Dynamic Analysis of Flexible Shaft and Elastic Disk Rotor System Based on the Effect of Alford Force
Taken the flexible shaft and elastic disk rotor system as the research object, vibration differential equations are established by the modal synthesis method. Based on the Runge–Kutta method of direct integration, dynamic equations are solved successfully. Then, the effects of Alford force caused by the airflow excitation are researched on the stability and dynamic response. The results show modal vibration of the disk takes a great effect on the coupled natural frequency of shaft transverse vibration in the high rotational speed region. The effect of shaft-disk coupling makes rotor frequency to decrease, and different rotational speeds have little effect on rotor frequency. Eccentric distance is sensitive to the amplitude of rotor response, but it has little effect on rotor stability, motion law, and relative relationship of two response frequencies. The proportion of rotor responses caused by aerodynamic excitation force becomes larger with the increase of rotational speed and disk radius. However, the proportion decreases with the second support moving to the right, and it is helpful for the stable operation of the rotor system.
As a kind of large rotating machinery, a gas turbine is widely applied in industrial fields such as aviation, aerospace, transportation, and electric power. It takes an important role in national economy.
A gas turbine works under the high-speed environment, and various nonlinear factors are shown on strong coupling, high dimension, and unsteady state. There will be hidden serious problems in operation, and it is often huge and even disastrous if the accidents occur.
For a discontinuous system of the mass continuous distribution axis, it is difficult to express the dynamic characteristics of all parts in an equation. Gao and Meng  studied the vibration and stability characteristics of elastically supported cantilever beam by the transfer matrix method. But, this research was only on a continuous beam and shaft system, the effect of the disk was neglected on dynamic characteristics. By combining the modal synthesis method with the transfer matrix method, Han et al.  solved the dynamic characteristics of a complex rotor system.
With the appearance of optimization design techniques, the rotor structure is continuously improved. The wheel disk is made much thinner so that coupling vibration is more obvious between blade and disk. So elastic disk modeling is very important for the research on the gas turbine disk-shaft system. Zhang  established the perturbation motion equation of the flexible shaft-elastic disk rotor. Based on the finite element method, Meek et al. [4, 5] analyzed the dynamic characteristics of a rigid disc-flexible shaft rotor system. Hili et al.  developed an elastic disk-rigid shaft model with an elastic support, and the effects of elastic deformation and elastic support are discussed on rotor responses when the disk is under axial load.
In addition, fluid-solid coupling problems affect rotor stable operation seriously. It easily leads to the instability phenomenon by airflow excitation in gas turbine working [7, 8]. Because there is always the relative eccentricity between rotor blade and casing, the clearance of the blade tip is unevenly distributed along the circumferential direction. This makes the distributions of blade efficiency, pressure, and aerodynamic force different.
So except for a torque, blade circumferential aerodynamic force is also combined with Alford force, which is acting on the axis and perpendicular to eccentric shifting . Alford force is a cyclic force. Once it is larger than the external damping to maintain the system stable, positive work will be performed on the rotor system. That is inputting energy to promote the development of rotor whirl motion. The self-excited vibration will occur when its amplitude increases continuously.
A steam turbine was made by GE corporation, and a strong fluid gap vibration was produced when the load was increased . This is the first time that the phenomenon of aerodynamic excitation was discovered in a rotating machine. Soon after, Thomas also reported a similar vibration phenomenon . He first proposed the concept of gap excitation and suggested the calculation formula for turbine gap excitation force. However, the problem did not attract enough attention at that time.
In 1957, the first ultra-supercritical pressure units were operated at American Ohio Ferrer Power Plant. After that, the former Soviet Union, West Germany, and Japan had also developed and taken into operation the ultra-supercritical pressure units. In working operation, there are quite a few aerodynamic excitation problems on several supercritical units. Because of the higher steam parameters, aerodynamic excitation of a high-pressure rotor is increased on high-power ultra-supercritical pressure units.
