Study on Simulation Method of Virtual Material for Modal Characteristics of Steel-BFPC Joint Surface

: Basalt fiber polymer concrete (BFPC) machine tool has the properties of lightweight and high damping, and the BFPC machine tool has excellent vibration damping and anti-vibration performance, which can improve the processing performance of the machine tool. There are many steel-BFPC joint surfaces in the machine tool, and these joint surfaces have a key influence on the dynamic performance such as the modal performance of the machine tool. In order to establish a virtual material simulation analysis method for modal characteristics of steel-BFPC joint surface, the detection principle of contact parameters of steel-BFPC joint surface was studied by using forced vibration theory, and the contact parameter detection experiment of steel-BFPC joint surface was designed. The virtual material model of BFPC joint surface was established, and the theoretical formulas of equivalent elastic modulus and equivalent shear modulus were derived based on the identified parameters of the BFPC joint surface. For the correctness of the analysis theory, a BFPC bed was taken as an example to study the modal performance of the BFPC bed by means of experiment and virtual material simulation analysis, respectively. The results of simulation analysis were compared with the experimental results. The results show that the maximum error of the natural frequency is only 6.23%, and the modes of each order are consistent, which prove the effectiveness and accuracy of the virtual material


Introduction
High-speed cutting machine tools can significantly improve the quality of the machined surface, but high-speed machine tools are particularly sensitive to vibration due to the high speed of the spindle [1][2][3][4]. Traditional high-speed cutting machine tools are mostly made of metal materials (such as steel and cast iron), resulting in their poor vibration damping, vibration resistance and poor lightweight performance [5,6]. Therefore, designers and users are not completely satisfied with the performance of high-speed machine tools made of traditional materials. Studies have shown that the machine bed made of mineral composite materials can have excellent vibration damping and vibration resistance in a wide range of excitation frequency, which makes the overall performance of the machine tool more superior [7][8][9][10]. For example, CHO designed a concrete machine tool workbench, the quality of the machine tool workbench is reduced by 36.8% and the structural rigidity is increased by 16%, which can effectively improve the low-order natural frequency of the machine tool to avoid resonance [9]. SUH and LEE have manufactured a polymer concrete high-speed CNC milling machine, which has high damping (2.93%-5.69%) in a wide frequency range [10].
In traditional material machine tools, 60% to 80% of the total machine stiffness is provided by the joint surface [11][12][13]. 80%~90% of the total damping of the machine tool comes from the joint surface [14][15][16]. More than 60% of machine tool vibration problems are related to the dynamic performance of the machine tool joint surface [17][18][19][20][21]. In mineral composite machine tools, there are a large number ·3· of mineral composite material-metal joint surfaces, such as the joint surfaces of metal guide rails and mineral composite machine tool foundation parts, and the joint surfaces of metal embedded parts and mineral composite machine tool foundation parts. These joint surfaces have a key impact on the dynamic performance of mineral composite material machine tools. This type of joint surface is quite different from the bimetal joint surface of traditional material machine tools in terms of constituent materials and the static and dynamic properties of the joint surface. In order to explore the contact parameter detection method of the joint surface of the mineral composite machine tool and the simulation analysis method of the dynamic performance. In this paper, a typical mineral composite material machine tool (basalt fiber polymer concrete machine tool) is used as the research object. The contact parameters of the steel-BFPC joint surface are tested through experiments. The modal analysis model of the steel-BFPC joint surface is established using virtual material theory.
The full paper consists of 5 sections. In Section 2, the principle of dynamic parameter detection of steel-BFPC joint surface is studied and the detection method of dynamic parameters of steel-BFPC joint surface is designed. Section 3 establishes a virtual material simulation model of the steel-BFPC joint surface and derives the equivalent parameter calculation method of the virtual material. In Section 4, a BFPC bed containing a steel-BFPC joint surface is studied, and the modal performance of the BFPC bed is investigated by means of experimental study and virtual material simulation analysis respectively to verify the correctness and accuracy of the study. Two key conclusions are summarized in Section 5. The machined surfaces of steel and BFPC specimens are rough, and there are many micro-peaks on their surfaces. The steel and BFPC specimens contact and extrude to form a steel-BFPC joint surface, as shown in Figure 1(a). The dynamic characteristics of the joint surface are determined by the contact characteristics of these peaks.
Each group of contacting peaks can be regarded as a set of spring damping units [7,22,23], so the steel-BFPC joint surface can be equivalent to the theoretical model in Figure 1(b).

