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ROLLER BEARING FAILURES IN REDUCTION GEAR CAUSED BY INADEQUATE DAMPING BY ELASTIC COUPLINGS FOR LOW ORDER EXCITATIONS

~by Herbert Roeser, Trans Marine Propulsion Systems, Inc. Seattle

Flexible couplings provide a convenient means for modifying and improving the natural frequency characteristics of a system, often having significant damping effect under resonant conditions. The ability to reduce resonant vibration amplitudes is usually confined to modes of vibration having dominant relative angular deflections across the coupling elements, since the actual damping torque is proportional to the rate of change of angular displacement in the coupling. This means that the frequency of the vibration has a large effect on the damping torque in an elastic coupling. This is the reason why the lower order vibrations are more difficult to dampen out. The power loss in the coupling is related to the amount of damping provided by the coupling. Due to this, it is often necessary to tune the system such that there is no resonant condition within the operating range of the system that can be excited by the low engine orders. If this is not accomplished, low order vibration amplitudes can be transmitted to the reduction gears by the coupling due to insufficient damping at resonance condition.



1. INTRODUCTION
The problem an engineer faces today when commencing a torsional vibration analysis of a diesel engine propulsion plant is the ever increasing complexity of these plants. One aspect of the greater complexity is the absolute increase in the calculations that are required due to the use of PTI and PTO gears with attached generators and multiple geared diesel engines. With respect to the diesel engine development, in order to achieve a better fuel economy, the trend is lower running speeds; longer stroke-to-bore ratios and higher combustion pressures.
In order to predict the behavior of any physical system, a model suitable for mathematical analysis is of great importance. The requirement of the model is to be able to predict the behavior of the system with sufficient accuracy. In the early days of torsional vibration analysis, a sufficient accurate model meant, a model which is able to predict the first few modes of vibration and the vibratory stress amplitudes in the shaft system at the calculated critical speeds. This type of simple analysis is no longer sufficient. The ever ongoing development of higher output diesel engines and increased complexity in operation and design of propulsion plants have proven the need for mathematical models capable of predicting behavior at non-resonant conditions. This includes the effects of damping, vibratory torques in reduction gears and elastic couplings, and heat losses in the elastic couplings and torsional dampers. The calculations must involve all expected operating parameters of the analyzed plant. In this paper, I shall describe two cases where the damping in the elastic coupling was not sufficient to dampen the low frequency vibration which eventually led to early roller bearing failure in the gears in question. The author’s intension is not to describe the method of mathematical modeling, but to use simple mathematical models to better explain the problem described in this paper.

2. EQUATION OF MOTION
Mass-elastic properties of the components in a propulsion system are often specified by each manufacture. In addition to the stiffness and inertia values, the shaft and mass damping are part of a complete mass-elastic system. The total mass-elastic system of a propulsion plant constitutes the mathematical model to be analyzed. By applying Newton’s second law for a rotating body, the so-called equation of motion is derived. If the general case with damping and an external torque is considered, the equation of motion for a mass becomes,





The first term in this equation is the inertia torque; the second and third terms are the shaft (relative) damping torque and the mass (absolute) damping torque, respectively. The fourth term is the spring or stiffness torque and the last term is the external or excitation torque.

A propulsion system consists of several masses connected by shafts and the equation of motion is applied to each mass in the system. The result is a set of linear differential equations where the number of equations is equal to the number of masses in the system. The free modes of vibration and the corresponding natural frequencies are found by setting the damping torque and external torque equal to zero and solving the resulting eigen value problem by Holzer tabulations or a root finding technique.

3. METHOD OF ANALYSIS
A torsional vibration analysis consists of two parts. The first part is natural frequencies and mode shape calculations, and the second part is the forced-damped calculations. After the mass-elastic system has been established, the analysis is in general performed in the following manner. The equation of motion is setup for each mass in the system. The result is sets of linear differential equations that are transformed into a characteristic system of linear algebraic equations. These are solved as an eigen value problem for the natural frequency calculations, and as a system of simultaneous equations for the forced damped calculations.
The natural frequency and mode shape calculations will determine the resonant or critical speeds and the relative deflection curve of each mode of vibration. The results should be presented in such a way that the resonant speed at each mode of vibration is identified, including the vector summation of each order. The vector sum is calculated based on the firing order and deflection curve and indicates the relative significance of each order for a specific mode of vibration.
The conventional Holzer tabulation for undamped free vibrations with the natural frequencies, relative deflections, inertia torques, and phase vector summations, still serves as a way of obtaining an overall view of vibration characteristics of the propulsion system, and forms the basis for the detailed forced-damped vibration analysis.