In the 1960s, the problem of aerodynamic excitation occurred again. At this period, the gap excitation mainly took place in the turbine of high-power ultra-supercritical pressure units which were manufactured by the Federal Republic of Germany. In the study of aeroengines stability, American scholar J. S. Alford achieved the important progress in theory. In his research, the theory of gap-flow excitation was proposed for the first time, and the calculation formula of turbine excitation force was given. Then, the problem of gap excitation was given enough attention.
For experimental verification of Alford’s force, Urlichs and Wohkab [12, 13], respectively, developed the experimental research on turbine engines. Vance and Laudadio  performed the test of lateral force acting on the blower impeller and confirmed the presence of Alford force. In the sealing force models proposed by Thomas and Alford, circumferential flow velocity is ignored at the inlet, which is the primary factor of crosscut forces.
Soviet scholars studied the spiral effect of aerodynamic flow in a high-pressure turbine rotor and found that the inertial effect of aerodynamic flow is far greater than the friction effect in the sealing gap . At the end of the twentieth century, more researchers began to study aerodynamic excitation forces and the interaction forces between impeller and flow section [16, 17]. Shanghai Power Equipment Research Institute conducted an experiment on the aerodynamic oscillation of a gas turbine . Considering the excitation of aerodynamic force, a small-scale modeling test rig was established by the single-span flexible rotor. Then, a simplified calculation method was proposed for the airflow excitation stiffness. The stability and unstable rotation speed were analyzed on the 200 MW turbine power generators.
Chai et al.  developed a further research on the airflow exciting force. Taken from the fluid dynamics, the Alford force formula of the uniform flow field was derived on a straight blade and a twisted blade. Chen and Sun  proposed an entirely new method of oscillatory fluid dynamics, and it was used for solving the airflow stability and dynamic coefficient of air seal.
Lu et al.  analyzed aerodynamic excitation in the cylinder of 200 MW high-pressure turbine and discussed the effect of aerodynamic excitation on the stability of the rotor system. For modifying the formula of Alford force, Yang et al.  decomposed rotor displacement into static displacement and dynamic displacement. This provided a method for adjusting rotor static displacement to reduce the exciting force.
In this paper, vibration differential equations of an elastic disk-flexible shaft rotor are established by the modal synthesis method. Based on the Runge–Kutta method of direct integration, dynamic equations are solved successfully. Then, the effects of Alford force caused by the airflow excitation are researched on the stability and dynamic response.
2. Vibration Characteristics of Disk-Shaft Coupling System
Figure 1 is the model of the elastic disk with inner diameter a and outer diameter b. For keeping the disk orthogonal to the shaft in all modes, it is assumed that the contact between disk and shaft is rigid. The differential equation of transverse elastic vibration can be derived by Hamilton’s principle. The free vibration equation of the disk is expressed as
In order to simulate dynamic characteristics of a flexible shaft and an elastic disk rotor system, the elastic circular plate with an inner clamp edge and outer freedom edge is adopted in the research.
Bending displacement of the elastic disk is expressed aswhere is the orthogonal mode of the disk and is the time-related part of generalized coordinates. Here, it is the simple harmonic vibration, so , where is the i-order natural frequency of the elastic disk. can be expressed aswhere is indicated as the orthogonal mode with m nodal diameters and n circles. can be expressed as by the reordering of natural frequencies increasing. is the radial distribution of mode shapes; and are constants which determine the initial angle of the disk combined mode.
For a full symmetric disk, theoretical analysis and experiments have shown there is no fixed initial angle for the disk mode whether it is uniform or not. To simplify the following formula, it is generally assumed that the initial angle is zero. When boundary conditions are certain, has the inherent form.
Inertia force of the vibrating disk is expressed as
In the formula,
The direction of inertia moment at point is perpendicular to the axis. It is expressed as
In the formula,
From equations (4) and (6), it can be found that the zero-diameter mode of the disk is coupled with longitudinal vibration of the shaft by the effect of inertia force . In addition, one pitch diameter mode of the disk is coupled with bending vibration of the shaft by the effect of inertia moment. However, other pitch diameter modes of the disk are not coupled with bending vibration of the shaft. So the bending coupled vibration is only discussed, and one pitch diameter mode of the disk is selected.
