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Dynamic Characteristics of Rotor System with a Slant Crack Based on Fractional Damping
Chinese Journal of Mechanical Engineering volume 34, Article number: 27 (2021)
Abstract
The traditional modeling method of rotor system with a slant crack considers only integerorder calculus. However, the model of rotor system based on integerorder calculus can merely describe local characteristics, not historical dependent process. The occur of fractional order calculus just makes up for the deficiency in integerorder calculus. Therefore, a new dynamic model with a slant crack based on fractional damping is proposed. Here, the stiffness of rotor system with a slant crack is solved by zero stress intensity factor method. The proposed model is simulated by RungeKutta method and continued fraction Euler method. The influence of the fractional order, rotating speed, and crack depth on the dynamic characteristics of rotor system is discussed. The simulation results show that the amplitude of torsional excitation frequency increases significantly with the increase of the fractional order. With the increase of the rotating speed, the amplitude of first harmonic component becomes gradually larger, the amplitude of the second harmonic becomes smaller, while the amplitude of the other frequency components is almost invariant. The shaft orbit changes gradually from an internal 8type shape to an ellipsetype shape without overlapping. With the increase of the slant crack depth, the amplitude of the transverse response frequency in the rotor system with a slant crack increases, and the amplitude in the second harmonic component also increases significantly. In addition, the torsional excitation frequency and other coupling frequency components also occur. The proposed model is further verified by the experiment. The valuable conclusion can provide an important guideline for the fault diagnosis of rotor system with a slant crack.
Introduction
Since the longterm transverse bending fatigue, the shaft is easy to form traversing cracks during the service period or beyond the service period. At present, the dynamic characteristics of rotor system with traversing cracks have been widely studied [1,2,3,4,5,6,7,8]. However, in engineering practice, many rotating shafts can also be used to transfer torque, which will inevitably produce slant cracks. In the 1880s, Ichimonji et al. [9] proposed the model of rotor system with a slant crack under the transversal steady state. He thought the response frequency of rotor system consists of frequency multiplication components and coupled frequency caused by the torsional excitation. Later, Sekhar et al. [10] used the similar solution method of transverse crack stiffness to solve the stiffness of the rotating shaft with a slant crack, and obtained the response frequency of rotor system in the transverse vibration base on the dynamic equations constructed by the finite element method. Darpe [11, 12] proposed the threedegreeoffreedom and fourdegreeoffreedom equations of rotor system with a slant crack respectively. Based on these proposed equations, the influence of the slant crack angle on the stiffness of rotor system was studied. Considering the response frequency of rotor system in the transverse vibration, he also compared the torsional excitation frequency of transverse crack with that of the slant crack. Based on the relationship between the stress intensity factor (SIF) and the strain energy release rate (SERR) in fracture mechanics, Lin et al. [13] derived the coupled stiffness of rotor system with an opentype crack of 45degree, and further used the RungeKutta method to solve the dynamic equations with coupled stiffness, and analyzed the influence of the crack depth, eccentricity, and rotating speed on the dynamic characteristics of rotor system with a slant crack. On the basis of material mechanics and fracture mechanics, Li et al. [14, 15] used the strain energy release rate (SERR) to obtain the stiffness matrix of rotor system with a slant crack, and discussed the influence of crack angle, slender ratio of shaft and crack depth on the stiffness of rotor system with a fully open crack. In addition, Refs. [14, 15] also analyzed the variation of nonlinear output frequency response function value in rotor system. Yu [16] proposed a finite element model of the Timoshenko beam, used the crack closure line method to describe the breath effect of crack, and compared the stiffness of rotor system with different crack angles. The results showed that the stiffness of rotor system changed nonlinearly with the angle of slant crack. Jiao et al. [17] deduced the flexibility matrix and stiffness matrix of arbitrary spatial slantcrack and analyzed the variable stiffness characteristics of traversing crack, transverseslantcrack, and spatial slantcrack. The results showed the shear stiffness of spatial slantcrack in the axial and vertical direction was very obvious. Lu et al. [18] used the energy principle to derive the stiffness matrix of an element with a slant crack, and used the crack closure line method to simulate the breathing behaviour of slant crack, and used the finite element method to construct the coupled dynamic equations of rotor system with a breathing crack base on the static unbalance and eccentricity distance.
