Reliability analysis of dynamic accuracy for the heavy-duty machine tool segmented beam

The dynamic reliability of the heavy-duty CNC machine tool beam is directly related to the machining accuracy which is challenging to maintain because of the gantry machine structure. It is important to evaluate the reliability and accuracy of gantry machine components, especially the tool beam. This research investigated the dynamic accuracy of a segmented beam of gantry heavy machine tool by studying the surface morphology of the joint surface. The mechanical model of a single asperity is established based on the finite element method. The elastic and the plastic stage deformation boundary values are obtained, and the contact model of the full joint surface is established by using the Gaussian distribution characteristics. The finite element model of the sectional beam is developed by transforming the contact parameters of the joint surface. The accuracy of the model is verified by the field modal experiment. Additionally, the proposed model is used to analyze the impact of vibration on machining accuracy. The results revealed that the proposed model is highly reliable.


Introduction
The reliability of computer numerical control (CNC) machine tools directly affects machining quality, productivity, and efficiency and enhances customers' confidence [1,2]. The reliability modeling method is applied to evaluate the reliability of CNC machine tools. Cheng et al. [3] proposed a reliability analysis method for determining machining accuracy based on the fast Markov chain simulation. Li et al. [4] proposed an improved reliability modeling method for a four-parameter non-homogeneous Poisson process considering element action. Liu et al. [5] suggested an inherent reliability modeling method by employing an imprecise Dirichlet model in data fusion based on the imprecise probability theory and the Bayesian theory. Xiong et al. [6] proposed a new multi-performance multi-sequence hidden Markov model for evaluating the reliability of heavy CNC machine tools.
The reliability of the CNC machine tools is affected by many factors due to their complex electromechanical structure. In terms of mechanical structure, reliability is evaluated from several aspects such as geometric error, vibration error, and thermal error. Xia et al. [7] proposed a reliability evaluation method of machining accuracy by employing Monte Carlo simulation of directional importance sampling considering geometric and vibration errors. Zhang et al. [8] proposed a geometric error budget method considering geometric error, thermal error, and tool wear. Niu et al. [9] presented a research method by analyzing the crosscorrelation studies of geometric error parameters to improve and promote the accuracy and reliability of CNC machine tools.
Reliability must be considered at the concept design stage of machine tools to ensure that the manufacturing of such tools meets the reliability goals. Reliability must also be established during the remanufacturing of CNC tools. Du et al. [10] combined a neural network with a remanufacturing coefficient and proposed an improved reliability allocation method for remanufacturing of machine tools. Gu et al. [11] proposed a reliability allocation method considering the correlation between influencing factors and subsystem failure. Cheng et al. [12] investigated the geometric accuracy distribution of multi-axis CNC machine tool based on the sensitivity analysis and reliability theory. Du et al. [13] performed a detailed analysis of the remanufacturing of machine tools and developed a reliability allocation method based on fuzzy evaluation and failure effects of components. Cheng et al. [14] established a new method of machine tool reliability allocation by employing intuitionistic trapezoidal fuzzy numbers and (a technique for order of preference by similarity to ideal solution) TOPSIS.
The reliability and dynamic accuracy of the key components of machine tools are related to the stability of machining precision. Lee et al. [15] evaluated the reliability of the turning machine tool holder with the highest failure rate. Yan et al. [16] proposed a new method for predicting the spindle reliability of CNC machine tools based on the ADAM algorithm-optimized cascade feedforward neural network. Yue et al. [17] developed a thermal error modeling and analyzed the machining accuracy reliability for the spindle. Sun et al. [18] established the reliability analysis model for milling shaft fatigue strength based on the universal generating function (UGF). Li et al. [19] proposed an improved random thermal network model with dynamic thermal excitation to estimate the positioning accuracy and reliability of the ball screw system.
In this article, a precise mechanical model is developed for the beam of the sensitive part of the CNC machine tool to study the reliability and dynamic accuracy. First, the finite element theory is used to establish the micro-macro contact model of the joint surface. Then the precise mechanical model of the beam is constructed by implanting the contact parameters of the joint surface. Subsequently, the model is applied to analyze the dynamic precision reliability of the beam. The results provided the basis for the formulation of machine tool processing and promising theoretical and practical importance.

