Modal Analysis for a Rod-Fastened Rotor considering Contact Effect Based on Double Fractal Model

+e rod-fastened rotor is the core component of the gas turbine. It comprises several discs and tie rods.+e flexural stiffness of the contact interface is the key factor for rotordynamic analysis. +e contact interfaces of the discs are usually manufactured by grinding.+emeasured contour curve of the contact interfaces of an experimental rod-fastened rotor is analyzed by the structural function method, which shows that the contact interfaces can be well described by the double fractal model with fractal dimensions D1 and D2 and the fractal roughness parameters G1 and G2. +e Hertz model is used to analyze the contact of the single asperity on the contact interface. On this basis, the flexural stiffness of the contact interface considering the pretightening force and the bending moment is derived. Modal frequencies of the experimental rod-fastened rotor under different pretightening forces and the bending moment (caused by gravity) are obtained by three-dimensional finite element analysis and experimental modal tests. It is observed that the modal frequencies increase with the nominal pressure of the contact interface, and the experimental results are consistent with the calculated results.


Introduction
e rod-fastened rotors are widely used in aeroengines and heavy-duty gas turbines (see Figure 1).In order to avoid the damage caused by flexural vibration, it is necessary to accurately calculate the flexural vibration modal frequencies of the gas turbine rotor at the design stage.Because the rotor is not a continuous whole, the contact effect between two discs has a great influence on the dynamic characteristics.ere are some researches about the dynamic performance of such structures.Klompas [1] analyzed the influence of the bending moment on the dynamic characteristic of the rotor joints in the turbomachine.Lee and Lee [2] investigated the effect of the central tie rod on the rotordynamic performances of an auxiliary power unit gas turbine rotor.Lu et al. [3] calculated the modal frequencies of a rod-fastened rotor subjected to different pretightening forces by the threedimensional finite element method.Liu et al. [4] studied the dynamic stability of a rod-fastened rotor bearing system considering the nonuniform pretightening forces.However, the effect of the contact stiffness on the dynamic characteristic of the rod-fastened rotor was not considered.
For the contact stiffness of the contact interface, some researchers have studied the contact stiffness in microscale.Rao et al. [5] calculated the contact stiffness using the Hertz elastic contact theory and Greenwood-Williamson statistical model.He et al. [6] studied the contact stiffness of rough contact interfaces using Greenwood-Williamson model and the elastic-plastic contact model.Zhang et al. [7] identified the contact stiffness by modal test and obtained the relationship between the contact stiffness and the pretightening force.ere are also some researches to analyze the flexural stiffness considering the bending moment.Isa et al. [8] established a bilinear stiffness model to calculate the flexural stiffness of a rod-fastened rotor subjected to the bending moment.Gao et al. [9] studied the effect of the bending moment and the pretightening force on the flexural stiffness of the contact interface in the rod-fastened rotor.Several researchers calculated the equivalent stiffness of bolted structures by the finite element method.Lehnhoff et al. [10,11] calculated the equivalent stiffness of the bolted joints using the axisymmetric finite element model.Yuan et al. [12] calculated the contact stiffness and flexural stiffness of the gas turbine rotor with curvic couplings using the three-dimensional finite element model.
In the above studies, the contact stiffness is calculated using Greenwood-Williamson statistical model.However, the contact interfaces of the discs in the actual gas turbine are manufactured by grinding.Due to the different scales of plastic deformation and microfracture during processing, such machined surfaces generally have double fractal characteristic [13].Jiang et al. [14] calculated the normal contact stiffness of the joint surface based on the fractal model and compared the theoretical calculation results with the experimental data.Buczkowski et al. [15] used the Weierstrass-Mandelbrot function to express the fractal features of the rough surface and calculated the normal contact stiffness of the isotropic rough surface during extrusion deformation.
However, there are few investigations that focus on the analysis of the flexural stiffness of the rod-fastened rotor, considering the pretightening force and the bending moment based on double fractal model.In this paper, an experimental rod-fastened rotor is designed.
e contact interfaces of the rotor are grinding.e contour curve of the contact interfaces is analyzed by the structural function method, which shows that the contact interfaces have double fractal characteristic.
e normal contact stiffness of the contact interface is obtained by the Hertz contact theory and the double fractal model.e flexural stiffness of the contact interface considering the pretightening force and the bending moment is also derived.Finally, the modal frequencies of the rotor under different pretightening forces are obtained by the three-dimensional finite element method and the experimental modal method, respectively.Comparing the experimental test results with the calculated results, it is found that the double fractal model is reasonable to calculate the contact stiffness of the rod-fastened rotor.