The above proposed models of rotor system with a slant crack are based on integerorder calculus. However, in these dynamic models of rotor systems with a slant crack, the influence of fractional order characteristics of damping in rotor system is ignored. Therefore, it is necessary to introduce fractional damping into modeling of rotor system with slant crack. Compared with integerorder calculus, fractionalorder calculus has the following advantages.

(1)
Fractional calculus has global correlation and can better reflect the historydependent process of system function development. However integerorder calculus has locality and is not suitable for describing historydependent process.

(2)
The fractional calculus model overcomes the serious shortcoming, i.e. the theory does not agree well with the experimental results in the classical integerorder differential model, and can use fewer parameters to obtain good results.

(3)
Compared with the traditional nonlinear model, the physical meaning of the fractional model is clearer and the expression is more concise when the fractional calculus model is used to describe complex physical and mechanical problems.
In recent years, the fractional calculus has been successfully applied to the nonlinear dynamic analysis of rotor system [19,20,21]. For example, Xue et al. [22, 23] used fractional damping to proposed a transverse crack model of Jeffcott rotor system. However, the proposed equations only considered the transverse crack, not considered a slant crack. Here, fractional order calculus is introduced into the modeling of rotor system with a slant crack. The fractional damping model of rotor system with a slant crack is proposed, and the dynamic characteristics of the proposed model are studied.
The structure of this paper is as follows. In Section II, the first stress intensity factor (FSIF) of rotor system is calculated. In Section III, the flexibility of rotor system with a slant crack is calculated, and the variation of the additional flexibility of rotor system with a slant crack from opening to closing is studied. In Section IV, a dynamic model of rotor system with a slant crack based on fractional damping is proposed and solved by the fourorder RungeKutta method. According to the axis orbit and doublefrequency diagram, the influence of fractional order, rotating speed, and crack depth on the dynamic characteristics is discussed. In Section V, the experiments are completed on a rotor test rig in order to analyze the dynamic characteristics of rotor system with a slant crack at different rotational speeds. In Section VI, some valuable conclusions of this paper are given.
Basic Principle of SIF = 0
In the theory of fracture mechanics, the first stress intensity factor K_{I}, the second stress intensity factor K_{II}, and the third stress intensity factor K_{III} can be used to represent by the cracktip of openingtype crack (i.e., typeI crack), sheartype crack (i.e., typeII crack) and teartype crack (i.e., typeIII crack) respectively [24]. The opening and closing degree of the crack is related to the stress distribution at the edge of the crack, that is, to the value of the first stress intensity factor K_{I} at the edge of the crack. When K_{I} > 0, the corresponding crack is in the open region, and when K_{I} < 0, the corresponding crack is in the close region. Therefore, the line of the first stress intensity factor K_{I} = 0 can be used to distinguish the open and closed regions of the crack.
The diagrammatic sketch of the parameters of cracksection is shown in Figure 1. In Figure 1, ξ'O'η' is the coordinate axis fixed at the shaft center, a is the depth of the crack, dη' is the thickness of a sheet with a certain crack depth, h is the width of the sheet, α_{s} is the specified depth of the sheet. At any position of the crack area on the crack surface, the total first stress intensity factor K_{I} can be expressed as follows
where K_{Iξ'} and K_{Iη'} are the first stress intensity factor FSIF on the ξ'axis and η'axis respectively [25] and can be calculated by Eqs. (2) and (3):
where F_{2}(\(\frac{{\alpha_{s} }}{h}\)) and F_{1}(\(\frac{{\alpha_{s} }}{h}\)) are two influence coefficients, Q_{ξ} and Q_{η} are the sums of internal forces on the ξ'axis and θ'axis respectively, l is the length of the shaft, R is the radius of the shaft, η is the distance from the crack edge to the shaft center.
