2.1. The Experimental Rod-Fastened Rotor

The experimental rod-fastened rotor consists of discs, a front shaft end, a rear shaft end, and nine tie rods (see Figure 2). The rotor is clamped together by the circumferentially arranged tie rods. A rough contact interface is existed between two discs of the rotor. As shown in Figure 3, there are six rough contact interfaces of C1–C6 in the rotor. The structure of the rotor is bilaterally symmetrical. The material of the rotor is 40Cr.

The elastic modulus E is 206 GPa, the density ρ is 7870 kg/m³, and the Poisson ratio v is 0.3.

2.2. The Mechanical Model of the Contact Segment in the Experimental Rod-Fastened Rotor

The flexural stiffness correction coefficient η represents the stiffness reduction of the contact interface to the continuous shaft. For the contact segment, η is mainly related to the pretightening force and the bending moment.

2.3. The Equivalent Flexural Stiffness of the Contact Interface K_cc

2.3.1. The Double Fractal Model of the Contact Interface in the Experimental Rod-Fastened Rotor

A rough surface profile with double fractal characteristic can be represented by Weierstrass–Mandelbrot (W-M) function [16]:

$$\begin{array}{}z\left(x\right)=L{\left(\frac{G_1}{L}\right)}^{D_1-1}\sum _{n=0}^{n_{\text{c}}}\cos \left(2\pi {\gamma }^{n}x{L}^{-1}\right){\gamma }^{\left(D_1-2\right)n}+L{\left(\frac{G_2}{L}\right)}^{D_2-1}\sum _{n=0}^{n_{\max }}\cos \left(2\pi {\gamma }^{n}x{L}^{-1}\right){\gamma }^{\left(D_2-2\right)n},\end{array}$$ (4)

where L is the sampling length, G₁ and G₂ are the fractal roughness parameters which determine the profile height, D₁ and D₂ are the fractal dimensions of the surface topography (1 < D₁ < 2, 1 < D₂ < 2), which determine the distribution ratio of high frequency and low frequency in the surface contour height, and γ is the contour spectral density.

The contact interfaces of the experimental rod-fastened rotor are machined by grinding, and the rough contact interfaces can be regarded as isotropic. The contour curves of the contact interfaces are measured by the SJ201P surface topographer. According to the contour curve, the fractal parameters can be obtained by the structural function method.

When τ ≤ τ₁₂, the structural function of the W-M function is

$$\begin{array}{c}{S}_1\left(\tau \right)=〈{\left[z\left(x\right)-z\left(x+\tau \right)\right]}^2〉=C_1{G}_1^{22D_1-}{\tau }^{42-D_1}.\end{array}$$ (5)

When τ ≥ τ₁₂, the structural function of the W-M function is

$$\begin{array}{c}{S}_2\left(\tau \right)=〈{\left[z\left(x\right)-z\left(x+\tau \right)\right]}^2〉=C_2{G}_2^{22D_2-}{\tau }^{42-D_2},\end{array}$$ (6)

where τ is the displacement along the x direction, C₁ and C₂ are the coefficients related to the fractal dimension, and < > indicates spatial average. The structural function diagram of the contour curve is shown in Figure 5. The structural function diagram is divided into two parts according to τ₁₂ which represents the boundary displacement of the two fractal regions along the x direction. The region I corresponds to the fractal structure formed by the microfracture of the smaller scale, and the region II corresponds to the fractal structure formed by the plastic deformation of the larger scale. The contact of two rough surfaces can be equivalent to the contact between a rough face and a rigid face, and the structural function is the sum of the two rough face structure functions. According to the structural function diagram, the fractal parameters of the contact interface are obtained and shown in Table 1.

Table 1. The fractal parameter values of the contact interface.

D1	D2	G1 (m)	G2 (m)	τ12 (m)	Re (μm)
1.299	1.730	1.065 × 10−10	1.811 × 10−7	2.80 × 10−5	1.57

2.3.2. The Contact Model of the Double Fractal Surface

The contact of two rough surfaces can be equivalent to the contact between a rough face and a rigid face. Figure 6 shows the microcontact established between an asperity of the rough surface and opposing rigid plane. The contact asperity is assumed to be a sphere, when the small contact asperities are in the state of full plastic.