3. Dynamic Modeling of Flexible Shaft and Elastic Disk Rotor System
3.1. Rotor Dynamics Model of Flexible Shaft and Elastic Disk
The model of the flexible disk-flexible rotor system is shown in Figure 2. The disk is located at the leftmost end of the shaft. The distance from the first spring support to the disc is , the distance between the two spring supports is , and the distance from shaft end to the second spring support is .
For describing motion state of the shaft, four variables , and , are taken in two planes which are relatively vertical. Among them, and indicate small lateral deformation by the effect of bending, and and indicate small lateral deformation by the effect of shearing. Therefore, total deformation of the shaft is expressed as
In the rotating coordinate system, potential energy of the shaft due to bending and shearing can be expressed aswhere is the shape factor which is a constant related to the shaft cross-sectional shape. It can characterize the distribution of uneven shear stress in the cross-sectional area. , , and are axis Young’s modulus, shear mode, and cross-sectional area, respectively. is the axis inertia moment related to diameter.
Due to translation and rotation, kinetic energy of the axis can be expressed aswhere is the density of the shaft. and denote the small-angle rotation of the shaft cross section due to bending, and they can be expressed as
Kinetic energy of the disk is expressed aswhere is the mass of the disk, and are the disc inertia moments of axis and radial direction, and are the shaft displacements at the disc location, and and denote the small rotation angle caused by shaft bending. The tiny rotation angle is ignored which is caused by the rigid body displacements because it will be eliminated in the Lagrangian equation.
Strain energy of the disk can be expressed as
For elastic support in the rotor system, it can be calculated by the principle of virtual work. The formula is shown as follows:
The elastic disk-shaft rotor is approximately analyzed by the modal synthesis method. Ritz base of each substructure is obtained with the subsystem hypothesis morphological method. Eigenfunctions of transverse vibrations from flexible beam and shear beam with free ends are selected as the hypothetical function of continuous shaft transverse deflection induced by bending and shearing. Free vibration eigenfunction of the balanced disk is applied as the hypothesis function of the elastic disk in the system.
Transverse deformation of the elastic disk is expressed aswhere corner marker “1” indicates the first section diameter mode, corner marker “n” indicates the number of pitch circles (not including pitch circle subjected to the inner circle clamp). and are the Bessel functions of the first and second types, respectively, and and are the improved Bessel functions of the first and second types, respectively. , , , and are the constants which are determined by boundary conditions, the number of pitch circles, and the condition of orthogonality. is the dimensionless parameter related to natural frequency , and the relation is expressed as
Coefficient matrix determinant of linear equations is zero by the boundary conditions, so can be obtained.
According to the Lagrange equation:where , and the following is obtained:
As can be seen from Figure 2, each boundary or connection condition is shown as follows.
The boundary condition of O point is expressed as
Connection condition of A point is expressed as
The boundary condition of B point is expressed as
The boundary condition of free end is expressed as
By solving the equation set formed by the above equations, natural frequency and mode shape of the system can be obtained.
3.2. Differential Equations of Rotor System under Airflow Excitation Force
In order to analyze the effect of Alford force on rotor stability, the equivalent cross stiffness is applied to indicate the action of Alford force. Figure 3 illustrates the process of instability caused by equivalent cross stiffness. point represents the axis before the deformation, and point represents the axis after deformation. is the the radius of casing, and is the clearance between rotor and casing. Assuming that the rotor is precessing by the effect of unbalance, the direction of precession is shown by the arrow in Figure 3. is the inertia force, is the elastic recovery force, is the external damping force, and is induced by the Alford force. If is greater than the force of outer damping on the turbine, it will accelerate the positive precession. So the inertial force increases, the deflection increases, the unstable force increases, and the precession is faster. With this development, vibration amplitude of self-energized motion increases until it is limited by the structure or reaches the limit circle (or has caused damage before).