Additional Flexibility of Rotor System with a Slant Crack
Figure 2(a) is the diagrammatic sketch of rotor system with a slant crack of 45°, and Figure 2(b) is the diagrammatic sketch of crack section. In Figure 2, XOY is the coordinate axis fixed at the shaft center. The four loads subjected to the shaft are respectively the axial force q_{1}, the torque T, the bending moment q_{4} and the bending moment q_{5} in two directions.
In the Ref. [26], under different stresses, the stress intensity factor (SIF) of a unit thickness sheet with a certain crack depth was given. In order to obtain the stiffness of the cracked shaft, the shaft is divided into numerous thin sheets of thickness dY along the direction perpendicular to the yaxis to form the thin plates with oneside cracks of width \( h = 2\sqrt{{R}^2d{Y}^2} \), thickness dY, and crack depth a. Based on the formula of stress intensity factor (SIF) and the relationship between SIF and the strain energy release rate (SERR) [27], the energy release rate of slat body with onesided crack is obtained. The energy release rate is integrated on the whole crack cross section to obtain the strain energy release rate released by the whole crack body. Then, the additional flexibility of rotor system is calculated by Castigliano’s theorem. The flexibility of the cracked rotor is the sum of the additional flexibility and the flexibility of the shaft without crack. The stiffness of a patterned rotor is the inverse of the rotor compliance or the inverse of the rotor compliance matrix.
In order to obtain the additional flexibility of the cracked shaft, first, the stress intensity factor on the crack section should be calculated when the crack depth is a. For rotor system with a slant crack, the stress intensity factor generated by the four loads subjected to cracked shaft are respectively as follows:
where σ_{1}, σ_{4}, σ_{5}, and σ are respectively the maximum tensile stresses generated by q_{1}, q_{4}, q_{5,} and T, τ_{1} and τ_{5} are respectively the shear stresses generated by q_{1} and q_{5}. Therefore, the total stress intensity factor is
Suppose λ=πα/2h, according to Ref. [28], the three influence coefficients are respectively as follows:
The density function of strain energy release rate (SERR) is expressed as follows [29]:
where E' is the elastic modulus of the shaft, v is the crack opening displacement.
Suppose A' is the opening region of the crack (i.e., the first stress intensity factor K_{I} > 0), strain energy release rate (SERR) of rotor system can be expressed as
According to Castigliano’s theorem, the additional flexibility of rotor system can be obtained as follows:
Under the action of an unbalanced force, the axial force, and torque, the flexibility of the shaft without crack is
where I is the moment of inertia of the section, A is the area of the section.
The flexibility of the cracked shaft is the sum of the additional flexibility and the flexibility of the shaft without crack [30], i.e.,
where [G_{1}] and [G_{2}] are transformation matrices, which are mainly used to convert the rotation imbalance force, axial force, and torque to the two ends of the shaft in Figure 2. The transformation matrices are calculated as follows:
When calculating the additional flexibility of the shaft with a slant crack, the parameters are set as follows. The length of the shaft is L = 260 mm, the crack depth is a = d/2, and the diameter of the shaft is d = 9.5 mm. The variation of the slant cracked shaft with the additional flexibility of the crack from closing to opening is shown in Figure 3. In Figure 3, the ordinate axis represents the additional flexibility (unit: m/N), and the horizontal axis represents the boundary line of the crack opening area to the boundary line of the entire crack area. c(i, j) represents the additional displacement in the j direction under the unit force acting in the i direction, and 1, 2, 3, 4 represent two bending moment direction of the xaxis and yaxis, torque direction and axial direction respectively.
Dynamic Characteristics of Rotor System with a Slant Crack Based on Fractional Damping
In the rotor system, if the shaft is impacted by torsional vibration for a long time, it is easy to form a slant crack with an angle of about 45° in the shaft. Here, considering the influence of the transverse, torsional and axial motion on rotor system, the dynamic equation of the rotor system with a slant crack based on fractional damping is proposed, and the dynamic characteristics of rotor system with a slant crack are discussed.
In the case of torsional vibration, the Jeffcott rotor model is shown in Figure 4. In Figure 4, Ω represents the rotating speed, t represents time, Ωt+α is the rotating angle of the disc, φ is the torsion angle in the torsion direction.