4. FORCED-DAMPED CALCULATIONS
In order to determine the response of a complex propulsion system with several sources of damping, and to obtain a more realistic picture of the non-resonant conditions, a forced damped analysis is necessary. The system of equations is solved for each order by using the harmonic components of the tangential efforts as the external torque, including the propeller excitation. The calculated results for each order and mode shape are added as a synthesis, taking into account the phase relations in order to find the overall vibration amplitudes or power dissipation in the analyzed propulsion system.
In general, torsional vibration calculations are performed assuming there is equal excitation from each cylinder in the engine. However, based on experience and measurements it has been shown that the lower order harmonics are often more significant than calculated assuming equal excitation from each cylinder. This can be explained by the fact that the excitation from each cylinder is not equal or uniform, but shows a lot of variations due to changes in firing pressure and ignition characteristics. This irregularity will excite the low order vibration and must be considered as part of a torsional vibration analysis.
At this point it should be noted that it is very difficult to evaluate the resulting engine excitation torque, since the excitations will depend on the rate and magnitude of the irregularity of each cylinder and on the overall distribution of impulses throughout the engine. Due to this, in some cases, based on experience, it may be sufficient to consider two excitation modes; the first is a “normal” combustion mode with equal excitation from each cylinder, the other is a “misfiring” mode where one cylinder is operating without combustion (compression only). The misfiring condition is considered a failure mode, but in engines with large numbers of cylinders, this condition will simulate the normal stochastic irregularity reasonably well. On the other hand, for engines with less than 12 cylinders, the misfiring condition may be too unfavorable to simulate the irregularity and the actual vibratory response of the system is between the normal combustion mode (equal excitation) and the misfiring condition.

5. EXAMPLE 1: PREMATURE FAILURE OF ROLLER BEARINGS IN PTI GEARS
The propulsion plants in question consist of a large seven cylinder two-stroke engine, rated 20588 kW at 102 rpm. The five bladed controllable pitch propeller is directly driven by the main engine. In addition, an asynchronous electrical motor capable of delivering 3822 kW at 1200 rpm via a PTI gear through an elastic coupling to the propeller increases the total output to 24410 kW.
The propulsion system arrangement is shown in Figure 1.


Figure 1 Arrangement of Propulsion Plant

The propulsion plant arrangement shown in Figure 1 is similar to those installed on three (3) large container vessels in order for them to reach a service speed of 22 knots. After approximately one year of service all three vessels encountered PTI gear failures. During the investigation several different assumptions regarding the failures were discussed, due to the fact that in two vessels the failures related to roller bearings and on the third vessel it related to a gear tooth failure.
During our investigation, it became quite clear that the problem with these failures where directly related to the dynamic loading of the PTI gears in service. This dynamic loading must relate to the torsional behavior of the system that affected the roller bearing service life in a very negative way. Apart from the service restrictions specified by the engine manufacturer, based on their torsional vibration analysis, we now had to commence our own torsional vibration analysis. In particular, we were interested to evaluate the vibratory torque amplitudes transmitted through the PTI gear, since these torque amplitudes has a large influence on the dynamic loading of the roller bearings and gear teeth.
Based on our own calculations and in comparison with the engine manufactures calculated results (steady state calculations) we could not detect any parameters that should cause any concerns if the service restrictions specified by the engine manufacturer were followed. It was then decided that actual measurement should be commenced on one of the vessels which at that time had been repaired.

5.1 MEASUREMENT SETUP
Torsional vibrations were measured with a Bruel & Kjaer torsional vibration meter type 2523, which is a laser system for non-contact measurements of angular vibration velocity or angular vibration displacement.
Angular vibration displacements up to 17.2 degrees can be measured in the frequency range of 0.3 Hz to 1 KHz. The signal from the laser transducer is fed into a Bruel & Kjaer 2-channel real time analyzer type 2145. A tachometer is used to measure the engine speed and the output signal is used by the real time analyzer for order tracking. The measurements were recorded on tape for further analysis.

5.2 VERIFICATION OF THEORETICAL CALCULATIONS
In a case like this, it becomes important to verify the mathematical model used in the theoretical calculations, particularly since the natural frequencies and mode shape calculations do not include any damping in the system (Free vibration).
Based on our own experience the best way to verify the calculated natural frequencies and mode shape calculations is to measure the angular displacement and acceleration at the free end of the crankshaft.