Interface contact model
The contact stiffness of the joint surface in the machine tool accounts for about 60% to 80% of the total tool's stiffness [20], and the contact damping at the joint withstands more than 90% of the total tool's damping effect [21]. Studies have shown that more than 60% of the vibration problems in machine tool originates from the joint part [22]. As the gantry-type heavy CNC machine tool has the characteristic of large scale (as shown in Fig. 1), its beam is a bolted sectional structure, so the beam is a vibration-sensitive component with multiple joint surfaces.
The main goal of this paper is to analyze the dynamic accuracy and reliability of the three-section beam. The accurate mechanical model of the three-section beam is the basis for reliability research, and the joint surface parameters are the main factors affecting the mechanical properties of the machine tool. Therefore, the paper carried out research on the modeling of the joint surface contact model.
The joint surface contact mechanical model is developed to analyze the dynamic accuracy and reliability of the beam, as shown in Fig. 2. It consists of three parts: substructure A, substructure B, and joint surface.
The microstructure of the contact part of the bonded surface is shown in Fig. 3. The dimensions in the z-direction are much larger than those in the other two directions, and the external load is normal to the xy plane and does not change along the z-direction. Therefore, it can be assumed as a plane strain problem.
A single micro-convex body mechanical model is shown in Fig. 4.
Based on finite element theory, the element strain matrix is represented as: where Δ is the area of the triangular element;b i = y j − y m ,c i = −(x j − x m ),y i , and x i are the coordinate value corresponding to i, j, and m ( i, j, m rotation).
The elastic matrix of the plane strain problem is as follows: where E is the elastic modulus of the material; v is Poisson's ratio.
The stiffness equation is developed as follows: By substituting (1) and (2) into (3), the element stiffness matrix is transformed as: where is the unit height, l is the unit width, and t is the unit length.
By using Eq. 4, the overall stiffness matrix is given by Eq. 5: The static equation of the model is established as: The displacement of a single micro convex body node 3,4 under the same equivalent force is represented by d n . By introducing the constraint conditions, the static equation is converted into the overall stiffness matrix: can be simplified as: Therefore, the normal stiffness of a single micro-convex body is given by: Similarly, the tangential stiffness of a single microconvex body is: The boundary conditions for a single micro-convex body are shown in Fig. 5, where the uniform force is equivalent to the nodal force f n .
According to the static boundary conditions: In Eq. (13), y must be less than the yield strength c , therefore (11) x cos + xy sin = 0 xy cos + y sin = 2f n (12) The maximum deformation of the equivalent tangential load is as follows: Greenwood and Williamson [23] pointed out that the height of asperity on rough surfaces conforms to Gaussian distribution. Assuming that the distance between two planes is d, the contact occurs when the height of the convex point is greater than d. In the probability density distribution curve, the part where σ denotes the variance of the height Gaussian distribution of the asperity. When the rough surface is completely in a state of elastic contact, the normal k n and tangential k t contact stiffnesses can be expressed as: where N is the total number of surface asperities.

Determination of micro convex body
Mandelbrot developed Weierstrass-Mandelbrot functions with continuity, self-affine, self-similarity, and non-differentiability in infinite intervals applying molecular Brownian motion when only one scale is considered (i-e.,n = 1 ). The Weierstrass-Mandelbrot function is simplified as: According to formula (19), the height and radius of a single asperity are as follows: where l is the diameter of a single asperity matrix, which can be written as l = 1∕ . The size parameter of spectral density is taken as greater than 1 and for the random surfaces subject to a normal distribution, = 1.5.
A non-contact optical profilometer (NOP) can measure surface fractal parameters D and G. The roughness length method (RMS) is used to analyze the data collected by the ST400 profilometer. The roughness length method is based on the mean square error ( R q ) of the surface contour height distribution. Reference R q can be written as follows [24]: For the roughness length method, first, the mean contour height of a section is calculated, and then the square   Fig. 6). The data further confirmed that the height of the micro-convex body on the surface of the joint surface conformed to the Gaussian distribution.
The profiles D S = 1.411 and G S = 4.96 × 10 −12 m can be calculated with the help of Eq. (22). The distance between sampling points is 10 µm. Each profile included 400 data points; therefore, the average fractal parameters of 400 profiles are calculated separately.

Model validation
In order to verify the accuracy of the dynamic model of the beam considering the influence of the joint surface, firstly, the modal experimental analysis of the three-section beam of a heavy gantry CNC machine tool is carried out. Then, the contact parameters of the joint surface are calculated by using the contact model of the joint surface, and the finite element model of the three-section beam considering the influence of the joint surface is established. Based on the model, the finite element modal analysis is carried out. Finally, the first two natural frequencies of the experiment and simulation are compared to verify the accuracy of the theoretical model.
Let us take the section beam with over span and heavy load to verify the dynamic model, as shown in Fig. 1. The span is 10.5 m; the total length is 15 m, and the weight is 100 tons. The beam consists of three sections which are connected by 29 high-strength bolts. The beam is made of ball-milled cast iron. The material grades and properties are shown in Table 1.  To obtain the modal data of the beam, the acceleration sensor placed on the beam is used. The vibrator's placement mode and the arrangement of the measuring points are shown in Fig. 7.
The LMS (the data acquisition system) instrument is used for data acquisition and processing analysis. The excitation frequency range and the excitation force are 5 Hz-100 Hz and 120 N, respectively. It is observed that the frequency response function of all measuring points exhibits a peak at two frequencies (as shown in Fig. 8). The analysis of the frequency response function is obtained by the sweep frequency, and then the two order modes in the front and rear directions of the milling machine are attained as 21.782 Hz and 79.159 Hz, respectively.
The joint surface parameters are obtained by means of Eqs. (17) and (18), and the joint surface data are mapped into the three-dimensional finite element model of the beam with applied loads and constraints. The results using ANSYS software are presented in Table 2.  Table 2, by comparing the modal data with the experimental results, the frequency errors are estimated at 4.7% and 6.6%, respectively. The first two order array types are found to be consistent, which shows the accuracy of the proposed beam dynamic model considering the influence of the joint surface.

