Optimization of Square-shaped Bolted Joints Based on Improved Particle Swarm Optimization Algorithm

▪ Establish the dynamics model of square bolt connection based on fractal theory. ▪ An improved particle swarm optimization algorithm considering the influence of time-varying weight and contraction factor is proposed. ▪ Compared with the traditional algorithm, the improved optimization algorithm shows better operational performance. ▪ Meanwhile, the global stiffness optimization of the square bolt connection is realized. The bolted joint is widely used in heavy-duty CNC machine tools, which has huge influence on working precision and overall stiffness of CNC machine. The process parameters of group bolt assembly directly affect the stiffness of the connected parts. The dynamic model of bolted joints is established based on the fractal theory, and the overall stiffness of joint surface is calculated. In order to improve the total stiffness of bolted assembly, an improved particle swarm optimization algorithm with combination of time-varying weights and contraction factor is proposed. The input parameters are preloading of bolts, fractal dimension, roughness, and object thickness. The main goal is to maximize the global rigidity. The optimization results show that improved algorithm has better convergence, faster calculation speed, preferable results, and higher optimization performance than standard particle swarm optimization algorithm. Moreover, the global rigidity optimization is achieved.


Introduction
The heavy-duty CNC machine tools have a large number of joints, which not only break the continuity of machine structure, but also reduce the stiffness of the overall structure. According to statistics, the static stiffness of CNC machine is 30-50% depending on the joint stiffness, and 60% of the overall machine vibration is caused by joint parts. In the mechanical structure, joints exist in machined surfaces. Namely, there are many factors that influence the characteristics of joint surfaces, such as external load, material properties, processing methods, roughness, and structure type and size [1,2]. Therefore, the optimization of joint parameters is particularly critical for machine static and dynamic characteristics.
At present, the curve fitting method is used to determine joint parameters' numerical value. Many empirical formulas were proposed [3][4][5][6], but they are only suitable for small number of influencing factors, while they are intractable for multi-influencing factors, wherein complex changes are needed. H Yu et al. [7] addressed the dynamic balance of milling cutters, which cannot be neglected in high-speed milling. Based on the particle swarm optimization (PSO) algorithm to optimize the spiral edge shape of the end mill, the radius of curvature of the spiral edge curve is optimized. The unbalance of the milling cutter is effectively reduced and the dynamic performance of the milling cutter is further improved. D Wen et al. [8] proposed a fast backfire double-annealing particle swarm optimization (FBDAPSO) based method for motor parameter identification.
The accuracy and evolution speed are improved and the problem of redundant iterations of SAPSO is solved. SK Lu et al. [9] proposed a new method to calculate the stiffness of bolted joints. The influence of the joint surface stiffness on the overall stiffness was considered. An expression model of the stiffness characteristics of the joint surface was established. The stiffness of the bolted joint surface was effectively improved. W Liu et al. [10] constructed an AL-based ELM model based on fractal theory suitable for granular image features, and optimized the ELM model to obtain the optimal number of hidden layer neuron nodes for each mineral by predicting the feature parameters of granular images with the improved ELM algorithm. Based on RFPA2D and digital image processing techniques, L Zheng et al. [11] proposed a method to calculate the mesoscale fractal dimension of irregular rock particles. The effects of loading conditions and mesoscale heterogeneity on the damage of irregular sandstone particles were investigated. It is of great significance to explore the rock particle breaking and energy consumption laws, rock breaking mechanism, and to find an efficient and energy-saving rock breaking method. Liu et al. [12] proposed a nonlinear model of bolt pretightening force for optimizing the contact performance. Based on finite element simulation, Grzejda [13] studies the bearing capacity of a bolted connection fastener when a bolt is damaged. The quadratic regression mathematical model was employed by Xu et al. [14] to study the main working factors affecting attenuation of bolt preload under vibration. Wang et al. [15] used the spring damping element to simulate the contact characteristics of joint surface, and they established the finite element model of large thread grinder and optimized its static and dynamic characteristics. Cheng et al. [16] established the energy dissipation model of bolted joints based on the 2-dOF vibration differential mathematical model. Chen et al. [17] used the gradient method of unconstrained optimization to identify the stiffness and damping parameters of grinder fixed surface.
Wei et al. [18]  However, an optimization in the field of engineering design is more extensive, and a lot of joint parts of CNC machine need to be optimized in order to improve their overall stiffness and dynamic and static characteristics. At present, the most commonly used method is geometric optimization, wherein analysis follows modeling, and re-analyzing is involved according to requirements. However, this optimization process is mainly dependent on human experience, so there are many limitations. Moreover, the optimization process is complicated, which disables rapid obtaining of results.
In this paper, the fractal model of contact parameters of

