Dynamic Characteristics of Rotor System with a Slant Crack Based on Fractional Damping

The traditional modeling method of rotor system with a slant crack considers only integer-order calculus. However, the model of rotor system based on integer-order calculus can merely describe local characteristics, not historical dependent process. The occur of fractional order calculus just makes up for the deficiency in integer-order 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 Runge-Kutta 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 8-type shape to an ellipse-type 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 long-term 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 threedegree-of-freedom and four-degree-of-freedom 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 open-type crack of 45-degree, and further used the Runge-Kutta 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, transverse-slant-crack, and spatial slant-crack. The results showed the shear stiffness of spatial slant-crack 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 integer-order 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, fractional-order calculus has the following advantages.
(1) Fractional calculus has global correlation and can better reflect the history-dependent process of system function development. However integer-order calculus has locality and is not suitable for describing history-dependent 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 integer-order 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 four-order Runge-Kutta method. According to the axis orbit and double-frequency 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 crack-tip of opening-type crack (i.e., type-I crack), shear-type crack (i.e., type-II crack) and tear-type crack (i.e., type-III 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 ( α s h ) and F 1 ( α 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.  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.

Additional Flexibility of Rotor System with a Slant Crack
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 y-axis to form the thin plates with one-side cracks of width h = 2 √ R 2 − dY 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 one-sided 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 Figure 1 Diagrammatic sketch of the parameters of crack section 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 x-axis and y-axis, 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.
(22) mẍ + cD r x + k x x + k xy y + k xT α + k xu u = −mg + me(� + D r α) 2 cos(�t + α + ϕ) + meα sin(�t + α + ϕ), 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 x-axis and y-axis, 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). (23) mÿ + cD r y + k xy x + k y y + k yT α + k y u = me(� + D r α) 2 sin(�t + α + ϕ) − meα cos(�t + α + ϕ), (24) J pα + c T (� + D r α) + k xT x + k yT y+k Tu u= M(t)+mgsin(�t+α+ϕ) + mẍ(�t+α+ϕ) − mÿe cos(�t+α+ϕ), (25) mü + c u D r u + k xu x + k yu y + k Tu α + k u u = 0, (26)  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 8-type shape. In the transverse spectrum, there is only the first harmonic component caused by unbalanced and the second harmonic component caused 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 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 tension-bending coupling, bending-torsional coupling, and tension-torsional coupling, the coupling frequency 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 8-type 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 1/2-order 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 8-type shape to an ellipse-type 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 ellipse-type 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 ellipse-type 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 1/2-order 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 ellipse-type shape to an internal 8-type 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 ZT-3 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 first-order 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 first-order 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 8-type 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 8-type shape to an ellipse-type 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 ellipse-type shape to an internal 8-type shape with the increase of the crack depth. At the same time, the amplitude of the second harmonic also increased gradually.