surrogate model efficiently estimates the performance metrics of the composite structures.

Step 5: Return to Matlab to construct a fitness function based on the objective function values and calculate the fitness scores for each chromosome.

Step 6: Evaluate the fitness values to determine if the optimization should stop. If the criteria are met, output the results; otherwise, proceed with selection, crossover, and mutation operations to generate a new population and iterate the process until the objective function values converge.

After 100 iterations, the optimization converges to an optimal layup sequence of [+ 45°/—45°/90°/0°/0°/ + 45°/—45°]₈ repeated eight times. A comparison between the original layup scheme and the optimized one is provided in Table 2.

Table 2 Compares the strength ratios of the layup schemes before and after optimization.

After optimizing the layup sequence, there is a slight increase in the strength ratio, approximately 6.5%. This indicates that the sequence of the layup does influence the strength of composite laminates. Although the impact of changing the layup arrangement may be limited, it can still contribute to enhancing the strength of composite parts to a certain extent.

Design of structural connections

The main forms of connection between composite materials and metal materials include mechanical joining, adhesive bonding, welding, co—curing, and hybrid connections. Here, hybrid connections refer to a combination of the first four connection methods.

Mechanical joining

Typically, it involves bolts and rivets and is primarily used for transmitting high loads or in connection positions that emphasize reliability.

Adhesive bonding

It is achieved through interfacial bonding and is the only connection method that does not require penetration through the thickness of the material. It is generally used for connecting sections with smaller load—transmission requirements. Adhesives are often selected to be of the same material as the substrate to ensure material compatibility.

Welding

It offers advantages such as high—strength and high—stiffness weld joints, high fatigue strength, and no need for adhesives during the welding process, with minimal stress concentration.

Co—curing

It allows all components to be cured into a single part, eliminating the need for fasteners.

The connection design of different areas is the difficulty in composite base—plate modeling and subsequent processing. The overall model of the composite base plate is shown in Fig. 4, which mainly includes the main plate, conical basin, rib, socket, and hole.

Fig. 4

Overall model of composite base plate.

The hybrid connection is used for failure safety reasons and generally can achieve better connection safety and integrity than either a mechanical connection alone or an adhesive connection alone. Therefore, according to the structural stress characteristics of the composite base plate, mechanical connection should be preferred for strength in areas with large forces, such as the conical basin and the main plate. To ensure the safety of the structural connection, the hybrid connection “Adhesive & Mechanical” should be considered. To reduce the mass in the rib area where small loads are transferred and in the large—area part of the rib, the hybrid connection “Adhesive & Co—curing” is considered. Since the socket and the hole are still made of alloy steel materials, the connection with their adjacent structures is still achieved by means of welding. The specific design is listed in Table 3, and the corresponding positions are indicated in Fig. 4.

Table 3 Hybrid connection scheme of composite base plate.

Processing technology of composite base plate

In adhesive joints combined with mechanical hybrid connections, the mechanical connection can be regarded as an enhancement of the adhesive bond. The ideal design for the connection state is that both the adhesive layer and the bolts should bear the load simultaneously during the connection process. When the joint is approaching failure, both the adhesive layer and the bolts should reach their ultimate load—bearing capacity at the same time, or the adhesive bond should reach its ultimate load—bearing capacity first while the bolt connection still has some reserve load—bearing capacity.

Analysis of hybrid connection adhesive & mechanical

To achieve the ideal state of hybrid connections, the deformations of the adhesive bond and the mechanical connection must be coordinated. Specifically, the deflection deformation of the bolt in the direction of the applied force must be equal to the shear deformation produced by the adhesive layer at the bolt’s location. Under this condition, it is assumed that the ultimate load—bearing capacity of the adhesive joint combined with the mechanical hybrid connection is equal to the sum of the ultimate load—bearing capacity of the adhesive joint and the ultimate load—bearing capacity carried by the mechanical connection when the adhesive joint fails completely.

A method combining theoretical model analysis with finite element calculation comparison is adopted. The theoretical simplified model and finite element model of the titanium alloy frame connected to composite materials are shown in Fig. 5.

Fig.5

Analysis model of “Adhesive & Mechanical” hybrid connection.

