TRANSACTIONS
ON ELECTRICAL ENGINEERING
CONTENTS
Zablodskiy, N., Lettl, J., Pliugin, V., Buhr, K., Khomitskiy, S.:
Induction Motor Design by Use of Genetic Optimization
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 – 69
Homišin, J.: New Ways of Controlling Dangerous Torsional
Vibration in Mechanical Systems . . . . . . . . . . . . . . . . .
70 – 76
Folta, Z., Hrudičková, M.: Experience with Torque Measurement
on Rotating Shaft . . . . . . . . . . . . . . . . . . . . . . . . . .
77 – 81
Veleba, J.: Performance of Steady-State Voltage Stability
Analysis in MATLAB Environment . . . . . . . . . . . . . . .
82 – 88
Ondrášek, J.: Cross Slide Mathematical Model for Solving
Chatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89 – 96
Vol. 2 (2013)
No.
3
ERGO NOMEN
pp.
65 - 96
TRANSACTIONS ON ELECTRICAL ENGINEERING
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Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
65
Induction Motor Design by Use of Genetic
Optimization Algorithms
prof. N. Zablodskiy 1), prof. J. Lettl 2), doc. V. Pliugin 3), ing. K. Buhr 4), stud. S. Khomitskiy 5)
1)
Donbas State Technical University/Automation of electro-technical systems, Alchevsk, Ukraine, info@dmmi.edu.ua
Czech Technical University in Prague/Faculty of Electrical Engineering, Prague, Czech republic, lettl@fel.cvut.cz
3)
Donbas State Technical University/Automation of electro-technical systems, Alchevsk, Ukraine,
vlad.plyugin@gmail.com
4)
Czech Technical University in Prague/Faculty of Electrical Engineering, Prague, Czech republic, buhr@fel.cvut.cz
5)
Donbas State Technical University/Automation of electro-technical systems, Alchevsk, Ukraine,
stas.blitzkrieg@mail.ru
2)
Abstract — The problem of the automated calculation
and optimal design of an induction motor is presented . The
problem of optimization by use of genetic algorithms is set
and solved. The analysis of the obtained results is executed.
Keywords — induction motor, optimization, varied variables,
genetic algorithm, criteria, limitations, effective variant, EvoJ
library
I. INTRODUCTION
Neuron networks, being one of perspective trends of
researches in the artificial intelligence area, as a result of
watching processes going on in the nervous system of man
were created. Approximately by the same way genetic
algorithms were also «invented», but watched over the
man nervous system, by the process of living organisms
evolution.
Genetic algorithms - one of research trends in the
artificial intelligence area, engaging in creation of the
simplified evolution models of living organisms for the of
optimization task decision [1].
A classic genetic algorithm (GA) consists of following
steps:
1)
initializing, or choice of initial chromosomes
population;
2)
an estimation of chromosomes adjustment in a
population - calculation of adjusted function for
every chromosome;
3)
verification of algorithm stop condition;
4)
chromosomes selection - choosing of
chromosomes, participated in descendants for a
next population creation;
5)
application of genetic operators are mutations
and crossing;
6)
forming of new population;
7)
choosing of the «best» chromosome.
The block-diagram of GA is represented in Fig. 1.
A simple GA generates an initial population by a
random way. Working of the GA is an iteration process
which proceeds until the generations set number or some
another stop criterion will not be executed. On every
generation, a proportional selection on adjusted, crossing
and mutation is realized.
The simplest proportional selection is roulette. The
wheel of roulette contains one sector for every member of
population. The size of every sector is proportional to the
corresponding size of adjusted function. At such selection,
members of population with higher adjustment will be
chosen with greater probability than individuals with
subzero adjustment. The next step is using of crossing and
mutation.
A previous population, obtained after a mutation, is
overwritten and the cycle of one generation is completed.
Subsequent generations i.e. selection, crossing and
mutation obtained as a GA working result are processed
in the same way .
II. THEORY AND PROGRAM REALIZATION
In the examined task a GA provides one criterion of
optimality only, by virtue of the program realization of a
calculation function minimum search [2 - 4].
