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The influence of shaft bending on the coupling vibration of rotor-blades system is nonignorable. Therefore, this paper analyzed the influence of shaft bending on the coupling vibration of rotor-blades system. The vibration mode function of shaft under elastic supporting condition was also derived to ensure accuracy of the model as well. The influence of the number of blades, the position of disk, and the support stiffness of shaft on critical speed of system was analyzed. The numerical results show that there were two categories of coupling mode shapes which belong to a set where the blade’s first two modes predominate in the system: shaft-blade (SB) mode and interblade (BB) mode due to the coupling between blade and shaft. The BB mode was of repeated frequencies of (

In the compressor/turbine engineering equipment, the blade, the disk, and the shaft are assembled together by a certain connecting structure and forming a blade-disk-shaft coupling system which has some coupling characteristics. With the development of scientific research, many scholars in this field have carried out a lot of research on the vibration of blades rotor coupling system. However, the influence of shaft bending on the coupling vibration of rotor-blades system is nonignorable. And there are potential couplings between shaft bending and blade bending under the right condition in fact. Therefore, this present research focuses on the influence of shaft bending on the coupling vibration of rotor-blades system.

The flexible disk model is the most popular analytical model for such systems at first. Parker and Mote [

As the rotating systems become lighter and more flexible with higher operating speed for higher productivity and economical design, the rigid shaft-disk model cannot accurately predict the vibration characteristics of the system. And the shaft-blades model is considered, considering the coupling vibration among shaft bending and blade bending. Anegawa et al. [

Then the disk-blade model is used to predict the influence of stagger and twist on natural frequencies for stationary as well as rotating bladed disk systems [

From the preceding research it could be seen that there were two kinds of analysis method for the typical structure. One was regarding the support of blade as clamped or elastic support and analyzing the vibration of the blade separately and the other was regarding blade-disk system as a rigid body component and analyzing the dynamic characteristics of the rotor system separately. However, the research on blade-disk-shaft coupling system had gradually become the focus of research. Although some scholars had researched the coupling vibration of shaft torsion and blade bending, the influence of shaft’s bending is usually ignored. But we found that there is a new mode which is SB due to the coupling vibration of shaft’s bending and blade’s bending. And the present paper aim is to analyze the influence of shaft bending on the coupling vibration of shaft-disk-blades system. What is more, influences of the blades number, disk position, and support stiffness on the characteristic value of system, the Campbell diagram, and the mode shape of vibration were studied by numerical simulation.

A dynamic model of rotor-blade coupling system with elastic restraints is shown in Figure

Schematic diagram of the model of rotor-blade coupling system.

According to the relevant knowledge of elastic mechanics, the kinetic energy of the shaft can be expressed as

Regarding the disk as a mass point and ignoring the influence of the disk on the vibration mode of beam, the kinetic energy of disk can be represented in the following form:

The total kinetic energy of rotor system can be obtained as follows:

Analyzing the model of shaft and considering the torsion, bending, and shearing, the total potential energy of shaft can be given as follows:

In order to simulate the boundary condition of shaft more accurately, the shaft is simplified as an elastic support beam with springs at both ends and simplified model is shown in Figure

Schematic diagram of the simplified elastic support beam model.

Assuming the central axis of each section of beam has bending vibration at the same plane, applicating plane assumption in the process of vibration, excluding the effects of rotational inertia and shear deformation, and ignoring the rotation around the center axis of the cross-section and damping, the free vibration differential equation of beam has the following expression:

The solution of (

Equation (

As the elastic support beam is adopted, boundary conditions can be expressed as

Assuming

Substituting (

If there are nonzero solutions for

Vibration model functions of the direction of

Assuming that the shear force

Substituting (

Ignoring the member does not appear in the equation; it is written as follows:

Discretizing the kinetic energy of disk, the discrete kinetic energy of disk is shown in Appendix

The total kinetic energy of the rotor system is

In the same way, discretizing the potential energy of shaft, the discrete potential energy of shaft is shown in Appendix

In order to facilitate the representation and computation, it is written as

Substituting the kinetic energy equation (

Assuming there are

Schematic diagram of disk-blade system.

The displacement of arbitrary point

When

Specific expression is as follows:

Considering the effects of bending, circumferential compression, and centrifugal and normal-force-generating potential energy of blade, total potential energy of

Because the movement of end of the rigid disk and the blade is consistent, the modal function of cantilever beam is chosen for blade:

Vibration model functions of the displacement of blades

Substituting (

Substituting kinetic energy equation (

Discretize the differential equations of motion of the

Schematic diagram of assembly of matrices of the rotor-blade coupling system.

Mass matrix

Gyroscopic matrix

Stiffness matrix

Damping matrix

Assembled differential equations of motion of a rotor-blade coupled system are written as

In order to further study the vibration characteristics of the coupling system, it is necessary to analyze the critical speed. Specific parameters are shown in “Geometric and Material Properties of Rotor-Blade System.”

In order to reduce the length of article, only these effects on critical speed of rotor-blade coupling system are studied in this section: the number of blades

Figure

Frequency of blade changes due to the number of blades in a three-to-seven-blade rotor.

First-order frequency changing diagram

Second-order frequency changing diagram

Figure

Frequency of rotor changes due to the number of blades in a three-to-seven-blade rotor.

First-order frequency changing diagram

Second-order frequency changing diagram

Figure

Campbell diagram of blade changes due to the number of blades.

