Pressure control in band-saw hydraulic system using the fuzzy logic

Pressure control in band-saw
hydraulic system using the fuzzy logic
GRZEGORZ FILO
List of notations
A sil – Area of hydraulic cylinder piston, m 2
A sz – Area of piston gap in proportional relief valve,
m 2
B – Bulk modulus, MPa
c spr , c s – Spring stiffness, N/m
e – Normalized value of control error
∆e – Change of normalized control error
F el – Electromagnetic force, N
F h – Hydraulic cylinder force, N
F hs-s , F hs-p – Hydrostatic forces, N
F hd-s , F hd-p – Hydrodynamic forces, N
F s0 – Initial spring tension, N
G c – Force of gravity, N
g p – Width of cutting band, m
J – Moment of inertia, kg·m 2
K – Factor of proportionality, N/V
M spr – Moment of spring, N·m
m s , m p – Mass of piston and pilot, kg
M sc – Moment of gravity, N·m
M sil – Moment of hydraulic cylinder, N·m
M Py – Moment of normal component of cutting
force, N·m
M Pz – Moment of tangent component of cutting
force, N·m
N – Number of cutting teeth
P y – Normal component of cutting force, N
P z – Tangent component of cutting force, N
P y1 – Normal component of cutting force acting on
single tooth, N
P 23 – Distance between points, m
p sil – Hydraulic cylinder pressure, MPa
p vf – Vertical feed of single cutting tooth, mm/s
p zl – Pressure in return line, MPa
Q sil – Volumetric flow rate, m 3 /s
r 23 – Arm of a spring force, m
R c – Arm of force of gravity, m
R h – Arm of a hydraulic cylinder force, m
U – Control signal, V
V sil – Volume under hydraulic cylinder piston, m 3
V 0 – Volume of hydraulic pipes, m 3
x s , x p – Position of piston and pilot, m
x 0 – Initial spring tension
β – Resistance coefficient, Nm·s/rad
ϕ – Rotation angle of cutting head, rad
µ sz – Flow coefficient
ρ – Liquid density, kg/m 3
ω – Angular velocity of cutting head, rad/s
Currently, a trend in the development of machine
tools, such as cutting machines, grinders, percussive
hammers etc., towards a higher dynamical per-
Dr in˝. Grzegorz Filo jest adiunktem w Instytucie Informatyki
Stosowanej na Wydziale Mechanicznym Politechniki
Krakowskiej.
formance can be observed. It is a result of progress
which was achieved on the one hand in material
science and on the other hand in power transmission
and control systems. Electrical, pneumatic and hydraulic
drives can be now controlled with higher
speed and accuracy. In this work control of pressure
force in hydraulic system of band-saw machine was
taken into consideration. Band-saw machines became
very popular among tools for cutting steel, wood or
metal ceramics. It is known that during the cutting
process edges of teeth are subjected to an impulse
load. The value and time course of this load depends
on many parameters, among others: pressure force,
thickness of cut material, number of cutting teeth.
The parameter which has the greatest influence on
time and quality of cutting process is pressure force
acting on teeth of cutting band. In these machines
hydraulic systems for supporting rotational head with
cutting band are used. The pressure is commonly
controlled by throttle valve with adjusted area of gap.
It allows for changing outflow rate of working fluid
from the hydraulic cylinder. Although this kind of
control system is simple and reliable, it comes from
the practise that it is very difficult to secure teeth from
overload, especially when cutting profiles with
variable shape. In order to avoid this inconvenience,
in this work application of relief valve with microprocessor
controller and control algorithm based on
fuzzy logic was proposed.
Mathematical model of the system
Model of the system was built using Matlab 7.0 with
Simulink, Control System Toolbox and Fuzzy Logic
Toolbox. The model consists of three main parts: the
mechanical system, the hydraulic system and the
control system.
Fig. 1. Band-saw machine: 1 – head, 2 – band, 3 – control desk,
4 – hydraulic cylinder, 5 – electric motor
ROK WYD. LXVIII ZESZYT 5/2009 35

Model of the mechanical system
Schematic diagram of mechanical part of the system
is shown in fig. 2.
Differential equation of rotation of the head can be
formulated as follows:
In the model the following moments were taken
into consideration: of spring force, of gravity, of hydraulic
cylinder and of cutting forces. Length of spring
Fig. 2. Diagram of the band-saw: a) frame of the head, b) spring,
c) hydraulic cylinder, d) cutting band
and arm of spring force depends on the angle of head
rotation. Therefore, moment of the spring force can
be calculated as:
where distances between points 2 and 3 (fig. 2) were
calculated from geometrical dependencies. Formula
Fig. 3. Model of the system in Simulink
36
(1)
(2)
of moment of hydraulic cylinder force allows for
change hydraulic cylinder length and arm of the force:
ROK WYD. LXVIII ZESZYT 5/2009
(3)
Moment of gravity force was calculated as follows:
In the model two components of cutting force were
taken into consideration: normal to direction of the
band movement P y and tangent P z . It was assumed
that force P y is a result of band deflection. Deflection
is caused by difference between angular velocity
of the head and vertical feed of the band. The value
of this force was used to calculate two further parameters.
The first one was moment, assuming that
the arm is 0,405 m:
The second parameter was current thickness of
material cut by one tooth (vertical feed of the band
p vf ), according to Wojnarowski J. [1]
On the basis of p vf value tangent force was computed
P z (Wojnarowski J. and others [2], Wojnarowski
J., Lisowski E., Filo G. [3]):
Moment of this force was computed assuming that
the arm is 0,05 m:
(4)
(5)
(6)
(7)
(8)

