Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 8, No. 3, 2023 

 

167 

Structural	Reliability	Analysis	of	Microsphere	Focused	
Logging	Tool	Based	on	Response	Surface	Method	
Zixuan Zhao, Wen Sun, Junyuan Zheng 

School of Mechanical Engineering, Xi 'an Shiyou University, Xi 'an 710000, China 

 

Abstract: The push-pull system of microsphere focused logging tool is an important moving mechanism of this type of logging 
tool. The push-pull system will periodically bear large loads under working conditions and is prone to stress concentration. The 
structural strength reliability of the push-pull system is related to the logging accuracy and even the success or failure of the 
logging task. Based on ANSYS workbench software, this paper first conducts modeling and static simulation on the push-pull 
system of microsphere focused logging tool under working conditions, then conducts sensitivity analysis and experimental design 
on the basis of this analysis, and then conducts response surface analysis on the key stressed components of the push-pull system 
according to the analysis results. Finally, six sigma module is used to analyze the reliability of the push-pull system, and the 
structural strength reliability of the push-pull system is given by the probability distribution table obtained from the analysis. 
The analysis results show that the maximum equivalent stress occurs on the link arm of the push-pull system under working 
condition. The strength reliability of the link arm of the push-pull system under current working condition is high, and it can 
meet the daily logging requirements. However, there is still room for further design optimization. 

Keywords: Push-back system; Response surface analysis; six sigma module; Reliability analysis. 
 

1. Introduction 
Microsphere focused logging instrument is a very 

important oil logging equipment in the field of oil logging, 
and its push-pull system is a key system installed on the 
microsphere focused logging instrument to assist the 
microsphere focused logging instrument to complete 
downhole logging tasks[1]. When the push system is working, 
it will be subject to periodic load, and the load is large, which 
is easy to produce stress concentration, resulting in failure and 
failure. . 

At present, the research on the push-back system of 
microsphere focused logging tool mostly focuses on the 
design optimization of push-back system, the wear resistance 
design of push-back pad and the application of new materials. 
Based on the poor wear resistance of the push-back pad of 
microsphere focused logging tool, Dou Jinai improved the 
material and manufacturing process of the original pad, so 
that the wear resistance and insulation of the pad are better 
than that of conventional rubber pad[2]. Aiming at the 
minimum fluctuation between the transmission Angle of the 
main transmission mechanism and the root mean square of the 
expected transmission Angle, Ren Tao optimized the 
transmission performance and the motion stability of the push 
and lean system[3]. 

The microsphere focused logging tool push system is 
subject to a large periodic load in the working process, which 
is easy to produce stress concentration and lead to system 
failure and failure. The failure mode is usually that the 
structural strength of the link arm is not enough, so it is 
necessary to analyze the structural strength reliability of the 
microsphere focused logging tool push system. At present, 
there are few researches on this kind of analysis. Yang 
Baiqing discussed and analyzed the mechanical properties of 
the single-arm push-back device. By analyzing the working 
conditions of the push-back device and the movements of 
each mechanical arm, he established a theoretical basis for 
designing high-quality push-back devices[4]. Aiming at the 

reliability problems of logging instruments in the process of 
petroleum development, Zhang Guangzhou conducted 
specific research on the reliability design methods of logging 
instruments from multiple perspectives, and proposed that the 
framework strength and anti-impact reliability of instruments 
should be paid attention to in the design[5]. 

Based on the statics analysis of the microsphere focused 
logging tool push-pull system, the position of the maximum 
equivalent stress in the system is obtained, and the Six Sigma 
analysis module is used to analyze the reliability of the push-
pull system, which provides a reference for further 
optimization of the push-pull system. 

2. Introduction of Microsphere 
Focused Logging Tool Push-pull 
System 

The push-pull system of microsphere focused logging tool 
mainly includes: motor, self-locking device of transmission 
system, centripetal ball bearing group, push-pull upper arm, 
push-pull inner wall, push-pull pad, link arm and push-pull 
main body. The working state structure diagram of push-pull 
system is shown in Figure 1. 

 

 
Figure 1. Structure diagram of the working state of the 
microsphere focused logging tool push-pull system 

 
Note: 1- Motor; 2- transmission system self-locking device; 

3- radial ball bearing group; 4- Push the upper arm; 5- Push 
against the inner wall; 6- Push the pad; 7- Link arm; 8- Push 



 

168 

on the subject 
During logging, the motor drives the thrust rod and thrust 

plate to move forward in a straight line through the lead screw, 
thus compressing the spring and opening the push arm, thus 
driving the push plate to move until the pad is attached to the 
well wall[6]. When the pad is completely attached to the well 
wall, the motor is powered off, and the spring provides the 
push force of the pad. 

3. Statics Analysis of Microsphere 
Focused Logging Tool by System 

3.1. Model Import and setting 
According to the actual working conditions, the push-pull 

system of the microsphere focused logging tool was modeled 
and assembled in CREO modeling software. In order to 
reduce the amount of calculation, this paper only modeled the 
upper arm of unilateral push-pull, inner wall of push-pull, pad 
of push-pull and link arm, and imported the simulation model 
into ANSYS workbench simulation software. The simulation 
model of microsphere focused logging tool push-on system is 
shown in Figure 2. 

 

 
Figure 2. Simulation model of microsphere focused 

logging tool push-pull system 

3.2. Grid Division 
Tetrahedral method was used to divide the grid of the 

pushing system, the overall mesh size was set to 2mm, and 
the whole model was refined by Body Sizing. The number of 
grid units, nodes and grid quality after processing were shown 
in Table 1. 

 
Table 1. The number of model grid elements, nodes and 

grid quality 
Name Number of 

elements 
Number 
of nodes 

Mesh  
quality 

Push against upper 
arm 

79128 122432 0.8341 

Push against inner 
arm 

98426 147887 0.8391 

Push against plate 202911 295535 0.8474 
Link arm 7789 12756 0.8259 

3.3. Add contact and load 
The connection mode of the moving pairs between the parts 

of the push-pull system is consistent with the actual 
connection mode, as shown in Table 2. According to the 
actual working conditions and reference 7, the load added to 
the push-back system is as follows: the driving force of the 
pin hole at the lower part of the link arm is 225N, and the wall 
pressure on the upper surface of the plate is 240N. 

 
 

Table 2. Connection information table of the moving pair 
of the microsphere focused logging tool push-pull system 

Number 
Kinematic pair 

type 
part 1 part 2 

1 Moving pair Link arm ground 

2 Rotating pair Link arm 
Push against 

inner arm 

3 Rotating pair 
Push against 

inner arm 
ground 

4 Moving pair 
Push against 

inner arm 
Push against 

plate 

5 Rotating pair 
Push against 

plate 
Push against 
upper arm 

6 Rotating pair 
Push against 
upper arm 

ground 

3.4. Analysis of simulation results 
The maximum equivalent stress and deformation cloud 

diagram of each part of the statics simulation of the 
microsphere focused logging tool under the loading condition 
of the system are shown in Figure 2 (a) -(d). 

 

 
(a) nephogram of the maximum equivalent stress of the 

pusher plate 

 
(b) Nephogram of the maximum equivalent stress of the 

inner arm 

 
(c) Nephogram of the maximum equivalent stress on the 

upper arm 

 
(d) Cloud map of the maximum equivalent stress of the link 

arm 
Figure 3. The maximun equivalent stress and deformation 

cloud map of each part 
 



 

169 

The simulation results of each part of the push system are 
shown in Table 3. 

 
Table 3. Maximum equivalent stress of each component 

of the microsphere focused logging tool push-back system 
Name Maximum equivalent 

stress (MPa) 
Push against upper arm 66.548 
Push against inner arm 74.118 

Push against plate 15.593 
Link arm 163.51 

 
It can be seen from Table 3 that under working conditions, 

the equivalent stress on the link arm is the largest, which is 
163.51MPa, while the equivalent stress on the other parts is 
much less than that on the link arm, among which the 
equivalent stress on the upper arm of the push is the least, 
which is 15.593MPa. Therefore, the strength of the link arm 
should be considered and the strength reliability analysis of 
the link arm should be carried out when designing the 
microsphere focused logging tool push-in system. 

4. Reliability Analysis of Link Arm 
Strength Based on Response Surface 
Method 

4.1. Link arm strength reliability analysis 
process 

Link arm strength reliability analysis steps are divided into 
the following two points: 

1) Parameter response surface analysis 
Response surface method is an optimization method that 

integrates experimental design and mathematical modeling. It 
fits the functional relationship between design variables and 
output results within the global scope through local test point 
regression, and has the characteristics of fewer test times, 
high precision and good prediction performance[8]. In this 
paper, Kriging method is used to fit the response surface. 
Kriging model can be regarded as an interpolation model, 
which combines non-parametric random process with 
parametric linear regression model[9,10]. The expression of 
the model is as follows: 

 
𝐺 𝑥 Γ 𝛽, 𝑥 𝑧 𝑥 𝑓 𝑥 𝛽 𝑧 𝑥       (1) 

 
Type: 𝑓 𝑥 𝑓 𝑥 , 𝑓 𝑥 , . . . , 𝑓 𝑥 ， 𝛽  is the 

regression coefficient vector，𝛽 𝛽 , 𝛽 , . . . , 𝛽 ，Γ 𝛽, 𝑥  
Is a polynomial regression model，𝑧 𝑥  is a random Gaussian 
process with variance 𝜎  and mean zero. Classical Kriging 
model Common functional models include Gaussian model, 
exponential model, spline model, linear model, etc. The most 
commonly used one is Gaussian model (y) : 

 
ℝ 𝑥 , 𝑥 ; 𝜃 ∏ exp 𝜃 𝑥 𝑥      (2) 

 
Type: m first M-delements for the input vector, 𝜃 for the 

parameter vector, 𝜃 𝜃 , 𝜃 , . . . , 𝜃  , M for total 
dimension of input vector. 

Define the correlation matrix 𝑅 𝑥 , 𝑥 ; 𝜃  , then the 
estimated values of 𝛽 and 𝜎 are: 

𝛽
^

𝐹 𝑅 𝐹 𝐹 𝑅 𝐺               (3) 

 

𝜎
^

𝑎𝑟𝑔𝑚𝑎𝑥 𝑁 ln 𝜎
^

ln |𝑅|         (4) 
 
At this time, the variance and mean of the predicted value 

of the prediction point x are: 
 

𝜇 𝑥 𝛽
^

𝑟 𝑥 𝑅 𝐺 𝛽
^

𝐹

𝜎 𝑥 𝜎 𝑥 1 𝜇 𝑥 𝐹 𝑅 𝐹 𝜇 𝑥 𝑟 𝑥 𝑅 𝑟 𝑥
(5) 

 
Type: r(x)= ℝ 𝑥 , 𝑥 ; 𝜃, 𝑥 , 𝑥 ; 𝜃, … , 𝑥 , 𝑥 ; 𝜃  , 

𝜇 𝑥 is treated as the predicted value of point X. 
2) Reliability analysis 
The calculation of strength reliability is usually based on 

the stress-strength interference model[11], and its schematic 
diagram is shown in Figure 4: 

 

 
Figure 4. Schematic diagram of interference model 

 
Its functional functions are: 
 

G z 𝑓 𝑥 𝑔 𝑦 0              (6) 
 
Type：f(x) is the strength of the part, and g(y) is the stress 

on the part. If f(x) and g(y) both follow a normal distribution, 
then: 

 

R
√

exp               (7) 

 
Then the reliability can be expressed as: 
 

R P z 0
√

exp dz    (8) 

 
If μ z μ /σ , then: 

 

R P z 0
√

exp 𝑢 du ϕ β     (9) 

 
β                  (10) 

 
Type：β is the strength reliability index, 𝜇  and 𝜎  are 

the mean and standard deviation of input variables, and 𝜇  
and 𝜎   are the mean and standard deviation of output 
variables. 

4.2. Parameter response surface analysis 
Taking the maximum equivalent stress of the link arm of 

the microsphere focused logging tool pushing system as the 
output variable, the three parameters of link arm rod length 
(p1), pin hole diameter at the lower end of link arm (p2) and 
the x-axial force on the pin hole at the lower end of link arm 
(p12) were selected as the input variables, and the rod length, 



 

170 

pin hole diameter and load on the link arm were selected as 
the input variables. The influence of input variables on output 
variables is shown in Figure 5: 

 

 
(a) p2, p12 and p8 response surfaces 

 

 
(b) p1, p12 and p8 response surfaces 

 

 
(c) p1, p2 and p8 response surfaces 

Figure 5. Response surface of random variables to output 
parameters 

 
It is easy to see from the response analysis diagram of the 

maximum equivalent stress that the above three random 
variables have different degrees of influence on the maximum 
equivalent stress of the link arm. The length of the link arm 
(p1) is negatively correlated with the maximum equivalent 
stress of the link arm, while the diameter of the pin hole at the 
lower end of the link arm (p2) and the x-axial force on the pin 
hole at the lower end of the link arm (p12) are positively 
correlated with the maximum equivalent stress of the link arm. 
The maximum equivalent stress is minimum when the length 
of the link arm is minimum and the diameter of the pin hole 
at the lower end of the link arm and the axial force of the pin 
hole X are maximum. 

4.3. Reliability Analysis 
In the Six Sigma module of ANSYS Workbench, the 

probability value corresponding to a certain fixed value can 
be intuitively obtained by analyzing the probability 
distribution table. The probability distribution table of the 

maximum equivalent stress of the link arm is shown in Figure 
6. 

 

 
Figure 6. Table of maximum equivalent stress distribution 

of link arm 
 
It can be seen from Figure 6 that the probability that the 

maximum equivalent stress of the link arm is less than 
177.83MPa is 0.99993. According to the experience, the 
design maximum allowable stress of the link arm is 175MPa, 
and the strength reliability of the link arm is 0.9981. Take the 
variation coefficient of the maximum equivalent stress of the 
link arm as 0.02, and bring it into equation (9) and (10) to 
calculate β=1.946, R=0.99813, which is basically consistent 
with the corresponding probabilities in the probability sub-
table, indicating that the simulation results are correct. 

4.4. Closing remarks 
Aiming at the microsphere focused logging tool push-pull 

system, this paper conducts static analysis of the push-pull 
system under working conditions, and on this basis carries out 
uncertainty analysis. Based on the analysis results, reliability 
analysis is carried out and the maximum equivalent stress 
probability distribution table of the link arm of the push-pull 
system is obtained. The research shows that: 

1. The maximum equivalent stress occurs in the link arm of 
the microsphere focused logging tool push-back system under 
working condition; 

2. When the maximum allowable stress is 175MPa, the 
strength reliability of the link arm is 0.9981, indicating that 
the strength reliability of the link arm is high and can meet the 
current logging requirements, but there is still much room for 
improvement. 

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