By the action of rotational dragging, the fluid will generate a circular flow in the sealing gap, which plays a significant role in rotor vibration. In a certain conditions, it will cause the rotor instable and generate self-excited lateral vibration. In 1987, Muszynska and Bently  proposed the Muszynska model based on a large number of experiments. This model introduces the average circumferential speed ratio τ to characterize the overall motion of fluid in the gap between the bearing and sealing. Rotational effect of seal excitation force is the main factor of rotor instability. The reaction force caused by sealing excitation force rotates around the rotor at a fixed angular velocity. In fixed coordinates, this reaction force can be described as the following mathematical equation:where , , and indicate the stiffness, damping, and inertia effects of fluid’s disturbing motion on the rotor. It is generally taken that cyclic force is rotor tangential driving force constituted by the cross stiffness term . According to the equation, larger fluid damping is easy to make rotor instable. Reducing the average circumferential velocity ratio can improve the stability of the rotor-seal system. A large number of experiments  have proved the assertion that the application of antiswirling structure can improve the stability of the sealing system. In addition, the main stiffness term in the model, that is, rotor radial stiffness , can become a negative value at the effect of fluid centrifugal inertia force.
Perturbation analysis and experimental data show that  the effect of fluid centrifugal inertia force becomes very obvious, when the ratio of seal gap and rotor radius is small. This has a great influence on the stiffness matrix. So the radial stiffness term can easily reach a negative value, and it is also a factor that causes rotor instable. The Muszynska model correctly reflects the basic dynamic characteristics of sealing force. Equation (23) not only is applied in the condition of small displacement at the rotor center position but also can indicate the nonlinear sealing force at larger displacement. Experimental and numerical results confirm that  , , and are the nonlinear functions of disturbed displacements and . It can be expressed as follows:where is rotor relative partial displacement and is the sealing clearance. Specific seal is approximately described by selecting the appropriate exponents n and b. Due to the effects of axial flow, surface roughness, secondary flow, and other factors, in general . , , and can be calculated by the combination of Childs seal dynamic coefficient equation and equation (24):
The Muszynska sealing force model is mainly based on a large number of experiments and abstracted from the model. It has a strong generality and grasps the main mechanical characteristics of sealing force. The correctness of the Muszynska model has been widely recognized in the application. The Muszynska sealing force model is adopted as the model of airflow excitation force, and the dynamic differential equation of the elastic disc-flexible shaft rotor is established by the modal synthesis method.
Applying the principle of virtual work, virtual work caused by the Alford force can be obtained aswhere
According to the modal synthesis method, the dynamic equation of the rotor can be expressed aswhere is the imbalance force caused by eccentric mass and is the force of bearing oil film. is expressed aswhere
So , , , and in equation (29) are expressed as
Assuming that , , and , equation (28) is expressed as
4. Analysis and Discussion of the Results
In order to explain the effects of different parameters on rotor vibration responses, rotor parameters are taken as follows if no special description is given. The geometric parameters of rotor structure are indicated as , , and . The inner and outer radii, thickness, density, elastic modulus, and Poisson’s ratio of the elastic disk are, respectively, indicated as , , , , , and . The radius, density, Poisson’s ratio, and elastic modulus of the flexible shaft are indicated as , , , and . The stiffness coefficient of elastic support is indicated as . The damping coefficient of the support is indicated as .
In the model of aerodynamic excitation force, the circumferential loss coefficient of sealed inlet airflow is indicated as . The seal length is indicated as . The radial seal clearance is indicated as . Fluid average axial velocity in the sealing cavity is indicated as . The sealing radius is indicated as . The coefficient of fluid viscosity is indicated as . The empirical coefficients are indicated as , , , and . Axial seal pressure drop is indicated as , where is the density of gas.