First-order Campbell diagram

Second-order Campbell diagram

Campbell diagram of rotor changes due to the number of blades are shown in Figure

Campbell diagram of rotor changes due to the number of blades.

First-order Campbell diagram

Second-order Campbell diagram

Figure

Frequency of blade changes due to the position of disk.

First-order frequency changing diagram

Second-order frequency changing diagram

Figure

Frequency of rotor changes due to the position of disk.

3-blade rotor-blade coupling system changing diagram

7-blade rotor-blade coupling system changing diagram

Figure

Campbell diagram of blade changes due to the position of disk.

First-order Campbell diagram

Second-order Campbell diagram

Campbell diagram of rotor changes due to the position of disk is shown in Figure

Campbell diagram of rotor changes due to the position of disk.

First-order Campbell diagram

Second-order Campbell diagram

In order to make the effect of support stiffness on critical speed more obvious, in this section

Figure

Frequency of blade changes due to the support stiffness of shaft.

First-order frequency changing diagram

Second-order frequency changing diagram

Figure

Frequency of rotor changes due to the support stiffness of shaft.

First-order frequency changing diagram

Second-order frequency changing diagram

Figure

Campbell diagram of blade changes due to support stiffness of shaft.

First-order Campbell diagram

Second-order Campbell diagram

In order to further analyze the influence of the change of support stiffness, it is analyzed that the critical speed of blade changes due to the support stiffness of shaft when

First two-order critical speeds of blade change due to the support stiffness of shaft are shown in Figure

Critical speed of blade changes due to the support stiffness of shaft.

First-order critical speed changing diagram

Second-order critical speed changing diagram

Figure

Campbell diagram of rotor changes due to support stiffness of shaft.

First-order Campbell diagram

Second-order Campbell diagram

In order to further analyze the influence of the change of support stiffness and make the change more obvious, it is analyzed that the critical speed of rotor changes due to the support stiffness of shaft when

Figure

Critical speed of rotor changes due to the support stiffness of shaft.

First-order critical speed changing diagram

Second-order critical speed changing diagram

In order to further analyze the vibration characteristics of rotor-blade coupling system and verify the above phenomena, the influence of different factors on mode shapes of system is analyzed in detail. In order to make the analysis more reasonable, the mode shape of first-order natural frequency of the blade is analyzed prior to the formal analysis. Here, the number of blades

Figure

Mode shape of 4-blade rotor-blade coupling system.

SB mode of blade

BB mode of blade

First-order mode of rotor

Second-order mode of rotor

Based on the conclusion that has been drawn, when discussing the influence of different factors on the mode shape of coupling system, the mode shape of blade is represented by the first-order mode shape of SB mode. Similarly, the mode shape of rotor is represented by the first-order mode shape of rotor.

Mode shapes of rotor-blade coupling system changes due to the number of blades are shown in Figure

Mode shapes of rotor-blade coupling system change due to the number of blades.

Mode of blade (4 blades)

Mode of blade (7 blades)

Mode of rotor (4 blades)

Mode of rotor (7 blades)

Figure

Mode shapes of rotor-blade coupling system due to the position of disk.

Mode of blade (at 1/6)

Mode of blade (at 1/2)

Mode of rotor (at 1/6)

Mode of rotor (at 1/2)

Figure

Mode shapes of rotor-blade coupling system due to the rotational speed.

Mode of blade (100 rad/s)

Mode of blade (1000 rad/s)

Mode of rotor (100 rad/s)

Mode of rotor (1000 rad/s)

The model of rotor-blade coupling system model is established, the mode function of elastic support beam is deduced, the differential equation of system is deduced by the semianalytical method, and the influences of blades number, disk position, and support stiffness on system characteristic value, Campbell diagram, and mode shape of vibration are studied by numerical simulation.

Because of the coupling between shaft and blade, there are SB and BB modes dominated by blade’s first two-order modes which could be found and the BB modes are of repeated frequencies of (

With the increase of the number of blades, the frequency of SB mode increases linearly, that of BB mode is constant, and that of rotor decreases linearly. The changing trend of Campbell diagram is the same as natural frequency.

Frequencies of SB mode of blade and rotor mode are affected by the position of disk and change symmetrically with the center of shaft. What is more, the changing trend of first- and second-order frequency is different. But that of Campbell diagram is the same as natural frequency.

It could be found that there is no obvious difference between the mode shape of SB mode and BB mode of blade. Furthermore, the mode of blade has no effect on the deformation of rotor but the mode of rotor has an obvious effect on the deformation of blade. The number of blades has no significant effect on the mode shape of blade and rotor but the position of disk and rotational speed have an obvious effect on the mode shape of rotor.

(1) The specific meaning of each transformation matrix of blade can be obtained as follows:

(1)

(2)

(3)

Density (7850 kg/m^{3})

Shear modulus (75 Gpa)

Shaft length (0.6 m)

Radius (0.015 m).

Density (7850 kg/m^{3})

Young’s modulus (200 Gpa)

Outer radius (0.2 m)

Thickness (0.015 m)

Poisson ratio (0.3).

Density (7850 kg^{3})

Young’s modulus (200 Gpa)

Blade length (0.6 m)

Cross-section (^{2})

Area moment of inertia (^{2}).

The authors declare that they have no competing interests.

The project is supported by the China Natural Science Funds (no. 51575093) and Fundamental Research Funds for the Central Universities (nos. N140304002 and N140301001).