Stiffness Calculation Model of Thread Connection Considering Friction Factors

In order to design a reasonable thread connection structure, it is necessary to understand the axial force distribution of threaded connections. For the application of bolted connection in mechanical design, it is necessary to estimate the stiffness of threaded connections. A calculation model for the distribution of axial force and stiffness considering the friction factor of the threaded connection is established in this paper. The method regards the thread as a tapered cantilever beam. Under the action of the thread axial force, in the consideration of friction, the two cantilever beams interact and the beam will be deformed, these deformations include bending deformation, shear deformation, inclinationdeformation of cantilever beam root, shear deformation of cantilever beam root, radial expansion deformation and radial shrinkage deformation, etc.; calculate each deformation of the thread, respectively, and sum them, that is, the total deformation of the thread. In this paper, on the one hand, the threaded connection stiffness was measured by experiments; on the other hand, the finite element models were established to calculate the thread stiffness; the calculation results of the method of this paper, the test results, and the finite element analysis (FEA) results were compared, respectively; the results were found to be in a reasonable range; therefore, the validity of the calculation of the method of this paper is verified.


Introduction
The bolts connect the equipment parts into a whole, which is used to transmit force, moment, torque, or movement. The bolt connection is widely used in various engineering fields, such as aviation machine tools, precision instruments, etc. The precision of the threaded connection affects the quality of the equipment. Especially for high-end CNC machine tools, the precision of the thread connection is very high. Therefore, it is important to study the stiffness of the threaded connection to improve the precision of the device. Many researchers have conducted research in this area.
Dongmei Zhang et al. [1], propose a method which can compute the engaged screw stiffness, and the validity of the method was verified by FEA and experiments. Maruyama et al. [2] used the point matching method and the FEM, based on the experimental results of Boenick and the assumptions made by Fernlund [3] in calculating the pressure distribution between joint plates. Motash [4] assumed that the pressure distribution on any plane perpendicular to the bolt axis had zero gradient at r = ℎ /2 and r = , where it also vanishes. They mainly use numerical methods to calculate the influence of different parameters on the stiffness of bolted connections. Kenny B, Patterson E A. [5] introduced a method for measuring thread strains and stresses. Kenny B [6] et al. reviewed the distribution of loads and stresses in fastening threads. Miller D L et al. [7] established the spring model of thread force analysis and, combined with the mathematical theory, analyzed the stress of the thread and compared with the FEA results and experimental results to verify the correctness of the spring mathematical model. Wang W and Marshek K M. et al. [8] proposed an improved spring model to analyze the thread load distribution, compared the load distributions of elastic threads and yielding thread joints, and discussed the effect of the yield line on the load distribution. Wileman et al. [9] performed a two-dimensional (2D) FEA for members stiffness of joint connection. De Agostinis M et al. [10] studied the effect of lubrication on 2 Mathematical Problems in Engineering thread friction characteristics or torque. Dario Croccolo et al. [11][12][13] studied the effect of Engagement Ratio (ER, namely, the thread length over the thread diameter) on the tightening and untightening torque and friction coefficient of threaded joints using medium strength threaded locking devices. Zou Q. et al. [14,15] studied the use of contact mechanics to determine the effective radius of the bolted joint and also studied the effect of lubrication on friction and torque-tension relationship in threaded fasteners. Nassar S. A. et al. [16,17] studied the thread friction and thread friction torque in thread connection. Nassar S. A. et al. [18,19] also investigated the effects of tightening speed and coating on the torque-tension relationship and wear pattern in threaded fastener applications in order to improve the reliability of the clamping load estimation in bolted joints. Kopfer et al. [20] believe that suitable formulations should consider contact pressure and sliding speed; based on this, the contribution shows experimental examples for main uncertainties of frictional behavior during tightening with different material combinations (results from assembly test stand). Kenny B et al. [21] reviewed the distribution of load and stress in the threads of fasteners. Shigley et al. [22] presented an analytical solution for member stiffness, based on the work of Lehnhoff and Wistehuff [23]. Nasser [24], Musto and Konkle [25], Nawras [26], and Nassar and Abbound [27] also proposed mathematical model for the bolted-joint stiffness. Qin et al. [28] established an analytical model of bolted disk-drum joints and introduced its application to dynamic analysis of joint rotor. Liu et al. [29] conducted experimental and numerical studies on axially excited bolt connections.
There are also several authors that, starting from the nature of thread stiffness, from the perspective of thread deformation, established a mathematical model of the calculation of the distribution of thread axial force. The Sopwith method [30] and the Yamamoto method [31] received extensive recognition. The Sopwith method gave a method for calculating the axial force distribution of threaded connections. Yamamoto method can not only calculate the axial force distribution of threads but also calculate the stiffness of threaded connections. The assumption for Yamamoto method is that the load per unit width along the helix direction is uniformly distributed. In fact, for the three-dimensional (3D) helix thread, the load distribution is not uniform. Therefore, based on the Yamamoto method, Dongmei Zhang et al. [1] propose a method which can compute the engaged screw stiffness by considering the load distribution, and the validity of the method was verified by FEA and experiments. The method of Zhang Dongmei does not consider the influence of the friction coefficient of the thread contact surface. In fact, the friction coefficient of the contact surface of the thread connection has an influence on the distribution of the axial force of the thread and the stiffness of the thread. Therefore, we propose a new method which can compute the engaged screw stiffness more accurately by considering the effects of friction and the load distribution. The accuracy of the method was verified by the FEA and bolt tensile test. The flow chart of the article is shown in Figure 2. Yamamoto [31], the thread is regarded as a cantilever beam, and the thread is deformed under axial force and preload. These deformations include the following (shown in Figure 3): thread bending deformation, thread shear deformation, thread root inclination deformation, thread root shear deformation, radial direction extended deformation, or radial shrinkage deformation.