4.1. Natural Vibration Analysis of Flexible Shaft and Elastic Disk Rotor
4.1.1. Comparison of Results at Different Modes
Take the shaft bending mode number, shaft shear mode number, and the disk modal number are, respectively, as 25, 25, and 4. The first eight natural frequencies of the rigid disk-flexible shaft rotor system can be obtained based on the boundary and connection conditions. Similarly, the first eight natural frequencies of the elastic disk-flexible shaft rotor system can be obtained, as shown in Table 1.
The corresponding mode shapes are shown in Figure 4.
From modal vibration shapes in Figures 4(a) and 4(b), it can be seen that the vibration of the disk is not obvious at the low-order frequency. As the frequency increases, the vibration of the disk gradually becomes prominent.
The above derivation is performed in the rotating coordinate of shaft and the rotating coordinate of the disc, so the obtained natural frequencies are also relative to the rotating coordinate system. In order to compare with the experimental data, it is necessary to convert the obtained natural frequency into an inertial coordinate system. For the transverse vibration of the shaft and disk, the conversion relationship is different between rotating coordinate system and inertial coordinate system. Hence, there are two different natural frequencies in the inertial coordinate system.
4.1.2. Analysis of Coupled Natural Frequency in Transverse Vibration
For transverse vibration of the shaft, conversion relation is expressed aswhere is the natural frequency in the rotating coordinate system, is the angular velocity of the shaft, and is the natural frequency in the inertial coordinate system.
For transverse vibration of the disk, the transformation is difficult from the rotating coordinate system to the inertial coordinate system. Compared with the rotation angle , declination angles and of transverse vibration can be ignored in the transformation of the coordinate system. So, the relationship between two angular coordinate systems can be expressed as
Excitation force of the disc is , and vibration response at and is expressed aswhere is the natural frequency in the rotating coordinate system and is the number of mode nodal diameters in the disk. It can be seen from the above equation that there are two resonant frequencies in the inertial coordinate system:
Plotting the chart of frequency-rotational angular velocity, and are the continuous curves. Therefore, natural frequencies are generally divided into two parts when rotating. That is, at a certain rotational angular velocity, two different excitation frequencies will excite the same response mode in the inertial space.
For the specific problem, natural frequency of disk-shaft coupling is indicated as . The antiprecession mode of the rotating system may be negative, and only the first nodal diameter mode of the disk will couple with the vibration of shaft, so . Equation (36) can be expressed as
According to equation (28), the coupled natural frequency of shaft transverse vibration is shown as Figure 5. Here, 5(a) is the diagram of coupled natural frequency in the flexible shaft and rigid disk, and 5(b) is the diagram of coupled natural frequency in the flexible shaft and elastic disk.
In Figure 5, the difference between Figures 5(a) and 5(b) is obvious when the rotational speed is large, and the difference between two models is smaller in the region with lower rotational speed. It is illustrated that modal vibration of the shaft occupies a dominant position in the low rotational speed region. Modal vibration of the disk has little effect on the coupled natural frequency of shaft transverse vibration, while it takes a great effect in the high rotational speed region. Hence, the difference between two models is quite obvious.
According to equation (37), the coupled natural frequency of elastic disk transverse vibration is shown as Figure 6. Here, 6(a) is the diagram of forward natural frequency, and 6(b) is the diagram of backward natural frequency.
Mass matrix and stiffness matrix of the rotor system will be affected by the coupling effect of shaft and disk, but the gyroscopic matrix and damping matrix are not affected. Without considering the physical background, the disk-shaft coupling term can be directly removed in the mass matrix and stiffness matrix. So, it is to analyze the effect of the disk-shaft coupling on rotor natural frequency.
From Figure 7, it can be seen that the effect of shaft-disk coupling makes rotor frequency decrease, and different rotational speeds have little effect on rotor frequency.
The effects of different structural parameters are discussed on rotor frequency. Figure 8(a) shows the frequency of the rotor system decreases with the increase of outer diameter. From Figure 8(b), the first-order frequency of the rotor system decreases when the first support moves to the right, but the third-order frequency increases. In Figure 8(c), the second-order frequency of the rotor system decreases obviously when the second support moves to the right, and it has little effect on the frequencies of other orders.