According to Timoshenko’s calculation, and considering the case where the length of the beam is $\frac{{t_{0} }}{2} + \eta \le x \le \frac{{t_{0} }}{2} + \eta + t_{i}$, its deformation curve is $y$, and the bending moment $M\left( x \right)$ and shear force $Q\left( x \right)$ can be expressed as:

$$M\left( x \right) = \frac{{P_{k} }}{2}x - \frac{{P_{k} }}{{2t_{i} }}\left( {x - \frac{{t_{0} }}{2} - \eta } \right)^{2} - m_{k}$$

(8)

$$Q\left( x \right) = \frac{{P_{k} }}{2} - \frac{{P_{k} }}{{t_{i} }}\left( {x - \frac{{t_{0} }}{2} - \eta } \right)$$

(9)

where,$P_{k}$ represents the load borne by the k-th bolt in the hybrid connection, and $m_{k}$ represents the end moment of the k-th bolt when simplified as a simply supported beam. From Eqs. (8) and (9), we can derive:

$$\frac{{d^{2} y}}{{dx^{2} }} = - \frac{{P_{k} }}{{E_{b} I_{b} }}\left[ {\frac{1}{2}x - \frac{1}{{2t_{i} }}\left( {x - \frac{{t_{0} }}{2} - \eta } \right)^{2} } \right] + \frac{{m_{k} }}{{E_{b} I_{b} }} - \frac{{P_{k} }}{{\lambda_{b} G_{b} A_{b} }}\frac{1}{{t_{i} }}$$

(10)

where, $E_{b} I_{b}$ represents the bending stiffness o f the connecting bolt,$\lambda_{b} G_{b} A_{b}$ represents the shear stiffness of the bolt ($\lambda_{b} = \frac{{6\left( {1 + \nu } \right)}}{7 + 6\nu }$), and $\nu$ represents the Poisson’s ratio of the bolt. By performing a double integration on the above equation, we can obtain:

$$y = - \frac{{P_{k} }}{{E_{b} I_{b} }}\left[ {\frac{1}{12}x^{3} - \frac{P}{{24t_{i} }}\left( {x - \frac{{t_{0} }}{2} - \eta } \right)^{4} } \right] + \frac{1}{2}\left( {\frac{{m_{k} }}{{E_{b} I_{b} }} - \frac{{P_{k} }}{{\lambda_{b} G_{b} A_{b} }}\frac{1}{{t_{i} }}} \right)x^{2} + kx + b$$

(11)

where, k and b are undetermined coefficients determined by the boundary conditions. The boundary conditions for the fixed beam are:

$$\left\{ \begin{gathered} y\left( 0 \right) = 0 \hfill \\ \frac{dy\left( 0 \right)}{{dx}} = \frac{1}{2}\frac{{P_{k} }}{{\lambda_{b} G_{b} A_{b} }} \hfill \\ \end{gathered} \right.$$

(12)

This gives:

$$P_{k} = m_{k} /\left( {\frac{L}{8} - \frac{{t_{i}^{2} }}{24L}} \right)$$

(13)

Then the ultimate load-bearing capacity of the bolt $P_{2}$ is expressed as:

$$P_{2} = \sum\limits_{k = 1}^{n} {P_{k} }$$

(14)

Therefore, the ultimate load P that the adhesive-bolt hybrid connection can bear is expressed as:

$$P = P_{1} + P_{2}$$

(15)

Finite element analysis is conducted on the adhesive-bolt hybrid connections of composite materials and titanium alloy to assess the feasibility and reliability of the adhesive-bolt hybrid connection. The material parameters for the adhesive layer, bolts, and nuts are listed in Tables 4 and 5.

Table 4 Material parameters of the adhesive layer.

Table 5 Material parameters of bolts and nuts.

In the model, composite materials, titanium alloy plates, adhesive layers, and bolts are all modeled using the Solid85 element. A three—dimensional contact element, CONTACT174, and its corresponding TARGET170 are used to establish contact relationships between the composite material and the adhesive layer, between the adhesive layer and the titanium alloy plate, and between the bolt hole and the bolt so as to simulate the connection between the structures. According to the simplified form shown in Fig. 5, a fixed constraint is applied to one end of the titanium alloy plate. To simulate the stress situation of the mortar plate, a uniform load is applied to the other end of the titanium alloy plate, with the load being equivalent to the maximum base pressure of the mortar shell, 112 MPa. The simulation results of the bolts are shown in Fig. 6.

Fig. 6

Simulation results of the bolt.