Characteristic analysis of bolted joints
The bolt connection is very critical for the heavy duty machine tools, which plays a prominent role in fixation and support.
Since, the joints have a considerable influence on the overall performance of CNC machines [26], it is very important to analyze joint surfaces. The relaxation characteristics of a highstrength bolted structure are affected both macroscopically and microcosmically by multi-factor coupling effects, which make the relaxation mechanism of high-strength bolted structure very complicated. Due to the rapid development of heavy duty CNC machine tools in terms of high precision, high reliability, multifunction and intelligence, the research on relaxation mechanism of high-strength bolted structures and its influence on precision of heavy duty machine tools are very imperative because these mechanisms have become basic and urgent problem for advanced heavy duty CNC machine tools manufacturing in terms of higher grade. This paper focuses on analysis of high-strength bolted structure in heavy-duty machine tools, Fig. 1.

The influencing factors of joint
As it is well known, there are many factors that influence machine joints. A certain number of scholars have conducted numerous studies on dynamic characteristics of bolted joints.
Among these dynamic characteristics, the typical unit parameters play a very important role in the machine structure dynamics [27]. The influencing factors of joints are mostly nonlinear, and due to the diversification of use conditions the analysis becomes more complicated. The main influencing factors are: (1) joint material and its properties ( , , , ), (2) joint processing method, (3) surface roughness between bonding surfaces, (4) joint static load (i.e. working surface pressure), (5) joint structure, type, shape and size, (6) joint dynamic force, and (7) vibration frequency.
The above-mentioned factors can be summarized into three categories: the structure-related factors, such as joint type, shape and size; the inherent-characteristics factors, such as materials, construction methods and surface roughness; and, the relevant factors, such as combination state and vibration frequency.
In summary, the contact parameters of joint surfaces have a great influence on joint dynamic characteristics. Moreover, joint thickness, preload, number of bolts, distribution of bolts and contact surface morphology also have an effect on joint dynamic characteristics. However, this paper mainly analyzes the contact parameters of joint surface.