4.2. Stability Analysis of Elastic Disk-Flexible Shaft at Aerodynamic Force
The effects of different parameters are discussed on rotor time-domain chart, axis orbit, and spectrum graph. Based on the variation of rotor responses, the stability of rotor system is analyzed. Here, the dimensionless displacement in the y direction is denoted as Y, the dimensionless displacement in the z direction is denoted as Z, the ratio of average circumferential speed is denoted as τ, the frequency of rotor response is denoted as f, and the amplitude of rotor response is denoted as A.
4.2.1. The Effect of Eccentric Distance and Rotational Speed
Figures 9 and 10 show the responses of the rotor system at different imbalance and rotational speed. According to the spectrum graph, the responses of the rotor system include the frequencies caused by aerodynamic excitation force and rotor imbalance. Summarizing the above figures, rotor frequency caused by aerodynamic excitation force can be obtained, and it is about 50 Hz.
Comparing with rotor responses at different eccentric distances, there is no clear change in rotor motion law and relative relationship between two response frequencies. It illustrates eccentric distance has little effect on rotor stability, and only the amplitude of rotor response is changed.
With the increase of rotational speed, the value of vibration frequency caused by the imbalance increases, and the proportion of vibration response caused by aerodynamic excitation force becomes larger. Ultimately, the rotor system is caused unstable when rotational speed increases to be a certain value.
4.2.2. The Effect of Disk Radius
Figure 11 shows the responses of the rotor system at different disk radii. According to spectrum graphs, the proportion of rotor responses caused by aerodynamic excitation force becomes larger with the increase in the disk radius and gradually covers the response caused by the imbalance. Comparing with the charts of the time domain, vibration amplitude of the rotor system increases rapidly with the increase in the disk radius. The rotor system becomes unstable when the disk radius increases to be a certain value.
4.2.3. The Effect of Support Position
Figure 12 shows the responses of the rotor system at different positions of the second support. According to spectrum graphs, the proportion of rotor responses caused by aerodynamic excitation force decreases with the second support moving to the right. Comparing with the charts of the time domain, vibration amplitude of the rotor system becomes smaller rapidly with the second support moving to the right. It is because that support center of the whole rotor system is also moving to the right; the mass of disk and Alford force can be balanced. The rotor system would be stable when the second support moves to the right at a certain position.
Vibration differential equations of the elastic disk-flexible shaft rotor are established by the modal synthesis method. Based on the Runge–Kutta method of direct integration, vibration characteristics of the elastic disk and the flexible shaft rotor were analyzed successfully. Then, the effects of Alford force caused by the airflow excitation are further researched on the stability and dynamic response. The results show the following:(1)Comparing with vibration characteristics of different rotor models, modal vibration of the disk takes a great effect on the coupled natural frequency of shaft transverse vibration in the high rotational speed region. The effect of shaft-disk coupling makes rotor frequency decrease, and different rotational speeds have little effect on rotor frequency. The frequency of the rotor system decreases with the increase in outer diameter. The first-order frequency decreases with the first support moving to the right, but the third-order frequency increases. When the second support moves to the right, the second-order frequency of the rotor system decreases obviously.(2)Analysis of rotor responses at different structure parameters was performed, and eccentric distance is sensitive to the amplitude of rotor response, but it has little effect on rotor stability, motion law, and relative relationship of two response frequencies. The proportion of rotor responses caused by aerodynamic excitation force becomes larger with the increase of rotational speed and disk radius. The rotor system becomes unstable when rotational speed or disk radius increases to a certain value. However, the proportion decreases with the second support moving to the right, and it is helpful for the stable operation of the rotor system.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The work was supported by the Natural Science Foundation of Liaoning Province (Grant no. 20180550880), Education Department Series Project of Liaoning Province (Grant no. L201746), and Young Doctor Scientific Research Foundation of College (Grant no. 17YB04).
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