The maximum stress on the composite material plate is 130.02 MPa, which is primarily concentrated in the contact area between the composite material and the bolt. From the overall stress distribution map, there is no significant stress concentration is observed. The maximum stress in the bolt is approximately 258.65 MPa, mainly concentrated in the contact area between the bolt and the adhesive layer. The theoretically calculated ultimate load value is about 286.93 MPa, and these two values are relatively close. The maximum displacement of the composite material plate is approximately 2.64 mm, and the maximum deformation of the bolt is about 0.07 mm, which is consistent with the displacement distribution trend of the composite material plate. Based on these numerical simulation results and previous design experiences with similar structures, it can be preliminarily concluded that the proposed hybrid connection structure meets the design requirements.

Process treatment method of hybrid connection Adhesive & Co—curing

The area of the strengthened region occupies more than half of the total surface area of the base plate. By using adhesive bonding, the need for mechanical drilling and mechanical joints can be effectively reduced, thus ensuring the strength of the entire material. To maximize the efficiency of adhesive bonding, specialized design for the adhesive bonding or co—curing of joints is essential because adhesive bonding and co—curing can overcome many limitations in joint design. Before adhesive bonding, the titanium alloy must be mechanically ground and surface—treated to increase surface roughness and enhance bonding strength.

In the preparation of single—lap adhesive joints, the thickness of the adhesive layer has a significant impact on the bonding strength. Scholars both at home and abroad have conducted in—depth research on the adhesive layer thickness. To obtain the optimal adhesive layer thickness of an adhesive, tensile experiments were conducted on single—lap specimens with different adhesive layer thicknesses. Given that the optimal adhesive layer thickness for commonly used adhesives in current research ranges from 0.1 mm to 0.3 mm, experiments were conducted using thicknesses of 0.1 mm, 0.2 mm, and 0.3 mm respectively. The experimental results are listed in Table 6, and the maximum difference in the average failure load of the single—lap adhesive joints reaches 2000 N, indicating that the adhesive layer thickness has a significant impact on the tensile shear strength of adhesive joints and is one of the factors influencing joint strength.

Table 6 Experimental results of bonding joint strength under different adhesive layer thickness.

It can be seen that when the thickness of the adhesive layer is 0.2 mm, the tensile—shear load of the bonded joint reaches its maximum. According to the adsorption theory of adhesives, mechanical interlocking occurs between the surface of the bonded object and the adhesive. When the thickness of the adhesive layer is small, it cannot fill the depressions on the metal surface, which weakens the mechanical interlocking effect. Additionally, when the adhesive layer is too thin, its thickness is less than the depth of the depressions, making it prone to breakage and reducing the tensile—shear strength.

On the other hand, when the thickness of the adhesive layer is too large, curing—related defects may occur within the adhesive layer during the curing process, leading to discontinuities in the adhesive layer. Furthermore, during the tensile test, the single—lap joint is subjected to both tensile and shear stresses. As the thickness of the adhesive layer increases, the shear stress also increases, which can easily lead to peeling stress. The peeling strength of the adhesive layer is much lower than its tensile strength. Once peeling defects occur in the adhesive layer, they can quickly spread throughout the entire layer, resulting in a decrease in the bonding strength of the joint.

Thus, it can be concluded that as the thickness of the adhesive layer increases, the tensile—shear strength of the single—lap bonded joint first increases and then decreases. When the thickness of the adhesive layer is 0.2 mm, the failure load of the single—lap joint reaches its maximum.

Process treatment method of connection Co—curing

Based on engineering experience, it was decided to lay a stainless—steel wire mesh on the carbon—fiber woven fabric to enhance the material strength. Experiments were conducted to verify the feasibility of the co—curing process treatment. The material preparation and experimental procedure are shown in Fig. 7.

Fig.7

The illustration of the material preparation and experimental process.

First, the carbon fiber was heated in a high—temperature furnace at 550 °C for 1 h for surface oxidation treatment. Then, the carbon fiber woven fabric was laid, and PEEK powder was injected into the carbon fiber fabric. The mixture was cold—extruded into blocks using a press. Subsequently, it was hot—pressed at 330 °C under a pressure of 20,000 N for 10 min. After that, the temperature was lowered to room temperature for demolding and sampling.

Tensile performance tests were conducted on carbon fiber—reinforced PEEK composite laminates with and without the stainless—steel wire mesh using an electronic universal testing machine. The specimens after testing are shown in Fig. 7. It can be seen that laying a stainless—steel wire mesh between the layers of carbon fiber—reinforced PEEK composite laminates significantly improves the material’s toughness. Further testing revealed that the tensile strength of the carbon fiber—reinforced PEEK composite material with the stainless—steel wire mesh is 110 MPa, and the impact toughness is 35 J/cm², which meets the material strength requirements.

