Optimizing Multi-Row Cam Roller Bearing for Long Fatigue Life of Hydraulic Motors

Cam-lobe radial-piston hydraulic motors are widely used as rotation driving units for various marine machinery owing to their ultrahigh output torque (more than 100 kN m). A multi-row cam roller bearing (MCRB) is the key component that directly determines the fatigue life of a cam-lobe radial-piston hydraulic motor. However, compact geometry and complex loads render MCRB susceptible to fatigue failure, highlighting the need for an optimized MCRB to achieve longer fatigue life and higher reliability. Therefore, this study proposes an innovative geometry optimization method for an MCRB to improve its fatigue life. In this method, a quasi-static model was developed to calculate the load distribution, with the fatigue life of the MCRB calculated using both basic dynamic loading and load distribution. Subsequently, a genetic algorithm was used to obtain the optimized geometry parameters, which significantly improved the fatigue life of the MCRB. Finally, a loading test was conducted on a hydraulic motor installed with both the initial and optimized MCRB to validate the effectiveness of the proposed optimization method. This study provides a theoretical guideline for optimizing the design of MCRB, thereby increasing the fatigue life of hydraulic motors


Introduction
Cam-lobe radial-piston hydraulic motors are low-speed and high-torque hydraulic motors widely used as rotation driving units in various large machinery such as deck winches, tunnel boring machines, and dredgers [1][2][3].In marine applications, particularly those involving tough winches, this type of hydraulic motor is renowned for delivering superior torque control at any speed, thereby minimizing the risk of seizures and malfunctions.As the most important power transmission component in a hydraulic motor, a multi-row cam roller bearing (MCRB) can directly convert hydraulic energy into rotating mechanical energy.However, an MCRB operates with a compact geometry and high load, rendering it prone to fatigue failure [4,5].Generally, the MCRB is the first to fail in a hydraulic motor, which directly determines the fatigue life of the hydraulic motor.The fatigue life of an MCRB mainly depends on the basic dynamic load rating and load distribution [6]-all determined by geometric parameters.As a result, designing an MCRB with an optimized geometry to achieve a long fatigue life and high reliability of hydraulic motors is crucial.
The finite element method (FEM) and analytical method are two widely used methods to calculate the load distribution in rolling element bearings [7].The FEM can easily accommodate all the components of the bearings, shafts, and supporting structures and obtain accurate results.Kania [8] used the FEM to calculate the roller deformation in a slewing bearing.Demirhan and Kanber [9] obtained stress and displacement distributions in cylindrical roller-bearing rings and experimentally validated their FEM solutions.Lostado et al. [10] determined the contact stresses in double-row tapered roller bearings using the FEM, which was validated and adjusted using an analytical method and contact pressure sensor experiment.However, FEM incurs high computational costs and requires a denser mesh in the contact area, leading to extremely low computational efficiency.Moreover, FEM has poor flexibility, making it difficult to adjust the geometry and process the results immediately.As a result, the FEM is unsuitable for the continuous optimization of the MCRB.In contrast to the FEM, the analytical method based on quasi-static analysis has higher computational efficiency and greater flexibility.Jones [11] developed a general theory for the load-deflection analysis of bearings based on the Hertz theory.Harris and Kotzalas [12] improved the model by applying a widely known slicing technique; as a result, the model could analyze the roller contact along the axial direction under misalignment and heavily loaded conditions.Analytical methods have been used to calculate the load distributions in cylinder roller bearings [13], tapered roller bearings [14,15], double-row tapered roller bearings [16], and ball bearings [17][18][19].
Although numerous studies have focused on load distribution in various rolling bearings, research that considers the influence of geometric variations on load distribution when optimizing rolling bearings remains limited.Most studies focused on maximizing the basic dynamic load rating to prolong fatigue life [20][21][22].Various types of optimization algorithms, including genetic algorithms [23,24], evolutionary algorithms [25][26][27], metaheuristic algorithms [21], and hybrid algorithms [20,28,29], have been utilized to achieve optimal geometry parameters for higher computing efficiency.In addition, the EHL film thickness [22], friction torque [30], and bearing stiffness [31] are considered constraints or other objective functions.Compared with traditional rolling element bearings, the outer rings of MCRB are prone to free tilting and flexible deformation, which can significantly alter the load distribution [32].In this study, a geometry optimization method that considers both the dynamic load rating and load distribution is proposed to obtain a longer fatigue life for MCRB.
The remainder of this paper is organized as follows.Section 2 introduces the working principles and failure modes of the MCRB in hydraulic motors.Subsequently, a geometry optimization method is proposed in Section 3. Based on this method, the optimized geometric parameters of the MCRB were obtained and discussed in Section 4. Finally, a comparative experiment between the initial and optimized MCRB was conducted on a hydraulic motor test bench to validate the effectiveness of the proposed optimization method.

Working Principle and Failure Analysis of MCRB
Figure 1 shows the structural diagram of the cam-lobe radial-piston hydraulic motor.In this motor, the cylinder block and guide plates are fixed, whereas the piston assemblies reciprocate in the cylinder block to convert hydraulic power into mechanical torque.The working principle of the hydraulic motor is as follows.When the piston chamber is connected to the high-pressure oil port on the oil distributor, the high-pressure oil enters the piston chamber to push the piston assembly outward to rotate the cam ring.When the piston chamber passes the high-pressure oil port and connects to the low-pressure oil port on the oil distributor, the cam ring rotates to press the piston assembly inward to drain the hydraulic oil from the low-pressure oil port.The hydraulic motor can output continuous rotation when the cyclic movement is repeated.The MCRB on the piston assembly is a crucial transmission component that endures complex loads owing to its composition of two guide rollers and two cam rollers.The cam and guide rollers are all full complement needle bearings and possess the same pitch diameter d m and needle diameter d r .The outer ring diameter of the guide roller D 1 is smaller than that of the cam roller D 2 .
A typical failed piston assembly disassembled from a failed hydraulic motor is shown in Figure 2. The failed motor served in the deck winch for approximately three years.The red dashed rectangles marked on the piston pin represent severe fatigue wear regions in contact with the needles of the cam rollers.The needle in the cam roller in Figure 2 exhibits a large amount of fatigue pitting, indicating that the fatigue severity on the needle is comparatively lower than that observed on the piston pin.The outer ring of the cam roller in Figure 2 shows almost no wear scarring, suggesting that the contact pressure on the outer ring is much smaller than that on the piston pin, thus indicating that the fatigue of the MCRB causes the failure of the piston assembly, with the cam roller most susceptible to fatigue failure in the MCRB.Hence, optimizing the geometric parameters of MCRB for a longer fatigue life is necessary.

Geometry Optimization Method of MCRB
In this section, the geometry optimization method, which comprises quasi-static and optimization models, is introduced.A quasi-static model was developed to calculate the load distribution in the MCRB, and an optimization model was established to obtain the optimized geometric parameters for a longer fatigue life of the MCRB.

Quasi-Static Model
The quasi-static model was developed based on the static assumption, where the load-deformation relationship in the MCRB was considered.A schematic of the piston assembly load is shown in Figure 3(a).The piston assembly is subjected to a hydraulic force on the bottom of the piston (F h ), contact force by the guide plate (F 1,n ), and contact force by the cam ring (F 2,n ).The pressure angle between the cam roller and cam ring is β 1 = β n , whereas the pressure angle between the guide roller and guide plate β 2 can be considered as π/2.Based on the load balance relation- The moments acting on the piston pin were calculated using the deflection angle.According to the Timshenko beam theory, the deflection angle of an arbitrary section of the piston pin in the yoz plane (θ x ) can be expressed as [33]: (1) where L r is the outer ring width of the cam and guide rollers.L h denotes the piston width.E and I are the elastic modulus and moment of inertia of the piston pin, respectively.Considering the antiderivatives of Eq. ( 1) and the continuity condition of the piston pin deflection, the moment acting on the piston pin in the yoz plane can be expressed as [12]: where θ 2,px is the deflection angle at point o 2 , as shown in Figure 3(b).Based on the load-balance relationship of the piston pin, Figure 4(a) shows the overall load schematic of a cam roller or guide roller under a normal contact load (F R,n , M R,n ) with a pressure angle β R , where R = 1 refers to the guide roller and R = 2 refers to the cam roller in the MCRB.The reaction loads generated on the piston pin are F R,x , M R,x , F R,z , and M R,z .Figure 4(b) shows the cross-section of the outer ring along the normal direction.As the outer ring is in line with the cam ring or guide plate, the normal contact load (F R,n , M R,n ) can be calculated using the slicing technique.The outer ring is divided into n r slices; thus, the length coordinate of the k n th slice is Then, the contact force on the k n th slice of the outer ring can be calculated using Hertzian contact theory: where c r denotes the contact stiffness between the outer ring and the cam ring/guide plate [6].δ R,kn is the contact interference between the outer ring and the cam ring/ guide plate, which is related to the microdisplacement of the outer ring.Then, the normal contact load (F R,n , M R,n ) can be calculated as: Figure 4(c) shows the cross-section of the jth needle at an azimuth angle of ψ j , where ψ j = 2π(j−1)/n b and n b is the total number of needles.Considering the cam and guide rollers are both fully complemented needle bearings, the total needle number can be calculated as follows [34]: The function fix(•) rounds the argument to the nearest integer towards zero.The needle is divided into n s slices; thus, the length coordinate of the kth slice is y k = − l r /2 + (k − 1/2)l r /n s ; l r is the needle length.Then, the contact force on the kth slice of the jth needle can be calculated using the Hertz contact theory: (4) where c s is the contact stiffness between the needle and outer ring/piston pin.δ R,ijk and δ R,ojk are the contact interferences between the needle and the outer ring/piston pinrelated to the microdisplacements of the piston pin and needle.This calculation method is described in detail elsewhere [32].The contact forces and moments on the jth needle can be calculated using y k q R,ijk , and M R,oj = n s k=1 y k q R,ojk .Then, the equilibrium equation of the jth needle can be expressed as: where F R,cj is the centrifugal force of the jth needle on the cam and guide rollers, respectively.For the load equilibrium of the entire cam or guide roller, the equilibrium equation can be expressed as: In a cam or guide roller, the outer ring deformation, which can significantly affect the load distribution, can be calculated as: where K r is the Fourier series estimated using [35].φ is an arbitrary positional angle of the outer ring.
The microdisplacements of the piston pin, needles, and outer ring in the cam and guide rollers were obtained by solving Eq. ( 4) and Eqs. ( 8)- (10).The iterative Newton-Raphson method is used.The contact forces on each slice of the needles (q R,ijk , q R,ojk ) in the cam and guide rollers can then be obtained.Based on the Hertz theory, the contact pressure between the needles and the piston pin/outer ring (p R,ijk , p R,ojk ) can be calculated as: where E′ is the equivalent elastic modulus.a = i is the upper sign, indicating the contact between the needle and piston pin. a = o is the lower sign, referring to the contact between the needle and outer ring.

Optimization Model
The objective of the MCRB optimization method is to prolong the fatigue life by optimizing the geometric parameters.The fatigue life of the MCRB was calculated using the load distribution and basic dynamic load rating.The basic dynamic load rating of the cam and guide rollers in the MCRB can be expressed by the following geometric parameters [28]: with where i denotes the number of rows of the cam and guide rollers.b m is a modification factor for improving the quality of the bearing steel; b m = 1.1.γ is an auxiliary parameter, defined as γ = d r / d m .
The dynamic load rating of the piston pin or outer ring can be calculated as follows: where α is the nominal contact angle and α = 0.
Then, the basic dynamic load rating of a single contact slice of the raceway is: (1 + γ ) The equivalent contact pressure concentrated on the kth slice of the jth needle is [13]: where w = 4 for the rotating ring, i.e., the outer ring, and w = 4.5 for the nonrotating ring, i.e., the piston pin.
Finally, the fatigue life of the guide and cam rollers in the MCRB was calculated based on ISO/TS 16281 [6].
where R = 1 refers to the guide roller and R = 2 refers to the cam roller.
According to the load analysis in Section 3.1, the external contact force on the cam roller (F 2,n ) was generally smaller than that on the guide roller (F 1,n ).Therefore, the fatigue life of the cam roller is generally shorter than that of the guide roller.Therefore, the fatigue life of the cam roller was set as the objective function.The needle diameter (d r ) and pitch diameter (d m ) are the basic geometric parameters of the MCRB that determine the fatigue life and can be selected as design variables.The other geometric parameters and working conditions are listed in Table 1.Thus, the objective function can be expressed as: The constraints of the optimization model are as follows: First, the fatigue life of the guide rollers L 1,10r should be no less than that of the cam rollers L 2,10r during (16) . ( L R,10r = n s k=1 q cik q R,eik the geometry optimization of the MCRB.In addition, variations in the geometric parameters can lead to a high contact pressure between the needles and raceways, possibly resulting in some degree of permanent deformation at the contact surfaces.To avoid this type of failure, the contact pressure should be constrained to less than 4000 MPa [28].Furthermore, variations in the needle diameter (d r ) and pitch diameter (d m ) can also change the thickness of the outer rings and the diameter of the crosshead pin, which significantly influences the strength of these parts.The maximum stress in the outer ring of the cam roller S 1,r , maximum stress in the outer ring of the guide roller S 2,r , and maximum stress in the piston pin S c should be less than the permissible yield stress [σ m ].Finally, the circumference clearance C between the needles should be larger than the limited value C LV to guarantee the smooth running of the needles without collision-generally one degree between needles [28].
In summary, the optimization constraints are: A computer program was developed to calculate the objective functions and constraints using MATLAB and then linked to commercial optimization software.A genetic optimization algorithm was applied to determine the optimal solution.A flowchart of the optimization procedure is presented in Figure 5.The principle of the genetic optimization method can be expressed as an analogy between natural selection and the survival of fitness.Genetic optimization starts with a randomly generated population of individuals.Individuals in the initial population are characterized by genomes containing a string of chromosomes (randomly generated design variables).The evaluation process follows, in which each individual is evaluated based on the fitness function expressed in terms of the objective function.Using natural genetic operators, selection, crossover, and mutation are used to update a population [36].Finally, the optimized design variables were obtained.

Optimization Results and Discussion
The initial design and ranges of the design variables are listed in Table 2.In total, 1000 designs were calculated, and the computation time for a single design was approximately 0.2 s.The optimization curves of the design variables and objectives are plotted in Figure 6(a) and (b), respectively.First, random designs of experiment (DOE) were created as the initial population for the optimization (20) algorithms.Then, the variation in the design variables and the objective tends to stabilize after approximately 400 generations.After optimization, the fatigue of the cam rollers in MCRB increased from 0.66×10 6 to 1.31×10 6 r (approximately 98.5% increase), while the fatigue life of the guide rollers in MCRB increased from 4.29×10 6 to 6.82×10 6 r (approximately 59.0% increase), thus indicating that the proposed optimization method can effectively improve the fatigue life of MCRB.Generally, the MCRB is the first to fail in hydraulic motors.Thus, this optimization method can effectively prolong the fatigue life of hydraulic motors.Figure 7(a) and (b) show the contact pressure distributions in the initial and optimized guide rollers, respectively, and Figure 7(c) presents the contact pressure along the length of the most loaded needle.The contact pressure distributions of the initial and optimized guide rollers were similar, indicating that the optimization method did not increase the contact pressure on the guide rollers.Figure 8 shows the contact pressure distributions in the initial and optimized cam rollers.The maximum contact pressure in the optimized cam roller is significantly lower than that in the initial one.Therefore, the optimization method can reduce the contact pressure in the MCRB.According to Eq. ( 12), the basic dynamic load rating of the optimized guide and cam rollers in the MCRB increased from 89 to 109 kN.As a result, the fatigue life of both the guide and cam rollers increased.In summary, the optimization method increased the fatigue life of the MCRB by reducing the contact pressure and increasing the dynamic load ratings.

Experimental Validation
The full fatigue life test of an MCRB in a hydraulic motor is time-consuming.It is difficult to identify when the MCRB starts fatiguing because it is assembled in the motor and difficult to observe during the test.However, the fatigue life of an MCRB depends on its basic dynamic load rating and distribution.The dynamic load rating of the optimized MCRB, which was calculated using geometric parameters, improved from 89 to 109 kN.Therefore, a reduction in the contact pressure is an alternative way to prove the effectiveness of the proposed geometry optimization method.In this study, a loading test was conducted on a hydraulic motor installed with both initial and optimized MCRB.The wear losses of the initial and optimized MCRB were compared to demonstrate the contact pressure reduction in the optimized MCRB, thereby validating the proposed optimization method indirectly.Four piston assemblies each, with the initial MCRB and the optimized MCRB, were manufactured.Figure 9 presents the material preparation procedure used in the experiment.The initial and optimized cam and guide rollers are shown in Figure 9(a).Then, the corresponding piston and piston pins were prepared and assembled, as shown in Figure 9(b) and (c).Next, the cam rollers, guide rollers, and other parts were mounted on the piston pin, as shown in Figure 9(d).Finally, four piston assemblies with the initial MCRB and four with the optimized MCRB were assembled in the hydraulic motor, as shown in Figure 9(e).The initial and optimized piston assemblies are arranged alternately in the tested hydraulic motor, whereas the optimized piston assemblies are marked by red circles in Figure 9(e).The hydraulic motor test bench is shown in Figure 10.The output shafts of the two hydraulic motors with similar displacements were rigidly coupled.The tested motor was actuated using the first pump with a high-pressure oil output.The loading motor serves as a pump, where low-pressure oil is pumped by the second pump, which then outputs high-pressure oil.The outlet pressure of the loading motor and the inlet pressure of the tested motor were adjusted using a proportional relief valve.The rotational speeds of the two motors were adjusted based on the displacement of the first and second pumps.
Before and after the experiment, the circular profile of the prepared piston pins was measured using a CMM, as shown in Figure 9(c).The CMM used in this study is Zeiss CMM Micura 5/5/5 with a measuring accuracy of less than 1 μm.During the measurements, five uniformly spaced circles along the contact length between the cam roller and the piston pin in each piston assembly were selected and measured.
The experimental operating conditions are shown in Figure 11(a).The duration of the experiment was approximately 6 h.The inlet pressure of the tested motor was varied from 15 to 25 MPa (equivalent to a hydraulic force F h of 85 to 141.8 kN), and the rotation speed was varied from 15 to 10 r/min.After the hydraulic motor test bench experiment, the piston pin was measured again to calculate the circular wear loss.The wear loss results are shown in Figure 11(b).The wear loss in the optimized MCRB was much smaller than that in the initial MCRB, indicating significant reductions in the contact pressures in the optimized MCRB.The average reduction rate was approximately 60%.Consequently, the fatigue life of the MCRB was improved using the proposed optimization method.

Conclusions
In this study, an innovative geometry optimization method for the MCRB to improve the fatigue life of hydraulic motors was proposed.A loading test was conducted on a hydraulic motor installed with both the initial and optimized MCRB to validate the effectiveness of the proposed optimization method.The following conclusions were drawn.(1) The proposed geometry optimization method can effectively increase the fatigue life of the MCRB in hydraulic motors: Approximately 98.5% increase for the cam roller and approximately 59.0% increase for the guide roller.(2) The fatigue life of an MCRB is based on its basic dynamic load rating and distribution.The proposed geometry optimization method can reduce the contact pressure and improve the basic dynamic load rating of the MCRB, thereby increasing the fatigue life.
(3) The experimental results showed that the wear loss of the piston pin on the optimized MCRB could be reduced by approximately 60%, indicating that the contact pressures in the optimized MCRB were significantly smaller than those in the initial one.Thus, indirectly validating the improvement in the fatigue life of the optimized MCRB is possible.
While geometry optimization is an effective method for improving the fatigue life of MCRB in hydraulic motors, it has limitations.The geometry optimization design can approach material performance limit continuously, but can not exceed it.In future work, a fatigue life test rig of the MCRB will be established, particularly focusing on material surface strengthening in the MCRB to further improve the fatigue life.

1 2Figure 1
Figure 1 Structure diagram of cam-lobe radial-piston hydraulic motor and MCRB can be deduced.As shown in Figure 3(b), the component forces of the cam and guide rollers acting on the piston pin are F R,x = F R,n sin(β 1 ) and F R,z = F R,n cos(β 2 ), where R = 1 refers to the guide roller and R = 2 refers to the cam roller in the MCRB.

Figure 2 Figure 3
Figure 2 Failure piston assembly

Figure 4
Figure 4 Load analysis on the MCRB: (a) Load on the whole MCRB, (b) Load on the outer ring, (c) Load on the needle

Figure 5 2 Figure 6
Figure 5 Flowchart of the geometry optimization method

Figure 7 Figure 8
Figure 7 Comparison of contact pressure distribution in the guide roller: (a) Initial guide roller, (b) Optimized guide roller, (c) Most loaded needle in the initial and optimized guide roller

Figure 9
Figure 9 Material preparation for the experiment: (a) Initial and optimized cam roller and guide roller, (b) Piston, (c) Piston and piston pin, (d) Optimized piston assembly, (e) Hydraulic motor

Figure 10 Figure 11
Figure 10 Hydraulic motor test bench: (a) Front view of the test bench, (b) Side view of the tested motor, (c) Schematic of the test bench hydraulic system

Table 1
Basic parameters in geometry optimization