The above experiment results show that the
solution for determining part’s surface roughness
through grinding wheel’s wear value has achieved
satisfactory results. This solution is based on application
of the partial linear interpolation method. Grinding
wheel’s wear value is measured directly in the grinding
process on the basis of the application of pneumatic
measuring probe. By combining two pneumatic
measuring probes, the system determines online wear
value at the top and at the edge of the curving shape of
grinding wheel. Then, on the basis of application of
linear interpolation method, the software system has
calculated surface roughness’s value at each point
corresponding on part’s surface to give online warning
signals for users. Thanks to that, users determine the
resonable time to dress grinding wheel 3. In addition,
these results can be applied as the basis of adaptive
control for automatic compensation of grinding wheel’s
wear.
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Journal of Science & Technology 122 (2017) 041-047
41
Applying Pneumatic Probe System to Monitor Part’s Surface Roughness
and Grinding Wheel’s Wear in Profile Grinding
for Ball Bearing's Inner Ring Groove
Nguyen Anh Tuan1*, Vu Toan Thang2, Nguyen Viet Tiep2
1 University of Economics and Technical Industries, No. 456 Minh Khai Str., Hai Ba Trung, Ha Noi, Viet Nam
2 Hanoi University of Science and Technology, No. 1, Dai Co Viet Str., Hai Ba Trung, Ha Noi, Viet Nam
Received: September 19, 2017; Accepted: November 03, 2017
Abstract
The* article presents the results of research and apply pneumatic measuring probe to monitor part’s surface
roughness and grinding wheel’s wear in profile grinding for the ball bearing's inner ring groove. On that basis,
it makes an online warning for limiting dress grinding wheel according to the given surface roughness threshold
or wear threshold. The signals obtained from pressure sensors of pneumatic measuring probe system are
processed to calculate grinding wheel’s wear and part’s surface roughness. The output values of this probe
system is very close to experimental data values. The results demonstrate ability to monitor the real status of
grinding wheel’s wear and part’s surface roughness.
Keywords: Profile grinding, online monitoring, grinding wheel’s wear.
1. Introduction
In the profile grinding process, besides part’s
surface roughness indicator, abrasive wear and
grinding wheel durability are also important technical
and economic output parameters. The surface
roughness is one of the most important features to
assess the quality of product, while grinding wheel
wear directly impacts the cutting ability of grinding
wheel, productivity, quality and efficiency of profile
grinding’s whole process. For profile grinding, area of
contact between grinding wheel and workpiece surface
is large, thus cutting force and cutting heat are high.
Therefore, grinding wheel is worn continuously and
unequally over different cross-sections, the cuting
ability of grinding wheel is reduced, the original shape
accuracy of grinding wheel is changed during grinding
process. It leads to increase surface roughness and
form tolerances of product. Thus, the precision in
general and the surface roughness of part achieved for
profile grinding are strongly related to dress grinding
wheel in grinding process. Monitoring the amount of
wear to determine the appropriate time for dressing
grinding wheel will prevent making poor quality
product, using too wear grinding wheel, and changing
grinding wheel while machining product.
To ensure economic and technical efficiency of
profile grinding process, one of the industrial
requirements is online monitoring grinding wheel’s
wear and part’s surface roughness. This is the basis
* Corresponding author: Tel.: 0964.945.889
Email: natuan.ck@uneti.edu.vn
of adaptive control and the indispensable
components of an innovative and intelligent
grinding system.
Both part’s surface roughness and grinding
wheel’s wear are functions of the cutting mode. Thus,
they must have a relationship with each other [9, 10].
This relationship has both positive and negative impact
on the monitoring process. The negative influence is
that the change of cutting mode to improve one
parameter would worsen other parameters. The
positive side is the ability to caculate hard-to-measure
parameters via easy-to-measure parameter. This ability
has been utilized to set up online monitoring surface
roughness in grinding process.
Part’s surface roughness can not be measured
directly in the machining process. On the other hand,
other parameters such as cutting force (F), grinding
wheel’s wear (δi), etc could be. Therefore, part’s
surface roughness value (Rai) will be caculated
through grinding wheel’s wear value (δi) measured by
pneumatic measuring probe system.
From the above analysis, the authors have
researched to apply pneumatic measuring probe to
perform online monitoring part’s surface roughness
and grinding wheel’s wear for profile grinding ball
bearing's inner ring grove.
Journal of Science & Technology 122 (2017) 041-047
42
2. Content of the study
2.1. Model for monitoring grinding wheel’s wear and
part’s surface roughness in profile grinding process
The basic features of the machining process are
complex and nonlinear. It is often influenced by
unforeseen factors. In traditional machining, control
and monitoring processes often rely on well-training
workers. They have ability to recognize situations and
handle them in a flexible manner. However, if control
is based on worker’s senses and handling, it will not be
able to guarantee objectivity, reliability and stability in
the machining process. Hence, in order to reach a
automatic production system, the roles of human in
monitoring and controling have to be replaced by
machines. Therefore, it is necessary to build a system
monitoring automatically technological system’s
elements in machining processes [9, 10].
In previous studies, some online monitoring and
adaptive control systems have introduced [9, 10]. This
is a complex mechatronic system involving many
different areas. The task of designing these systems is
generalized into five main areas and represented in
Fig. 1.
Fig. 1. The structure diagram of an online monitoring
system [10]
This diagram shows the general structure of an
online monitoring system. This system includes
elements as follows: (1) Sensors measuring cutting
forces, pressure, temperature, etc; (2) hardware (DAQ)
collecting data; (3) software (DSS) processing data,
extracting samples and supporting decision-making.
However, the system merely performs task of
monitoring, supporting decision-making and
providing warning signals to users. It has not control
functions.
In order to have a smart machining system, an
adaptive controllers (AC) or adaptive controller
optimization (ACO) should be added to the model
as Fig. 2 [9, 10]. In essence, this adaptive
controller is a decision-making system having
function of calculating, identifying, classifying
events and giving control signals. Therefore, AC
and ACO already contain module DM (Dicision
Making).
Two control models show that online monitoring
is an indispensable system of adaptive control
process. In order to make adaptive control system, the
first requirement should be to perform online process
monitoring. Although the highest goal is grinding
process’s adaptive control, in the area of research the
authors only focus on the direction to build an system
monitoring online grinding wheel’s wear and part’s
surfaces roughness for profile grinding the ball
bearing's inner ring grove. The system gives warning
signal for users. It indicates the right time needed to
dress grinding wheel.
Fig. 2. The general structure of an adaptive control
system [9, 10]
Derived from the general structure diagram of an
online monitoring system, on the basis of reference to
the previous online monitoring and adaptive control
systems, the author proposes a functional scheme for
system to monitor online grinding wheel’s wear and
part’s surfaces roughness as shown in Fig. 3 and Fig.
4.
Fig. 3. The functional diagram of system to monitor
online grinding wheel’s wear and part’s surface
roughness for profile grinding ball bearing's inner ring
grove [9, 10]
In the system, 3MK136B profile grinding machine is
fitted with pneumatic measuring probe to measure grinding
wheel’s wear. The computer connected to DAQ hardware
and DSS software is responsible for receiving, processing
of measured signals to caculate grinding wheel’s wear (δi)
and part’s surface roughness (Rai). These values are
displayed on the computer screen to inform users. In
addition, an adaptive controller could be inserted into the
processor to adjust the cutting mode. However in the
research of this paper, the system does not have adaptive
controller, it consists only of continuous line’s elements.
Signal
Processing
Signal
Processing
User
interface
DAQ Computer DSS
From the real world
To the real world
Adjustment
Adjustment
rules
Profile grinding
machine mounting
pneumatic measuring
head system
DAQ, DSS
Online-monitoring system
Warning
Observed variables δi, Rai
Rai
δi
Output
Parameters
Measurement
Parameters V’đ, V’ct,
S’hk, t’, N’ct
ΔV’đ, ΔV’ct, ΔS’hk,
Δt’, ΔN’ct
Vđ Vct Shk Nct t
Adjustment Vđ, Vct, Shk, t, Nct
Adaptive
controller system
Control
rules
δ, Ra
requirement
CNC
Machining
process
S2
V2
t2
Sensors
Data
acquisition
(DAQ)
Data
processing
ACO
adjustment
s, v, t
Cutting force
Sound
Vibration
Surface roughness
Productivity
Tool wear
S2 V2 t2
Coding X, Y, Z
NC
program
S1, V1, t1
Optimal
indicator
Constraints
u3 u2 u1
Journal of Science & Technology 122 (2017) 041-047
43
The dash-line elements belong to adaptive control system.
It can be developed in the future.
Fig. 4. Online-monitoring model of grinding wheel’s
wear and part’s surface roughness for profile grinding
ball bearing's inner ring grove [9, 10]
In this model, the system can perform two tasks:
(1) Caculating grinding wheel’s wear value and part’s
surface roughness value; (2) Warning on-line
limitation of grinding wheel dressing under the given
surface roughness threshold or wear threshold.
2.2. Theoretical basis of monitoring grinding
wheel’s wear in grinding profile process
The authors apply pneumatic measuring probe to
monitor online grinding wheel’s wear for grinding profile
ball bearing's inner ring grove. The diagram of online
monitoring system is shown on Fig. 5. In profile grinding
process, the amount of wear at various points on the
curving edge shape of grinding wheel will not be the
same. Therefore, in order to monitor grinding wheel’s
wear, the author uses two pneumatic measuring probe
systems to evaluate wear at the top of the curving edge
shape and at the edge of the curving edge shape of
grinding wheel. These are two points having the largest
difference in the amount of wear compared to all the other
points on the whole curving edge shape of grinding
wheel’s working surface. Also, to measure pressure in the
measuring chamber of each pneumatic measuring probe,
two pressure sensors (SEU-31 of the Pisco) with
measurement range 1-10 Bar, resolution 0.001 Bar have
been used and taken signal continously during measuring
process. Especialy, in order to perform online monitoring
grinding wheel’s wear, a hardware and software system
connected to computer is designed and manufactured to
continuously receive pressure signals from pressure
sensors. On that basis, the system will calculate radial
wear value at the peak point and at the edge point of the
curving edge shape of grinding wheel after each time a
part grinded completely.
In this monitoring model, the authors have
chosen a STM32F4 microcontroller (No.11 part) for
processing and transmission signals of two pressure
sensors measuring pressure of measuring chamber in
each measuring probe system. The interface software
is built to transmit control signal between computer,
connection circuit and pressure sensors. Matlab
software is used as a tool to connect, handle data,
store results because programme language on Matlab
is quite simple [8, 9].
Fig. 5. The system diagram for online monitoring grinding
wheel’s wear in profile grinding process [1-5]
In the above system diagram:
1: Pneumatic probe measuring wear at the top of the
curving edge shape of grinding wheel (The top probe).
2: Grinding wheel.
3: Pressure sensor for measuring pressure in measure
chamber of the top probe.
4: Adapter DC -12V; 5: Power supply - 220V
6: Pressure sensor for measuring pressure in measure
chamber of the margin probe.
7, 10: Amplifier; 8, 9: Analogue to digital converter
11: STM32F4 processor; 12: Computer (PC)
13: Pneumatic probe measuring wear at the edge of the
curving edge shape of grinding wheel (The margin
probe); 14, 19: Source pressure gauges.
15, 18: Constant-pressure valve.
16, 17: Air filter; 20: Air compressor.
171819
141516
01
02
R
d
03 04
05
06
07 08 09 10
20
12
11
Computer
Interface RS 232
Processor
Amplifier A D
Pressure
Sensor
AC-220V
DC-12V
A D
Pressure
Sensor
Amplifier
Air
compressor
A
ir
su
pp
ly
Z
13
Profile grinding
machine mounting
pneumatic
measuring probe
systems
Computer with a
hardware and
software system
(DAQ, DSS) to
receive and
process measured
signals
Display grinding wheel’s
wear and part’s surface
roughness
Warning online time to
dress grinding wheel
Chamber pressure (P)
Vđ, Vct, Shk, t, Nct
δi,Rai
Dressing
Journal of Science & Technology 122 (2017) 041-047
44
The process of retrieving data takes place as Fig.
6: Signals from two sensors will be transmitted to the
IN0 and IN1 pins of ADS1256 module. After
microcontroller configurates for ADS1256 module, it
will send a 16-bit data frame. Each bit is transmitted on
the DIN pin of microcontroller, 1 bit will be received
from module on the DOUT pin. Therefore with 16 bit is
transmitted, DATA register of ADS1256 will be
received 16 bit [6, 7]. After pressing START in software
interface, the data receiving flag is equal to 1. The
program will set time t1 at this time as the starting point.
At the same time, analog values in both pressure sensors
will be transmitted consecutively from the
microcontroller onto PC via RS232 connection. They
are stored alternately in A1[i] and A2[i] array. At this
moment, the data receiving flag will be equal to 0 for
pausing transmittation data from microcontroller to PC.
Then, the program calculates the conversion from
analog values to pressure values. Next, these pressure
values will be saved into P1[i] and P2[i] array. At this
time, the program will update the current time t2 to
identify program run-time (Runtime=t2-t1). Based on
that, it draws analog value graph and pressure value
graph from time to time. These graphs are displayed on
the screen of PC. Simultaneously, the data receiving
flag at the moment will be transferred to 1. Thus,
microcontroller continues sending next analog values
from both pressure sensors onto the PC. Therefore, after
each such sampling, the program will determine
program run-time. It will always compare the above
program run-time with the initial installation dressing
time and grinding time between two consecutive parts
to find out minimum pressure value (Pmin) during
dressing time or grinding time. This is also pressure
value of measuring chamber at the time a part grinded
completely. Based on dynamic characteristic lines of
each pneuamatic measuring probe, it will find the
corresponding transfer functions (with the top
measuring probe:
2
3.5 ;
1 5.4613 0.4*
p
e z
=
+ −
with the
margin probe:
21.86
3.5
1 0.55 4*
p
e z
=
+ −
) [3]. Therefore,
from minimum pressure value (Pmin) it counts gap (Z)
to calculate the amount of change of gap (Z). This is
radial wear value of grinding wheel after every dressing
or after every a part grinded completely. Next, the
program will compare this wear value with the given
wear threshold to make signal “warning” to users, as the
algorithm diagram in Fig. 8. If wear value at the top or
at the edge of grinding wheel is greater than the wear
threshold, the program will give a signal "warning" to
users. Thanks to that, users will know that this is the
time to dress grinding wheel in order to ensure the
accuracy of parts in the grinding process.
Fig. 6. The diagram of control circuit principle
2.3. Theoretical basis for monitoring part’s surface
roughness in profile grinding process
Although parameters in grinding process have
relations with each other, the relations between them are
very intricate and complex. It is difficult to identify
explicitly relations between them by mathematics. So
far it has not seen any announcement about
mathematical relations between them, such as between
durability and cutting force, durability and cutting
temperature, etc. It can only build the experimental
parameters. On the other hand, the relationship between
parameters is nonlinear. It affected by many unforeseen
disturbances factors. In addition, it is also not easy to
analyze multi-variables functions in mathematics. This
explains why traditional approximation methods, such
as the smallest square method, are not suitable for online
monitoring issue.
However, when the optimal input parameters of the
technology system are already known, their values will
be permanently installed on the machine during the
grinding process. With this case, relation between part’s
surface roughness (Rai) and grinding wheel’s wear (δi) is
determined. Therefore, by caculating part’s surface
roughness values (Rai) through grinding wheel’s wear
values (δi), part’s surface roughness (Rai) can be
monitored in the grinding process. However,
establishment of the relation between grinding wheel’s
wear (δi) and part’s surface roughness (Rai) by
mathematical formula is very difficult and infeasible.
Thus, the determination of this relation based on
approximation interpolation methods by experiment
shows more effectiveness than other methods.
Concretely, from experimental results a set of empirical
value pairs (δi, Rai) is identified. A set of discrete
experimental points is defined as shown in Fig. 7.
Fig. 7. Construction characteristic line Ra(δ) between
grinding wheel’s wear and part’s surface roughness
based on experiment
The first pressure
sensor
The second
pressure sensor
Journal of Science & Technology 122 (2017) 041-047
45
START
A1[i]=get(COM1)
A2[i]=get(COM1)
RS232 is available
P1[i]=f(A1[i])
P2[i]=f(A2[i])
Get time t1
Get time t2. The run time is
time=t2-t1
Time to dress grinding wheel
>Time
flag1=0
P1[i]=P1min
P2[i]=P2min
Time between 2 parts+t1min >time
Time between 2 parts+t2min>time
flag1 == 1
t1=time
t2=time
SUM1[k]=M1[k]+M1[k-1]
SUM2[k]=M2[k]+M2[k-1]
k == number of parts in one cycle
i=0,j=0; flag1=0;
k=0
Number of cycle = Number of
cycle +1
Stop button is pressed
END
C1[j]=f(P1min)
C2[j]=f(P2min)
Draw 2 axes (A1[i]A2[i]),time and
(P1[i],P2[i]), time
flag1 = 1
Q1[k] > Ra requirement 1
Q2[k]> Ra requirement 2
Warning
i=i+1
T1min=t1
T2min=t2
i=i+1
A1[N],A2[N],P1[N],P2[N],T1[N]
C1[N’],C2[N’],M1[N’],M2[N’]
SUM1[N’],SUM2[N’], Q1[N’], Q2[N’]
i=0,j=0; flag1=0;t1min=0;t2min=0
k=0
P1[i]<P1min
P2[i]<P2min T
T
P1min
P2min
F
P1[i]=P1min
P2[i]=P2min
t1=time
t2=time
P1[i]<P1min
P2[i]<P2min
T
P1min
P2min F T
M1[k]=f(P1min)
M2[k]=f(P2min)
j=j+1
k=k+1
T1min=t1
T2min=t2
S
(1)
F
Q1[k]=f(SUM1[k])
Q2[k]=f(SUM2[k])
Fig. 8. Algorithm for interface software program to find grinding wheel’s wear value
Fig. 9. Experimental system measuring grinding wheel’s wear for profile grinding ball bearing's inner ring grove after
mounting two probes onto profile grinding machine 3MK136B [2, 3]
Caculating part’s surface roughness value (Rai) at
grinding wheel’s wear value (δi) is based on an
approximation function constructed by interpolation
method from a set of empirical value pairs (δi, Rai).
Several methods can be listed such as partial linear
interpolation, quadratic part interpolation, spline
interpolation In this paper, the authors apply partial
linear interpolation method. Therefore, after each a part
grinded completely, from wear values (δi) measured
online in the grinding process, the software system will
be used to interpolate to caculate surface roughness
values (Rai) respectively. The wear values (δi) measured
at the top probe will be used to interpolate to caculate
surface roughness values (Rai) respectively at the
bottom on the shape of ball bearing's inner ring grove.
The wear values measured at the margin probe are used
to interpolate to caculate values of corresponding
surface roughness at the edge on the shape of ball
Computer Processor ADC The top probe The margin probe
Grinding
wheel
Journal of Science & Technology 122 (2017) 041-047
46
bearing's inner ring grove. Then, the program will
always compare this surface roughness value (Rai) with
the given surface roughness threshold (Rarequirement) to
indicate warning signals for users as shown in the
diagram of the algorithm Fig. 8. In machining process,
if surface roughness value at the top or at the edge of the
part’s shape (Rai) is larger than the given surface
roughness threshold (Rarequirement), the program will give
a “warning signal” to users. Thanks to that, users know
the right time to dress grinding wheel to ensure part’s
surface roughness.
2.4. Data processing algorithm
The algorithm of data processing program is
represented as Fig. 8. Inputs include the following
signals: Analog signals from two pressure sensors in
two pneumatic probes are stored in two data arrays
A1[N] and A2[N]; Pressure values in the measuring
chamber at two probes are stored in two data arrays
P1[N] and P2[N]; Grinding time is stored in the data
array T1[N]. Outputs include the following parameters:
Wear values measured at two probes after each dressing
grinding wheel are stored in two data arrays C1[N’] and
C2[N’]; Wear values measured at two probes after each
grinding done one part are stored in two data arrays
M1[N’] and M2[N’]. Total wear values caculated at two
probes after each grinding done one part are stored in
two data arrays SUM1[N’], SUM2[N’]. Surface
roughness values at the bottom and at the edge are stored
in two data arrays Q1[N’] and Q2[N’].
2.5. Experimental and Results
The experimental process is performed on
3MK136B profile grinding machine to grind 6208 ball
bearing's inner ring grove. The experiment system is set
up as shown in Fig.s 5 and 9. Two pneumatic measuring
probe are used to measure grinding wheel’s wear at two
different points on its working surface. The top probe
has the following parameters: Source pressure P0=4 Bar;
Control orifice d1=0,85; Measuring nozzle d2=1,5. The
margin probe has parameters as follows: Source
pressure P0=4 Bar; Control orifice d1=0,65; Measuring
nozzle d2=1,6.
Thirty parts are grinded in the experimental
process with technological parameters as follows:
- Grinding wheels: 500x8x203A/WA100xLV60
- The speed of grinding stone: nw= 2500 rpm
- The speed of part: Vs = 6 m/min
- The depth of cutting: traw = 0.2 mm; tfine= 0.01 mm
- The rate of normal feed: Sraw = 30 µm/sec; Sfine =
5 µm/sec
- Time to dress grinding wheel: tdress = 20 seconds.
- Time to grind a part: tgrind = 22 seconds
Experiment results are exported to 1 txt file and 1
image file. Grinding wheel’s wear values (δi) and part’s
surface roughness values (Rai) are calculated after each
time one part grinded completely as shown in software
interface in Fig. 10. In the software interface has 3
graphs as follows:
- A graph shows pressure change in the measuring
chamber in each probes over grinding time. This graph
is located at the third quadrant on the right hand side.
While the above graph corresponds to the top probe, the
below graph corresponds to the margin probe.
- A graph shows the total amount of grinding
wheel’s wear (δi) over the number of grinding parts.
This graph is located at the first quadrant on the left hand
side. While the above blue graph corresponds to the
margin probe, the below red graph corresponds to the
top probe.
- A graph shows surface roughness value change
(Rai) over the number of grinding parts. This graph is
located at the fourth quadrant on the right hand side.
While the above blue graph corresponds to the margin
probe, the below red graph corresponds to the top probe.
From these graphs, the authors find that: Grinding
wheel’s wear values and part’s surface roughness values
at the edge of the curving edge shape of grinding wheel
is higher than those of at the top of the curving edge
shape of grinding wheel. The unequal distribution of
grinding stock is the main reason for this. Mechanical
surplus at the edge is greater than those of at the top. In
addition, the grinding wheel at the initial parts (the first
part and the second part), after just dressing grinding
wheel, will be worn more than at the later parts. The
amount of grinding wheel’s wear at the initial parts
corresponds to the initial wear phase of grinding wheel.
The amount of wear in the following parts tends to
decrease. It corresponds to the steady wear rate stage of
grinding wheels.
In particular, at the time 19th part grinded
completely, the program gives a "warning" signal. At
that time, surface roughness value at the edge of of the
curving shape of part has surpassed the value of
requirement surface roughness (Rarequirement = 0.42). It is
time to must dress grinding wheel to ensure requirement
quality of part’s surface roughness. Therefore, the
amount of grinding wheel wear at this time have to pass
the given wear threshold. Thus, grinding wheel wear
threshold can be determined through the requirement
surface roughness value of the operation. From the
requirement surface roughness value of the profile
grinding operation for 6208 ball bearing's inner ring
grove, using the method of partial linear interpolation, the
corresponding wear value will be determined (Hz17 =
9.405 µm). This value is grinding wheel wear threshold
for the profile grinding operation with the cutting mode
being investigated.
Journal of Science & Technology 122 (2017) 041-047
47
Fig. 10. Software interface after finished grinding 30 parts in one cycle for both the top probe and the margin probe
3. Conclusion
The above experiment results show that the
solution for determining part’s surface roughness
through grinding wheel’s wear value has achieved
satisfactory results. This solution is based on application
of the partial linear interpolation method. Grinding
wheel’s wear value is measured directly in the grinding
process on the basis of the application of pneumatic
measuring probe. By combining two pneumatic
measuring probes, the system determines online wear
value at the top and at the edge of the curving shape of
grinding wheel. Then, on the basis of application of
linear interpolation method, the software system has
calculated surface roughness’s value at each point
corresponding on part’s surface to give online warning
signals for users. Thanks to that, users determine the
resonable time to dress grinding wheel 3. In addition,
these results can be applied as the basis of adaptive
control for automatic compensation of grinding wheel’s
wear.
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Trường Học Viện Kỹ Thuật Quân Sự, Hà nội 02/2014
Warning! Times (s)
Pr
es
su
re
v
al
ue
(B
ar
)
The number of parts (part) The number of parts (part)
W
ea
r v
al
ue
(u
m
)
Su
rfa
ce
ro
ug
hn
es
s
(u
m
)
Các file đính kèm theo tài liệu này:
- applying_pneumatic_probe_system_to_monitor_parts_surface_rou.pdf