Mode analysis of the composite base plate

Mode experiment

In order to grasp the vibration characteristics of the base plate and check the accuracy of the finite element model, a modal experiment of the composite base plate was carried out using the pulse excitation method to obtain modal data such as natural vibration frequency and vibration mode.

Before the modal experiment, several excitation points were marked on the measured object. The excitation device was a force hammer. Each excitation point was hammered several times in turn. The experiment system automatically recorded the input excitation signal and acceleration signal. Based on the input excitation signal and acceleration signal, the modal data such as natural vibration frequency and mode shapes were obtained. In order to ensure that the free boundary conditions of the computational mode and the experimental mode are consistent, the free boundary was simulated by hanging in the experimental mode analysis. The composite base plate is a welded structure of titanium alloy and carbon fiber, which has good linear dynamic characteristics. So hammer excitation was adopted. According to the simulation calculation, it can be seen that the stress on the composite base plate around the standing mortar is the largest. The test points were arranged in such areas for easy measurement and easy identification. Sixteen excitation points were arranged in the experiment, and the modal experimental site was shown in Fig. 8.

Fig. 8

Mode experiment of the composite base plate.

Mode analysis

The essence of modal analysis is to solve the vibration characteristic equation of an undamped elastic system with a finite number of degrees of freedom, which can get the solution of the eigenvalues and eigenvectors of the system (corresponding natural frequencies and modes). The free vibration motion equation of the undamped multi-degree-of-freedom linear system is as follows:

$$M\ddot{d}\left( t \right) + Kd\left( t \right) = 0$$

(16)

Assuming that the system vibrates simply, then:

$$d\left( t \right) = X\sin \left( {\omega t + \theta } \right)$$

(17)

where,$X$ represents the feature vector or mode; $\omega$ represents the angular frequency; $\theta$ is the initial phase angle. By substituting the above formula into formula (16), we get:

$$- \omega^{2} MX\sin \left( {\omega t + \theta } \right) + KX\sin \left( {\omega t + \theta } \right) = 0$$

(18)

Considering the arbitrariness of $\sin \left( {\omega t + \theta } \right)$, the above formula can be written as:

$$\left( {K - \omega^{2} M} \right)X = 0$$

(19)

For solving the above formula, the commonly used methods mainly include Lanczos method and subspace method³³. When the scale of the model is large and multi-order modes need to be extracted, the Lanczos method is faster. When the extracted modes are less than 20 orders, the subspace method is faster. Because the practical mode of the base plate mainly exists in the low frequency part, the high order natural frequency has little significance to the dynamic characteristic analysis. So the subspace method is chosen as the solution algorithm of the eigenvalue problem in this paper. The general solution procedure of the subspace method is as follows:

1) The spatial model of the modes is constructed and expressed by the formula:

$$\left\{ {\begin{array}{*{20}c} {x_{k + 1} = Ax_{k} + \omega_{k} } \\ {y_{k} = Cx_{k} + v_{k} } \\ \end{array} } \right.$$

(20)

where,$x_{k}$ is the discrete time state of the vector; $y_{k}$ is the response of the structure; $\omega_{k}$ and $v_{k}$ are the measured noise;$A$ is the feature matrix; $C$ is the output matrix.

2) Construct a Hankel matrix, which is expressed by the correlation function $R$:

$$T_{{{1 \mathord{\left/ {\vphantom {1 i}} \right. \kern-0pt} i}}} = \left[ {\begin{array}{*{20}c} {R_{i} } & {R_{i - 1} } & \cdots & {R_{1} } \\ {R_{i + 1} } & {R_{i} } & \cdots & {R_{2} } \\ \vdots & \vdots & \ddots & \vdots \\ {R_{2i - 1} } & {R_{2i - 2} } & \cdots & {R_{i} } \\ \end{array} } \right] = O_{i} C_{i}$$

(21)

where,$R_{k}$ denotes the correlation function;$O_{i}$ and $C_{i}$ represent the observable and controllable matrices in the discrete space equations, respectively. Specifically, it is expressed as:

$$R_{k} = E\left[ {y_{k + 1} y_{k} } \right]^{T}$$

(22)

$$O_{i} = \left[ {CCA \cdots CA^{i - 1} } \right]^{T}$$

(23)

$$C_{i} = \left[ {A^{i - 1} GA^{i - 2} G \cdots G} \right]^{T}$$

(24)