e Experimental Rod-Fastened Rotor.
e experimental rod-fastened rotor consists of discs, a front shaft end, a rear shaft end, and nine tie rods (see Figure 2).e 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.e structure of the rotor is bilaterally symmetrical.e material of the rotor is 40Cr.
e elastic modulus E is 206 GPa, the density ρ is 7870 kg/m 3 , and the Poisson ratio v is 0.3.

e Mechanical Model of the Contact Segment in the Experimental Rod-Fastened Rotor.
ere are some contact asperities on the rough contact interface.e asperities come into contact under the pretightening force F pre .According to the fractal model, a small plastic contact point first appears on the contact interface.As the pretightening force increases, the small plastic contact points merge into large contact points.When the contact area exceeds the critical value, the large contact points begin to make elastic contact.As shown in Figure 4, each asperity on the rough contact interface can be seen as a spring with bending stiffness, and all asperities on the contact interface can be characterized by a spring with equivalent flexural stiffness K cc .
en, the equivalent flexural stiffness K eq of the contact segment is a series connection of the flexural stiffness K cc of the contact interface and the flexural stiffness K s of the continuous shaft with the length L s .K eq is given by For the continuous shaft with the length L s , the flexural stiffness is where E is the elastic modulus, I is the moment of inertia, and L s is the length of the contact segment.
Considering the effect K cc on the K eq , the flexural stiffness correction factor of the contact segment η is introduced by e 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.

e Equivalent Flexural
where L is the sampling length, G 1 and G 2 are the fractal roughness parameters which determine the profile height, D 1 and D 2 are the fractal dimensions of the surface topography (1 , which determine the distribution ratio of high frequency and low frequency in the surface contour height, and c is the contour spectral density.e contact interfaces of the experimental rod-fastened rotor are machined by grinding, and the rough contact When τ ≤ τ 12 , the structural function of the W-M function is ( When τ ≥ τ 12 , the structural function of the W-M function is where τ is the displacement along the x direction, C e 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.

2.3.2.
e Contact Model of the Double Fractal Surface.e 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.e 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 where H is the hardness value of the material, is the yield stress, E is the elastic modulus, ] is the Poisson ratio, and a π(r t ) 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- Shock and Vibration 3 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] δ According to the Hertz contact theory, the normal elastic force of a single contact asperity can be expressed as where E * is the equivalent elastic modulus.
where v 1 , v 2 and E 1 , E 2 are Poisson's ratio and elastic modulus of the contact interfaces, respectively.Because the same material is used for each part of the experimental rodfastened rotor, In Figure 6, the relationship between r t and δ is where the radius of curvature R e is general much larger than δ.en, equation ( 11) can be expressed as Substituting equations ( 8) and ( 12) into (9) yields e critical truncated area a c is given by [17] a c 2 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 τ 12 corresponds to the truncated area a 12 .e asperities deformation of the double fractal area can be divided into four cases according to the relationship among a c1 , a c2 , and a 12 : 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 < a 12 are elastically deformed.Case 2: when a c1 > a 12 , all contact asperities in region I are plastically deformed.Case 3: when a c2 < a 12 , all contact asperities in region II are elastically deformed.Case 4: when a c2 < a 12 , the contact asperities satisfying a 12 < 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 a 12 can be obtained, respectively, and the relationship is a 12 < a c2 < a c1 .According to the relationship, the double fractal area of the contact interface satis es the case 2 and case 4.
For the truncated area a L of the maximum contact asperity, when a L < a 12 , the frequency distribution function of the contact asperities is given by [18] When a L > a 12 , the frequency distribution function of the contact asperities is expressed as   Shock and Vibration e 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 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 sti ness calculation.