Axial Load Distribution. According to
For the ISO thread, the axial deformation of the thread at at the axial unit width force z is thread bending deformation 1 , thread shear deformation 2 , thread root inclination deformation 3 , thread root shear deformation 2.1.1. Bending Deformation. In the threaded connection, under the action of the load, the contact surface friction coefficient is , when the sliding force along the inclined plane is greater than the friction force along the inclined plane, the relative sliding occurs between the two inclined planes, and the axial unit width force (shown in Figure 3) is ; if the influence of the lead angle is ignored, the force per unit width perpendicular to the thread surface can be expressed as The force per unit width perpendicular to the thread surface can be decomposed into the x-direction component force and the y-direction component force, respectively cos = cos sin + cos (2) and sin = sin sin + cos The friction generated along the slope is w ; i.e., The force w is also decomposed into x-direction force and y-direction force, which are sin and cos , respectively. sin = sin sin + cos (5) cos = cos sin + cos (6) In the unit width, the thread is regarded as a rectangular variable-section cantilever beam. Under the action of the above-mentioned force, the thread undergoes bending deformation, and the virtual work done by the bending moment on the beam section is According to the principle of virtual work, the deflection 1 (see Figure 3(a)) of the beam subjected to the load is where is the bending moment of the unit load beam. is the bending moment of the beam under the actual load. I(y) is the area moment of inertia of the beam at . is Young's Modulus of the material. c is the length of the beam. Here, the forces are assumed as acting on the mean diameter of the thread.
As shown in Figure 5, the height ℎ( ) of the beam section per unit width and the area moment of inertia ( ) of the section can be expressed by using the function interpolation.
where h is the beam end section height; b is the beam section width; 1 is the beam root section height and the beam end section height ratio; see Figure 5.
From Figure 5, the bending moment of the beam is related to the y-axis component of and w , and these components cause the beam to bend; therefore, the analytical solution shows that the bending moment of the unit width beam subjected to the friction force and the vertical load of the thread surface is Substituting (10) and (9) to (8) and integrating to obtain the analytical expression of the deflection 1 (shown in Figure 3(a)) of the cantilever beam with variable cross-section under load one has ( 1 2 1

Shear Deformation.
Assume that the distribution of shear stress on any section is distributed according to the parabola [31] and the deformation 2 (see Figure 3(b)) caused by the shear force within the width of unit 1 is

Inclination Deformation of the Thread Root.
Under the action of the load, the thread surface is subjected to a bending moment, and the root of the thread is tilted, as shown in Figure 3(c). Due to the inclination of the thread, axial displacement occurs at the point of action of the thread surface force, and the axial displacement can be expressed as

Deformation due to Radial Expansion and Radial
Shrinkage. According to the static analysis, the thread is subjected to radial force sin − cos (shown in Figure 4), and it is known from the literature [31] that the internal and external thread radial deformation (shown in Figure 3 sin + cos (14) and