Deformation of the normal contact force is given by

$$\begin{array}{}{F}_{\text{p}}=Ha,\end{array}$$ (7)

where H is the hardness value of the material, H = (2Y/3){1 + ln[E/(2(1 − ν)Y)]}, Y is the yield stress, E is the elastic modulus, ν is the Poisson ratio, and $a=\pi {\left(r_{\text{t}}\right)}^2$ is the truncated area of the contact asperity, r_t is the radius of the truncated area.

When the pretightening force is increased, the small contact asperities are merged into a large contact asperity, and the large contact asperity begins to make elastic contact. When a_L ≤ a_c, the contact interface is in a fully plastic contact state. When a_L > a_c, the contact interface enters the elastic-plastic contact state, where a_L is the truncated area of the maximum contact asperity and a_c is the critical truncated area where the elastic contact occurs.

For the contact asperity, the normal deformation δ can be expressed as [17]

$$\begin{array}{}\delta =2G^{\left(D-1\right)}{\left(\ln \gamma \right)}^{12/}{\left(2r_{\text{t}}\right)}^{\left(2-D\right)}.\end{array}$$ (8)

According to the Hertz contact theory, the normal elastic force of a single contact asperity can be expressed as

$$\begin{array}{}{F}_e\left(\delta \right)=\frac{4E^{\ast }{R}_{\text{e}}^{12/}{\delta }^{32/}}{3},\end{array}$$ (9)

where E^∗ is the equivalent elastic modulus.

$$\begin{array}{}{E}^{\ast }={\left[\frac{\left(1-v_1^2\right)/E_1+\left(1-v_2^2\right)}{E_2}\right]}^{-1},\end{array}$$ (10)

where v₁, v₂ and E₁, E₂ are Poisson’s ratio and elastic modulus of the contact interfaces, respectively. Because the same material is used for each part of the experimental rod-fastened rotor, E₁ = E₂ = E and v₁ = v₂ = v.

In Figure 6, the relationship between r_t and δ is

$$\begin{array}{}{\left(R_{\text{e}}-\delta \right)}^2+r_{\text{t}}^2=R_{\text{e}}^2,\end{array}$$ (11)

where the radius of curvature R_e is general much larger than δ. Then, equation (11) can be expressed as

$$\begin{array}{}{r}_{\text{t}}^2=2R_{\text{e}}\delta .\end{array}$$ (12)

Substituting equations (8) and (12) into (9) yields

$$\begin{array}{}{F}_{\text{e}}\left(\delta \right)=\frac{2^{\left(92-D\right)/2}}{3{\pi }^{\left(3-D\right)/2}}{\left(\ln \gamma \right)}^{12/}{G}^{\left(D-1\right)}{E}^{\ast }{a}^{\left(3-D\right)/2}.\end{array}$$ (13)

The critical truncated area a_c is given by [17]

$$\begin{array}{}{a}_{\text{c}}={\left[\frac{2^{\left(92-D\right)}{\pi }^{\left(D-3\right)}{G}^{\left(22D-\right)}\ln \gamma }{9}{\left(\frac{E^{\ast }}{H}\right)}^2\right]}^{1/\left(D-1\right)}.\end{array}$$ (14)

For the double fractal contact interface, region I is corresponding to the critical truncated area a_c1, region II is corresponding to the critical truncated area a_c2, and the boundary displacement τ₁₂ corresponds to the truncated area a₁₂. The asperities deformation of the double fractal area can be divided into four cases according to the relationship among a_c1, a_c2, and a₁₂:

•  

  Case 1: when a_c1 < a_c2, the contact asperities satisfying a < a_c1 in region I are plastic deformation, and the contact asperities satisfying a_c1 < a < a₁₂ are elastically deformed.

•  

  Case 2: when a_c1 > a₁₂, all contact asperities in region I are plastically deformed.