e Flexural Sti ness of the Contact Interface of the
Experimental Rod-Fastened Rotor.From the Hertz contact theory, the normal contact sti ness of each microcontact pair can be written as Neglecting the contact sti ness of the asperities in the state of plastic deformation, the normal contact sti ness K n of the entire contact interface is equal to the sum of the normal contact sti ness of all elastic contact pairs of the contact interface.e contact sti ness K n can be expressed as e value of K n can be obtained by the method of numerical integration.For the annulus contact interface, the exural sti ness of contact interface K cc can be given by where A is the nominal contact area of the contact interface and y is the distance along the axis.e value of K cc is determined by the normal contact sti ness 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 As shown in Figure 7, the contact interface of the rodfastened rotor is a concentric annular surface and the outer and inner radius of the annulus are R 1 and R 2 , respectively.When the contact interface is in the full contact state, the contact pressure is a linear function of y, then where a and b are the constant coe cients.e 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 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 coe cients a and b can be obtained.Substituting them into equation ( 22) yields where P 0 F pre /A.When the whole zone of the contact interface is in full contact, y c 0. Equation (25) can be written as where ξ is the exural dimensionless load coe cient, ξ MR 1 /IP 0 , and β y/R 1 .
For the annulus contact interface, Combining equations ( 21) and ( 27), the K cc is expressed as (28)

Equivalent Flexural Sti ness K eq of the Contact
Segment. e exural sti ness 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 exural sti ness of the contact segment K eq can be obtained.K eq will be a ected by the pretightening force, Shock and Vibration the bending moment, the contact area, and the length of the contact segment.To analyze the e ect of above factors on K eq , the contact segment corresponding to the contact interface C2 is selected to conduct the analysis.

E ect of the Pretightening Force and the Bending
Moment on K eq .e pretightening force and the bending moment applied on the experimental rod-fastened rotor will a ect the contact pressure of the contact interface.In equation ( 26), the exural dimensionless load coe cient ξ is introduced to represent the in uence of the pretightening force and the bending moment.e e ect of the contact interface on K eq is denoted by the exural sti ness correction coe cient η (see equation ( 3)).For the contact interface C2, the value of R 1 and R 2 is 110 mm and 70 mm, respectively.
e nominal contact pressure P 0 can be obtained under di erent pretightening forces.Figure 8 shows the e ect of the pretightening force and the bending moment on η.When ξ is constant, η increases with P 0 , but the increasing trend gradually slows down.When P 0 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.

E ect of the Contact
Area on K eq .According to equation (28), the exural sti ness K cc is also in uenced by the contact area.e outer radius R 1 is given as 110 mm, and the ratio of the inner radius and outer radius α R 2 /R 1 is de ned.
e change of the contact area is obtained by changing α. Figure 9 shows the e ect 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 sti ness with the increase of the contact area.

E ect of the Length of Contact Segment L s on K eq .
According to equations ( 1) and (3), η is in uenced 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 in uence on η.When L s reaches a certain value, η no longer increases with L s .erefore, 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.