Model of the system was implemented as block diagram
in Matlab-Simulink system. The diagram is presented
in fig. 3.
Differential equation of head rotation is implemented
in subsystem Band-saw_head. Subsystems
designed for computing moments acting on the
head are as follows: Mom_spring, Mom_gravity,
Mom_cylinder, Mom_Pz1, Mom_Py. Output signals
of these subsystems are summed and used as input
for Band-saw_head subsystem. These signals are
also written into variables M_spring, M_gravity,
M_cylinder, M_Pz1 and M_Py1 for creating time
courses. Angle of rotation of head is used for computing
number of teeth currently used in cutting
process for various shapes of material in subsystem
Teeth_number. Block Control_system consists of
models of regulators with algorithms: PID and Fuzzy
Logic. There is also a possibility of carrying out
simulations without any regulator. In subsystem
PLOTS are collected all time courses of parameters
created using blocks Scope.
Model of the hydraulic system
The pressure force was controlled by a hydraulic
system which consists of proportional relief valve.
The system is shown in fig. 4. This system allows for
obtaining the proper run of force holding down head
to cut material. It was realized by controlling pressure
in line between supply unit and hydraulic cylinder.
Fig. 4. Scheme of the hydraulic system: 1 – supply unit, 2 – control
valve, 3 – hydraulic cylinder, 4 – proportional relief valve,
5 – controller
The hydraulic system consists of supply unit 1, control
valve 2, hydraulic cylinder rod 3, proportional
relief valve 4 and control unit 5. The controller generates
control signal for the valve on the basis of signal
from transducer installed on cutting head. Flow
Fig. 5. Block diagram of subsystem Control_system
balance of hydraulic cylinder chamber was formulated
with assumed bulk modulus of working fluid:
In the hydraulic system a pilot operating proportional
relief valve with two movable elements is used:
a pilot at first stage and a piston at second stage.
Equation of the piston motion is as it follows:
ROK WYD. LXVIII ZESZYT 5/2009 37
(9)
(10)
while equation of pilot motion can be formulated as:
(11)
The electromagnetic force F el acting on valve pilot
is proportional to control signal U: F el = k · U.
The main flow rate through the piston gap can be
described by formula:
(12)
where width of piston gap A sz can be computed on the
basis of valve geometry.
Controller design
Models of regulators were implemented in model of
main system (fig. 3) in subsystem Control_system.
Block diagram of this subsystem is shown in fig. 5.
As it arises from fig. 5, controller input signal is
current value of force acting on single cutting tooth
Fig. 6. Control surface of FLC controller

Fig. 7. Block diagram of subsystem for computing number of cutting teeth
Py1. This parameter is time-quantized with zero order
hold (Brzózka J. [4], Franklin G. F. [5]) with assumed
sampling frequency. Then is computed and normalized
control error value e. The normalized control
error is an input value for controllers in blocks:
Fuzzy_Controller and PID_Controller. It is also used for
computing value of ISE parameter which determines
control quality. Subsystem proximity_detector is used
for decreasing speed of head while approaching to
material, which prevents from impact load.
Model of Fuzzy Logic Controller (FLC) was built
using Fuzzy Logic Toolbox, which is one of tools
available in Matlab system. FLC has two input signals
38
and one output signal, which are analogous with
incremental PI controller Driankow D., Hellendoorn H.,
Rainfrank M. [6], Piegat A. [7]. After adjusting parameters,
control surface was obtained as shown in
fig. 6. The figure shows dependency between input
signals: e (value of control error) and ∆e (change of
control error from last step) and output: change of
control signal ∆U.
Simulation
Simulations of cutting process were carried out
with several different shapes of cross-sections. In
Fig. 8. Time courses of tooth force at the beginning of cutting
process: a) without controller, b) with PID controller, c) with
fuzzy controller
each simulation controller maintained an optimal
value of tooth force, independently from angle of
head and shape of material. Main results of simulations
were time courses of parameters like: rotation
angle of head, tooth force, number of cutting teeth,
control signal, moments etc.
Models of cut shapes
Computing number of cutting teeth was carried out
in subsystem Teeth_number. Block diagram of this
subsystem is shown in fig. 7. The input signal in this
subsystem is current rotation angle of the cutting
head. Three example cross-sections: triangular, circu-
ROK WYD. LXVIII ZESZYT 5/2009

