Research Article  Open Access
Effect of the Supports’ Positions on the Vibration Characteristics of a Flexible Rotor Shafting
Abstract
In this study, we evaluated the effect of changing supports’ position on the vibration characteristics of a threesupport flexible rotor shafting. This dependency was first analyzed using a finite element simulation and then backed up with experimental investigations. By computing a simplified rotor shafting model, we found that the firstorder bending vibration in a forward whirl mode is the most relevant deforming mode. Hence, the effect of the supports’ positions on this vibration was intensively investigated using simulations and verified experimentally with a housemade shafting rotor system. The results demonstrated that the interaction between different supports can influence the overall vibration deformation and that the position of the support closer to the rotor has the greatest influence.
1. Introduction
The shafting is an indispensable part of transmission systems, such as helicopter tails, turbines, and ship propellers, and research constantly focusses on improving its design and performance. Most of these rotor shafting systems are designed to be flexible, because their working speed is generally higher than the critical speed [1]. When the rotor exceeds the critical speed, vibrations are generated due to mass imbalances, and severe vibrations can cause damage to the bearing and rotor structures and in turn affect the normal operation of the mechanical system [2, 3]; a properly designed support can effectively reduce the vibrations of the rotor shafting at the critical speed and even avoid that the working speed reaches the resonance zone.
In shafting systems, it is commonly recognized that the position and features of the shaft’s supports have a role in the system’s overall stability. Xiao et al. [3], taking a magnetic levitation highspeed motor as their model, used a numerical method to study the influence of changing the bearing position on the vibration characteristics near the critical speed of a rotorbearing system. They observed greater effects on the amplitude by changing the radial position, but less effects on the axial position. Liu et al. [4] investigated the influence of a radially displaced bearing on the vibration characteristics of a marine shafting system. The results demonstrated that changing the displacement of the bearing had an effect on the shafting vibration amplitude; however, the effect was not significant. Li et al. [5] studied the influence of bearing spacing on the acoustic vibration characteristics of underwater structures, finding that the bearings’ spacing has little effect on underwater acoustic radiation; however, it can easily influence the vibration state of the propulsion shafting. Xu and Li [6] used the finite element method to analyze the influence of the bearings’ position and aperture on the static stiffness and natural frequency of the rotorbearing system of an electric spindle. They concluded that increasing both the span and aperture can improve the static stiffness of the spindle. Changing the bearing position at the rear end was found to significantly affect the frequency and mode of the spindle. With the same analysis method, Su et al. [7] and Qin et al. [8] observed that changing the bearings’ stiffness and position can influence the natural frequency of vibration in ship shafting systems. Ma et al. [9] investigated the relationship between dynamic stiffness, damping coefficient, and rotor speed of different bearings. Kumar and Somnath [10] theoretically studied the effects of different bearing lengths, rotor speeds, and unbalanced forces reported in the literature on the dynamic characteristics of rotorbearing systems. Generally, the effects of bearing stiffness on the critical velocity and vibration response characteristics of shafting systems have been widely investigated [11–14], as well as the methods to optimize the bearing stiffness, damping, length, gap, and diameter to improve the overall stability of a shafting [15–17]. Yücel and Saruhan [18] also used the Taguchi method to test the vibration of a rotor system under different coupling configurations, disk positions, and speeds in order to determine the combination of parameters that minimize deformation due to vibration. However, little is known on the effect of simultaneously changing different supports’ positions, and how the overall supports’ array can be configured to minimize overcritical vibrations.
It is common that support configurations, especially those of multibearing rotors, are designed so that they can guarantee optimal working strengths, without preliminarily paying attention to the magnitude of deforming vibrations, especially those that generate over the critical speed. However, attention needs to be paid to the final configuration of these rotors, to guarantee optimal performances at all conditions.
In this paper, a threebearing rotor shafting system was chosen as the research object. The multifactor orthogonal test method was used to analyze how the interaction positions of the three supports affect the vibration characteristics at the critical speed and determine the optimal array. We then verified the computational results with experiments, testing a housebuilt flexible rotor shafting system.
2. Flexible Rotor Shafting Model
Our flexible rotor shafting system was composed of three bearing supports (I, II, and III), a turntable, a coupling, and two flexible shafts. Support I, a rigid support, was arranged on the leftend side of the shaft, between the end of the shaft and the turntable. Another rigid support, support III, was located at the right end of the shafting system, while the elastic support II was arranged at an intermediate position between supports I and III (an example is depicted in Figure 1(a)), the initial position of each support is zero (Figure 1(b)). The bearing of support II was equipped with a rubber ring containing a bushing, and the position of the bearing support was changed by moving the bushing. Several reference parameters are listed in Table 1. The UG software was used to computationally develop the solid model of each component, which could be assembled into the final rotor system by inputting its desired position and size parameters (Figure 1).
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The ANSYS Workbench software was used to conduct our calculations; however, due to its limitations when applied to assembly simulations, we first extrapolated the supports’ stiffness and damping parameters separately and then computed them in the model. Specifically, after obtaining the natural frequency and natural mode of vibration of the system, we extrapolated the radial vibration response parameters of the supports, which are listed in Table 2. For rigid supports I and III, the damping was set as zero, since their bearings, accurately lubricated with grease, were considered to be exempt from the effects of vibration. Afterwards, we pondered that, since the shape and structure of the model affect both the finite element meshing and the computational efficiency, the model needed to be appropriately simplified. The processed UG 3D model of the flexible rotor shafting system was imported into ANSYS, and the surface mapping grid and local size grid were used for mesh division and refinement. The bonded configuration was set for all contact zones. The bearing of body ground is added to the outer ring of the bushing to simulate the effect of the bearing on shafting. A remote displacement constraint was added to the outerend surface of shaft I, to limit its axial rotation and movement. Figure 2 depicts the finished model of the rotor shafting system.

3. Shafting Modal and Harmonic Response Analyses
After optimization of our model, we first analyzed its modal response and harmonic response. Initially, we employed modal analysis to extrapolate the first six natural frequencies of the shafting system. The first four were selected for analysis and are reported in Table 3.
 
Frequencies are reported in ascending order 1 to 4; = natural frequency of mode N; FW = forward whirl; BW = backward whirl. 
Figure 3 shows the modeled shapes of the shafting subject to the four natural vibrations in Table 3. It can be seen that the first four modes are mainly bending vibrations, and that only a small difference in frequency is present between the first and second, and between the third and fourth modes. Since it has been demonstrated that a shaft mainly vibrates in a forward whirl mode [19], we decided to only consider the firstorder bending vibration of the rotor shafting in our next analyses.
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Since the natural frequency of a rotor shafting depends on the rotational speed, it is essential to evaluate its trend, as well as the critical speed. In the ANSYS software, during modal analysis, it is possible to compute different speeds and obtain a Campbell diagram. This plot is able to predict the speed of a rotating system at different frequencies and will report the critical speed relative to each natural frequency (Figure 4). The intersection of the isokinetic curve and each natural frequency curve is the critical speed point for that frequency. As can be seen in Figure 4, the firstorder critical speed of our rotor shafting was 3221.8 r/min and the secondorder critical speed was 6597.4 r/min.
In physical shafting systems, installation errors, manufacturing errors, and material unevenness are only some of the issues that distance their performance from ideality. Shafting systems are more or less eccentric, and eccentricity is one of the main factors that cause vibration [4]. Hence, we obtained the vibration deformation of the rotor shafting system at the critical speed by harmonic response analysis. Since this analysis takes into consideration the gyroscopic effect, it can only be solved by the full solution method. The meshing method and boundary conditions were the same as in the modal analysis, and the “rotating force” mode was used to simulate the eccentricity of the disk. First, we derived the rotational force generated by the eccentric mass, and obtained a model; then, we inputted the firstorder natural frequency to obtain the corresponding mode, as shown in Figure 5. The position of the maximum vibration deformation on the shafting varies with the change in the supporting position. In the experiment, it is impossible to predict the actual maximum vibration position, so it is a good choice to take the fixed position for vibration analysis. Therefore, for the convenience of the experiment, the independent variable was chosen to be the position between support I and support II. As an example, for a distance of 428 mm, the deformation was calculated to be 0.9616 mm.
4. Orthogonal Simulation Test Design
As mentioned, the goal of our study was to evaluate the effect of the interaction position of the three supports of our flexible rotor shaft system on the vibration characteristics near the critical speed. To account for the interaction between factors (in an experiment, not only are the various factors at work but also the factors sometimes combine to influence a certain indicator, which is called interaction), we used the L_{27} (3^{13}) orthogonal table, with the level design factors shown in Table 4. The direction of the coordinate axis is marked in Figure 2. The (+) direction of shifting the support’s positions is consistent with the direction of the coordinate axis. To reduce the variability of the possible supports’ positions, we restricted their values based on the movement limitations of the physical model prepared in our laboratory, making sure to reflect the exact adjustment ranges as much as possible. Tables 5 and 6 list the orthogonal test’s scheme and report different calculated parameters that account for the varied bearing positions. By comparing different matches, we were able to find the optimal combination of supports’ positions and, in turn, to find the minimal value of vibration deformation at the critical speed of our shafting system. Analyzing the direct correlation between single positions (I, II, or III) and the impact of the interaction on the vibration, we were able to determine which parameter has the greatest contribution.
 
= support I position; = support II position; = support III position. All the positions refer to the distance from the reference positions listed in Table 1. 
 
n = test number, = support I position; = support II position; = support III position; = interaction support I × II; = amount of vibration deformation for the set of parameters reported in the n^{th} row; = the maximum vibration deformation for the set of parameters reported in the n^{th} row. All the positions refer to the distance from the reference positions listed in Table 1. 
 
, , = test levels; , , = sums of the vibration deformation amounts at the corresponding level and for the relative position ; , , = average values of the test factor indexes at the corresponding level at the relative position ; = the range of each . 
5. Analysis of Simulation Results
Tables 5 and 6 show different parameters, useful to evaluate the influence of the supports’ position on the amount of vibration deformation under the firstorder bending vibration mode. In Figure 6, for a quicker evaluation, we plotted the vibration deformation and the maximum vibration deformation values for each of the set of parameters in Table 5. The two curves basically have the same trends; consequently, we could safely use the vibration deformation amount as the only target parameter. From the values in Table 6, we found that the position of support I had the greatest influence on the vibration deformation, and the effect of the interaction support I × II was slightly larger than that of the position of support II. In short, the following influence trend was observed: support I > interaction support I × II > support II > support IIIinteraction support I × III > interaction support II × III. Therefore, we obtained a set of conditions that allowed us to determine the optimal combination of supports’ positions and modulate it as needed according to its effect on the overall vibration. Figure 7 shows the effect of changing the positions of the three supports on the vibration deformation. If support I is positioned at 0 mm, the deformation is the highest; the farther the position of support III is from the shaft’s end, the smaller the vibration deformation is. From our results, we found that the minimum deformation could be achieved when supports I, II, and III were placed at 40 mm, 0 mm, and −60 mm, respectively. If we ignored the interaction between the factors, Figure 7 indicates that 40 mm, −40 mm, and −60 mm were the optimal positions of supports I, II, and III, respectively. However, we know that the interaction of the three supports also play a role, which makes this set of conditions unsuitable for practical applications. Hence, to determine the actual optimal positions, we had to first determine the position of support I, which is the parameter that influenced vibration the most; from there, the optimal position of support II could be determined, according to the interaction support I × II. Finally, we could set the optimal position of support III (it was not necessary to first determine the interaction support I × III; its influence on vibration is similar to that of ). Hence, a total of six parameters (three absolute positions and three distances) can be determined by only obtaining three (, , and ). To better evaluate the influence of , Table 7 and Figure 8 summarize the results presented in Tables 5 and 6. Figure 8 shows that setting the position of support II to 0 mm, support I should be placed at 40 mm to achieve the lowest deformation; however, if the system shifts by −40 mm (support I at 0 mm and support II at −40 mm), the vibration deformation would double. In summary, 40 mm, 0 mm, and −60 mm are excellent positions for supports I, II, and III, respectively.

6. Experimental Device and Testing Scheme
To apply the computational model in practice, we built a test platform consisting of a threesupport rotor test bench, a signal acquisition device (uT3408FRSDY, Youtai Electronics, China), a speed controller, a computer, and a displacement sensor (Figure 9).
The threesupport rotor test bench consisted of a base, a drive motor, two flexible shafts, a counterweight disc, two couplings, and three supports. The parameters of the drive motor are shown in Table 8. We made sure to configurate the test platform to allow for a convenient adjustment of the supports’ positions: two parallel rows of convex grooves were opened on the base and a slider with a threaded hole was placed on the lower part of the convex groove. The bearing seat and the slider were fixed to the base with bolts. Supports I and III were rigid supports, while the movable support II was elastic. Before each test, the desired position of each support was marked on the base of the test bench; the lock nut and the bolts on the bearing seat were loosened, the supports were moved, and then fixed in the desired positions. Then, the lock nut and the bolts were tightened back.

The combination of the signal acquisition device and the eddy current displacement sensor allowed us to monitor the vibration amplitude and time dependency, the axial trajectory, and other information, in real time, and to input each collected signal into the signal acquisition device. After testing the samples under highfrequency generating stimuli, we were able to obtain the timedomain response data of the bending vibrations of the shafting and import them into the computer through the acquisition system. The data could be saved offline in the computer and later analyzed to comprehensively investigate the dynamic characteristics of the shafting.
The speed controller could be either adjusted manually or programmed for automatic speed regulation. The latter allowed to select amongst a range of speed adjustment modes, achieved by inputting the target speed and acceleration time. For our experiments, we chose automatic speed regulation, with a steady speed of 4800 rpm and a linear acceleration of 5 rps, starting from a stationary motor.
The eddy current displacement sensor was placed at an intermediate position between supports I and II. The sensor was used to measure the radial bending vibration deformation of the shafting in a range between 0 and 3 mm. The output voltage was set to be zero when the probe was 2 mm from the shaft. Before the test, the sampling frequency was set to 512 Hz. The specific parameters of the current displacement sensor are shown in Table 9.

To verify our computational results, we initially screened the effect of the positions reported in Figure 7 on the vibration displacement at the relative critical speeds. As mentioned, these positions were obtained without considering the interaction between two supports. The results are shown in Table 10.

As mentioned, the second and third most relevant parameters in defining the effect on vibration displacement are the interaction support I × II, and the position of support III, and especially, the interaction support I × II is even more relevant at transcritical speeds than the absolute position of support II. Hence, we proceeded and evaluated the effect of and on vibration displacement. By fixing the position of support I, we varied the position of supports II and III; the results are shown in Table 11.

7. Analysis of Test Results
The results reported in Table 10 are summarized in Figure 10. By comparing Figure 7 (simulation) with Figure 10 (test), we gladly noticed that the trends of the influence of each set of positions on the vibration deformation were basically the same. However, the test bench certainly suffers from axial misalignment, bearing load effects, bolt preload, and manufacturing errors, which are difficult to be estimated and eliminated. Therefore, the numerical values of the test and the simulation were not compared.
Figure 11 shows the case where the positions of support II and support III were kept constant and the position of the support I was changed. The deformation was measured at a precise position (between supports II and I); hence, the obtained deformation value refers to that position, and was measured when the critical speed was reached. Figure 11(a) shows the vibration waveform obtained when the three supports I, II, and III were positioned at −40 mm, 0 mm, and 0 mm, respectively. In this configuration, the maximum vibration deformation across the firstorder critical speed was 0.38 mm. Figure 11(b) shows the vibration waveform when the all the three supports were placed at their reference position, with a corresponding vibration deformation of 1.26 mm. Figure 11(c) shows the vibration waveform of the three supports at the 40 mm, 0 mm, and 0 mm positions, where the maximum vibration deformation decreased to 0.299 mm.
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Figure 12 shows the case where only the position of support II was changed, and the shafting was subject to vibration deformation at the critical speed. For positions of 0 mm, −40 mm, and 0 mm, the maximum vibration deformation across the firstorder critical speed was 0.772 mm (Figure 12(a)), while at 0 mm, 40 mm, and 0 mm, the deformation increased to 0.848 mm (Figure 12(b)).
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Figure 13 shows the waveform of the vibration in the case where only the position of support III was changed. Figure 13(a) shows the vibration waveform of the three supports at the 0 mm, 0 mm, and −60 mm positions, with the maximum of 0.975 mm. If support III was moved to −30 mm, the deformation reached 1.06 mm (Figure 13(b)). As can be seen from the test results, changing the position of support I to its optimal value reduced the vibration deformation by 76.27%; in comparison, for supports II and III, the improvements were only 38.73% and 22.62%, respectively. As a result, our experiments showed that the position of support I has the greatest influence on the vibration of the shafting over the first order of critical speed, while the position of the support III has the smallest influence. This is consistent with the simulation results.
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Both numerical and experimental results show that when support II is at the 0 mm position, the vibration deformation at the first critical speed is the smallest. Due to the interdependence of different supports’ positions, the interaction support I × II had a greater effect on the vibration deformation than the absolute position of support II. As can be seen from the comparison between (a) and (b), or between (c) and (d), or between (e) and (f) in Figure 14, the amount of vibration deformation when support II is at 0 mm is lower than that at 40 mm (with the same support I and III positions). That is, the interaction support I × II (support II at 40 mm) has a larger deformation effect than support II (support II at the 0 mm). These results are consistent with the conclusions of the simulation.
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Amongst all the examined cases, we can conclude that the minimum vibration deformation across the firstorder critical speed could be achieved with the 40 mm, 0 mm, and −60 mm combination (Figure 15), with a value of only 0.174 mm. When the positions of the three supports were left unchanged (as the reference) a maximum vibration deformation of 1.26 mm was observed. The difference between these two aforementioned cases is an astonishing 86.19%.
8. Conclusion
In this paper, the finite element simulation method was used to investigate the effect of the supports’ positions of a threesupport rotor shafting on the vibration characteristics near the critical speed. The numerical data were backed up by experimental observations, and the following conclusions were drawn. (a) The positions of the three supports have a great influence on the amount of vibration deformation near the firstorder critical speed of the rotor shafting. In particular, the position of support I has the greatest influence. (b) Within a certain range, the vibration deformation first increased, and then decreased, when the positions of support I and support II were shifted to the right (Figure 10). For the same movement, the vibration deformation gradually and steadily increased for support III. (c) The effect of the position of support II on the vibration characteristics of the shafting was influenced by the position of support I position, which played a leading role.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research work was supported by the Fundamental Research Funds for Central Universities (nos. NT2018015) and the National Natural Science Foundation of China (nos. 51505215).
Supplementary Materials
The Excel sheet named Table10_1 represents the test data under test number 1 in the manuscript table 10 (Figure 11(a)). Similarly, the Excel sheet named Table10_2 represents the test data under test number 2 in the manuscript table 10 (Figure 11(b)), and the Excel sheet named Table10_3 represents the test under test number 3 in the manuscript table 10 (Figure 11(c)), the Excel sheet named Table10_4 represents the test data under test number 4 in the manuscript table 10 (Figure 12(a)), and the Excel sheet named Table10_5 represents the test data under test number 5 in the manuscript table 10 (Figure 12(b)), the Excel sheet named Table10_6 represents the test data under test number 6 in the manuscript table 10 (Figure 13(a)), and the Excel sheet named Table10_7 represents the test data under test number 7 in the manuscript table 10 (Figure 13(b)), the Excel sheet named Table11_8 represents the test data under test number 8 in the manuscript table 11 (Figure 14(f)), the Excel sheet named Table11_9 represents the test data under test number 9 in the manuscript table 11 (Figure 14(e)), and the Excel sheet named Table11_10 represents the test data under test number 10 in the manuscript table 11 (Figure 14(d)), the Excel sheet named Table11_11 represents the test data under test number 11 in the manuscript table 11 (Figure 14(c)), the Excel sheet named Table11_12 represents the test data under test number 12 in the manuscript table 11 (Figure 14(b)). The Excel sheet named Table11_13 represents the test data under test number 13 in the manuscript table 11 (Figure 14(a)). The Excel sheet named figure 15 represents the test data when the critical speed vibration amount is the smallest (Figure 15). (Supplementary Materials)
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Copyright © 2020 Miaomiao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.