The order of optimization will be different from
considered Carthesian product (CP) in the previous article
[5]:
1)
2)
3)
4)
setting the range of the varied variables;
setting limitations;
choosing the criterion of optimality;
calling the CA function for optimization and
getting the optimal varied variables set;
5) calling the function of the induction motor (IM)
automatic calculation for the found set.
We will consider an example of the IM optimization
programmatic realization using GA on Java in NetBeans
IDE [2]. For a decision the problem we will use free Javalibrary EvoJ (Evolution Java) [3]. A project EvoJ is
designed as upgradable framework of Java classes for the
decision of various optimization tasks by use of
evolutional (genetic) algorithms. For the use of EvoJ a
programmer must implement one simple interface,
consisting of one method only. All other steps undertake
EvoJ algorithm.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
66
START
Initialization – choosing of initial
chromosome population
Estimation of chromosomes and
population adjustment
YES
NO
Algorithm stop condition
achieved?
The best
chromosome
choosing
Chromosome
selection
END
Applying of
genetic
operators
Creating of new
population
Fig. 1. Magnetization as a function of applied field.
In the example two varied variables will be considered:
the internal diameter of the stator core and the length of
the stator core.
We shall create Java-interface with the name “Solution”
in which we set the range (minimum and maximum
values) of the varied variables. The code of the Solution
interface is down in the text.
EvoJ is able to change the variables without a setting of
a range. However if it is needed to implement an own
mutation strategy, one have to declare setters. In other
case we shall not have a possibility to change the variable
range.
Pay attention to annotation @of Range - it sets the
range of values which a variable can accept. Variables of
random values from the set range are initiated.
package MotorClasses;
import net.sourceforge.evoj.core.annotation.MutationRange;
import net.sourceforge.evoj.core.annotation.Range;
public interface Solution {
//Diameter of stator core
String smin1 = "165";
String smax1 = "205";
//Length of stator core
String smin2 = "115";
String smax2 = "145";
@Range(min = smin1, max = smax1)
double getX(); //return the optimal diameter
@Range(min = smin2, max = smax2)
double getY(); //return the optimal length
}
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
However as a result of mutation they potentially can go
out from the indicated range. It can be prevented using the
parameter of strict=«true», that will not allow a variable to
take on an impermissible value, even if we make an effort
to propose them using setter-method.
Another case on which it is needed to pay attention
here, is that all parameters of all limitations in EvoJ are
strings. It allows both to specify the value of parameter
directly and to specify the name of property instead of
concrete value, to specify the value of limitation
parameters at the compile-time.
Now we have an interface with variables and we shall
write a fitness-function. Fitness-function in EvoJ is
implemented as the following interface:
public interface SolutionRating <T> {
Comparable calcRating(T solution);
}
67
@Override
public Comparable doCalcRating(Solution solution){
double fn = calcFunction(solution);//call motor function
boolean flag = mot.control();//control of limitations
if (Double.isNaN(fn) | flag == false){
return null;//sift-out false variant
} else {
return - fn;//return an effective variant
}
}
}//end of class
All code lines above are obviously enough. We simply
take and count our function, using variables from the
Solution interface:
double fn = calcFunction(solution);//call motor function
Here parameter <T> is our interface with variables. The
greater value return (according the contract Comparable)
the more suitable solution is considered. Null can be
returned and it is the smallest value of a fitness-function.
It is recommended to implement this interface as
mediated using helper-classes. They undertake some
service functions: elimination of the old decisions (if the
maximal life term of decision is set), cashing of function
value for decisions which were not sifted from in the
previous GA iteration.
A Fitness-function for our case will look like the
following (we create a new class with the name Rating):
package MotorClasses;
import
net.sourceforge.evoj.strategies.sorting.AbstractSimpleRating;
public class Rating extends AbstractSimpleRating <Solution> {
static AMotor mot;//motor object
static int krit;//index of optimality criterion
static int iter_numb;//number of iterations
//Constructor
public void set_motor(AMotor mot, int krit){
this.mot = mot;
this.krit = krit;
this.iter_numb = 0;
}
//reception of iterations number
public int get_iter(){return this.iter_numb;};
public static double calcFunction(Solution solution){
iter_numb++;//increase of iterations count
double x = solution.getX();//reception of new diameter
double y = solution.getY();//reception of new length
mot.stator.set_D(x/1000);//setting of new diameter
mot.stator.set_ld(y/1000);//setting of new length
double fn = mot.auto(krit);//automatic motor calculation
return fn;//return a criterion of optimality
}
Because we search minimum and the contract of class
supposes that the best decisions must have a greater
rating, we return the value of function, multiplied by 1:
return - fn;//return an effective variant
The population with the highest value of variable fn will
be considered as the most close to optimum result.