Fig. 9. Time courses of tooth force while obtaining stabilized
value of tooth force: a) without controller, b) with PID controller,
c) with fuzzy controller
lar and tubular are implemented in the model. Three
blocks: Triangular, Round and Tubular allow for computing
number of cutting teeth for each shape. Shape
can be chosen using manual switches Sw1 and Sw2.
Block Vibrations allows for adding step change of
teeth force which appears when tooth enters and material
exits. The output signal is written into variable
Profil for using in other part of the model.
Simulations of cutting tubular profile
As an example results of simulation obtained with
tubular profile are presented. This is one of more
difficult cases, because while cutting this type of profile
there are rapid changes of number of cutting teeth.
Outer radius of profile was assumed as r = 30 mm,
Fig. 10. Time courses of tooth force at the end of cutting process:
a) without controller, b) with PID controller, c) with fuzzy
controller
ROK WYD. LXVIII ZESZYT 5/2009 39

wall thickness g = 5 mm. Time courses of tooth force
while cutting tubular profile with various control
methods are presented in fig. 8 – 10. The optimal
value of tooth force was assumed as F yp_opt = 60 N.
The time courses of angle and rotational speed
of the head while cutting tubular profile are presented
in fig. 11.
As it arises from fig. 8 – 11, application of FLC controller
reduces changes of tooth force amplitude at the
beginning and at the end of the process (fig. 8 – 10).
Cutting without controller (fig. 8a) causes strong impact
with tooth force over 200 N when cutting band
first time touches profile. An application of PID controller
(fig. 8b) reduces maximum impact to 160 N, and
an application of FLC (fig. 8c) reduces it to 140 N.
While using PID controller stabilized value of tooth
force 60 N is obtained after 3.0 seconds, with FLC
– after 2.8 seconds (fig. 9b, 9c). The cutting time is
reduced from 18 seconds without controller (fig. 10a)
to 14 seconds with FLC (fig. 10c).
40
Conclusion
Solution proposed in this work consists in upgrading
of tooth force control system in band-saw
machine using the PID controller and the FLC controller.
Applied controllers allow for maintaining value
of tooth force during the cutting process at the level
close to optimal. Value of this force is independent
from rotation angle of cutting head, shape and size of
material profile. Using such controllers shortens cutting
time and causes less wear of teeth.
Proposed control system can be applied to existing
hydraulic system of band-saw machines. It requires
only small changes in a hydraulic system. The main
change is using electromagnetically controlled proportional
relief valve in exchange for manually controlled
throttle valve. Digital modules of controllers
and transducers for measuring rotation angle of
cutting head and pressure in hydraulic cylinder or
pressure force acting on cut material must be also
added.
Simulations were carried out with two types of controllers:
PID and FLC. It arises from the results, that
application of both types of controllers allowed for
obtaining similar effects. Cutting time was 2 – 10%
Fig. 11. Time courses obtained while cutting tubular profile with FLC: 1 – angle of cutting head (α), 2 – rotational speed of cutting
head (ω)
shorter while using FLC controller, while ISE index
was 1 – 8% lower with PID controller. Application of
FLC controller reduced better oscillations at the beginning
of cutting process when the band first time contacts
material. There are also greater possibilities of
adjusting controller parameters when using FLC. In
this case the number of fuzzy sets can be easily increased
and more input signals can be added, for
example rotation angle of cutting head or band-saw
driving moment. Further extension of FLC and additional
tests can lead to obtain even better results.
REFERENCES
1. Wojnarowski J.: Model komputerowy przecinarek z pi∏ami
taÊmowymi bez koƒca w badaniu zjawisk dynamicznych
w procesie ci´cia. Silesian University of Technology, Gliwice
2002.
2. Wojnarowski J. and others: Ograniczanie drgaƒ przecinarek
z taÊmowymi pi∏ami bez koƒca. Monograph, Silesian University
of Technology, Gliwice 2007.
3. Wojnarowski J., Lisowski E., Filo G.: Sterowanie hydraulicznym
uk∏adem nacisku pi∏y bez koƒca przy zastosowaniu
regulatora FLC. XX Konferencja Naukowa Problemy Rozwoju
Maszyn Roboczych, Zakopane 2007.
4. Brzózka J.: Regulatory cyfrowe w automatyce. MIKOM, Warszawa
2002.
5. Franklin G. F.: Digital control of dynamic systems. Addison
Wesley Longman Inc, Menlo Park CA 1998.
6. Driankow D., Hellendoorn H., Rainfrank M.: An Introduction
to Fuzzy Control. Springer-Verlag, Berlin 1993.
7. Piegat A.: Modelowanie i sterowanie rozmyte. Akad. Oficyna
Wydawn. EXIT Warszawa 1999.
ROK WYD. LXVIII ZESZYT 5/2009