Modeling of joint surface
The characteristics of fixed joint surface relate to elastic and damping characteristics, energy storage and energy consumption. In establishment of dynamic model of bolted joints, the quality of contact element is ignored and only its stiffness and damping are considered. Many scholars have proposed a lot of models that analyze contact stiffness of joint surface from a microscopic point of view [28][29], which mainly considers the normal contact stiffness. In this paper, the M-B fractal theory is used to study the nonlinear model of rough contact surface from the microscopic point of view. Namely, we analyze fractal characterization of rough surface, asperity contact model, and cross-sectional distribution function of contact asperity. The profile curve of rough surface can be expressed by Weierstrass-Mandelbrot function: where is the contour height of rough surface, denotes the coordinates of surface sampling length, is the fractal dimension of contour curve, is the feature scale coefficient of rough surface, is the size parameter of spectral density, and is a random phase.   The corresponding mathematical formula is: where is the fractal modulus, which stands for the frequency of high and low composition of contour and it is in the range 1-2, is the fractal roughness, which represents the contour height, the greater is, the rougher surface is, is the contact length of a single asperity in the range of and it is from − 2 to 2 .
If we label the cross-sectional area of asperity as ′ , then deformation is obtained from 0 ( ) when is equal to 0: At the microscopic scale, it can be considered that ′ is equal to 2 . The apex radius of asperity is defined by: According to the Hertz theory, the critical conditions for deformation are defined by: where is the yield strength of softer material, is the hardness of softer material, is the scale factor, and is the equivalent elastic modulus of contact materials, and it is defined by: where 1 , 2 and 1 , 2 represent the elastic modulus and Poisson's ratio of contact materials, respectively.
If is equal to , the critical contact area can be obtained by: It can be seen in Eq.7 that ′ is a fixed value in the case we get: In M-B model, the relationship between number of asperity contact points and cross-sectional area is defined by: where ′ is the maximal contact area, and ( ′ ) is the density function of contact area distribution.
Then, the total elastic contact load and plastic contact load of joint surfaces can be expressed by (11) and (12), respectively: Consequently, the total surface normal load model is Assuming that pressure between joint surfaces has a uniform distribution, the equivalent pressure of joint surface can be expressed by: The dynamic parameters of bolted joint include stiffness and damping, wherein stiffness and damping are calculated in , and directions. According to the definition of stiffness, the normal contact stiffness of a single asperity can be obtained by: If (15) is combined with the cross-sectional area distribution function, the total stiffness of joint is defined by: The ANSYS software was used to simulate static parameters for different preloading conditions, and contact pressure of each node was extracted. Then, according to the contact pressure of joint surfaces, normal contact stiffness and contact damping of greater the contact stiffness is, the better the effectiveness of contact is, and the higher natural frequency is.

Improved particle swarm optimization algorithm
The Particle Swarm Optimization (PSO) algorithm mimics the foraging behavior of birds, wherein the search space problem is equated to the flying space of birds, and each bird is abstracted as a mass-free, volume-free particle. The PSO is used to express a candidate solution. The optimal solution is equivalent to the objects that birds are looking for. The basic idea is to determine
During each iteration, the particles are updated by searching for optimal solution; the first is the optimal solution found by particle itself -the individual extreme = ( ,1 , ,2 , ⋯ , , ) , and the other is the optimal solution found within the whole population -the global optimal solution = ( ,1 , ,2 , ⋯ , , ) . During the searching for these two optimal solutions, the particle is iterated by: , +1 = , + 1 1 ( , − , ) + 2 2 ( , − , ) , Where 1 and 2 are the positive learning factors, 1 and 2 are evenly distributed random numbers in the range 0-1, , is the velocity of particle d in the ℎ iteration, , is the position of particle in the ℎ iteration, , is the position of individual in D-dimensional space of particle, and , is the position of global extreme value of particle in D-dimensional space.

Improved particle swarm optimization algorithm with time-varying weight
The Where, is the inertia weight. However, the fixed weight is commonly used, so is the constant between 0 and 1. However, when problem becomes more complex, the performance of fixed weight algorithm related to the searching for optimal solution decreases. Thus, the improved optimization algorithm with time-varying weight is introduced: decreases linearly, and when gradually decreases the particle speed also decreases, which accelerates the particle convergence speed and improves the optimization performance.

Improved particle swarm optimization algorithm with shrinkage factor
Clerc introduced the concept of shrinkable factor in the basic particle swarm algorithm, which can ensure the convergence of particle swarm algorithm by controlling the inertia weight and value of learning factor. The improved algorithm is defined by: where the contraction factor is the function of 1 and 2 , and it can be expressed by: In the Clerc's shrinkage factor method, is usually equal to 4.1, thus, is equal to 0.729.

Improved particle swarm optimization with combination of time-varying weight and shrinkage factor
Considering the advantages of above-mentioned PSO algorithm improvements, a new combinatorial improved optimization algorithm, which can both accelerate convergence and improve solution quality, is proposed. In order to guarantee the convergence of particle swarm algorithm, we introduce the algorithm presented in Fig. 5, which is defined by: (1) Set the necessary settings for APDL file generated by ANSYS and code file generated by MATLAB.
(2) Use the integrated optimization algorithm and set objective function, boundary conditions and parameter range.
(3) Start the optimization algorithm and run it until the termination condition is met.
The above processes should be repeated several times to get the desirable optimization results.  Fig. 6. Optimization flow chart.

