#### 1. 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.

#### 2. Mathematical Model

According to 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.

For the ISO thread, the axial deformation of the thread at at the axial unit width force is thread bending deformation , thread shear deformation , thread root inclination deformation , thread root shear deformation , and radial direction extended deformation (nut) or radial shrinkage deformation (screw), and calculate these deformations of the thread, respectively, and then sum them, that is, the total deformation.

###### 2.1.1. Bending Deformation

In the threaded connection, under the action of the load, the contact surface friction coefficient is μ, 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, respectivelyand

The friction generated along the slope is wμ; i.e.,

The force wμ is also decomposed into x-direction force and y-direction force, which are and , respectively.

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 (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 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; 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 wμ, 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 (shown in Figure 3(a)) of the cantilever beam with variable cross-section under load one has

###### 2.1.2. Shear Deformation

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

###### 2.1.3. 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 [31]

According to the static analysis, the thread is subjected to radial force (shown in Figure 4), and it is known from the literature [31] that the internal and external thread radial deformation (shown in Figure 3(d)) areand

###### 2.1.5. 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]

For ISO internal threads, the relationship between a, b, c, and pitch isSubstituting (17) into (9), (10), (11), (12), (13), (14), and (16) one gets the relation

For ISO internal threads, the relationship between a, b, c, and pitch is

Substituting (24) into (9), (10), (11), (12), (13), (15), and (16) type one gets the relation

By adding these deformations separately, the total deformation (shown in Figure 4) of screw thread and nut thread can be obtained under the action of force .

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 isand

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, β is the lead angle of the thread shown in Figure 6, and then

Assume

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, , and .

As shown in Figure 1(a), the screw is subjected to the tensile force , 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 , , , and is (see 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

Let

Then

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 =-()/ and =, 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

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

Substituting (44) and (55) into (57), the stiffness of the bolt connection is expressed as

#### 3. 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 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 μ of the thread contact surface is set first. In FEA, the contact algorithm used is Augmented Lagrange. Figures 11(a)11(c) are the force convergence curves for FEA of threaded connections. Figures 12(a)12(c) are the effect of the reciprocal of the mesh size on the axial force obtained by FEA. We can see from Figures 12(a)12(c) 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

#### 4. 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

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].

#### 5. Results and Discussion

##### 5.1. 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 aluminum-aluminum 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 = =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 aluminum-aluminum 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 specification is M6×1×6.1 and the friction coefficients are 0.01, 0.05, 0.1, 0.2, 0.25, and 0.3. (as shown in Figures 1520)

Figures 21 and 22 show the results of stiffness calculations. The thread size is M10×1.5×9 and M6×1×6.1, the material is steel, Poisson’s ratio of the material is 0.3, Young’s Modulus of the material is 200 GPa, and the thread surface friction coefficient is taken as 0.01, 0.05, 0.1, 0.2, 0.25, and 0.3. Calculated using the theories of this paper, FEA and Yamamoto, respectively, and from Figures 21 and 22, we can see that the results of FEA are very similar to the results of the theoretical calculations of this paper, the variation trend of stiffness with friction coefficient is the same, and it increases with the increase of friction coefficient, and the results of the FEA are in good agreement with those of the FEA; however, Yamamoto theory does not consider the influence of the friction coefficient on stiffness, and this is obviously unreasonable.

##### 5.2. Effect of Friction Coefficient on Axial Force Distribution

Take the thread size as M6×0.75×6.1, the axial load 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.