Shear Deformation of the Root.
Assuming that the shear stress of the root section is evenly distributed, the displacement of the point in the direction caused by the shear deformation (shown in Figure 3(e)) is the same as the displacement of the thread in the direction; this displacement can be expressed as [31] 4
The unit force per unit width of the axial direction can be expressed as Under the action of unit force of axial unit width, the total deformation of external thread and internal thread is and For threaded connections, at the x-axis of the load F, the axial deformation of screws and nuts can be expressed as Here is the length along the helical direction, and the relation between the axial height and the length along the helix direction can be represented by the following formula according to the geometric relation shown in Figure 6.
Here, is the lead angle of the thread shown in Figure 6, and then Here, (x) and (x) represent the stiffness of the unit axial length of the nut and the screw, respectively, for the unit force.
The axial total deformation of the threaded connection at is denoted as The stiffness of the unit axial length of the threaded connection is expressed as As shown in Figure 1(a), the threaded connection structure includes a nut body and a screw body. The nut is fixed, the screw is subjected to pulling force, the total axial force is , and the axial force at the threaded connection screw is F(x). If the position of the bottom end face of the nut is the origin 0 and, at the position, the axial force is F(x), the screw elongation amount and the nut compression can be obtained from the following: where (x) and (x) are the vertical cross-sectional areas of screws and nuts at the position. and are, respectively, Young's modulus of the screw body and Young's modulus of the nut body. Find the displacement gradient for the expression, which is, respectively, expressed as Here, As shown in Figure 1(a), the screw is subjected to the tensile force b , with the bottom of the nut as the coordinate origin, and the force at the x position is , and then the elongation of the screw at x is ∫ ( ) = , and the compressed shortening amount of the nut at x is ∫ ( ) = . The relationship between b , n , b , and Figures 7 and 1(a),), and the partial derivative of this relation can be obtained by the following formula: Substituting (45), (46), (47), and (48) into (49) and simplifying it From mathematical knowledge, the equation is a differential equation. The general solution of the equation can be expressed as As can be seen from Figure 1, the axial force at the first thread at the connection surface of the nut and the screw is , and the axial force at the last thread at the lower end of the thread joint surface of the nut and screw is 0; that is, the boundary condition is F(x=0)= and F(x=L)=0. Taking these boundary conditions into the equation will give 1 =-(cosh( ))/sinh( ) and 2 = , so we get the expression of the threaded connection axial load as Therefore, the axial force distribution density of the thread connection along the direction can be expressed as

Thread Connection Stiffness.
The stiffness in the axial direction of the bolted connection is equal to the axial force distribution of the threaded connection multiplied by the unit stiffness; i.e., The overall stiffness of the bolt connection can be expressed as

FEA Model
A 3D finite element model (shown in Figure 10) was established, and FEA was performed to analyze the influence of various parameters of the thread on the thread stiffness. These parameters include material, thread length, pitch, etc. The FEA software ANSYS 14.0 was used for analysis. During the analysis, the end face of the nut was fixed (shown in Figure 9), the initial state of the model is shown in Figure 8, and an axial displacement Δ was forced to the end face of the screw. Then, the axial force of the screw end face was extracted. The axial stiffness of the threaded connection was calculated by the FEM. The friction coefficient of the thread contact surface is set first. In FEA, the contact algorithm used is Augmented Lagrange. that as the mesh size decreases, the resulting axial force gradually decreases, but when the mesh size is small to a certain extent, the resulting axial force will hardly decrease. The axial force at this time is the axial force required by the author. With known displacements and axial force, the stiffness of the threaded connection can be calculated using the formula = Δ (59)

Tensile Test of Threaded Connections [1]
In order to verify the effectiveness of this paper method, the experimental data of the experimental device in [1] are used. In [1], the electronic universal testing machine is used to measure the load-defection data of samples, and the test sample is made of brass. The tension value can be read from the test machine. The axial deflection of thread connection can be represented by the displacement variation between two lines as shown in Figure 14, which can be measured by a video gauge [1]. In [1], in order to obtain the most accurate data possible, each size of the thread is in a small range of deformation during the tensile test, and each size of the thread tensile test is performed 10 times, and the average value is calculated as the final calculated data. Some samples in the experiment are shown in Figure 13. The stiffness calculation formula is = . (60) Mathematical Problems in Engineering Figure 7: Illustration of the elastic deformation of the screwed portion of the threaded connection. The materials used to make nuts and screws are brass. Young's modulus of brass is 107GPa, and Poisson's ratio is 0.32 [1].