Figure 2 Average Vibratory Amplitudes

The graph in Figure 2 shows a very good correlation between the calculated and measured results. The graphs include the synthesis and the first seven engine orders of vibratory angular amplitudes. The measured values are average values, measured over a time interval of two minutes. At the time of these measurements the main engine load was 80% of MCR, and the booster load was 100%. As can be seen, the measured amplitudes are higher than the theoretical calculated amplitudes for normal firing (equal excitation from each cylinder). This is expected due to the normal stochastic irregularities between the cylinders as explained earlier in this paper. Additionally, it is important to notice that the larger the stochastic irregularities between the cylinders, the more significant is the influence from the engine orders, in this case, 1st, 2nd, 3rd and 4th.

5.3 TORSIONAL VIBRATION MEASUREMENTS ON THE PTI GEAR
These measurements were very important in this case, and had to be planned carefully. Since we were concerned about the lower engine orders and their influence on the PTI gear, it was determined to measure the lower engine orders across the gear tooth coupling between the asynchronous electrical motor and the high-speed pinion gear. Figure 3 shows a picture of the measurement setup.



Figure 3 Vibration Measurement Setup

Figures 4 and 5 show the measured vibratory amplitudes at the booster motor and gear side respectively.


Figure 4 Measured Angular Displacements
on the Booster Motor Side


Figure 5 Measured Angular Displacements
on the Gear Side

As can be seen from the measured results, the vibratory amplitudes across the gear tooth coupling are independent of booster motor load. The engine order that gives the highest contribution is the 1st, 2nd, and 3rd engine orders. In general, the amplitudes are somewhat higher on the gear side than on the motor side, this could be related to the damping in the gear tooth coupling, even though this parameter is not important. In this case, the important parameters are the vibratory amplitudes on the PTI gear side.

6. DYNAMIC ROLLER BEARING LOAD CALCULATIONS
Based on our findings, particularly by our measured data, it became important to determine how large the variation in vibratory amplitudes was over a certain time interval for the lower engine orders. As can be seen from Figure 6, the lower engine orders show a relative large variation in vibratory amplitudes over the measured time intervals, particularly the 1st engine order.


Figure 6 Time Capture of Measurement

As we attempted to evaluate the actual dynamic loading of the roller bearings involved, and account for the large variations in vibratory amplitudes, particularly from the 1st engine order, it became clear that we could not account for these large variations by simply applying the measured average vibratory amplitudes. The significance of the dynamic bearing load from the high measured transient peaks will be reduced only when applying the average measured vibratory amplitudes. It was decided to use a factor in the calculations that accounts reasonably well for the high transient peaks measured. Additionally it was decided to use a dynamic factor in order to account for the solid torque reaction at the gear tooth coupling, caused by the low engine orders being transferred through the PTI gear, since the constant transmitted torque from the booster motor will resist the vibratory torque transferred through the PTI gear. A sketch of PTI gear bearing arrangement is shown in Figure 7.


Figure 7 PTI Gear Bearing Arrangement

Bearing life is generally defined in hours of operation before the bearing fails under given loading conditions. It is very difficult to predict the life of an individual bearing under given loading conditions because it depends on several factors like lubrication, loading and speed of the bearing. In practice, apparently similar bearings exhibit different bearing life under exactly similar loading conditions. Hence, bearing life is described statistically in terms of life. life of a bearing is the number of hours, 90% of exactly similar bearings run under the given load conditions before failing. 10% of the bearings will fail before the life is reached. The most common way to calculate b₁₀ life of a bearing is by using the formula given by ISO.



Roller bearing life calculations were commenced for two different loading conditions, case #1 and case #2.

#1. Based on mean transmitted torque at 3822 kW@ 1200rpm.
#2. Based on mean transmitted torque at 3822kW@1200rpm and in addition, the contribution from the measured vibratory torques transmitted through the gears including the correction factors used.

Bearing Item#	Calculated bearing life in hours
	Load Case#1	Load Case#2	% Reduction
#103 Cylindrical Roller Bearing	35554	24025	32.5
#104 cylindrical Roller Bearing	27206	10169	62.6
#105 Spherical Roller Bearing	355881	13412	62.6
#106 Angular Contact Bearing	64933	43830	32.5

Based on the calculated results and as expected, the roller bearing’s B10 life is drastically reduced when the dynamic loads due to low order engine excitations are included. From the calculated results, it can clearly be seen that all roller bearings are very highly loaded, and as a result, reliability and service life are severely compromised.

7. EXAMPLE 2: PREMATURE THRUST BEARING FAILURE IN TWO REDUCTION GEARS
The thrust bearing failures described occurred on two large fishing vessels with identical propulsion plants.
The failures occurred within a month from each other, the bearings that failed are spherical roller thrust bearings and they were so-called pre-loaded bearings. The pre-loading was accomplished with 12 springs acting on the outer race.
The reason for the pre-loading is to insure satisfactory operation of the bearing in service. In order to insure that one must always make sure the bearing is subjected to a minimum given load condition specified by the manufacturer. If this is not accomplished, then due to the inertia forces of the rollers and cage and the friction in the lubricant can have a detrimental influence on the rolling conditions in the bearing that eventually will cause damaging sliding movements between rollers and raceways.
The vessels in question had the following service profile, 12 hours of towing, 10 hours of idling and 2 hours in transit. This service profile was very common for the vessels in question for approximately 280 days per year.
The failed forward thrust bearings from both vessels showed the same damage, the outer race was broken in three sections, the rollers were severely damaged, the inner race was cracked and the thrust cover was damaged due to the failed outer race.

8. FAILED THRUST BEARING LIFE ANALYSIS
At first it was adequate to simply calculate the basic bearing life for the bearing in question. The failed forward thrust bearing is a FAG 29436E Bearing with a dynamic load rating of 2076KN. The actual calculations commenced include the following parameters:

- Actual axial loading at full transmitted torque caused by the helix angle=31.4 KN

- Average thrust load from the propeller based on 12 hours of towing, 10 hours of idling and 2 hours in transit per 24 hours of service is 264KN.

The basic bearing life for the bearing in question was calculated to be 60,300 hours. By using the adjusted life factors published by the bearing manufacturer, where they considered the dimensions of the bearing and the lubricant condition, the adjusted bearing life is equal to 150,800 hours. Based on the service profile of the vessels, the thrust bearing should have lasted for approximately 23 years, but the bearing failed after 10 years of service.

9. TORSIONAL VIBRATION OF THE PROPULSION PLANT
The source of the premature failure of the forward thrust bearing was suspected to be related to the torsional response of the propulsion plant. By evaluating the initial torsional vibration analysis results, it was determined that the method of calculations used was inadequate, consequently a new model was constructed using modern calculation methods as described earlier in this paper.
Based on the calculated results, we can see that the first engine order, second mode of vibration and the first propeller order coincide with each other. The reason for this is the fact that the gear ratio is 4:1 and the propeller is a 4-Bladed propeller.


Figure 8 Vibratory Torques in Main Gear

The idling speed of the engine in question is set at 350 to 375 rpm. The first engine order, second mode of vibration, and the first propeller order have a resonance condition at 416 rpm. Due to the damping capacity of the elastic coupling a certain quantity of the vibration energy is transferred into heat in the coupling elements. As the elements heat up, the torsional stiffness will change (lower torsional stiffness). Consequently, with the existing coupling the resonant peak will move from 416 to 369 engine rpm, which is very close to the idling speed of the engine.
Based on the above, we were concerned about this condition because the vessel’s service profile dictated approximately 10 hours of idling with zero pitch every 24 hours of service. At idling condition the mean transmitted torque through the reduction gear is 1.96 kN-m (calculated), and as can be seen from the calculated graph the vibratory torque between the pinion and bull gear is between 1.8 kN-m to 2.5 kN-m (synthesis). This became a concern particularly regarding the pre-loaded forward thrust bearing, since the reduction gear in question uses single helix gears.


Figure 9 Modified System with a New Coupling Element

10. FORWARD THRUST BEARING FAILURE MODE
Under normal operating conditions the forward thrust bearing will be loaded predominately in the axial direction by the propeller thrust force, plus the axial force created by the helix angle. At idling operation with zero propeller pitch, the axial force produced by the helix angle momentarily becomes zero, due to the relative large vibratory torque amplitudes transmitted through the gears in relation to the mean transmitted torque. This becomes a major problem and this will occur at a frequency equal to the input speed. When the axial force momentarily becomes zero, only the set pre-loading of the bearing exist. Over time the pre-loading springs will lose their elasticity and results in the loss of their pre-loading capability. When the pre-loading reaches a value below the recommended minimum pre-load value, the bearing runs into major problems. Due to the loss of pre-loading the rollers will start the sliding movements between the rollers and race, and this eventually led to the reduced bearing life and failure experienced in the above mentioned case.

11. CONCLUSION
In both cases described, the failures are directly related to the damping capacity of the elastic couplings at low frequencies. In the first case, particularly the PTI gear manufacturer should have considered this, since the roller bearings service life was adversely affected by these particular parameters.
In the second case, the damping capacity of the coupling was ignored. If that would have been evaluated in more depth, one would with certainty, make sure to tune the system such that there is no resonance within the operating range that can be excited by the low engine orders. This was not accomplished; consequently, low order vibrations were transmitted to the reduction gear by the coupling due to insufficient damping at resonance condition.