DOI: 10.5545/sv-jme.2025.1394 219
© The Authors. CC BY 4.0 Int. Licencee: SV-JME  Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 7-8   ▪  Y 2025
Cam Ring Curve Optimization for Radial Hydraulic Motor 
Based on Seagull Algorithm
Bowen Zhang − Shaopeng Kang
 
 – Runze Zhou − Hongbin Qiang − Jing Yang − Kailei Liu − Yunkai Zhou
Jiangsu University of Technology, School of Mechanical Engineering, China
 ksp@jsut.edu.cn
Abstract With the accelerated adjustment of the global energy structure, the intensity of strategic mineral resource development continues to rise. The 
growing demand for construction machinery and equipment has led to increased use of radial piston hydraulic motors, known for their low-speed and high-
torque characteristics, in heavy-duty mining machinery and equipment. However, the cam ring curve design of a radial piston hydraulic motor still faces critical 
challenges, including significant contact stress, and noticeable output pulsation. To address these issues, this paper proposes a multi-step composite curve 
optimization design method based on the seagull optimization algorithm (SOA). This method integrates the concepts of stepped and trapezoidal acceleration 
curve design, utilizing the SOA algorithm to tackle these challenges. The SOA algorithm is more adaptive than the genetic algorithm and the particle swarm 
optimization algorithm, maintaining population diversity even in the later stages of iteration. This effectively overcomes the limitations of the particle swarm 
optimization and genetic algorithm in multi-peak problems. A cam ring curve optimization model is established to minimizing contact stress through theoretical 
modeling and dynamics analysis. Using the combined global search and local exploitation capabilities of SOA, we achieved multi-constraint optimization on 
key parameters such as the amplitude-angle ratio in the acceleration zone and the amplitude-angle in the zero-speed zone. This results in a composite cam 
ring curve characterized by reduced stress, low pulsation, and shock-free operation. The effectiveness of this method was validated through bench tests, which 
showed that the maximum contact stress of the optimized multistep composite curve at a speed of 100 rpm is reduced by 5.4 % and 18.3 % compared to the 
conventional equal acceleration curve and trapezoidal curve, respectively. Additionally, the speed pulsation rate decreases by 10.81 % and 25.73 % under 20 
MPa and 30 MPa conditions, respectively, with no sudden change in reaction force and a reduction in pulsation shock during operation.
Keywords radial piston hydraulic motor, cam ring optimization, seagull optimization algorithm, contact stress, speed pulsation
Highlights
 ▪ Proposed a multi-step composite curve optimization design method based on the Seagull Optimization Algorithm.
 ▪ Demonstrated the effect of cam ring curve on motor performance.
 ▪ Revealed superior adaptability and population diversity preservation in multi-peak problems.
 ▪ Confirmed optimized smoothness and energy efficiency of hydraulic motors.
1 INTRODUCTION
Radial piston hydraulic motors are the core driving elements of 
medium- and large-sized machinery and equipment. Due to their 
high stability at low speeds and excellent starting efficiency, they are 
widely used in construction machinery, ships, and other applications 
[1-3] . The cam ring is the key element of the in-curve radial piston 
hydraulic motor, as it governs the macroscopic motion behavior of 
the roller-plunger-cylinder assembly. Different theoretical design 
approaches can lead to different component motion patterns [4-5]. 
Assuming that the input hydraulic fluid oil pressure remains constant 
and the instantaneous output torque of the motor is the sum of the 
tangential component forces generated by each plunger pair on 
the rotary axis [6-7], synthetic torque pulsation can be reduced by 
optimizing the cam ring curve without considering external factors.
As the component consistently interacts with the cam ring during 
operation, it is susceptible to shock and vibration under high oil 
pressure and high-speed conditions. Optimizing the design method 
of the cam ring curve can effectively reduce both the output torque 
pulsation and speed pulsation of the motor. In the early stages, due 
to the limitation of processing technology and machining equipment, 
the cam rings were primarily designed using simple and easy-to-
machine curves, such as arcs and straight lines [8]. However, the 
discontinuous curvature at the joints caused sudden changes in the 
acceleration of the piston, leading to periodical fluctuations in the 
output torque of the hydraulic motor and resulting in prominent 
pulsations. With the development of high-precision machining 
techniques and equipment, the design methods for curves have 
become diversified [9], contribution to reduced speed pulsation 
and torque pulsation in hydraulic motors [10-11]. Many scholars 
and engineers have proposed various solutions aimed at achieving 
smoother operation, longer service life, and lower pulsation rates. 
Giljen et al. [12] used the variable speed technology effectively 
optimizing the performance of water pumps under different operating 
conditions. The internal transient flow is extremely complex, causing 
unstable phenomena such as eddy currents and backflow. Research 
on these flow characteristics is of great significance for ensuring 
the safe and stable operation of the unit and improving efficiency. 
Tang et al. [13] used surface texturing technology to regulate the 
friction interface characteristics and explore the influence of groove 
texturing on vibration characteristics at different speeds. Gao et al. 
[14] established a life assessment model, the effectiveness of the life 
assessment method with crankshaft bearings as key components, 
providing a reference for contact stress analysis in this study. Chen 
et al. [15] designed a conical gear pair with a vertical tooth profile to 
improve its transmission performance in microfabrication and refine 
the design theory. Zhang et al. [16] proposed a genetic algorithm-
based design method for optimizing the high-order cam ring curves, 
effectively reducing pulsation rate, and ensuring smooth, shock-free 
transitions between speed intervals. Li et al. [17] used an improved 

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particle swarm algorithm to optimize the partition amplitude angle 
to mitigate fatigue wear and reduce impact effectively. Zhou et 
al. [18] employed the Fourier series to describe the displacement 
function of the follower in a high-speed cam mechanism, which 
effectively reduced vibration and shock velocity. Chao et al. [19] 
proposed an optimal design method for pre-compensation of cam 
ring deformation, which couples the effect of elastic deformation 
during the design stage to reduce the motor pulsation thus enhancing 
the stability of the motor. Despite the progress made in reducing the 
pulsation in radial piston motors, further progress is needed to better 
optimize the impact and contact stresses.
Despite advancements in cam ring curve design, radial piston 
hydraulic motors continue to exhibit shocks, pulsations, and limited 
cam ring service life. The study proposes a multi-step composite 
curve design that combines the low-stress characteristics of the 
stepped curve with the shock-free advantages of the trapezoidal 
curve, introducing the seagull optimisation algorithm (SOA) 
optimization algorithm to achieve global parameter optimization and 
reduce shocks and pulsations during motor motion. First, a typical 
radial piston motor’s motion, force, and deformation characteristics 
are analyzed in detail. Secondly, a multi-amplitude angular partition 
optimization method is established to optimize the cam ring curve, 
thereby reducing pulsation of the motor. Finally, the effectiveness 
of the multi-step composite curve optimization design method is 
verified through bench tests. This study combines SOA with cam 
ring design for the first time, providing a new practical strategy for 
reducing shock and improving operational stability in radial piston 
hydraulic motors, while contributing significantly to the broader 
optimization of fluid transmission systems.
2 METHODS & MATERIALS
Internal curve radial piston hydraulic motors can operate in multiple 
modes of action and be switched to various functions according to the 
load conditions and the application environment to meet the needs of 
diverse industries. The operating principle of the motor is shown in 
Fig. 1. During operation, the hydraulic pump extracts hydraulic oil 
from the tank and conveys it through the pipeline to the inlet port 
of the motor. The high-pressure hydraulic oil entering motor then 
flows into the piston chamber where the pressure acts on the bottom 
of the piston, pushing the piston outward. During the return stroke, 
the piston assembly presses the low-pressure fluid from the piston 
chamber to the return port. Due to the geometry of the inner curve 
combined with the piston-roller assembly, the assembly moves at a 
certain pressure angle, which ensure torque transmission and power 
output. Under the continuous action of hydraulic oil, the piston drives 
the roller to complete a lift-and-return motion along the cam ring rail. 
One complete lift and return constitutes an action interval, and the 
cam ring curve consists of several action intervals.
Fig. 1. Schematic diagram of motor structure;  
1. front cover assembly, 2. bearing, 3. cam ring, 4. rear cover assembly, 5. distributor valve,  
6. main shaft, 7. roller, 8. shaft tile, 9. piston, and 10. cylinder block assembly
Since the roller continuously interacts with the cam ring during 
operation, shock and vibration are likely to occur under high oil 
pressure and high speed. Over time, it can cause damage to the 
cam ring, thereby affecting the overall service life of the motor. 
To solve this problem, this study aims to reduce the output torque 
pulsation, speed pulsation, shock, and vibration. The feasibility and 
effectiveness of the proposed design method is validated by bench 
testing.
2.1 Force Characterization of Cam Ring-Roller Component
2.1.1 Reaction Force Solving
To calculate the overall torque pulsation, speed pulsation, and the 
service life of the cam ring guide for the motor, it is necessary to 
determine the maximum contact stress experienced by the roller 
during motion. This stress is related to several factors, including 
radius of curvature of the cam ring, friction, hydraulic pressure, and 
reaction forces. Therefore, a detailed force analysis of the cam ring-
piston assembly is required. The schematic of the component force is 
shown in Fig. 2.
Fig. 2. Schematic of the force analysis of piston assembly
A reaction force analysis of the piston, roller, and cam ring is 
conducted, as shown in Fig. 2. Taking O
2
, the center of a roller, as 
a reference point, the force and moment balance equations can be 
obtained as follows:
MMMM
1 234
0  , (1)
FNFN F
tf g
 
12
cos,  (2)
FFFFFFF
pGaLRrf
sin.  (3)
Where M
1
 torque exerted by the frictional force F
f
  between the 
cam ring rail and the roller, acting at the center O
2 
of the roller, M
2
 
is the resultant torque at the center O
2
 of the roller, equivalent to the 
combined sliding friction forces F
L
 and F
R
 between the piston and 
cylinder block assembly, also acting at the center of the roller, M
3
 
represents the torque acting on the center of the roller O
2
 due to the 
forces N
1
 and N
2
 distributed along the contact line in a triangular 
pattern between the generated between the piston and the inner wall 
of cylinder block assembly, M
4
 is the inertia force F
g
 generated by the 
Gothic acceleration a
g
 at the croller’s center O
2
. F
p
 is the hydraulic 
pressure, while F
a
 is the inertia force generated by the acceleration 
of the piston relative to the cylinder block assembly, and F
G
 is the 

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centrifugal force generated by the rotation of the piston relative to 
the cylinder block assembly. Assume the length of the contact line 
between the piston and the inner wall of the cylinder block assembly 
be L
c
, which is expressed as follows:
LLLL
c

120

 min
.
 (4)
Here, L
0
 is the contact length when the roller is positioned at the 
minimum pole diameter, and ρ
min
 is the minimum pole diameter.
The magnitudes of each individual moment is:
MF R
f 1
= , (5)
MF
d
F
d
RL 2
22
  , (6)
MN LLNL LL L
c 31 12 11
1
3
1
3







 

  







 min
, (7)
MF L
gg 4
 . (8)
Here F
f
 iS the friction force exerted on the roller by the cam ring, 
R is the radius of the roller, and d is the diameter of the bottom of 
the piston. By substituting Eqs. (5) to (8) into Eq (1), the parameter 
L
1
 can be determined. Due to the motor lubrication at the roller and 
cam ring rail contact, the resulting friction is minimal. Furthermore, 
based on existing literature and industry experience, friction torque 
is much smaller than the main transmission torque and therefore has 
a negligible impact. Experimental results also revealed no abnormal 
speed fluctuations or unstable motor operation caused by roller 
friction. Therefore, M
1
 can be ignored. Similarly, since the distance 
from the roller center to the center of mass L
g
 is small compared to 
the polar diameter, and the speed pulsation of the motor under normal 
operating conditions is also small, the inertia force F
g
 generated by 
the corresponding acceleration a
g
 is negligible. Thus, M
4
 can also be 
ignored. The equations can then be solved as follows:
L
LL Lf dL
Lf dL
cc c
c
1
2
64 3
12 66











min
min
. (9)
Substitute Eq. (4) into Eq. (3) to obtain N
1
 and N
2:
NF FF
L
LL L
fN g
cc
1
1
2
1
2
 


coss in , 
 (10)
NF FF
LL
LL L
fN g
c
cc
2
1
2
1
2
 

 

coss in .  (11)
Substitute Eqs. (10) and (11) into Eq. (2) yields:
F
FFFF
ff
N
Gpag




 


22
coss in cosc os sin
,
  
 (12)


 
2
1
11
2
1
2
LL L
fL LL
cc
c
. (13)
The mechanical analysis conducted by this research study is 
based on quasi-static processes, taking into account inertial forces, 
centrifugal forces, and reaction forces during motion. This approach 
is suitable for analyzing force variation trends under stable operating 
conditions but is not applicable to transient limit states occurring at 
high-speeds or under unstable conditions.
To ensure the physical validity and applicability of the force 
balance equations and moment balance equations established in this 
study, the derived formulas are based on the following assumptions: 
1. Rotational speed range: in the analysis applies to low- to medium-
speed operation of radial piston hydraulic motors (typically below 
350 rpm). Within this speed range, inertial forces generated by 
motor components are relatively small, and the imbalance effect 
that occur at high speeds can be ignored.
2. Pressure range: The working pressure of the hydraulic system is 
assumed not to exceed the rated pressure value of 35 MPa. Pressure 
beyond this limit would cause significant elastic deformation 
of key components such as the cylinder block assembly and the 
piston assembly.
3. Friction conditions: It is assumed that sufficient lubrication exists 
between the rollers and the cam ring, as well as between piston 
and cylinder block assembly. Under these conditions, frictional 
forces are minimal, allowing the friction torque M1 and inertial 
torque M4 to be ignored.
Under the above operating conditions, the mechanical model 
derived in this paper provides a reasonable approximation of the 
actual operating conditions of the motor.
2.1.2 Contact Stress Analysis
In a radial piston motor, the roller acts as a rolling element that moves 
along the cam ring curve. As shown in Fig. 3, when the roller comes 
into contact with the cam ring surface, the contact area experiences 
localized elastic deformation due to the applied pressure.
In the contact between the roller and the cam ring, the radius of 
curvature of the cam ring is much larger than the radius of curvature 
of the roller. Therefore, it can be regarded as a problem of contact 
between a cylinder and a plane. The contact between the roller and 
the cam ring guide is thus treated as a line contact. According to the 
Hertzian contact theory, the maximum contact stress σmax at the 
center of the contact surface is expressed as follows:


max

2F
lb
N
, (14)
b
FR
lE
N

4
*
*
,

 (15)
Fig. 3. Simplified model of Hertzian contact of cam ring with roller

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222   ▪   SV-JME   ▪   VOL 71   ▪   NO 7-8 ▪   Y 2025
1
11
1
2
1
2
2
2
EE E
*
, 






 (16)
where l is the length of the contact line between the two elastic solids, 
which in this case equals the width of the cylindrical roller, and b is 
the width of the contact zone. Parameters μ
1
 and μ
2
, and E
1
 and E
2
 are 
the Poisson’s ratios and modulus of elasticity of the two contacting 
elastic solids, respectively. For a hydraulic motor with a cylindrical 
roller, the equivalent (integrated) curvature is given as:
111
RR R
c
*
,  (17)
where R is the radius of curvature of the roller, and R
c
 is the radius 
of curvature of the cam ring rail at the point of contact between the 
roller and the rail curve. Substituting Eqs. (15) to (17) into Eq. (14) 
yields:

 
max

 








FEERR
lRRE E
Nc
c
12
12
2
21
2
11
. (18)
2.2 Methods for Designing Multi-Step Composite Curves
The traditional iso-acceleration curve produces rapid changes in 
piston acceleration during the acceleration and deceleration phases. 
This leads to flexible shocks, and large instantaneous torque 
fluctuations. which in turn increase transient contact stresses, induce 
significant impact loads, and decrease the service life of the cam ring. 
To mitigate these effects, the stepped iso-acceleration curve divides 
the single-stage acceleration of the traditional iso-acceleration curve 
design into two stages. This approach reduces instantaneous torque 
fluctuations during both acceleration or deceleration, helping to 
maintain relatively smooth piston motion. However, flexible impacts 
still occur at the beginning and end of each section of the stepped 
acceleration curve, which limits its overall effectiveness.
To eliminate the problems of flexible impacts and excessive 
contact stress, this study proposes a multi-step composite curve, 
which integrates the “stepped” and “trapezoidal” acceleration curve 
design concepts. The original equal-acceleration zone is divided into 
two acceleration segments, while a transition zone is added to the 
deceleration phase. The result is a composite cam ring curve that is 
shock-free, exhibits reduced pulsation, and generates lower contact 
stress .
Fig. 4. Cam ring curve schema
In radial piston hydraulic motors, the cam ring guide determines 
the motion pattern of the piston-roller assembly. The cam ring 
guide curve schema is shown in Fig. 4. Curve B represents the ideal 
theoretical contour, i.e., the trajectory of the roller center, while 
curve A is the outer envelope of curve B, i.e., the actual curve of the 
cam ring. Depending on the theoretical design, the B curve leads to 
different laws of motion of the plunger assembly. In this context: the 
distance between the roller center O and the roller center O
2
 is the 
pole diameter ρ, the minimum value of the pole diameter is ρ
0
, the 
roller stroke is h, the maximum value of the pole diameter is ρ+h, the 
angle between ρ and the initial position ρ
0
 is φ, the amplitude angle of 
the zero-speed zone is φ
0
, the angle of rotation per unit time dt is dφ, 
α is the pressure angle, abd F
n
 is the reaction force of cam ring guide 
on the roller. The reaction force of F
n
 is the reaction force of the cam 
ring guide on the roller, which can be decomposed into tangential 
force F
r
 and radial force F
t
.
The piston-roller assembly motion can be divided into the 
telescopic motion and the rotary motion of the piston assembly 
relative to the cam ring. The angular velocity v
φ
 and angular 
acceleration a
φ
 of the piston assembly relative to the cylinder block 
assembly are expressed as follows:
v
d
d



 , (19)
a
d
d




2
2
. (20)
Fig. 5. Plot of amplitude distribution of multi-step composite curves
As shown in Fig. 5, the multi-step composite curve with single 
lift consists of 13 zones. These include: the zero-velocity zone θ
0
, 
the first-order iso-accelerated pre-compensation zone θ
1
, the first-
step acceleration zone θ
2
, the second-step equal-acceleration front 
compensation zone θ
3
, the second-step acceleration zone θ
4
, second-
step post-compensatory zone θ
5
, the uniform-velocity zone θ
6
, the 
pre-compensation zone before the first-step iso-deceleration θ
7
, the 
first-step iso-deceleration zone θ
8
, the second-step equal-deceleration 
front compensation area θ
9
, the second-step deceleration zone θ
10
, the 
second-step post-compensation zone θ
11
, and the zero-velocity zone 
θ
12
. The displacement ρ(j) at the center of the roller, angular velocity 
n(  j), and angular acceleration a(  j) are shown in Figs. 6 to 8.
The design process for each section is as follows: θ
0
 provides 
a buffer for the piston when transitioning from rising to falling, 
avoiding impact between the roller and the cam ring guide rail due 
to sudden speed reversal. This zero-speed zone reduces hydraulic 
shock and energy loss, helps accurate oil inlet and return cycles, and 
improves distribution efficiency. θ
1
 serves as a cushioning zone for 

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the piston, mitigating “flexible impact” caused by sudden changes 
in acceleration. The transition zone allows continuity of curvature 
between the zero-speed and acceleration sections. θ
2
 represents the 
first-stage acceleration. In traditional constant-acceleration designs, 
excessive acceleration leads to high reaction forces and contact stress. 
By dividing the original single-stage acceleration into two stages 
( θ
2
 and θ
4
), acceleration in this phase is reduced, lowering reaction 
force and contact stress at the transition point. θ
3
 connects the first- 
and the second-step acceleration zones, eliminating acceleration 
discontinuities and avoiding “flexible impact” caused by sudden 
acceleration changes. θ
4
 denotes the second acceleration zone of the 
piston. Excessive acceleration in this zone can also cause excessive 
reaction forces and contact stresses. θ
5
 provides a buffer when the 
piston enters the constant-speed section, reducing motor vibration 
caused by sudden acceleration reduction. θ
6
 is the uniform-velocity 
zone, where piston speed remains and the acceleration is 0. θ
7
~ θ
11
 
are symmetrical design relative to the acceleration phase. This region 
consists of deceleration compensation zone, deceleration zone, and 
transition compensation zone. θ
12
 functions identically to the zero-
velocity zone θ
0
, ensuring a smooth transition without impact.
The corresponding values are computed as follows:
Zero-velocity zone (0 ≤ φ < φ
0
):
 
0
. (21)
First-order iso-accelerated pre-compensation zone ( φ
0
 ≤ φ < φ
1
):



10
10
3
1
6

  a
, (22)
First-step acceleration zone ( φ
1
 ≤ φ < φ
2
):
 
 
21 1
11 11 1
2
22
  
 

  aa
. (23)
Second-step equal-acceleration front compensation zone ( φ
˝
 ≤ φ < φ
3
):
   

 
32 2112 2
12
2
12
3
1
2
1
2
1
  






 
  
  
a
a
a
6 6
3

.
 
 (24)
Second-step acceleration zone ( φ
3
 ≤ φ < φ
4
):
    
 
43 311233 3
13
2
1
2
1
2
1
1
2
     






 
 
a
a . . (25)
Second-step post-compensatory area ( φ
4
 ≤ φ < φ
5
):
  



54 41 4
2
4
3
5
11 23
1
23
1
2
1
2
    
 








 
a
a    






  1
34 4
   . (26)
Uniform velocity zone ( φ
5
 ≤ φ < φ
6
):
 
    
65 5
11 23 34 55
1
2
1
2
1
1
2
 
    






  a . (27)
Compensation zone before the first-step iso-deceleration ( φ
6
 ≤ φ < φ
7
):
 


   
76 6
16
3
7
11 23 34 5
6
1
2
1
2
1
1
2
  
 
    



a
a
 


  
6
. (28)
First-step iso-deceleration zone ( φ
7
 ≤ φ < φ
8
):
  
    
87 717
2
12 33 45
1
2
1
2
1
1
2
   
   







a
a
7 7
 . (29)
Second-step equal-deceleration front compensation zone ( φ
8
 ≤ φ < φ
9
)
     

98 81 33 45 8
18
2
1
2
1
1
2
1
2
     






 
 
a
a  
    

1
6
18
3
9
a
. (30)
Second-step deceleration zone ( φ
9
 ≤ φ < φ
10
):
   
10 99 14 59 9
2 1
2
2        

a . (31)
Second-step post-compensation zone ( φ
10
 ≤ φ < φ
11
):
   


11 10 10 15 10 10
2
10
3
11
1
23
      
 








a . (32)
Zero-velocity zone ( φ
1˝
 ≤ φ < φ
12
):

12 0
 h, (33)
where a
1
 is the first-step iso-acceleration zone , and μ is the ratio of 
the second-step acceleration to the first-step acceleration.


110221 332443
554665 7
   
  
,,,,
,,
7 76887
9981 01 09 11 11 10 12 12 11
 
   

   
,,
,,, ,, . a
h
x
1
12 34 5
1
2
1
2
1
2
1
1
2

   






    
2.3 Optimized Design Based on SOA Algorithms
Intelligent optimization algorithms based on SOA algorithms 
demonstrate significant advantages in addressing complex 
engineering problems. Classical methods, such as genetic alorithm 
(GA) and particle swarm optimisation (PSO) are commonly used 
for parameter optimization. However, these algorithms still face 
limitations in tackling high-dimensional and nonlinear challenges. 
GA requires predefined parameters such as crossover rate and 
variance rate, which are highly experience-dependent and greatly 
influence the quality of the solution. This reliance often results in 
premature convergence or computational inefficiency, as its local 
development capability is inadequate, leading later iterations to 
become trapped in local optimal solutions [20-21]. In addition, PSO is 
prone to converge prematurely in complex scenarios due to a loss of 
population diversity, particularly in multi-peak function optimization. 
Its parameters, including inertia weights and learning factors, 
are highly sensitive and must be dynamically adjusted; improper 
configurations significantly diminish robustness [22-23]. SOA, by 
contrast, is a relatively new heuristic swarm intelligence algorithm 
inspired by the foraging and migratory flight patterns of seagulls 
[24-26]. This algorithm optimizes complex nonlinear problems by 
simulating the seagull flocking behavior , making it particularly well-
suited for continuous optimization and multi-peak scenarios. Unlike 
PSO, which requires manual adjustment of inertia weights, SOA 
demonstrates greater adaptability. By incorporating spiral flight and 
random perturbation strategies, it maintains population diversity even 
in later iterations, thereby effectively addressing the limitations of 
PSO and GA in multi-peak problems.

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224   ▪   SV-JME   ▪   VOL 71   ▪   NO 7-8 ▪   Y 2025
Fig. 6. Roller center displacement
Fig. 7. Roller center angle velocity
Fig. 8. Roller center angular acceleration
SOA is inspired by two characteristic behaviors of seagulls−
foraging and migration− which jointly enable local exploitation near 
promising solutions and global exploration of the search space: 
1. Foraging behavior (local search): During foraging, seagulls 
hover, circle and gradually approach the target before diving to 
capture the prey. This behavior reflects intensification around the 
current best solution. The position of the i
th
 seagull is updated as:
Xt Xt eR XX t
ii
t
best i
() () (),   

1

 (34)
 where X
i
(t+1) is the position of the i
th
 seagull at iteration t+1, β is 
a parameter controlling the speed of convergence, R is a random 
value, which can be used to increase the randomness, and X
best
 is 
the current best position in the population.
2. Migratory behavior (global search): During migrations, 
seagulls travel long distances and stop at target migration points, 
which corresponds to diversification over a broader region to 
escape local optima. A simple migration-inspired update that 
introduces a directional (spiral/circling) drift is: 
Xt XX tX
ib esti best
() sin( ),    1  (35)
 where α is a scaling factor to control the scope of the local search, 
and θ is a random angle to simulate the flight path of a circling 
seagull.
2.4 Contact Stress Suppression Strategy Based on SOA 
Algorithm
The trajectory and velocity of the piston assembly significantly 
affect the distribution of contact stress. As acceleration increases, 
the resulting inertia force is transmitted to the cam ring guide via 
the piston roller, leading to higher contact stresses. This effect is 
particularly pronounced during curve transitions, where abrupt 
acceleration changes can lead to sharp stress spikes, leading to 
increased friction and wear. Such conditions may also lead to 
localized material yielding or surface damage. By contrast, an 
optimized design of the cam ring can effectively extend the motor’s 
service life.
In designing the inner curve of the cam ring, a zero-speed zone θ
0
 
is typically established both at the beginning and the end of the curve 
to mitigate the mechanical shock caused by sudden speed changes in 
the piston assembly during transitions and to reduce the leakage effect 
resulting from misalignment of the mating phase. Improperly setting 
the zero-speed zone shortens the effective working stroke of the 
piston and decreases the motor’s output displacement, consequently 
reducing its output power and overall efficiency. θ
1
 and θ
3
 provide 
cushioning for the piston, improving the continuity and stability of 
the curve. If θ
1
 and θ
3
 are too small, acceleration discontinuities 
occur, which can easily lead to “flexible impact”. Conversely, if they 
are too large, they compress effective acceleration interval, reducing 
the acceleration angle and increasing contact stress and reaction 
forces, which reduces the motor operating efficiency. θ
2
 controls 
both the timing and magnitude of acceleration. A too small θ
2
 yields 
excessive acceleration, causing peak contact stresses and accelerating 
rail wear, while an overly large θ
2
 prolongs the acceleration cycle, 
affecting the output response delay. Segmented acceleration control 
allows smoother acceleration transitions, greatly reducing stress 
concentrations. The ratio of the acceleration in the second-stage to 
that in the first is determined by μ. If μ is too large, it causes excessive 
differences in acceleration, resulting in sudden velocity jumps. If μ is 
too small, it reduces the optimization effect.
To address the issues of local material yielding, surface damage, 
and reduced cam ring life caused by excessive contact stress, the 
SOA algorithm has been selected to optimize the design of multi-step 
composite curves. The fitness function is defined as the minimum 
value of the maximum contact stress within the deformation. 
Additionally, since the acceleration and deceleration zones of the cam 
ring curve are symmetrical, the zero-speed zone amplitude angle θ
0
 is 
optimized alongside the three acceleration zone amplitude angles θ
1
, 
θ
2
, θ
3
, as well as the ratio of the amplitude angles of the two equal-
acceleration zones, μ.
Minimize:(),
[
FX
FEERR
lRRE E
X
Nc
c

 









12
12
2
21
2
11  
x xxxxx
1 2345 0123
,,,,][,,,,],   (36)
 Subject to: 0.5 ≤ θ
0
 ≤ 1.5, 0.6 ≤ θ
1
 ≤ 1, 1.2 ≤ θ
2
 ≤ 9.6, 0.6 ≤ θ
3
 ≤ 1, 
1.3 ≤ μ ≤1.7.

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Where F(X) represents the maximum value of contact stress 
generated in the roller throughout the motion process in contact with 
the guide curve, θ
0
 is the magnitude of the zero-speed zone, θ
1
, θ
2
, 
and θ
3
 are the angular magnitudes of the three acceleration intervals. 
The size of the μ = θ
4
/ θ
2
 is defined as the ratio of the angular width of 
the first constant-acceleration section to that of the second constant-
acceleration section.
Based on the SOA algorithm construction and optimization 
method, the overall process of cam ring curve optimization for 
a radial piston hydraulic motor is shown in Fig. 9. The main steps 
include: 
Step 1: Start the algorithm, and define necessary parameters, 
including population size, iteration limits and others.
Step 2: Initialize the population (the “seagulls”) and generate a 
random candidate solution within the defined boundaries.
Step 3: For each candidate solution in the population, calculate its 
corresponding maximum value of contact stress using the fitness 
function, and assign this as the fitness value.
Step 4: Update the positions of each sub-population using the SOA 
motion rules, simulating foraging and migratory behavior of 
seagulls to perform global and local search.
Step 5: Apply all constraints, such as boundary conditions, linear 
constraints, to ensure that the solution remains feasible.
Step 6: Compare the solutions in the current population to identify 
a new global optimal solution and record the improvement 
compared to the previous optimal solution.
Step 7: If the maximum number of iterations is reached, or the 
improvement in the solution is below a specific tolerance, continue 
to the next step; otherwise, return to the position update step.
Step 8: Once the stopping condition is satisfied, output the current 
optimal solution.
Step 9: Finish the optimization process and terminate the algorithm.
To evaluate the impact of key control parameters on the 
optimization results of the SOA algorithm, a sensitivity analysis was 
performed, namely examined the effect of factors, such as population 
size and maximum iteration count, comparing the optimization results 
and convergence behavior under different parameter configurations. 
The results are shown in Table 1.
Table 1. Algorithm settings and optimized results
Population 
size
Number 
of 
iterations
Optimal 
contact stress 
value [MPa]
Number of 
iterations
Dimension
Adaptive 
parameters
50 200 2414.9 125 5 2
100 200 2414.3 94 5 2
150 200 2414.6 98 5 2
200 200 2414.8 95 5 2
As shown in Table 1, when the population size increases from 50 
to 100, the convergence speed of the algorithm improves significantly, 
while also the optimal fitness value decreases. When the population 
Fig. 9. Flowchart of SOA algorithm

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226   ▪   SV-JME   ▪   VOL 71   ▪   NO 7-8 ▪   Y 2025
size is further increased beyond 100, both the optimization result and 
the required number of iterations remain essentially stable. Therefore, 
a population size of 100 can be selected as the optimal balance point.
3 RESULTS AND DISCUSSION
3.1 Analysis of Optimization Results and Experimental Validation
At the end of the optimization run, the optimal solution set with the 
maximum value of the optimized contact stress is obtained. θ
1
 and θ
3
 
are taken as 0.65 and 1.12, respectively, while the minimum contact 
stress occurs when θ
0
 is 0.72, θ
2
 is 7.82, and μ is 1.46. The specific 
iteration results are shown in Fig. 10. The algorithm converges 
the fastest when the number of population size is 100, and reaches 
convergence at the 94
th
 iteration. Under these conditions the optimal 
solution yield a minimum contact stress of 2414.3 MPa.
To further validate the optimization capabilities of SOA, this 
study compared it against the GA and PSO algorithms under identical 
conditions, such as population size, fitness function, and constraints. 
Key performance indicators of the three algorithms, such as the best 
fitness value, optimization time, and average number of iterations, 
were recorded. The comparison results are shown in Table 2.
Fig. 10. Algorithm iteration results 
Table 2. Comparison of the optimization performance using different algorithms
Algorithm
Optimal contact 
stress value [MPa]
Average convergence 
time [s]
Average number 
of iterations
GA 2456.1 10.36 105
PSO 2434.5 11.87 98
SOA 2414.3 8.89 94
As shown in Table 2, SOA achieves the lowest contact stress, 
outperforming PSO and GA, by 0.8 % and 1.7 % respectively. 
Furthermore, SOA converges faster than both GA and PSO, 
demonstrating higher computing efficiency. These results confirm 
that SOA provides superior adaptability and for complex multi-peak 
optimization problem.
The variation in angular acceleration of the cam ring guide curve 
greatly influences the motor’s output characteristics, such as torque, 
speed, and noise. It also directly affects the contact stress between the 
roller and cam ring guide, which largely determines the overall service 
life of the motor. The maximum value of the angular acceleration 
of the traditional iso-acceleration curve is smaller than that of the 
curve designed in this study. However, the lack of transition between 
the zero-speed zone and the acceleration zone results in significant 
flexural impact during operation, which is prone to cause severe wear 
and tear of the guide rail at the beginning of the acceleration zone. 
The trapezoidal acceleration curve avoids this issue by introducing 
the transition zones before and after the acceleration and deceleration 
zones. Nevertheless, the transition zone comes at the expense of 
increased maximum acceleration compared to iso-acceleration curve.
In this study, amplitude optimization is carried out using the SOA. 
To validate the effectiveness and reasonableness of the proposed 
design method, the results were compared with those obtained from 
the the iso-acceleration curves and trapezoidal iso-acceleration 
curves. The angular acceleration characteristics of each curve are 
shown in Fig. 11 and Table 3.
Fig. 11. Degree acceleration of each curve
Table 3 Maximum values of degree acceleration for each curve
Curve
Iso-acceleration 
curves
Trapezoidal 
iso-acceleration 
curves
Multi-step  
composite  
curves
Angular acceleration 
maxima [mm/rad
2
]
188.7 242.1 237.4
As shown in Fig. 12 and Table 4, curve’s contact stress 
characteristics of each curve differ due to the different geometrical 
design, velocity, and acceleration distribution, radius of curvature, 
contact area, and stress distribution of different curves. At the 
beginning of the acceleration zone, the motion of the piston-roller 
assembly rapidly changes from low speed to high speed. During 
this process, angular acceleration increases sharply in a short 
period, causing a rapid increase of inertial forces. These forces are 
transmitted to the cam ring guide through the roller, resulting in a 
sudden stress peak in the contact area. The maximum contact stress 
of the iso-acceleration curve is 2557.2 MPa, while, the peak of 
the trapezoidal acceleration curve is 2959.7 MPa. In contrast, the 
maximum contact stress of the multi-step composite curve designed 
in this study achieves a significantly lower peak of 2414.3 MPa. This 
represents a reduction of 545.7 MPa compared to the peak of the 
maximum contact stress of the trapezoidal acceleration curve by, and 
142.9 MPa compared to the peak of the iso-acceleration curve. 
The reaction force of the cam ring guide to the roller directly 
affects both the stress distribution and the size of the contact area. 
Excessive reaction force can cause uneven distribution of contact 
stress, stress concentration, and excessive force in critical areas, 
thereby increasing the risk of fatigue damage. Moreover, the reaction 
force affects the friction at multiple interfaces; between the cam ring 
guide and the roller, between the roller and the piston, and between 
the piston and the cylinder block assembly. The increased friction 
leads to more wear, especially under poor lubrication conditions, 
ultimately shortening the service life of the parts. When the reaction 
force is irregular or experiences sudden changes during operation, 
the piston assembly or roller of the hydraulic motor may exhibit an 

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unstable motion trajectory. The uneven rail reaction force can trigger 
intermittent impacts between the roller and the rail, resulting in 
increased noise. Therefore, the reaction force is a critical evaluation 
index for assessing the smoothness and rationality of the hydraulic 
motor.
Fig. 12. Contact stresses for different curves
Table 4. Maximum contact stresses for each curve
Curve
Iso-acceleration 
curves
Trapezoidal 
iso-acceleration 
curves
Multi-step 
composite  
curves
Maximum contact 
stress [MPa]
2557.2 2959.7 2414.3
The reaction force of different curves in the oil-feeding stage is 
shown on Fig. 13. It can be seen that the reaction force of the iso-
acceleration curve occurs at four reaction force mutation points at 
(the four circles in the picture), where the sudden peak jumps are 
represented in difference of 126.9 N, 130.2 N, 127.7 N, and 126.8 N. 
The mutation of the reaction force is mainly caused by the angular 
acceleration mutation. As indicated in Eq. (18), the sudden changes 
in angular acceleration will cause the mutation of R, which makes the 
sudden change of the maximum contact stress. In contrast, the multi-
step composite curve designed in this study eliminates reaction force 
mutation point entirely. Its reaction force in the oil feeding stage 
remains lower and smoother compared to both iso-acceleration curve 
and trapezoidal curve.
Fig. 13. Reaction force for different curves
To verify the effectiveness of the optimized multi-step composite 
curve design proposed in this study, the optimized multi-step 
a) 
b) 
Fig 14. Test stand; a) front side view of motor under test and test stand, and b) rear side view of test stand

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228   ▪   SV-JME   ▪   VOL 71   ▪   NO 7-8 ▪   Y 2025
composite curve cam ring and iso-acceleration cam ring were 
manufactured and installed on a hydraulic motor test bench to test 
the motor speed, as shown in Fig. 14. The experimental validation 
analysis by the test bench shows that the cam ring processed with 
multi-step composite curve has improved the wear and impact 
problems compared with the equal acceleration curve cam ring, 
which can further meet the engineering requirements.
3.2 Bench Test Validation
To validate the effectiveness of the new proposed curve optimization 
design proposed in this study, a radial piston motor test bed is used 
for conducting the experiments. The radial piston hydraulic motor 
test bench mainly comprises the flow sensor, torque sensor, speed 
sensor, integrated valve manifold, oil tank, and drive motor. The 
industrial control computer sets the operating parameters, while the 
drive motor provides the initial incentive to drive the motor under 
stable test rotation. During the experiment, data from the tested motor 
is collected in real time through various sensors. Since hydraulic oil 
temperature significantly influences the experimental results, the 
system was pre-run until the hydraulic oil stabilized at 50 °C before 
measurements were taken. The optimized multi-step composite curve 
cam ring and iso-acceleration cam ring were fabricated, and the both 
were applied to the hydraulic motor test bench to test the motor speed 
condition. The main structural parameters of the radial piston motor 
are shown in Table 5, while the photographs of the motor under test 
and the test bench are shown in Fig. 14.
Table 5. Main structural parameters of radial piston hydraulic motors
Itinerary 
[mm]
Piston 
diameter 
[mm]
Roller 
diameter 
[mm]
Max 
pressure
[MPa]
Effective 
length of 
piston[mm]
Min pole 
diameter
[mm]
11.73 30 20 35 22.18 112.44
The motor speed pulsation can be expressed as:
 


nn
n
avg
maxm in
, 100 (37)
where δ is speed pulsation expressed as a percentage, n
max
 is the 
maximum value of instantaneous speed, n
min
 is the minimum value of 
instantaneous speed, and n
avg
 is the average speed.
Fig. 15. Pulsation rate at experimental working pressure
The speed pulsation rates of the equal-acceleration cam ring curve 
and multi-step composite curve were obtained based on Eq. (37), 
and the relative pulsation reduction rates of the two curves were 
also calculated, as shown in Figs. 15 and 16. The speed pulsation 
of the multi-step composite curve is significantly lower than that 
of the equal-acceleration curve, and the relative pulsation reduction 
rate increases with operating pressure. At the working pressure of of 
20 MPa, the pulsation reduction rate is 10.81 %, while at 30 MPa, 
it reaches 25.73 %. The calculation results show that the design 
method of the multi-step composite curve can significantly reduce 
the pulsation of the radial piston motor.
Fig. 16. Reduction of pulsation at different working pressures
4 CONCLUSIONS
In this study, we addressed the issues of high contact stress, shock 
vibration, and output pulsation in the design of radial piston hydraulic 
motor cam ring curves by constructing a constrained optimization 
model aimed at minimizing contact stress. By combining the global 
search and local refinement capabilities of SOA, we obtained optimal 
parameters for the amplitude angle ratio μ in the acceleration zone, 
the amplitude angle θ
0
 in the zero-speed zone, and the amplitude 
angles θ
1
, θ
2
, and θ
3
 in the acceleration zone. The optimized 
multi-step composite curve yields a maximum contact stress of 
2414.3 MPa, representing reductions of 5.4 % and 18.3 % compared 
to the traditional equal-acceleration and trapezoidal curves at a speed 
of 100 rpm, respectively. This indicates a significant advantage 
in reducing wear and prolonging the cam ring service life. Bench 
tests further showed that the speed pulsation rate of the multi-step 
composite curve under 20 MPa and 30 MPa working conditions is 
reduced by 10.81 % and 25.73 %, respectively. Furthermore, the 
elimination of abrupt reaction force changes, confirms that optimized 
design positively impacts the smoothness of the hydraulic motor 
operation and energy conversion efficiency.
The cam ring curve optimization for radial hydraulic motors based 
on the seagull optimization algorithm (SOA) proposed in this study 
demonstrates both versatility and scalability, and can be applied to 
radial piston hydraulic motors with varying structural parameters. 
Optimization results show stable performance under different 
pressure conditions, highlighting its robustness. Looking ahead, this 
method holds strong potential for broader application in the design of 
diverse types of internal curve motors.
Nomenclature
O
2
 center of the roller, [-]
O
1
 center of the piston, [-]
F
p
 hydraulic pressure, [N]
F
a
 inertia force generated by the acceleration of the piston relative 
to the cylinder block assembly, [N]
F
G
 centrifugal force generated by the rotation of the piston relative 
to the cylinder block assembly, [N]
F
g
 generated by the gothic acceleration a
g
 on the center of the 
roller, [N]
F
R
 friction between the piston and the cavity, [N]
F
L
 friction between the piston and the cavity, [N]
F
f
 friction of the roller by the cam ring, [N]

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L
0
 contact length when the roller is at the minimum pole diameter, 
[N]
M
1
 friction force F
f
 of the cam ring rail in contact with the roller 
on the center of the roller, [N·m]
M
2
 sliding friction F
L
 and F
R
 between the piston and cylinder 
block assembly on the center of the roller, [N·m]
M
3
 triangular distribution of force N
1
 and N
2
 along the contact line 
generated between the piston and cylinder block assembly wall 
on the center of the roller, [N·m]
M
4
 inertia force F
g
 generated by the Gothic acceleration a
g
 on the 
center of the roller, [N·m]
N
1
 the piston is subjected to positive pressure from the cylinder 
block assembly, [N]
N
2
 the piston is subjected to positive pressure from the cylinder 
block assembly, [N]
L
1
 triangular normal force N
2
 acting on the piston length, [mm]
L
2
 triangular normal force N
2
 acting on the piston length, [mm]
ρ
min
 minimum pole diameter, [mm]
R radius of the roller, [mm]
d diameter of the bottom of the piston, [mm]
L
g
 center of mass to the center of the roller distance, [mm]
ρ
φ
 the polar radius of the piston center at the φ position, [mm]
L
c
 the length of the contact line between the piston and the inner 
wall of the cylinder block assembly, [mm]
a
g
 gothic acceleration, [mm/rad
2
]
α pressure angle, [-]
f
1
 sliding friction coefficient between piston and cylinder block 
assembly, [-]
f
2
 sliding friction coefficient between roller and cam ring, [-]
l length of the contact line between the two elastic solids, [mm]
b width of the contact zone, [mm]
μ
1
, μ
2
 Poisson’s ratios, [-]
E
1
, E
2
 modulus of elasticity, [MPa]
R radius of curvature of the roller, [mm]
R
c
 radius of curvature of the rail at the contact between the roller 
and the rail curve, [mm]
θ angle between the piston and its initial position, [rad]
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Acknowledgements This work is supported by the National Natural 
Science Foundation of China (51805228), the Natural Science Foundation 
of the Jiangsu Higher Education Institutions of China (22KJB460021, 
23KJA460006) and Changzhou Leading Innovative Talents Introduction and 
Cultivation Project the (CQ20210093, CQ20220089).
Received: 2025-05-16, revised: 2025-07-16, accepted: 2025-08-27  
as Original Scientific Paper.

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230   ▪   SV-JME   ▪   VOL 71   ▪   NO 7-8 ▪   Y 2025
Data availability The data supporting the findings of this study are included 
in the article.
Author contribution Bowen Zhang: Conceptualization, Formal Analysis, 
Methodology, Writing - original draft, Writing - review & editing; Shaopeng 
Kang: Conceptualization, Funding Acquisition, Resources, Supervision, 
Validation, Writing - original draft, Writing - review & editing; Runze Zhou: Data 
Curation, Supervision; Hongbin Qiang: Resources; Jing Yang: Visualization; 
Kailei Liu: Underlying Acquisition; Yunkai Zhou: Validation.
Optimizacija krivulje odmikalnega obroča  za radialni hidravlični 
motor na osnovi algoritma obnašanja galebov
Povzetek S pospešenimi prilagoditvami globalne energetske strukture 
se intenzivnost izkoriščanja strateških mineralnih virov še naprej povečuje. 
Zaradi rastočega povpraševanja po gradbenih strojih in opremi se radialni 
batni hidravlični motorji, znani po nizkih vrtilnih hitrostih in visokih navorih, 
vse pogosteje uporabljajo v težkih rudarskih strojih in opremi. Zasnova 
krivulje odmikalnega obroča radialnega batnega hidravličnega motorja 
še vedno predstavlja ključne izzive, kot so visoke kontaktne napetosti, 
izrazite pulzacije izhodne hitrosti. Ta članek predlaga metodo večstopenjske 
optimizacije sestavljene krivulje, ki temelji na algoritmu optimizacije 
obnašanja galebov (SOA). Metoda združuje koncepte stopničaste in trapezne 
zasnove pospeševalne krivulje ter uporablja SOA algoritem za obvladovanje 
omenjenih izzivov. Algoritem SOA je prilagodljivejši od genetskega algoritma 
in algoritma rojev delcev, saj ohranja raznolikost populacije tudi v poznejših 
fazah iteracije. S tem učinkovito premaguje omejitve drugih algoritmov pri 
reševanju problemov z več lokalnimi optimumi. Zasnovan je bil optimizacijski 
model krivulje odmične obroče za zmanjšanje kontaktnih napetosti na 
podlagi teoretičnega modeliranja in dinamične analize. Izvedena je bila 
večparametročna optimizacija ključnih parametrov, kot so razmerje 
amplituda–kot v pospeševalni coni in amplituda–kot v coni ničelne hitrosti. 
Tako je bila pridobljena sestavljena krivulja odmikalnega obroča, za katero 
so značilne manjše napetosti, manjša pulzacija in obratovanje brez udarcev. 
Učinkovitost metode je bila potrjena z laboratorijskimi testi, ki so pokazali, 
da je največja kontaktna napetost optimizirane večstopenjske sestavljene 
krivulje pri hitrosti 100 vrt/min zmanjšana za 5,4 % in 18,3 % v primerjavi 
s tradicionalno enakomerno pospeševalno krivuljo oziroma trapezno krivuljo. 
Poleg tega se stopnja pulzacije hitrosti zmanjša za 10,81 % in 25,73 % pri 
tlaku olja 20 MPa oziroma 30 MPa, brez nenadnih sprememb reakcijske sile 
in z zmanjšanjem udarnih pulzacij med obratovanjem.
Ključne besede radialni batni hidravlični motor, optimizacija odmikalnega 
obroča, algoritem optimizacije obnašanja galebov, kontaktna napetost, 
pulzacija hitrosti