Modal Analysis of the Experimental Rod-Fastened Rotor.
e modal analysis of the rotor is conducted, considering the e ect of the contact sti ness.In this paper, the e ect of the contact interface on the exural sti ness is simulated by modifying the elastic modulus of the contact segment.According to equation (3), the exural sti ness correction coe cient η is obtained.en, where E eq is the equivalent elastic modulus of the contact segment.e modal analysis is performed, considering the e ect of the pretightening force and the gravity bending moment.
ere 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. e pretightening forces corresponding to tie rod set 1 and tie rod set 4 are kept equal and constant.e di erent pretightening forces applied on the rotor are listed in Table 2. e distribution of the gravity bending moment of the experimental rod-fastened rotor is shown in Figure 11.e distribution of the gravity bending moment is bilaterally symmetric.e gravity bending moments corresponding to the positions of C2 and C3 are selected as the bending moments of the contact interfaces.e length L s of the two contact segments is both 50 mm.According to the 6 Shock and Vibration pretightening force listed in Table 2, the exural sti ness correction coe cients η of the two contact segments are listed.
As shown in Table 3, η increases with the pretightening forces.
e modal analysis of the experimental rod-fastened rotor is conducted using ANSYS Version 11.0.e block Lanczos method is applied to solve the modal analysis.e three-dimensional nite element model is shown in Figure 12. e rst ve exural mode shapes of the experimental rod-fastened rotor are shown in Figure 13.It can be seen that the rst ve 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 rst ve orders of modal frequencies of the experimental rodfastened rotor under di erent pretightening forces.As the pretightening force increases, the modal frequency of the rotor gradually increases.Meanwhile, the in uence of the pretightening force on the 3rd and 5th order modal frequencies is greater than that of other orders.is 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 sti ness.

Experiment Validation.
To validate the calculation method considering the pretightening force and the bending moment, a simpli ed experimental rod-fastened rotor is manufactured (see Figure 2(a)), and the modal experiment is conducted.
e multipoint exciting method is used to measure the modal parameters of the rotor.
e rotor is excited by the force hammer.An acceleration sensor is used to acquire the response signals.e range of the measured frequency is 0-3200 Hz with a frequency resolution of 1.25 Hz. e 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 re ect the e ect of gravity bending moment, the experimental rotor is suspended by an elastic wire rope.e experimental results of the rst ve orders of modal frequencies are shown in Table 5. e 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 P 0 , 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 di erence 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 sti ness, which is determined by the pretightening force.
e 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 P 0 (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 P 0 , the relative errors decreased.According to the results of relative errors, it is found that the calculated results are correct.

Conclusions
In this paper, the contact sti ness of the rod-fastened rotor is analyzed considering the pretightening force and the bending moment.e results are summarized below:  Shock and Vibration 7 (1) e double fractal model and Hertz contact theory are used to calculate the normal contact sti ness of the contact interface.e e ect of the pretightening force, the bending moment, the contact area, and the length of the contact segment on the equivalent exural sti ness K eq are analyzed.When the exural dimensionless load coe cient ξ is constant, the exural sti ness correction coe cient η increases with the nominal contact pressure P 0 (pretightening force), but the increasing trend gradually slows down.
When P 0 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.
(2) e calculated and experimental modal analysis is conducted considering the pretightening force and the bending moment (due to gravity).As P 0 increases, the modal frequencies increase, and the experimental results are consistent with the calculated results.erefore, the double fractal model is reasonable to calculate the contact stiffness of the rod-fastened rotor.

Figure 1 :
Figure 1: Typical structure of the rod-fastened rotor.
1 and C 2 are the coe cients related to the fractal dimension, and < > indicates spatial average.e structural function diagram of the contour curve is shown in Figure 5. e structural function diagram is divided into two parts according to τ 12 which represents the boundary displacement of the two fractal regions along the x direction.e 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.

Figure 6 :Figure 5 :
Figure 6: Schematic of a microcontact established between an asperity of composite rough surface and opposing rigid plane.

Figure 7 :
Figure 7: Schematic for the contact interface.

Figure 8 : 2 Figure 9 :
Figure 8: E ect of the pretightening force and the bending moment on η.

Figure 11 :Figure 10 :
Figure 11: Distribution of the gravity bending moment of the experimental rod-fastened rotor.

Table 1 :
e fractal parameter values of the contact interface.

Table 2 :
Di erent pretightening forces applied on the tie rod set 2 and tie rod set 3.