•  

  Case 3: when a_c2 < a₁₂, all contact asperities in region II are elastically deformed.

•  

  Case 4: when a_c2 < a₁₂, the contact asperities satisfying a₁₂ < a < a_c2 in region II are plastically deformed, and the contact asperities satisfying a > a_c2 are in the elastic deformation state.

According to the fractal parameter values of the double fractal contact interface listed in Table 1, the value of truncated areas a_c1, a_c2, and a₁₂ can be obtained, respectively, and the relationship is a₁₂ < a_c2 < a_c1. According to the relationship, the double fractal area of the contact interface satisfies the case 2 and case 4.

For the truncated area a_L of the maximum contact asperity, when a_L < a₁₂, the frequency distribution function of the contact asperities is given by [18]

$$\begin{array}{}{n}_1\left(a\right)=\frac{D_1}{2}\frac{{\left(a_{\text{L}}\right)}^{D_1/2}}{{\left(a\right)}^{\left(D_1+2\right)/2}},\left(a<a_{12}\right).\end{array}$$ (15)

When a_L > a₁₂, the frequency distribution function of the contact asperities is expressed as

$$\begin{array}{c}{n}_1\left(a\right)=\frac{D_1}{2}{\left(\frac{a_{\text{L}}}{a_{12}}\right)}^{D_2/2}\frac{{\left(a_{12}\right)}^{D_1/2}}{{\left(a\right)}^{\left(D_1+2\right)/2}},\left(a<a_{12}\right),n_2\left(a\right)=\frac{D_2}{2}\frac{{\left(a_{\text{L}}\right)}^{D_2/2}}{{\left(a\right)}^{\left(D_2+2\right)/2}},\left(a_{12}<a<a_{\text{L}}\right).\end{array}$$ (16)

The total normal contact force of the contact interface F_N is the sum of the plastic contact force and the elastic contact force of the two regions. F_N can be expressed as

$$\begin{array}{}{F}_{\text{N}}={\int }_0^{a_{12}}{F}_{\text{p}}{n}_1\left(a\right)d\left(a\right)+{\int }_{a_{12}}^{a_{\text{c}2}}{F}_{\text{p}}{n}_2\left(a\right)d\left(a\right)+{\int }_{a_{\text{c}2}}^{a_{\text{L}}}{F}_{\text{e}}{n}_2\left(a\right)d\left(a\right).\end{array}$$ (17)

F_N is determined by the pretightening force F_pre. Combining with equation (17), a_L can be obtained. It will be applied to the following contact stiffness calculation.

2.3.3. The Flexural Stiffness of the Contact Interface of the Experimental Rod-Fastened Rotor

From the Hertz contact theory, the normal contact stiffness of each microcontact pair can be written as

$$\begin{array}{}{k}_{\text{n}}=\frac{d{F}_{\text{e}}\left(\delta \right)}{\delta }.\end{array}$$ (18)

Neglecting the contact stiffness of the asperities in the state of plastic deformation, the normal contact stiffness K_n of the entire contact interface is equal to the sum of the normal contact stiffness of all elastic contact pairs of the contact interface. The contact stiffness K_n can be expressed as

$$\begin{array}{}{K}_{\text{n}}={\int }_{a_{\text{c}2}}^{a_{\text{L}}}{k}_{\text{n}}{n}_2\left(a\right)d\left(a\right).\end{array}$$ (19)

The value of K_n can be obtained by the method of numerical integration. For the annulus contact interface, the flexural stiffness of contact interface K_cc can be given by

$$\begin{array}{}{K}_{\text{cc}}={\iint }_A\frac{K_{\text{n}}}{A}{y}^{2}dA,\end{array}$$ (20)

where A is the nominal contact area of the contact interface and y is the distance along the axis.

