Keywords

1 Introduction

When a rotor operates in a high-speed rotating system, there are certain specific rotational speeds at which, when the rotor operates at or near these speeds, the rotor shaft deflects and generates large unbalanced forces and moments that cause the rotor to vibrate violently. These speeds are known as critical speeds. Operating at these speeds, the rotor system may vibrate with serious consequences, such as failure of the rotating machinery to operate properly or damage to components. Therefore, it is important to accurately calculate the critical rotational speed of the rotor to control the vibration of the rotor system and to ensure the safe operation of the rotor.

Most of the current critical speed calculation methods are for structurally regular stepped shafts, and the transfer matrix method and the finite element method are used to calculate the critical speed of a rotor [1,2,3], and the analysis objects are mostly individual single rotors with continuous structure [4,5,6]. However, for most practical scenarios, the rotor system usually does not contain only one rotor, but at least the motor rotor as the power supply, and the output rotor to be driven, and the coupling as the connection between the input and output components, the stiffness characteristics of which have a non-negligible impact on the entire rotor system. Pirogova et al. [7] analyzed the intrinsic and critical frequencies of the turbocharger rotor and the starter-generator rotor when they are connected into a rotor system by means of an elastic coupling. Zeng et al. [8] took the slender tandem shaft system of submersible oil pump as the research object and analyzed the effect of torsional stiffness of coupling on the torsional vibration characteristics of the slender tandem shaft system. Kim et al. [9] developed a novel coupled modeling approach to analyze the effects of non-beam behavior on the dynamics of a rotor system brought about by local joint flexure in a preloaded coupling. Ren et al. [10] analyzed the flywheel rotor supported by electromagnetic bearings based on ANSYS Workbench software, and investigated the effect of damping and support stiffness on the critical speed of the flywheel rotor system, and calculated the optimum support stiffness range adjustment.

This paper takes the rotor system composed of two double diaphragm couplings coupling three rotors as the object, establishes the finite element model of the multi-span high-speed rotor system, carries out the solution of the critical speed, and analyzes the influence of the radial stiffness of the coupling and the bearing support stiffness on the critical speed of the rotor system.

2 Modeling of Multi-span Rotor Systems

The multi-span rotor system comes from the high-speed transmission part of a space-use gearbox load test stand, and consists of a motor rotor shaft, a speed and torque sensor shaft, a gearbox input shaft, two double-diaphragm couplings, and three shafts each with their own support bearings in a total of ten parts. The rotor shaft and couplings are modeled in Creo as shown in Fig. 1, from left to right, the motor shaft, diaphragm coupling 1, speed and torque sensor shaft, diaphragm coupling 2, and reducer input shaft. The length of the motor shaft is 400 mm; the length of the speed and torque sensor shaft is 164 mm; the input shaft of the reducer is 92 mm; the length of the coupling 1 is 76 mm and the diameter is 68 mm; the length of the coupling 2 is 50 mm and the diameter is 44 mm.

Fig. 1
An illustration of a rotor assembly system has interconnected cylindrical and spherical components.

Multi-span rotor system assembly

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Considering the structural complexity of the double diaphragm coupling, when performing the dynamic analysis of the rotor system, ANSYS Workbench is used directly to perform simulation calculations, ignoring the structures in the coupling that are not easy to calculate, such as chamfers, small holes, asymmetric cross sections, and screws used for locking in the bushings on both sides, and ANSYS Workbench can perform the dynamic analysis of the rotor for the axisymmetric rotor directly. The material of the three rotor shafts is 40Cr, while the materials of the diaphragm coupling are aluminum alloy (bushings), stainless steel (diaphragms), and alloy steel (pins). The material parameters are shown in Table 1.

Table 1 Rotor system material parameters
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For the model in this paper, the free mesh division mode is used, and appropriate mesh refinement is carried out at the coupling diaphragm as well as at the place where the rotor shaft and coupling are in contact, and the mesh division is carried out after the setup is completed, and the number of nodes is finally obtained to be 308,696, and the number of cells is 166,485, and the rotor system after the division of the mesh is shown in Fig. 2.

Fig. 2
A simulation window has a rotor assembly system with interconnected cylindrical and spherical components with mesh and grid patterns. A scale ranges from 0.00 to 200.00 millimeters.