The proposed dynamic equations of the rotor system with a slant crack is solved by the Runge Kutta method and continued fraction Euler method, and the steadystate responses in all directions are obtained. Then the Fourier transform method is used to process the obtained vibration signals. Finally, the vibration spectrum of the rotor system with a slant crack is obtained.
The Lagrangian equation is used to establish the dynamic equations of rotor system with a slant crack based on fractional damping in the transverse, torsion, and axial direction respectively.
where m is the mass of the disc, J_{p} is the moment of inertia of the disc, c is the damping coefficient in the transverse direction, c_{T} is the damping coefficient in the torsion direction. c_{u} is the damping coefficient in the axial direction, x and y is the displacement, D^{r} is the transformation coefficient of fractional order damping force, u is the displacement in the axial direction, e is the eccentric distance of the disc, M(t) is the torsional excitation. The stiffness matrix in fixed coordinate system is as follows:
where k_{x}, k_{y}, k_{T}, and k_{u} are respectively the stiffness in the two transverse directions of xaxis and yaxis, torsional direction, and axial direction, k_{xy}, k_{xT}, k_{yT}, k_{xu}, k_{yu}, and k_{Tu} are the coupling stiffness. In the most practical rotor system, the excitation force in the axial direction is small or even rare. So, the axial excitation force is not considered in Eq. (25).
The parameters required to solve the above equations are as follows. m = 0.595 kg, L = 260 mm, d = 9.5 mm, E = 210 GPa, c = 41.65 kg/s, c_{T} = 0.0091 kg·m^{2}/s, c_{u} = 146.2034 kg/s. The diameter of the disc is d_{p} = 76 mm. The phase angle of the disc is φ = π/6. The frequency of the torsional excitation is ω_{T} = 0.6 Ω. Poisson’s ratio is v = 0.3. The eccentricity of the disc is obtained by adding a screw with the weight of d_{m} = 3 g at a distance of 32.75 mm from the center of the disc.
Influence of the Fractional Order on the Dynamic Characteristics of Rotor System with a Slant Crack
In this section, the same crack depth and the same speed, we discuss the influence of the fractional order on the nonlinear dynamic characteristics of the rotor system with a slant crack based on fractional order damped. Here, crack depth is set to a = d/2, the rotating speed is set to Ω = 300 r/min.
From Figure 5 and Figure 6, the fractional order has a great influence on the dynamic characteristics of rotor system with a slant crack. When the fractional order is r = 0.2, the shaft orbit of the rotor system is a typical internal 8type shape. In the transverse spectrum, there is only the first harmonic component caused by unbalanced and the second harmonic component caused by the crack. In the axial direction, there is the 1/2order harmonic component, which is relatively weak than the first harmonic component. Because the shaft with a slant crack is subjected to torsion, tension, and bending moments, the stiffness of the shaft will occur tensionbending coupling, bendingtorsional coupling, and tensiontorsional coupling, the coupling frequency components will occur in the transverse spectrum diagrams. The inner circle in the shaft orbit of the rotor system gradually becomes smaller as the fractional order gradually increases. In the transverse spectrum diagram, the coupling frequency components become more and more obvious and their amplitudes gradually become larger. However, the amplitude of the first harmonic component and second harmonic component becomes smaller. The amplitude of the spectrum in the axial direction also gradually decreases.
Influence of Rotating Speed on the Dynamic Characteristics of Rotor System with a Slant Crack
In this section, under the same crack depth and the same fractional order, we discuss the influence of rotating speed on the dynamic characteristics of rotor system with a slant crack based on fractional damping. Here, the crack depth is set to a = d/2, the fractional order is set to r = 0.4.