The value of K_cc is determined by the normal contact stiffness per unit area k_n0 and the contact area. Since k_n0 is a function of the contact pressure P, K_cc also can be expressed as

$$\begin{array}{}{K}_{\text{cc}}={\iint }_A{k}_{\text{n}0}\left(P\right)y^{2}dA.\end{array}$$ (21)

As shown in Figure 7, the contact interface of the rod-fastened rotor is a concentric annular surface and the outer and inner radius of the annulus are R₁ and R₂, respectively. When the contact interface is in the full contact state, the contact pressure is a linear function of y, then

$$\begin{array}{}P\left(y\right)=ay+b,\end{array}$$ (22)

where a and b are the constant coefficients.

The rod-fastened rotor is subjected to the pretightening force and the bending moment during the operation. When the pretightening force F_pre and the bending moment M are applied on the rod-fastened rotor, F_pre and M can be written as

$$\begin{array}{}{F}_{\text{pre}}={\iint }_{A}P\left(y\right)dA={\iint }_A\left(ay+b\right)dA=a{S}_{\text{x}}+bA,\end{array}$$ (23)

$$\begin{array}{}M={\iint }_{A}P\left(y\right)ydA={\iint }_A\left(ay+b\right)ydA=aI+b{S}_{\text{x}},\end{array}$$ (24)

where S_x = Ay_c is the static moment of the contact interface and y_c is the coordinate of the centroid. Combining equations (23) and (24), the coefficients a and b can be obtained. Substituting them into equation (22) yields

$$\begin{array}{}P\left(y\right)=\frac{M-P_0{S}_{\text{x}}}{AI-S_{\text{x}}^2}\left(Ay-S_{\text{x}}\right)+P_0,\end{array}$$ (25)

where P₀ = F_pre/A.

When the whole zone of the contact interface is in full contact, y_c = 0. Equation (25) can be written as

$$\begin{array}{}P\left(y\right)=P_0\left(1+\xi \beta \right),\end{array}$$ (26)

where ξ is the flexural dimensionless load coefficient, ξ = MR₁/IP₀, and β = y/R₁.

For the annulus contact interface,

$$\begin{array}{}dA=2\sqrt{R_1^2-y^2}dy.\end{array}$$ (27)

Combining equations (21) and (27), the K_cc is expressed as

$$\begin{array}{}{K}_{\text{cc}}=2R_1^4{\int }_{-1}^1{k}_{\text{n}0}\left(P\right){\beta }^2\sqrt{1-{\beta }^2}d\beta .\end{array}$$ (28)

3.1. Equivalent Flexural Stiffness K_eq of the Contact Segment

The flexural stiffness K_cc of the contact interface in the experimental rod-fastened rotor can be obtained by the above calculation method. Combining with equation (1), the equivalent flexural stiffness of the contact segment K_eq can be obtained. K_eq will be affected by the pretightening force, the bending moment, the contact area, and the length of the contact segment. To analyze the effect of above factors on K_eq, the contact segment corresponding to the contact interface C2 is selected to conduct the analysis.

3.1.1. Effect of the Pretightening Force and the Bending Moment on K_eq

The pretightening force and the bending moment applied on the experimental rod-fastened rotor will affect the contact pressure of the contact interface. In equation (26), the flexural dimensionless load coefficient ξ is introduced to represent the influence of the pretightening force and the bending moment. The effect of the contact interface on K_eq is denoted by the flexural stiffness correction coefficient η (see equation (3)). For the contact interface C2, the value of R₁ and R₂ is 110 mm and 70 mm, respectively. The nominal contact pressure P₀ can be obtained under different pretightening forces. Figure 8 shows the effect of the pretightening force and the bending moment on η. When ξ is constant, η increases with P₀, but the increasing trend gradually slows down. When P₀ is constant, η decreases with the increase of ξ, which indicates that the contact interface will appear to be separated with the increase of the bending moment.

3.1.2. Effect of the Contact Area on K_eq

According to equation (28), the flexural stiffness K_cc is also influenced by the contact area. The outer radius R₁ is given as 110 mm, and the ratio of the inner radius and outer radius α = R₂/R₁ is defined. The change of the contact area is obtained by changing α. Figure 9 shows the effect of the contact area on η. When ξ is constant, η increases with the decrease of α, indicating that the bending moment weakens against the bending stiffness with the increase of the contact area.

