To analyze the torsional vibration of a diesel engine shaft, the torsional stiffness of the flexible coupling is a key kinetic parameter. Since the material properties of the elastic element of the coupling might change after a longtime operation due to the severe working environment or improper use and the variation of such properties will change dynamic feature of the coupling, it will cause a relative large calculation error of torsional vibration to the shaft system. Moreover, the torsional stiffness of the elastic coupling is difficult to be determined, and it is inappropriate to measure this parameter by disassembling the power unit while it is under normal operation. To solve these problems, this paper comes up with a method which combines the torsional vibration test with the calculation of the diesel shafting and uses the inherent characteristics of shaft torsional vibration to identify the dynamic stiffness of the elastic coupling without disassembling the unit. Analysis results show that it is reasonable and feasible to identify the elastic coupling dynamic torsional stiffness with this method and the identified stiffness is accurate. Besides, this method provides a convenient and practical approach to examine the dynamic behavior of the long running elastic coupling.
Due to the operation principle of a diesel engine as a complicated mechanical system, torsional vibration is inevitable. The classification society of all countries stipulates that the power of a diesel engine greater than 110 kw should be calculated and tested for torsional vibration of the shafting. During the calculation progress of the diesel shafting torsional vibration, the torsional stiffness of elastic coupling is an essential kinetic parameter. However, only the static stiffness of the couplings is provided when they are produced and the ratio of the dynamic stiffness to its static stiffness is usually 1.2. However, after the coupling is installed in the diesel shaft, the characteristics of the rubber parts of the coupling will be affected due to the high temperature under operation, aging, and alternating torque, and it can even lead to severe damage. As shown in Figure
Elastic coupling.
Francis and Avdeev aimed to obtain the accurate characteristics of the torsional stiffness of the flexible couplings, because allowing for small amounts of misalignment of the coupling may lead to equipment failure. A full 3D parametric finite element model of the coupling was developed and it was validated experimentally [
The elastic component of a coupling is the critical part to determine its characteristic. It cannot keep the same material properties after a period of time of operation, which will affect its mechanical behavior and lead to a calculation error of the shafting torsional vibration feature. Under these circumstances, the torsional stiffness of the elastic coupling is required. Therefore disassembling the unit to measure the coupling parameters is necessary, but this will cease the normal operation. To solve this problem, this paper comes up with a methodology which combines the torsional vibration test with calculation of the diesel shafting and it can identify the dynamic stiffness of the elastic coupling online by the inherent characteristics of shaft torsional vibration without disassembling it. The approach has also been applied to a typical diesel engine shaft. Apart from the above, the identified results have been verified offline to be feasible using a dynamic performance test station for elastic coupling.
The diesel propulsion shafting which is studied in this paper includes diesel engine, elastic coupling, reduction gear box, propulsion shaft, and propeller. According to the principle that the kinetic energy and potential energy remain the same after the system is simplified [
Lumped simplified model of a diesel propulsion shafting.
According to the vibration principle and as the torsional model of diesel propulsion shafting which is shown in Figure
Substituting (
For (
Equivalent simplified parameters.
Number  Name  Inertia 
Torsional stiffness 

1  Damping R  70 

2  Damping C  19 

3  Flange  11.8 

4  Piston 1  26.74 

5  Piston 2  26.74 

6  Piston 3  26.74 

7  Piston 4  26.74 

8  Piston 5  26.74 

9  Piston 6  26.74 

10  Piston 7  26.74 

11  Piston 8  26.74 

12  Cam  5.1 

13  Flange  3.25 

14  Flywheel  255 

15  Elastic coupling_1  41.3 

16  Elastic coupling_2  33.9 

17  Gear_I1  8.101 

18  Gear_I2_3  51.49 

19  Gear_I4_1  3.375 

20  Gear_I4_2  3.375 

21  Gear_I5_1  3.185 

22  Gear_I5_2  3.185 

23  Gear_I6  698.4 

24  Gear_I7  37.01 

25  I_shafting  19.8 

26  P_shafting  90 

27  Propeller  2713  — 
Based on the parameters shown in Table
Natural frequency calculation results (primary 4 orders).
Order  Natural frequency (Hz) 

1  5.29 
2  14.00 
3  37.03 
4  69.44 
Nowadays, the typical torsional vibration test is achieved through the torsion vibration test on shafting using the pulse counting principle. The magneticelectric sensor or encoder is used to pick up the torsional vibration signals. The signals are then collected by a noncontact torsional vibration meter [
Typical torsional vibration test system.
The natural frequency of the torsional vibration of the diesel engine is measured by increasing the speed continuously (from 450 r/min to 750 r/min within about 2 minutes with constant increment). By analyzing torsional angle amplitude at each harmonic order, the natural frequencies for the first four orders of shaft can be deduced and tabulated in Table
The test natural frequency values of the shafting.
Order  Test values (Hz) 

1  6.81 
2  14.71 
3  36.55 
4  68.75 
The calculation results in Section
Natural frequencies compared between calculation results (using static stiffness) and test results.
Order  Calculation results (Hz)  Test results (Hz)  Error (%) 