BFPC specimen
Steel specimen The dynamic parameters of the steel-BFPC joint surface are detected by means of sinusoidal excitation, as shown in Figure 2. The equivalent dynamic model of the steel-BFPC joint surface is shown in Figure 2(b), assuming that the total stiffness and total damping of the steel-BFPC joint surface are kc and cj, respectively. The mass of the BFPC specimen is m1. The total stiffness and total damping of the BFPC specimen itself and the joint surface formed by the contact between the BFPC specimen and the ground are k1 and c1, respectively. The mass and stiffness of the steel specimen are m2 and ks , respectively. The damping of steel is very small compared with BFPC material and steel-BFPC joint surface, so its influence is ignored in the analysis model of Figure 2 There is a sinusoidal excitation force Fsinωt in the center of the upper surface of the steel specimen. According to Figure 2(b), the vibration equation of the steel-BFPC joint surface piece can be obtained as Where k' is the equivalent stiffness of the steel specimen and the joint surface, because the stiffness ks of the steel specimen is much greater than the contact stiffness kj of the steel-BFPC joint surface, so ' By solving equation (1), the amplitude B1 and phase angle ψ1 of the BFPC specimen, the amplitude B2 and phase angle ψ2 of the steel specimen are obtained as Exciting force amplitude F and angular frequency  are set by the computer, which are known conditions. Then B1, B2, ψ1, ψ2, m1, m2 are measured through experiments. According to the above known conditions and equation (2), the stiffness kj and damping cj of the steel-BFPC joint surface can be solved.

BFPC specimen
Steel specimen Accelerometer  Figure 3 The principle of joint surface parameter detection The principle of detecting the parameters of the steel-BFPC joint surface is shown in Figure 3, and the experimental site is shown in Figure 4. The exciting force of the electromagnetic exciter acts on the center of the upper surface of the steel specimen. The frequency of the exciting force is controlled by a computer and a signal generator, and the amplitude of the exciting force is controlled by a power amplifier. The force sensor is used to detect the magnitude of the exciting force acting on the steel specimen. Two acceleration sensors are used to detect the vibration response of the steel specimen and the BFPC specimen. The data detected by the force sensor and acceleration sensor are transmitted to the dynamic signal test system for recording and processing. The quality of the steel specimen and the BFPC specimen is measured with an electronic scale. Finally, according to the experimental data, the stiffness kj and damping cj of steel BFPC joint surface can be obtained by solving equation (2) with MATLAB software.
When the exciting direction of the vibration exciter is the tangential direction of the joint surface and the signal picking direction of the acceleration sensor is set as the tangential direction, the corresponding tangential vibration of the joint surface can be detected, and then the tangential dynamic parameters of the steel BFPC joint surface can be solved according to equation (2 Figure 5 Virtual material model of steel-BFPC joint surface Although the equivalent spring damping element can be used to simulate the mechanical properties of the steel-BFPC joint surface, the equivalent spring damping element cannot simulate the thermal mechanical coupling contact properties of the steel-BFPC joint surface. Moreover, the layout of spring damping elements and the number of spring damping elements have great influence on the accuracy of the analysis results. Therefore, virtual material is used to simulate the contact performance of steel-BFPC joint surface. As shown in Figure 5, a thin virtual material layer is added between the steel specimen and the BFPC specimen, and the virtual material layer is connected with the metal specimen and the BFPC specimen by binding contact, as indicated by '×' in Figure  5 (b). Then the material performance parameters of the virtual material layer are defined to simulate the contact performance of the steel-BFPC joint surface. The different contact properties of the steel-BFPC joint surface are simulated by changing the material parameters of the virtual material. This method enables the theoretical analysis and finite element simulation to be better integrated, and contributes to the systematization of the joint surface research. The key material parameters of virtual material are modeled as follows.

Equivalent modulus of elasticity
The equivalent elastic modulus En of the virtual material is the ratio of its normal stress  to its normal strain  .
The normal contact stiffness of the steel-BFPC joint surface is Kn. When the normal load of the steel-BFPC joint surface is Fn, its normal deformation n  is Assuming the thickness of the virtual material is h, its normal strain is The average normal stress of the virtual material is Where Aa is the nominal contact area of the steel-BFPC joint surface.
Substituting equation (4) to equation (6) into equation (3), the equivalent elastic modulus is Where ' n K is the unit normal contact stiffness of the steel-BFPC joint surface.

Equivalent shear modulus of virtual material
The equivalent shear modulus Gt is the ratio of its shear The tangential contact stiffness of the steel-BFPC joint surface is Kt. When the tangential load of the steel-BFPC joint surface is Ft, the tangential deformation t  is When the shear strain is micro, the shear strain  is approximately equal to The average shear stress of the virtual material is Substituting equation (9) to equation (11) into equation (8), the equivalent shear modulus of the virtual material is Where ' t K is the unit tangential contact stiffness of the steel-BFPC joint surface.