From Figure 7 and Figure 8, the rotating speed has a great influence on the dynamic characteristics of the rotor system with a slant crack. When the rotating speed close to the critical speed or the subcritical speed, the noninear dynamic characteristics of the rotor system with a slant crack are very obvious. In Figure 7, when the rotating speed is relatively low, the shaft orbit is a typical internal 8type shape. In the transverse spectrum, the first harmonic and second harmonic are the main components. The amplitude of the second harmonic is greater than that of the first harmonic, the characteristic frequency of the crack is obvious. However, the other coupling frequency multiplication components are not prominent. In the axial spectrum, the 1/2order harmonic component is the main component, however the first harmonic is weak. The shape of the shaft orbit in the rotor system with a slant crack gradually changes from an internal 8type shape to an ellipsetype shape with the increase of the rotating speed. Because the shaft with a slant crack is subjected to the interference of the coupling stiffness, the ellipsetype shape in the shaft orbit is not overlapped together. In the transverse spectrum, the amplitude of the first harmonic gradually becomes larger. However, the amplitude of the second harmonic gradually becomes smaller or even disappears. In addition, the variation of the amplitude of the other coupling frequency multiplication components is almost obvious. In the axial spectrum diagram, the amplitudes of each frequency become gradually smaller.
Influence of the Crack Depth on the Dynamic Characteristics of Rotor System with a Slant Crack
In this section, under the same fractional order and the same rotating speed, the influence of crack depth on the dynamic characteristics of rotor system with a slant crack based on fractional damping is discussed. Here the fractional order is set to r = 0.4, the rotating speed is set to Ω = 300 r/min.
The occurrence of a crack will cause the variation in the stiffness of the shaft, the depth of the crack determines the amount of change in the stiffness of the shaft. Therefore, the depth of the crack has a great influence on the dynamic characteristics of the rotor system with a slant crack. From Figure 9 and Figure 10, when the crack depth is shallow, the crack has little effect on the stiffness of the shaft, and the crack information is almost obvious. Moreover, the shaft orbit of rotor system is an ellipsetype shape. In the transverse spectrum diagram, the first harmonic component caused by unbalance is the main component. The second harmonic and other coupling frequency components are relatively weak. In the axial spectrum diagram, the 1/2order harmonic is the main component, however the first harmonic is relatively weak. The influence of the crack on the stiffness of the shaft becomes more and more obvious with the increase of the crack depth. The shape of the shape orbit gradually changes from an ellipsetype shape to an internal 8type shape. In the transverse spectrum diagram, the amplitudes of the first harmonic and the other coupling frequency components are almost invariant. However, the amplitude of the second harmonic increases. In the axial spectrum diagram, the amplitude of the second harmonic also increases, however, this increase is relatively slow.
Experimental Study
The experiment data are collected on the ZT3 test bench. The vibrational displacement of rotor system in X and Y directions is measured by eddy current sensors. All cracked shafts in the experiment have the same geometric and material properties. The length of the shaft with aslant crack is L = 0.26 m, and the diameter of the shaft is d = 9.5 mm. A crack with a depth of 4 mm was obtained by wire cut electrical discharge machining (WEDM). The firstorder critical speed of the rotor system is 1754.6 r/min.
Figure 11 shows the diagrammatic sketch of bifurcation about the ratio of rotational speed to firstorder critical speed. From Figure 11, the bifurcation diagram of the rotor system with a slant crack is separated by three obvious change points, namely, 0.482, 0.577, and 0.871. Therefore, the rotational speed 846 r/min, 1012 r/min, and 1528 r/min are selected to evaluate the influence of the speed on dynamic characteristics of rotor system.
Figure 12 shows the shaft orbit and the spectrum of rotor system with a slant crack under the rotating speed is 846 r/min, 1012 r/min, and 1528 r/min, respectively. From Figure 12, the shaft orbit is an internal 8type shape. Under the rotating speed of 846 r/min, the first harmonic and second harmonic is the main components in the spectrum. In addition, the torsion frequency caused by the excitation of slant crack occurs, and relatively weak coupling frequency components also occur. The shaft orbit gradually changes from an internal 8type shape to an ellipsetype shape with the increase of the rotating speed. In the spectrum, the amplitude of the second harmonic gradually decreases, the amplitude of the first harmonic increases slowly, and the amplitude of the other frequency components is almost invariant. The experiment results are basically consistent with these of the numerical simulation in the rotor system with a slant crack.