3) The Hankel matrix is decomposed, and then the matrices $A$ and $C$ are found according to the matrices $O_{i}$ and $C_{i}$ Eigenvalue decomposition of matrix $A$:

$$A = \psi \Lambda \psi \&^{ - 1}$$

(25)

4) Eigenvalue $\lambda_{1}$ is solved by matrix $A$:

$$\lambda_{i} = e^{{\lambda_{i}^{e} \Delta t}}$$

(26)

5) Solve structural eigenvalues $\lambda_{i}^{c}$:

$$\lambda_{i}^{c} = \frac{{\ln \lambda_{i} }}{\Delta t} = \xi_{i} + \omega_{i} ,\;i = 1,2, \cdots ,n$$

(27)

where,$\omega_{i}$ is the natural frequency of the corresponding order; $\xi$ is the damping factor.

6) Calculate the frequency $f$ and mode shape $\phi$ of the structure:

$$f_{i} = \frac{{\left| {\lambda_{i}^{c} } \right|}}{2\pi }$$

(28)

$$\xi_{i} = \frac{{real\left( {\lambda_{i}^{c} } \right)}}{{\left| {\lambda_{i}^{c} } \right|}}$$

(29)

$$\phi = C\psi ,\;\phi = \left( {\phi_{1} ,\phi_{2} } \right)$$

(30)

Comparative analysis of mode results

After several tests, the signals of Sixteen points were collected. The mode shapes were estimated according to the least—squares frequency domain method. The free boundary conditions were set to obtain the mode shape contour of the first five modes of the composite base plate, as shown in Fig. 9. The frequency pairs of the calculated and experimental modes were listed in Table 7.

Fig. 9

A contour of the first five mode shapes of the computed modes.

Table 7 Frequency comparison between the calculated mode and the experimental mode.

On the one hand, the reason for the error is that the material properties used in the calculation mode assume that the structure is uniform and dense and there are no pore cracks, which are different from the actual situation of the product. Moreover, there is an anisotropic material structure in the composite base plate, which will bring errors in the calculation. On the other hand, the experiment is affected by the environment, equipment and other factors, which also cause relative errors. However, the overall relative error is less than 5%. It is within the acceptable range, and it can be considered that the two show good consistency.

Dynamic analysis of composite base plate

Given the good consistency between the calculated mode and experimental mode, the structural strength and stability of the composite Base Plate can be theoretically predicted by analyzing the dynamic response of the composite Base Plate model. The firing load is set to 112 MPa pressure generated by the cannon’s breech under full charge conditions with a mortar, and the working conditions are set at a 0° azimuth angle and an 85° elevation angle. The soil condition is set as common medium—hard soil, with material properties as shown in Table 8.

Table 8 Properties of the working condition soil.

Structural strength

The resulting stress cloud map and deformation cloud map are shown in Fig. 10. From Fig. 10, it can be seen that the force transmission path of the composite base plate clearly shows the titanium alloy skeleton structure. The stress distribution in the central region of the Base Plate is relatively dispersed, with few stress concentration points, and the stress concentration points mainly exist at the connections between the titanium alloy skeleton of the main plate and the composite laminate. The stress cloud map results indicate that the design scheme of the composite base plate is feasible, and, the titanium alloy skeleton, which underwent size design for the primary load—bearing structures and topology optimization for the non—primary load—bearing structures, fully exerts its load—bearing function. The deformation cloud map results show that the maximum deformation of the composite base plate mainly occurs in the composite material regions of the trunnion cup and the conical basin, indicating that high impact loads significantly influence the deformation of the central and surrounding regions of the composite base plate and have high demands on the connections between the titanium alloy skeleton and the composite laminate layers. The maximum stress of the composite base plate is approximately 315.9 MPa, and the maximum deformation is 4.391 mm. Under the same conditions, the maximum stress of the titanium alloy base plate is approximately 322.3 MPa, and the maximum deformation is 4.271 mm. It can be seen that there is no significant difference in the maximum stress and deformation between the two. The designed composite base plate meets the technical requirements. However, it is also noteworthy that the connection issues between the composite material and the alloy material remain critical to improving structural strength, which will be continuously refined in subsequent research.

Fig. 10

Dynamic response of composite base plate under medium-hard soil condition.