1  5.29  6.81  28.71% 
2  14.00  14.71  5.07% 
3  37.03  36.55  1.30% 
4  69.44  68.75  1.00% 
From Table
The vibration modes for the first two orders for this diesel shafting by torsional vibration calculation are presented in Figures
Firstorder vibration mode.
Secondorder vibration mode.
The torsional stiffness of an elastic coupling is a critical parameter for torsional vibration of a diesel engine propulsion shaft and should be tested before leaving the factory. However, the general test of the torsional stiffness only focuses on the STATIC torsional stiffness and puts no special attentions on the dynamic stiffness test and the dynamic stiffness is evaluated by the ratio between dynamic and static stiffness. In addition, after the coupling is used for a couple of periods, the material properties of the elastic component are easy to change. Hence, the dynamic behavior of the elastic coupling might alter during the operation process. Another problem is that once the coupling is settled in the unit, it is difficult to measure the parameters without dissembling the ship. According to the issue which is discussed in Section
Stiffness identification process online.
With the initial input for the shafting, the natural characteristics of the shaft torsional vibration can be calculated. Then the results are compared with the test data in order to determine whether the error is within the allowable range:
Once the maximum natural frequency error of the torsional vibration is less than 5%, the initial parameters do not need to be corrected. However, if the error is bigger than 5%, the natural frequencies and the mode of the system will be used to judge which output parameter effects the calculation results the most. As it can be seen from Section
Based on the identification method mentioned above, the natural frequencies, and the mode shape measured, the real dynamic stiffness value of the elastic coupling identified is 0.420 MNm/rad. This identified stiffness is used to recalculate the primary four orders of natural frequencies of the diesel propulsion shafting and the results are compared with the measured frequencies. The comparison results are shown as in Table
Natural frequencies compared between calculation results (using identified stiffness) and test results.
Order  Calculation results (Hz)  Test results (Hz)  Error (%) 

1  6.79  6.81  0.22 
2  14.89  14.76  0.88 
3  37.10  36.67  1.17 
4  69.46  68.33  1.66 
From Table
It should be mentioned that other parameters, such as the torsional vibration damper, have also been checked to identify whether they can influence the natural frequencies of the shafting. The calculation results show that all other parameters are not sensitive to the primary two natural frequencies except the elastic coupling. Therefore, it is quite reasonable to identify the torsional stiffness of the elastic coupling with the approach discussed in this paper.
Due to the hysteresis quality of the coupling elastic component, hysteresis loss will exist in a vibration circulation. Through specific measurement method and data processing system, the damping ellipse can be obtained to calculate the damping factor
Damping ellipse.
In Figure
The damping ratio
As
From the equation it can be seen that once the alternating torque and torsional angle are measured and the damping ellipse is obtained, the torsional stiffness can be computed from the equation above.
To validate the accuracy of this identification method for the dynamic torsional stiffness, the elastic coupling was settled on a dynamic performance test unit to measure its dynamic torsional stiffness, as shown in Figure
Elastic coupling dynamic stiffness measurement apparatus.
The maximum torsional torque of the test apparatus is 64 kNm and the total mass is 5800 kg. The right end of the coupling was connected to the test panel of the measurement apparatus with bolts and the left end of the coupling is applied with alternating torque by a hydraulic device. The relative angular displacement is measured by the angle encoder in order to calculate the dynamic torsional stiffness. This test method is simple, reliable, small energy consumption and is widely used for most torsional stiffness measurement apparatuses [
According to the actual operating condition of the diesel propulsion shafting, the loaded torque is set for the dynamic test apparatus. The test dynamic torsional stiffness of the elastic coupling is shown in Figure
Dynamic stiffness curve of the elastic coupling.
Comparing the two results of the identified stiffness which is suggested in this paper and the test result from the dynamic unit, the error is less than 5% which can be seen in Table
Comparison of the dynamic stiffness of the elastic coupling.
Test dynamic stiffness (MNm/rad)  Identified dynamic stiffness (MNm/rad)  Error (%)  

Elastic coupling  0.442  0.420  4.98 
This paper rises up an identification analysis idea to verify the dynamic torsional stiffness of an elastic coupling without tearing the coupling down from the power plant and provides reliable parameters for analyzing the properties for the elastic coupling even for the whole shaft. Besides, as the rubber parts of the elastic coupling always suffer from high temperature, oil, and alternating torque, its properties are easy to deviate from the original one. Hence, this method is provided for online identifying the dynamic behavior of the elastic coupling conveniently during the longterm operation.
In this paper, it states that the elastic coupling stiffness is difficult to be obtained in the torsional vibration calculation of the diesel propulsion shafting. Such problem has been solved using the natural characteristics of the torsional vibration of the shafting to identify the dynamic stiffness of the elastic coupling. Also the solution is applied to a typical diesel propulsion shafting. Using this elastic coupling dynamic stiffness identified with such method to calculate the torsional vibration of the engine shafting, the theoretical calculated result of the natural frequency has an error less than 5% compared with the test on the ship, which satisfies the relative specifications. Verifying the stiffness identified offline with a dynamic performance test apparatus for elastic coupling, the test result has an error less than 5%, which indicates that it is reasonable and feasible to identify the elastic coupling dynamic torsional stiffness with this method and the identified stiffness is accurate. As a result, this method provides a convenient and practical approach to examine the dynamic behavior of the elastic coupling online during a longterm operation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research work is supported by the National Natural Science Foundation of China (Grant no. 51375104), Heilongjiang Province Funds for Distinguished Young Scientists (Grant no. JC 201405), China Postdoctoral Science Foundation (Grant no. 2015M581433), and Postdoctoral Science Foundation of Heilongjiang Province (Grant no. LBHZ15038).