Accurate mechanical modeling of beams
The differential equations of motion with multiple degrees of freedom (DOFs) are as follows [25,26]: where [M] , [C] , and [K] are the total mass, damping, and stiffness matrix, respectively.
The stiffness and damping matrix considering the influence of the joint surface are: where K bi , K ii , and K ib are the structural stiffnesses of all measured parts; k b is the joint stiffness which is obtained by Eqs. (17) and (18);C ii , C ib , and C bi are the damping of all measured parts. C b is the damping of the joint surface, which is calculated by the Rayleigh damping [27] as: The M is the quality of the substrate virtual material [28], where the zero rough surfaces are located. The modal data of the beam obtained through the modal experiment is presented in Table 3 .
Using the formula (26), the final damping expression is as follows:

Reliability analysis of the z-direction start-stop condition
The beam of the gantry heavy-duty CNC machine tool produces large-amplitude vibrations under the action of starting and stopping inertial force, which affects the reliability of the machining accuracy of the machine tool. Therefore, the reliability analysis of the vibration error is conducted under the starting and stopping conditions (tool starting and c n,t = 0.514M + 0.000172k n,t stopping position d, starting and stopping acceleration value a). The initial conditions of the z-direction starting and stopping of heavy-duty machine tools are shown in Fig. 9.
The accelerations of 0.25 m∕s 2 , 0.3 m∕s 2 , and 0.35 m∕s 2 are used to start the machine tool in the z-direction. The relationship between distance d and the amplitude after moving to the surface of the machined part is shown in Fig. 10.
From Fig. 10, it can be observed that when the beam starts in the z-direction, the amplitude of the x, y, and z-axis increases with the distance to the surface of the machined parts, and the amplitude of the beam decreases gradually. When machined parts are at the same distance, the vibration due to the beam's inertia force is maximum along the x-axis (the maximum amplitude is 0.0552 mm) and minimum along the y-axis (the minimum amplitude is 0.01 mm). To avoid the loss of machining accuracy of CNC machine tools, the starting and stopping position of the machine tool beam should be controlled at 20 mm from the machined surface.

Reliability analysis of x-direction start-stop condition.
Due to the characteristics of the gantry machine tool beam, the starting and stopping of the x-axis feed also affect the reliability of the machining accuracy. Therefore, the accelerations of 0.25 m∕s 2 , 0.3 m∕s 2 , and 0.35 m∕s 2 are used to start the machine tool to feed in the x-direction. The relationship between distance d and the amplitude after moving to the upper surface of the machined part is also investigated. The initial conditions of the x-direction starting and stopping of the heavy machine tool are shown in Fig. 11.
From Fig. 12, it can be analyzed that when the distances from the machined parts are the same, the vibration generated by the inertial force of the beam shows the greatest impact on the x-axis (the maximum amplitude is 0.0541 mm), and the least impact on the y-axis (less than 0.01 mm). As compared to the start-stop effect in the z-direction, the start-stop effect in the x-axis is large, especially the vibration generated in the x-direction. The amplitude of the tool is also less than 0.01 mm when operated at 120 mm.

Reliability analysis of dynamic precision under impact load.
It is necessary to establish the relationship between the externally applied impact loads and the maximum amplitude to evaluate the reliability of the dynamic accuracy of the machine tool. Figure 13 is the displacement response nephogram of the tool due to an impact load. From Fig. 14, it is clear that the maximum amplitude of the beam increases with the increase of the impact load, especially the impact load in the x-direction. When the impact loads reach 1000 N and 2000 N, the maximum amplitudes are observed as 0.016 mm and 0.01 mm, respectively.

Conclusion
In this research, the precision modeling and dynamic precision reliability of a segmented beam of the gantry-heavy machine tool are studied. The main contributions of this research are: (1) A micro-macro collaborative joint surface modeling method is proposed by employing the finite element method. The model is easy to apply for analyzing the accurate dynamic reliability of the sectional beams. The field modal experiment verifies that the proposed model generates highly reliable results. 2) To improve the accuracy and reliability of the gantry heavy machine tool, the starting point should be more than 20 mm away from the upper edge of the machined part in the z-direction feed. In realizing the x-direction feed, the starting point should be more than 120 mm away from the machined part. When the load in the x-direction is greater than 1000 N, the load in the z-direction is around 2000 N. Due to the large-scale characteristic of heavy-duty CNC machine tools, the temperature field also contributes to machining accuracy. Due to the scope of this research, this factor is not studied in reliability analysis. The reliability of machine tools under thermo-mechanical coupling is planned for future research.
Funding This work was supported by the Scientific Research Fund of the Department of Education of Liaoning Province, China (LG201926).

Data availability
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.
Code availability Not applicable.