Grid division diagram of rotor

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The constraints are set after the meshing is completed. The three rotors are supported by their respective bearings, and the bearing support positions are shown in Fig. 3. The Stiffness K11 and K22 of the motor shaft bearings are defined as 4000 N/mm, and the Stiffness K11 and K22 of the speed and torque sensor shaft and the gearbox input shaft bearings are defined as 3000 N/mm. 2 N s/mm is defined for Damping C11 and C22. After lifting the five bearings to the remote points, remote displacement constraints are used to constrain the axial displacements of the five remote points and the axial displacement of the five remote points, use the remote displacement constraints to constrain the axial displacement as well as the axial rotation of these 5 remote points.

Fig. 3
A simulation window has a 3-D model of a rotor assembly system with cylindrical and spherical components. A scale below ranges from 0.00 to 200.00 millimeters.

Bearing support position diagram

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In the solution setup, set Damped in Solve Controls to Yes, Solver Type to Program Controlled; Coriolis Effect in Rotordynamics Controls to on, Campbell Diagram to on, Number of Points to 5; Stiffness Coefficient in Damping Controls to 8e-4.

When Workbench solves the critical speed of a rotor, the rotational speed must be defined. Insert five rotational speed points: 100, 1000, 5000, 10,000, 60,000 rpm. after defining the direction of rotation, modal analysis is carried out to solve the critical rotational speed, and the critical rotational speed of this multi-span rotor system is calculated to obtain the Campbell diagram as shown in Fig. 4, the first-order critical rotational speed is 7180 rpm, the second-order critical rotational speed is 12,919 rpm and the third order critical speed is 33,083 rpm.

Fig. 4
A line graph of frequency versus rotational speed plots trends for modes 1 to 15 B W and F W stable and ratio = 1, and plots for critical speed. The graph mostly plots increasing trends.

Campbell diagram

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3 Analysis of Critical Speed Influencing Factors

3.1 Influence of the Radial Stiffness of the Coupling on the Critical Speed

In order to quantitatively analyze the effect of the radial stiffness of the coupling on the critical speed of the rotor system, the coupling is simulated by using a bushing coupling, which can be used to simulate a flexible coupling to connect two shafts in ANSYS Workbench. The data in the parameter table in Fig. 5 can be changed to simulate different couplings with different properties. The first two elements of the diagonal can be changed to simulate couplings with different radial stiffnesses. To connect the end faces of two neighboring shafts by means of a Bushing coupling, the reference and moving surfaces must be carefully defined, the reference surface being the right end face of the first shaft and the moving surface being the left end face of the second shaft.

Fig. 5
A table with 7 columns and 6 rows is titled Stiffness Coefficients. The column headers are stiffness, per unit X, per unit Y, per unit Z, per unit theta x, per unit theta y, and per unit theta z. The values 400 for delta force X and delta force Y in the second and third columns are highlighted.

Table of stiffness coefficients for bushing connecting pairs

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The design value of radial stiffness of coupling 1 and 2 is 400, 300 N/mm respectively. The radial stiffness of coupling 1 and 2 are studied as variables, and 20, 40, 60, 80, 100, 150, 200, 250, 300% are set as adjustment factors. In addition, the radial stiffnesses of couplings 1 and 2 are adjusted proportionally at the same time. Calculate the critical rotational speed under each adjustment factor according to the above method, and plot the change curve as shown in Figs. 6 and 7.

Fig. 6
Two line graphs plot critical speed versus adjustment factor percent for first, second, and third order. Graph A has stable trends, and Graph B has stable trends except for the third order. The third order has an increasing trend.

a Critical rotor speed with radial stiffness variation curve of coupling 1; b Critical rotor speed with radial stiffness variation curve of coupling 2

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Fig. 7
A line graph plots critical speed versus adjustment factor percent. The first order passes through (20, 9000), (160, 9000), and (300, 9000). The second order passes through (20, 11000), (160, 11000), and (300, 11000). The third order passes through (20, 15000), (160, 40000), and (300, 55000).