3.1.3. Effect of the Length of Contact Segment L_s on K_eq

According to equations (1) and (3), η is influenced by the length L_s of the contact segment. As shown in Figure 10, when L_s is small, the length of the contact segment has a greater influence on η. When L_s reaches a certain value, η no longer increases with L_s. Therefore, it is necessary to select the appropriate length of the contact segment during the calculation. In this paper, L_s is equal to 50 mm.

3.2. Modal Analysis of the Experimental Rod-Fastened Rotor

The modal analysis is performed, considering the effect of the pretightening force and the gravity bending moment. There are six contact segments in the experimental rod-fastened rotor (see Figure 3). Due to the bilateral symmetrical structure, we only change the pretightening forces corresponding to tie rod set 2 and tie rod set 3, and the pretightening force of tie rod set 2 is equal to that of tie rod set 3. The pretightening forces corresponding to tie rod set 1 and tie rod set 4 are kept equal and constant. The different pretightening forces applied on the rotor are listed in Table 2.

The distribution of the gravity bending moment of the experimental rod-fastened rotor is shown in Figure 11. The distribution of the gravity bending moment is bilaterally symmetric. The gravity bending moments corresponding to the positions of C2 and C3 are selected as the bending moments of the contact interfaces. The length L_s of the two contact segments is both 50 mm. According to the pretightening force listed in Table 2, the flexural stiffness correction coefficients η of the two contact segments are listed.

As shown in Table 3, η increases with the pretightening forces.

The modal analysis of the experimental rod-fastened rotor is conducted using ANSYS Version 11.0. The block Lanczos method is applied to solve the modal analysis. The three-dimensional finite element model is shown in Figure 12. The first five flexural mode shapes of the experimental rod-fastened rotor are shown in Figure 13. It can be seen that the first five mode shapes are mainly the vibration of the front and rear shaft ends, and the mode shapes of the discs are not obvious. Table 4 shows the calculated values of the first five orders of modal frequencies of the experimental rod-fastened rotor under different pretightening forces. As the pretightening force increases, the modal frequency of the rotor gradually increases. Meanwhile, the influence of the pretightening force on the 3rd and 5th order modal frequencies is greater than that of other orders. This can be explained from the mode shapes of the rotor in Figure 13. Since the 3rd and 5th order mode shapes of the discs are larger than the other orders, the frequencies of the 3rd and 5th order are more sensitive to the change of the contact stiffness.

3.3. Experiment Validation

To validate the calculation method considering the pretightening force and the bending moment, a simplified experimental rod-fastened rotor is manufactured (see Figure 2(a)), and the modal experiment is conducted. The multipoint exciting method is used to measure the modal parameters of the rotor. The rotor is excited by the force hammer. An acceleration sensor is used to acquire the response signals. The range of the measured frequency is 0–3200 Hz with a frequency resolution of 1.25 Hz. The measurement system is shown in Figure 14.

In order to verify the calculated results, the pretightening forces applied on the experimental rotor are set according to Table 2. In order to reflect the effect of gravity bending moment, the experimental rotor is suspended by an elastic wire rope. The experimental results of the first five orders of modal frequencies are shown in Table 5. The comparison between the calculated and experimental results is shown in Figure 15. It can be seen that the modal frequencies gradually increase with the nominal contact pressure P₀, and the change trend of the calculated results is consistent with that of the experimental results. For the 3rd order and the 5th order, the difference between the calculated and experimental results is large. Because the 3rd and 5th order mode shapes of the discs are larger than the other orders, the frequencies of the 3rd and 5th order are more sensitive to the change of the contact stiffness, which is determined by the pretightening force.

The relative errors between the calculated and experimental results are listed in Table 6. For the 3rd and 5th order, the relative errors are large when the nominal contact pressure P₀ (pretightening force) is small. It could be explained by the relative errors of pretightening forces applied on the rotor being large when the pretightening forces are small. With the increase of P₀, the relative errors decreased. According to the results of relative errors, it is found that the calculated results are correct.