Stiffness of Threaded Connections.
Croccolo, D. [12], Nassar SA [19], and Zou Q [32] studied the coefficient of friction of the thread. According to the study by Zou Q and Nassar SA, in the case of lubricating oil on the thread surface, the friction coefficient of the steel-steel thread connection thread is 0.08, and the friction coefficient of aluminumaluminum thread connection thread is 0.1.
In order to verify the correctness of the calculation results of the theory presented in this paper, a variety of threaded connections were used to calculate an experimental test.
In the finite element analysis and theoretical calculations of this paper, Young's modulus of steel is b = n =200Gpa, and Poisson's ratio of steel is 0.3, and the friction coefficient [12,19,32] is set to 0.08. The experimental data, FEA data, and Yamamoto method data in Tables 1 and 5 are from literature [1]. As can be seen from Tables 1 and 5, the calculated values obtained in this paper are all higher than the experimental results. Perhaps the error is caused by the presence of a small amount of impurities on the surface of the thread and partial deformation of the thread inevitably and there is a slip between the threaded contact surfaces. The theoretical calculation results and FEA results in this paper have a small error.
In Table 2, the effect of thread length on stiffness is presented. It can be seen that when the same nominal diameter M10, the same pitch P=1.5, and the same material steel are taken, when the thread engaged length is taken as 14 mm, 9 mm, and 6 mm, respectively, it is found that the longer the thread engaged length, the greater the stiffness and the smaller the length of the bond, the smaller the stiffness.
In the FEA, the method of this paper and the Yamamoto method, Young's Modulus of aluminum alloy is E=68.9GPa; Poisson's ratio of the aluminum alloy is 0.34. The friction coefficient of the steel-steel threaded connection [12,21,22,32] is set to 0.08, and the friction coefficient of aluminumaluminum threaded connection [12,19,32] is set to 0.1. Table 3 shows the effect of different materials on the stiffness of threaded connections. The two types of threaded connections are made of two different materials, the steel and aluminum alloys. It can be seen from the table that, under the condition of the same pitch, the same nominal diameter, and the same engaged length, the stiffness of the steel thread connection is larger than that when the material is aluminum.
In Table 4, it also shows the influence of different pitches on the stiffness of the thread connection. It can be seen that with the same engaged length, the same material, and the same nominal diameter, the pitch is 1.5, 1.25, and 1, respectively, and we find that the smaller the pitch, the greater the stiffness.
When using FEM to analyze the influence of friction factors on the stiffness of threaded connections, the thread 10 Mathematical Problems in Engineering       however, Yamamoto theory does not consider the influence of the friction coefficient on stiffness, and this is obviously unreasonable.

Effect of Friction Coefficient on Axial Force Distribution.
Take the thread size as M6×0.75×6.1, the axial load b is taken as 100N, 350N, and 550N, respectively, and take the friction coefficients 0, 0.3, 0.6, and 1, respectively, to calculate the axial force distribution of the thread. As can be seen from Figure 23, when the friction coefficient is 1, the curve bending degree is the greatest, when the friction coefficient is 0, the curve bending degree is the lightest, the curve bending degree is greater, indicating that the more uneven the distribution of axial force, the smaller the curve bending degree, indicating that the more uniform the distribution of axial force. We can see that the friction coefficient of thread surface has an effect on the distribution of axial force.

Conclusion
This study provides a new method of calculating the thread stiffness considering the friction coefficient and analyzes the influence of the thread geometry and material parameters on the thread stiffness and also analyzes the influence of  (1) The results of the calculation of the thread stiffness calculated by the theoretical calculation method of this study are basically consistent with the results of the FEA. The results obtained by the test are smaller than the calculated results. This is due to the influence of the thread manufacturing on the experimental results.
(2) Thread-stiffness is closely related to material properties, pitch, and thread length. We can obtain higher stiffness by increasing Young's modulus of the material, increasing the length of the thread, and reducing the pitch. (3) We can also increase the friction coefficient of the thread joint surface to increase the stiffness of the thread connection, but we have found that using this method to increase the thread stiffness is limited.
(4) In order to make the axial load distribution of the thread uniform, we can reduce the friction coefficient of the thread surface, but we found that the use of this method to improve the distribution of the axial force of the thread has limited effectiveness.   The load on somewhere on the x-axis is F, where the screw thread axial deformation, mm :

Nomenclature
The load on somewhere on the x-axis is F, where the nut thread axial deformation, mm : Length along the helical direction, mm : Lead angle of the thread, degree ( ): Stiffness of the unit axial length of the screw, N/mm ( ): Stiffness of the unit axial length of the nut, N/mm 2 : Axial total deformation of the threaded connection, mm ( ): Stiffness of the unit axial length of the threaded connection, N/mm 2 : Total axial force (load), N : Atthe position, the axial force is ( ), the screw elongation amount : Atthe position, the axial force is ( ), the nut compression amount ( ): Vertical cross-sectional areas of screw at the position, mm 2 ( ): Vertical cross-sectional areas of nut at the position, mm 2 : Y oung'smodulusofthe screw body, N/mm 2 : Young'smodulusofthenut body, N/mm 2 ( ): Stiffness in the axial direction , N/mm : Overall stiffness of the threaded connection, N/mm Δ : Axial displacement, mm : Total axial force, N : Axialtensionload,N : Overall stiffness of the threaded connection, N/mm    : Axial deformation of the thread, mm : Thread original triangle high, mm.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.