Equivalent Poisson's ratio of virtual material
When the steel-BFPC joint surface is subjected to a normal load, micro convex peaks that contact each other on the rough surface will deform in the normal and tangential directions. As shown in Figure 1 (a), there are many gaps between the convex peaks that cause the joint surface to be incomplete contact. Therefore, the tangential deformation of the micro convex peak will fill each gap.
From the perspective of the entire joint surface, the overall tangential deformation of the steel-BFPC joint surface is zero, so it can be considered that the equivalent Poisson's ratio of the virtual material is zero [24,25]. its surface morphology is shown in Figure 6. According to Figure 6, the maximum value of micro convex peak is about 500 μm. Considering the deformation region of BFPC specimen, the thickness of the region is assumed to be 500 μm. So the equivalent thickness of BFPC specimen is hB=0.5+0.5=1 mm. The roughness of the steel ·6· specimen is Ra 3.2, which is about half of that of BFPC specimen. It is assumed that the sum of the thickness of the micro convex peak and the thickness of the deformation area is hs=0.5mm. The total thickness of the virtual material is h=1.5mm.

Equivalent density of virtual material
The equivalent density of the virtual material is the mass of the virtual layer divided by the volume of the virtual layer. Its equivalent density is The contact area of the micro convex peak of the steel BFPC joint surface is ' aR  = (16) Where '  is the deformation size of the micro convex peak; R is the curvature radius of the micro convex peak, and its size is Where G is the characteristic scale coefficient of W-M fractal function of the rough surface. When the micro convex peak is elastically deformed, that is, the contact area a is larger than the critical contact area ac. According to Hertz contact theory, it can be known that the elastic contact load of the micro convex peak is Where E is the composite elastic modulus, because the elastic modulus of BFPC is much smaller than that of steel, so E= EB/ (1-μB 2 ). EB, μB are BFPC's elastic modulus and Poisson's ratio.
Substituting equations (16) and (17) into (18) F a n a da F a n a da = +   (21) The relationship between the real contact area of the steel-BFPC joint surface and the load is obtained by integrating equation (21) and performing dimensionless processing.    and acceleration response signal [32]. The computer is used to set the parameters of modal analyzer and display the calculation results.
The first three natural frequencies of the BFPC bed are measured through experiments, and they are 537.7 Hz, 845.9 Hz, and 1100 Hz. Figure 11 shows the vibration mode of the BFPC bed from the first order to the third order. The first-order mode is the vibration of the left side of the BFPC bed, the second-order mode is the overall vibration of the BFPC bed, and the third-order vibration of the BFPC bed is the "S"-shaped vibration on the right side of the bed. (c) Third-order mode Figure 11 Experimental mode shape of BFPC bed

BFPC bed modal simulation analysis
The joint surface pressure of the BFPC bed is 0.8MPa.
The roughness of the steel guide rail is Ra 3. . Damping has almost no effect on the modal shape and frequency of the BFPC bed, so the damping effect is ignored in the simulation. The material of guide rails is C1045, its density is ρs=7890 kg/m 3 , elastic modulus is 200GPa, Poisson's ratio is 0.27 [33][34][35]. The material of the bed base is BFPC, its density is ρB= 2850 kg/m 3 , elastic modulus is 35 GPa, Poisson's ratio is 0.26. The equivalent performance parameters of the virtual material are obtained using formulas (7), (12), (13), (22), as shown in Table 1. The overall grid of the BFPC bed is shown in Figure 13 (a), and the grid of the virtual material layer is shown in Figure 13 (b). The BFPC bed is fixed on a horizontal plane, so a fixed constraint is added to the surface of the BFPC bed that contacts the ground. The modal analysis result of the BFPC bed is shown in Figure 14.  Figure 15 shows the relative error between the simulation frequency and the experimental frequency of the BFPC bed. The modal frequency of the simulation results is slightly higher than the experimental results. The relative error of the two methods is relatively small, ranging from 5.58%~6.23%. Therefore, the validity and accuracy of the virtual material method are verified. The main reasons for the error of virtual material method are as follows: (1) BFPC bed base is made by pouring technology.
During manufacturing, the BFPC base may have some pores or bubbles in the bed due to partial incomplete pouring. These reasons lead to the decrease of the overall rigidity of the BFPC bed, which reduces the experimental frequency and causes errors. (2) The modal performance of steel BFPC bed is studied by experiment and virtual material simulation method. The relative error between simulation results and experimental results is 5.58% ~ 6.23%, which proves the effectiveness and accuracy of the theory and simulation method of virtual material method.