Fatigue analysis

Furthermore, the stress curve of the impact load transferred to the composite material area of the main plate in the composite material base plate is shown in Fig. 11, with a maximum value of 198 MPa. A compression fatigue test was conducted on a certain laminate according to ASTM D7137/D7137M—2017 “Standard Test Method for Compressive Residual Strength Properties of Polymer Matrix Composite Laminates Containing Damage”. The fatigue S—N curve of carbon fiber T300 was obtained³⁴, as shown in Fig. 12. Among them, the number of cycles under a stress of 200 MPa exceeded 8000. The carbon fiber T700 selected in this paper shows better fatigue characteristics in engineering application scenarios compared to T300. Under the same stress, it may have a longer fatigue life and be more resistant to fatigue damage. Therefore, it can be preliminarily predicted that the number of cycles of the research object in this paper is not less than 8000, which can meet the actual application requirements. Of course, the overall impact fatigue resistance of the composite material base plate is also affected by the molding process, working environment, and loading conditions. Specific research still needs further investigation and verification.

Fig. 11

The stress curve of the composite material in the main plate area under impact load.

Fig. 12

The fatigue S–N curve of carbon fiber T300³⁴.

Firing stability

When a mortar is fired, its movement can be roughly divided into the accelerating recoil stage, the decelerating recoil stage, and the return stage. The firing stability of a mortar includes recoil stability and return stability. Recoil stability refers to the reliability that the gun barrel recoils along the axis direction of the gun bore during firing. Return stability refers to the limited rebound of the base plate after the recoil ends and the gun barrel returns to its original position, as well as the reliability that it returns to the original position. Therefore, to ensure the stability of the mortar during firing, the backward displacement, sinking amount, and jumping amount of the base plate are required to be within a relatively small range. Regarding these three parameters, a comparison between the composite base plate and the titanium alloy base plate is shown in Fig. 13.

Fig.13

Comparison of firing stability.

As can be seen from Fig. 13, when compared the backward displacement and sinking amount of the base plates, the composite base plate and the titanium alloy base plate generally show similar performance. The backward displacement and sinking amount of the composite base plate are slightly larger than those of the titanium alloy base plate, with the maximum increase not exceeding 5.89%. This is because the requirements for the backward displacement and sinking amount of the base plate conflict with each other with those for the structural dimensions of the base plate. After the mass of the base plate is reduced, due to the increase in the recoil kinetic energy of the mortar, the backward displacement and sinking amount of the base plate also increase. Among them, the sinking direction corresponds to the projection of the bore axis on the projection plane. The base plate is subjected to a large component force of the resultant bore force in the sinking direction, which is close to the backward displacement.

Shooting accuracy

To ensure the shooting accuracy of a mortar, it is required that the muzzle vibration state, namely the lateral displacement, longitudinal displacement, lateral angular displacement, and longitudinal angular displacement of the reference point at the muzzle center, be within a small range. The muzzle vibration conditions of the composite Base Plate and the titanium alloy Base Plate were calculated respectively. The variation curves of the muzzle lateral displacement $U_{x}$, longitudinal displacement $U_{z}$, lateral angular displacement $\theta_{x}$, and longitudinal angular displacement $\theta_{z}$ are shown in Fig. 14.

As can be seen from Fig. 14, the variation patterns of the corresponding parameters of the two types of base plates are quite similar. This indicates that the composite base plate not only reduces the weight but also does not significantly increase the muzzle vibration of the mortar. Under different operating conditions, all the vibration values of the composite base plate at the moment when the projectile exits the muzzle are smaller than those of the titanium alloy base plate. This shows that the optimization research on the composite base plate is conducive to reducing the muzzle vibration under comprehensive operating conditions, thus creating favorable conditions for improving the shooting accuracy of the mortar.

Fig. 14

Comparison of muzzle vibrations.

Conclusion

As research on the application of composite materials in the lightweight design of mortar base plates, the main conclusions are as follows:

• (1) A refined and efficient finite element modeling method is the foundation for model building. The layer design of composite materials is crucial for modeling, and a reasonable area connection pattern ensures the reliability of the model structure.

• (2) The subspace method has good applicability in the modal calculations of composite base plates. It can solve equations rapidly and yield results with high credibility.

• (3) Hybrid connections are an appropriate technique for connecting composite materials and alloy materials in the composite base plate. Incorporating a metal mesh within the composite lamination helps enhance the structural strength.

• (4) The experimental method that refers to the dynamic parameters of artillery indicates that the impact testing method is still applicable for the modal experiments of composite base plates, effectively obtaining the experimental modes of the structure.

In summary, this research provides valuable insights for the structural design, optimization, and multi—condition application analysis of composite base plates, contributing to the practical application of composite materials in the field of artillery.