Critical speed variation curves for simultaneous changes in radial stiffness of couplings 1 and 2

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Analyzing the data, it is learned from Fig. 6a that in the process of increasing the radial stiffness of coupling 1, the first three orders of critical speed of the rotor system are slowly increased with a low growth rate, and the overall critical speed remains stable; it is learned from Fig. 6b that in the process of increasing the radial stiffness of coupling 2, the first two orders of critical speed of the rotor system are almost unchanged, whereas the third order of the critical speed is significantly increased with a very high growth rate; From Fig. 7, when the radial stiffness of couplings 1 and 2 are adjusted proportionally at the same time, the first two critical speeds of the rotor system increase slowly with the increase of stiffness, and the overall tendency is stable, while the third critical speed is significantly increased with a high growth rate, and the overall value is higher than that under the influence of coupling 1 alone.

Therefore, it can be learned that for the coupled rotor system analyzed in this paper, the radial stiffness of coupling 1 mainly slightly affects the first two orders of critical speed of the rotor system, while the radial stiffness of coupling 2 significantly affects the third order of critical speed of the rotor system.

3.2 Effect of Bearing Support Stiffness on Critical Speed

Bearing support stiffness refers to the stiffness provided by the bearing in the radial and axial directions, which determines the stiffness characteristics of the bearing system. The larger the bearing support stiffness is, the stiffer the bearing system is, and the stronger the support ability for the rotor [11]. In high speed operation, a larger bearing support stiffness can reduce the deflection and deformation of the rotor, thus reducing the risk of resonance and improving the stability and reliability of the system.

In this paper, only the effect of bearing radial stiffness on the critical speed of the rotor system is considered, the initial stiffness of the two bearings of the motor shaft is 4000 N/mm, the stiffness of the bearings of the sensor rotor shaft is 3000 N/mm, and the stiffness of the bearings of the reducer input shaft is 3000 N/mm, as in the previous section, 20, 40, 60, 80, 100, 150, 150, 150, and 300% of the bearing radial stiffness are studied as adjustment factors, and the radial stiffness of the five bearings is adjusted synchronously and proportionally. 200, 250, and 300% as the adjustment coefficients are investigated to adjust the stiffness of the five bearings synchronously and proportionally. The variation curves of the first three orders of critical speed of the rotor system with bearing stiffness are calculated as shown in Fig. 8.

Fig. 8
A line graph of critical speed versus adjustment factor percent plots concave-down increasing trends. Some estimated values are as follows, first order, (20, 4000), (160, 6000), and (300, 8000), second order, (20, 6000), (160, 13000), and (300, 15000), and third order, (20, 24000), (160, 33000), and (300, 35000).

Variation curve of the first three orders of critical speed of rotor system with bearing stiffness

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From the data in the Figure, it can be seen that the first three critical speeds of the rotor system increase with the increase of bearing support stiffness, and the first two critical speeds of the rotor system in the range of the adjustment factor of 20–150%, the rate of change of the rotor system is relatively high; the adjustment factor of 20–80% of the range of the critical speed of the rotor system of the third order of the bearing stiffness of the influence of the rotor system is more significant; in the subsequent change of the bearing stiffness, the rotor system of the critical speeds of the first three orders has been increasing, but the overall growth rate slows down and gradually tends to be stabilized.

4 Conclusion

In this paper, a multi-span rotor system consisting of two double diaphragm couplings coupling three rotors is taken as the research object, and a finite element model of the multi-span rotor system is established, and the critical speed is solved using ANSYS Workbench to obtain the first three orders of the critical speed of the rotor system and Campbell’s diagrams, and meanwhile, the effects of the radial stiffness of the diaphragm couplings and the stiffness of the bearing supports on the critical speed of the multi-span rotor system are analyzed. The results show that:

  1. (1)

    For the coupled rotor model established in this paper, the radial stiffness of coupling 1 mainly slightly affects the first two orders of critical speed of the rotor system, while the radial stiffness of coupling 2 significantly affects the third order critical speed of the rotor system;

  2. (2)

    Within a certain range, the critical speed of the rotor system increases gradually with the increase of bearing support stiffness, and the third-order critical speed is more affected by the change of bearing stiffness than the first two orders;

  3. (3)

    This paper analyzes the influence factors of the critical speed of the coupled rotor, which can provide a reference for the lateral vibration of rotating machinery, and provide a reference basis for the selection of flexible couplings and bearings in engineering applications.