Figure 13 shows the shaft orbit and the spectrum of rotor system under the different depths of the crack. From Figure 13, the experimental results are similar to the simulation results, the shaft orbit changes from an ellipsetype shape to an internal 8type shape with the increase of the crack depth. At the same time, the amplitude of the second harmonic also increased gradually.
Conclusions
In this paper, the method of SIF=0 is used to solve the stiffness of rotor system with a slant crack. Then, the fractional order calculus is introduced into the modeling of rotor system with a slant crack. The dynamic model of rotor system with a slant crack based on fractional damping is proposed, and the simulation has been successfully completed by numerical methods. The experiment results verified the effectiveness of the proposed model. The influences of fractional order, rotating speed, and crack depth on the nonlinear vibration characteristics of rotor system with a slant crack are discussed. The obtained results are summarized as follows.

1.
The variation of fractional order has a great influence on the dynamic characteristics of rotor system with a slant crack. The frequency caused by the torsional excitation and the coupling frequencies increases significantly with the increase of fractional order. Therefore, when the dynamic model of rotor system with a slant crack is established, we can adjust the different fractional order to propose a more realistic rotor system with a slant crack, in order to better study the slant cracks.

2.
When the rotating speed close to the critical speed or the subcritical speed, the dynamic characteristics of rotor system with a slant crack are very obvious. We can analyze the dynamic characteristics near the 1/3order or 1/2order critical speeds, and extract the fault information of rotor system with a slant crack. When the rotor is running at a low speed, the second harmonic is the main component in the spectrum of rotor system with a slant crack. The amplitude of the first harmonic gradually becomes larger as the rotating speed increases. However, the amplitude of the second harmonic and other coupling frequency components is almost invariant. The shaft orbit gradually changes from an internal 8type shape to an ellipsetype shape without overlapping together.

3.
Due to the occurrence of cracks, the stiffness of the shaft changes. The depth of the crack determines the amount of change in the stiffness of the shaft. The vibration amplitude in the transverse direction increases with the increase of the crack depth, and the amplitude of the second harmonic significantly increases. The frequency components caused by the torsional excitation and other coupling frequencies also occur.
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Funding
This project is supported by National Natural Science Foundation of China (Grant Nos. 51675258, 51261024, 51265039), State Key Laboratory of Mechanical System and Vibration (Grant No. MSV201914), and Laboratory of Science and Technology on Integrated Logistics Support, National University of Defense Technology (Grant No. 6142003190210).
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ZL and DW were in charge of the whole trial, YL wrote the manuscript, and HW assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
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Zhinong Li, born in 1966, is currently a professor at Key Laboratory of Nondestructive Testing, Ministry of Education, Nanchang Hangkong University, China. He received his Ph.D. degree from Zhejiang University, China, in 2003. His research interests include mechanical dynamics and fault diagnosis.
Yunlong Li, born in 1993, is currently a master candidate at Key Laboratory of Nondestructive Testing, Ministry of Education, Nanchang Hangkong University, China. His research interests include mechanical dynamics and fault diagnosis.
Dong Wang, born in 1984, is currently an associate professor at State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiaotong University, China. He received his Ph.D. degree from City University of Hong Kong, China, in 2015. His research interests include mechanical fault diagnosis.
Zhike Peng, born in 1974, is currently a professor at State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiaotong University, China. He received his Ph.D. degree from Tsinghua University, China, in 2002. His research interests include mechanical dynamics and fault diagnosis.
Haifeng Wang, born in 1987, is currently a master candidate at Key Laboratory of Nondestructive Testing, Ministry of Education, Nanchang Hangkong University, China. His research interests include mechanical dynamics and fault diagnosis.
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Li, Z., Li, Y., Wang, D. et al. Dynamic Characteristics of Rotor System with a Slant Crack Based on Fractional Damping. Chin. J. Mech. Eng. 34, 27 (2021). https://doi.org/10.1186/s1003302100543w
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Keywords
 Fractional calculus
 Slant crack
 Rotor system
 Dynamic characteristics