Application examples
In this chapter, the square-shaped components connected by bolts in heavy duty CNC machine tools are studied.
First, ANSYS simulation software was used for static analysis in order to get the corresponding nodal pressure, and then contact stiffness of each node was calculated by MATLAB.
Using the combinatorial improved optimization algorithm, the parameters such as thickness, preload and number of bolt holes were optimized to achieve the maximal stiffness of joints.

Structure modeling and analysis
The square-shaped structure material was QT600-3, and machining precision of two contact surfaces was 1.6 μm. The material properties are shown in Table 1. We used two frame-like components connected by a plurality of bolts, which are shown in Fig. 7, wherein the main dimensions are provided.  In order to achieve a high rigidity of bolt joint, the square-

Finite Element Simulation
ANSYS was used to perform the finite element simulation.  Modal simulation based on static structure simulation. The deformation of each order of mode is shown in Fig. 11. The frequencies of each order of modes are shown in Table 2.
Eksploatacja i Niezawodność -Maintenance and Reliability Vol. 25  The simulation results of the static structure when the preload force is 10000N are shown in Fig. 12. According to the simulation results, the maximum stress is 79.934 MPa. The simulation results for each order of modalities are shown in Fig.   13. The frequencies of each order of modal is shown in Table 3.

Optimization results
The optimization results relate to the comparison of standard particle swarm (PSO) algorithm and improved particle swarm (MPSO) algorithm. The input parameters in both cases were preload force in 8 different positions 1 − 8 , fractal dimension , roughness and thickness , and the output was total stiffness of square-shaped structure joint. The boundary conditions for input parameters are shown in Table 2. Thickness( ) 30 50 60 For both standard particle swarm algorithm and improved particle swarm algorithm, all basic parameters were the same, namely, maximal number of iterations was set to 50, number of particles was 20, sum of iterations of particle swarm was equal to 1000, learning factors 1 and 2 were equal to 2, both global and local increments were 0.9 and maximal speed was 0.01.
Two important solution parameters were added to the improved particle swarm algorithm, namely the initial value of inertia weight and the contraction factor , which were 0.002 and 0.729, respectively. The iterative processes of standard and improved particle swarm algorithms are presented in Fig. 14 and Fig. 15, respectively.  According to the above two graphs, it can be concluded that improved PSO algorithm has better convergence than standard PSO algorithm. The improved algorithm found the optimal value near the 300 ℎ iteration, and the performance of searching for optimal solution is far higher than in the case of standard PSO algorithm. As it is shown in Table 5, the overall stiffness is the greatest when the preload and thickness are shown in Tab. 5, respectively. In order to present the particle swarm optimization process more intuitively, in Fig.16, preload is presented on -axis, thickness is presented on -axis, and total stiffness is presented on -axis. The optimization process of the entire particle group can be seen from the latitude and color in three-dimensional map (Fig. 16). The darker the color is, the closer the solution is to the optimal solution. At the beginning, only a few particles are far away from the optimal solution, and most of them iterate near the optimal value. In order to verify stability and accuracy of improved algorithm, the improved particle swarm optimization algorithm was used for another 1000 iterations, and final optimization of total particle swarm was obtained after 2000 iterations. The results are shown in Table 6.

Conclusion
A new improved particle swarm optimization algorithm based on the fractal theory of joint surfaces, which aims to the optimization of joints' parameters, is proposed. By changing component thickness, bolt preload, surface topography and other parameters, the overall joint contact stiffness is optimized.
According to the results, it can be concluded that improved algorithm has better convergence, faster searching speed, and better performance than basic particle swarm optimization algorithm. In simulations, the improved algorithm found optimal solution after 1,000 iterations, and then another 1,000 iterations were conducted in order to verify algorithm stability.
The optimal value of both algorithms was basically the same, which indicates that improved algorithm is stable and accurate, and that optimization of total stiffness of joints is successfully achieved.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: