Experiment and analysis of vibration modes of a certain type of marine transmission gearbox

The gearbox of a ship is not only required to have high power transmission, small size and light weight, but also to have low vibration and low noise [1]. Whether the gearbox can work properly will affect the working characteristics of the entire system. The vibration of the gearbox itself and the vibration of the gears transmitted by the shaft system are the main sources of ship radiation noise, which directly affects the combat effectiveness of the ship. The multi-input two-stage transmission gearbox of a certain type of ship has large vibration and noise, which is manifested as extremely large vibration level and whistling sound. This phenomenon is rare in other similar gearboxes. Through the modal comparison test of the gearbox body of this type of ship, the test results found the cause of the noise and vibration of the gearbox of a certain type of ship, and took corresponding measures to eliminate the fault.

1 Gearbox vibration signal analysis

Acquiring signals from the faulty gearbox and extracting fault information from them through digital signal analysis is an effective method for machine equipment status monitoring and fault diagnosis [2,3]. The structural components of the vibration signal reflect the vibration characteristics and fault nature of the gearbox. Therefore, by picking up and analyzing the vibration signals of two gearboxes of the same type, the main fault sources and their transmission paths of the gearbox are found. A

total of six measuring points are arranged on the gearbox, and the measuring points are arranged on the boss of the bearing thermometer of the gearbox cover, as shown in Figure 1.

 
Figure 1 Gearbox measurement point layout

At the same time, the sound level meter is used to test the air noise, analyze its spectrum, and compare its correlation with the vibration of the box. The main test instruments are: Kistler 8702250 acceleration sensor, Kistler 5124A amplifier, TEAC TD2135 T data recorder, HP25670 dynamic signal analyzer and QUEST MODEL 1800 sound level meter. From the vibration spectrum of the gearbox, the main frequency of its vibration spectrum is the meshing frequency of the secondary gear pair and its multiples. The main frequency of the air noise spectrum is consistent with the main frequency of the vibration spectrum, which is also the meshing frequency of the secondary gear pair. It can be concluded that the abnormal noise of the gearbox comes from the abnormal vibration of the gear unit. From the comparison of the acceleration vibration amplitude of the faulty gearbox (see Figure 2) and the acceleration vibration amplitude of the normal gearbox (see Figure 3), the vibration is strong and the noise is large at the 23# bearing on the faulty gearbox, which is the main monitoring point of the gearbox. The reason for the vibration of the gearbox may be that in the gear meshing transmission, when there are concentrated defects, distributed defects in the gears and bearings, or the shaft where the gears are located is bent, the rotation frequency will modulate the meshing frequency. If the shaft is severely bent or the gears or bearings are seriously faulty, resulting in abnormally large vibration energy, the abnormal vibration in the gear meshing transmission will excite the natural frequency of the transmission box. In addition, the vibration of the gearbox itself and the vibration of the gears transmitted by the shaft system are the main sources of radiated noise. It is necessary to perform modal testing and analysis on the gearbox.

Figure 2 Vibration acceleration of faulty gearbox

Figure 3 Normal gearbox vibration acceleration[page]

2 Theoretical model of modal test

Since vibration monitoring and analysis has the characteristics of fast diagnosis speed, high accuracy and the ability to realize online diagnosis, it is one of the most effective and commonly used methods for gearbox fault diagnosis. Among them, the application of modal test analysis method is an important way to perform fault diagnosis and condition monitoring. Usually, when a structure fails, such as cracks, looseness, component damage, etc., the physical parameters of the structure will change, and its characteristic parameters (natural frequency, modal damping, vibration mode, frequency response function, etc.) will change accordingly. According to the changes in these parameters, the type of fault can be determined, and sometimes the location of the fault can also be determined. Statistics on the failure of gearbox parts show that the failure of gears and bearings accounts for the largest proportion, 60% and 19% respectively [4]. Modal analysis of gearboxes and the use of modal parameters and other results for fault identification have become an increasingly effective fault diagnosis and safety detection method.

The vibration of the gearbox can be assumed to be a linear time-invariant system motion with n degrees of freedom, and its vibration differential equation is [5]:

Where: M, C, K are the mass, damping and stiffness matrices of the system respectively; X, F are the displacement response vector and the exciting force vector of each point of the system respectively.

Perform Laplace transform on both sides of equation (1). For a linear time-invariant system, whose poles are in the left half plane of the complex plane, the above process will be completely a Fourier transform process, and the obtained transfer function is the frequency response function, that is,

X (ω) = H(ω) F(ω) (2)

For a single input, when the excitation is at point p and the response is measured at point l, the displacement frequency response function is:

Theoretically, any row or column of the frequency response function matrix contains all the information of the system modal parameters, and the only difference is a constant factor. Therefore, in order to identify the mode, it is sufficient to measure one row or one column of the frequency response function matrix. In actual testing, it is more generally practical to use the power spectrum density to calculate the frequency response function of the system. The expression is:

H(ω) = Gf x (ω) / Gf f (ω) (4)

Where: Gf x (ω) is the input-output cross-power spectrum density; Gf f (ω) is the input-output self-power spectrum density.

The above formula uses the cross-spectrum analysis technology, which can greatly reduce the noise after multiple averaging. Since the least squares approximation method is used to estimate the frequency response function, the corresponding coherence function can be defined. It is a measure of the least squares error and is defined as:

Where: Gxx is the autospectrum of the response.

The coherence function γ2 represents the degree of linear correlation (or correlation coefficient) between the response and the force in the frequency domain. It varies between 0 and 1. The closer the coherence function is to 1, the better the linear relationship between the two compared signals (such as input and output) after all averages. After the unit impulse response function of the system is obtained, the single mode fitting method, that is, the least squares complex exponential method (LSCE) corresponding to single input and multiple output (SIMO) is used to estimate the modal parameters. Its basic idea is: first construct a polynomial, derive the autoregressive (AR) model of the system, and after solving the autoregressive coefficient, gradually identify the modal parameters of the system.

3 Gearbox modal test

3.1 Test instruments and analysis equipment

The impact hammer uses Kistler 9724A5000, 250 g counterweight, nylon hammer head, B & K8200 piezoelectric force sensor and B & K2635 charge amplifier; response test: use triaxial B & K4321 acceleration sensor, B & K2635 charge amplifier; recorder and analyzer: Belgium PIMEN TO8 channel dynamic signal acquisition and analysis system or American DP104 dynamic signal acquisition and analysis system and Belgium LMS company CADA2X structural modal test analysis software.

3.2 Measurement point arrangement and test plan

In order to test the gearbox modal, the gearbox is firstly analyzed and its geometric dimensions are measured, and preliminary finite element calculation and natural frequency distribution range estimation are performed. The estimated results show that the upper box of the gearbox composed of the upper and lower boxes has denser modes, so 216 response measurement points are arranged on the upper box, and 48 response measurement points are arranged on the lower box, totaling 264 response measurement points. The principle of point arrangement is to ensure that the modes of the gearbox can be stimulated. For important parts such as the bearing seat and parts that can cause relatively large noise, the principle of multiple response measurement points is adopted. The positions of each measurement point are marked on the box and numbered one by one.

According to the characteristics of the main transmission gearbox composed of the upper and lower boxes and the actual operating conditions, the test adopts the hammer method, which is a test method of fixing the knocking point and moving the response point. During the test, the force signal and the response signal obtained by the acceleration sensor enter the data acquisition device or portable computer through the amplifier, and the frequency response function and coherence of each measurement point at each knock are monitored on site with an analyzer. When the hammer strikes, the self-power spectrum of the impact force should be clean and flat within the selected frequency band, without continuous strikes, the force should be uniform and the test object should respond moderately, the average number of hammer strikes per point should be eight, and the signal size should meet the signal-to-noise ratio. When selecting the striking point, factors such as nodes and the geometric center of the approaching area should be avoided. In order to avoid modal leakage caused by improper selection of the response point, the response point should be selected on the asymmetric axis (or on the symmetric plane) and determined after multiple preliminary repeated tests. The gearbox adopts elastic vibration isolation with a vibration damping rubber device, and the original support method is used in the test. After the test, the recorded signal is sent to the modal analysis software for modal analysis. The block diagram of the test analysis system is shown in Figure 4.

Figure 4 Modal test and analysis system [page]

3.3 Data processing

The modal analysis adopts the real modal analysis method. According to the density of natural frequencies, the appropriate bandwidth is selected to make an initial estimate, and then the overall curve fitting is performed to obtain the frequency response function, and the modal vibration mode is comprehensively processed to eliminate the local mode and obtain the modal parameters of each order of the test box. Since the vibration mode vector is a relative value, the vibration mode vectors of different scales should be normalized and different generalized modal parameters are obtained. This test is normalized according to the modal mass of 1, and the modal parameters of the first 15 modes in Table 1 are obtained.

4 Modal analysis and conclusion of gearbox

By analyzing each vibration mode, the vibration of the upper gearbox is much greater than that of the lower gearbox, which is consistent with the result of finite element calculation. The bearing seat is located in the upper gearbox, so the large vibration of the upper gearbox causes the bearing seat to vibrate relatively large, which affects the alignment of the gear during operation, and then produces vibration and noise caused by the knocking of the tooth surface, which is an important source of vibration and noise in the gearbox.

Combined with the actual operation of the gearbox, the frequency of meshing of the high-pressure end and the low-pressure end gears in the gearbox under several working conditions can be obtained. From Table 1, we know that the 9th modal frequency of the faulty gearbox is 543.5 Hz, and the 9th modal frequency of the normal gearbox is 537.2 Hz. The meshing working frequency of the secondary gear at the high-pressure end is about 561 Hz when the main shaft speed is 105 r/min. Although the working frequency of 561 Hz does not fall on these two modal frequencies, for general engineering structures, it is required that each modal frequency is far away from the working frequency, or the working frequency does not fall within the half-power bandwidth of a certain mode (calculation shows that the half-power bandwidth of the faulty gearbox and the normal gearbox is 527.59～559.49 Hz and 522.5～551.9 Hz). In comparison, the 9th modal frequency of the faulty gearbox is closer to the working frequency than the modal frequency of the normal gearbox. In addition, from the analysis of the two adjacent modal frequencies of the 8th order (corresponding to 458.6 Hz and 470.1 Hz) and the 10th order (corresponding to 607.2 and 608.9 Hz) of the 9th order modal frequency of the faulty box and the normal box, the modal frequencies of these two orders (8th and 10th) are closer to the working excitation frequency, which is one of the reasons for the large vibration and noise of the faulty box when the main shaft speed is 105 r/min. As shown in Figure 2, the maximum vibration and noise of the faulty gearbox are at the bearing support of the secondary reduction gear 23# at the high-pressure end of the gearbox, which is consistent with the modal test results.

When the output speed is 150 r/min, the meshing frequency of the secondary gear at the high-pressure end is between 810 and 840 Hz, which varies with the fluctuation of the input speed. At this time, the 13th-order modal frequencies of the faulty and normal gearboxes are 845.1 Hz and 812.9 Hz respectively. Further calculations show that the half-power bandwidths of the 13th-order modal frequencies of the faulty and normal gearboxes are 828.2-861.9 Hz and 790.6-835.2 Hz respectively, both of which are near the operating frequency, which inevitably makes the vibration acceleration amplitude here larger. As shown in Figure 2, the vibration acceleration amplitude (RMS) of the 23# bearing of the faulty gearbox is 39.1 m/s2 at an output speed of 150 r/min, which is 2.6 times the acceleration amplitude of 15.1 m/s2 (RMS) of the 23# bearing of the normal gearbox (supporting the secondary gear at the high-pressure end). At this time, except for the 23# bearing (where the vibration is at its maximum value when the speed rises, the rest are relatively small) and the 24# bearing, the vibrations of the normal gearbox are relatively large (15.1 m/s2 and 13.6 m/s2 respectively).

From the above discussion, it can be found that the vibration and noise of the faulty gearbox are larger than those of the normal gearbox under the same working condition. By comparing the modal frequencies and damping ratios of the faulty gearbox and the normal gearbox in Table 1, the following analysis conclusions can be drawn:

(1) The meshing frequency of the secondary gear is related to the 9th and 13th order of the gearbox. The resonance caused by the overlap of the first-order modal frequencies causes the gearbox to vibrate and make a lot of noise. At the same time, it was found that the vibration and noise of the faulty gearbox were much greater than those of the normal gearbox, which was caused by the fact that the exciting force of the secondary gear of the faulty gearbox was much greater than that of the normal gearbox. The possible causes are: the sinking amount of the base of the faulty gearbox is greater than that of the normal gearbox (later tested to be the case);

the riveted and welded gearbox deforms over time, causing rotor imbalance, misalignment, shaft parallelism, bearing damage, gear surface damage, etc.;

(2) The modal damping ratio of the two gearboxes varies greatly. The greater the system damping, the greater the vibration attenuation. The maximum value of the damping ratio of the faulty gearbox is 5.38%, and the minimum value is 0.94%, while the damping ratio of the normal gearbox ranges from 5.38% to 1.86 %. Under the same order, the modal damping ratio of the normal gearbox is at least greater than or equal to that of the faulty gearbox, which is one of the reasons why the vibration and noise of the normal gearbox are smaller than those of the faulty gearbox. There are many reasons for the change in damping ratio, such as the clearance of the sliding bearing, the viscosity of the lubricating oil, the change in the tightening torque of the mounting bolts, etc.

5 Conclusion

According to the analysis conclusion, after disassembly and inspection, it was found that the alignment between the input end and the gearbox, the alignment between the gearbox and the coupling, and the base sinking amount were within the allowable range, but it was found that the bearing of the 23# sliding bearing was worn and there was damage on the surface of the secondary pinion. By repairing and balancing the secondary transmission gear and replacing the worn bearing, the vibration and noise of the faulty gearbox were greatly reduced. The above results show that the modal analysis method is a very useful tool for performance evaluation, fault diagnosis, maintenance and repair of gearboxes [6].

References:
[1] Chen Guojun, Zeng Fanming. Modern Ship Machinery Engineering [M]. Changsha: National University of Defense Technology Press, 2001.
[2] Jin Xianding. Progress and informatization of ship structural mechanics [J]. Vibration and Shock, 2002, 21 (4): 1-6.
[3] Cao Shuqian, Zhang Wende, Xiao Longxiang. Modal analysis of vibration structures - theory, experiment and application [M]. Tianjin: Tianjin University Press, 2002.
[4] Ding Kang, Zhu Xiaoyong, Chen Yahua. Vibration characteristics and diagnostic strategies of typical gearbox faults [J]. Vibration and Shock, 2003: 7-12
[5] Fu Zhifang, Hua Hongxing. Modal analysis theory and application [M]. Shanghai: Shanghai Jiaotong University Press, 2000.
[6] Cheng Guangli, Zhu Shijian, Huang Yingyun, et al. Gearbox vibration test and analysis [J]. Journal of Naval University of Engineering, (6): 83-88.