In addition, we sift-out false decisions (if NaN turned
out or motor limitations were not passed), returning null.
Override of function Comparable realize the
mechanism of genetic populations, using as the achieved
result the value returned by the function calcFuction().
In the IM class Motor we create the function of
automatic calculation, which accepts as an argument the
index of optimality criterion and returns the got criterion
after the motor calculation:
double auto(int krit){
int res = 0;
//Code body of motor calculation
//…
switch (krit){
case 1://1 is efficiency
res = 1/kpdnr;
break;
case 2://2 – power factor
res = 1/cosFinr;
break;
case 3://3 – starting current
res = I1pn;
break;
case 4://4 – starting torque
res = 1/Mpo;
break;
}
return res;//return a criterion depending on its index
}
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
68
Further in the motor class Motor we create the function
of GA realization (a code structure is explained in
comments):
It is obvious from Tab. I, an optimal motor efficiency
is higher than a base value and other parameters are
satisfying limitation range.
void optimization(int krit){
DefaultPoolFactory pf = new DefaultPoolFactory();
//creation of populations with the amount “populations”
GenePool<Solution> pool = pf.createPool(populations,
Solution.class, null);
Rating rtg = new Rating();//Constructor of Rating class
rtg.set_motor(this, krit);//getting of Motor object and criterion
//factory of initial decisions set generation
DefaultHandler handler = new DefaultHandler(rtg, null, null,
null);
//implementation of iterations number “iterations”
//over the population “populations”
handler.iterate(pool, iterations);
//reception of the best found decision
Solution solution = pool.getBestSolution();
D_opt = solution.getX()/1000; //optimal diameter
L_opt = solution.getY()/1000; //optimal length
int iter = rtg.get_iter(); //reception of iterations count number
}
TABLE I
GENETIC ALGORITHM: TABLE OF PARAMETERS BEFORE AND
AFTER OPTIMIZATION
Name
Base value Optimal value
Induction in the air-gap, T
0.748
0.807
Internal diameter of stator core D, mm
185
194
130
115
Length of stator core Lδ, mm
0.895
0.755
Relative size λ = Lδ/τ (τ = πD/2p,
where p - number of pole pairs)
Height of stator slot, mm
21.9
14.6
Height of rotor slot, mm
32.2
33.2
Width of the upper line of stator slot,
7.7
7.8
mm
Width of the down line of stator slot,
10.2
9.3
mm
Upper diameter of rotor slot, mm
7.9
7.8
Down diameter of rotor slot, mm
3.7
3.4
Efficiency
0.885
0.891
Power factor
0.893
0.9
Starting current relative value
5.84
6.52
Starting torque relative value
1.4
1.62
Overload torque capability
2.65
2.88
Overall stator winding temperature, C
93.25
95.69
From a NetBeans form, the code of the GA
optimization implementation consists of two lines:
motor.limits();//setting of limitations
motor.optimization(krit);//optimization with the criterion “krit”
Functions limits() consist restrictions on motor
geometric sizes, temperature limits, starting currents and
etc. In the fitness-function limitations are checked by the
control function
boolean flag = mot.control(); //control of limitations
This function return false if even one restriction will be
broken. In control() function there are 16 motor variables
limits have been set. In particular we can avoid high
temperatures of the stator that is probably resulted because
of increasing of the stator winding cross-section.
If a decision will not arranged (do not satisfy to motor
restrictions according to limitation function) it is possible
to continue the GA iterations (increasing the populations
number “populations” and iterations “iterations”), while
the desired quality of decision will not be attained.
So to solve a GA task using EvoJ it is necessary:
1) to create an interface with variables;
2) to implement the interface of the fitness-function;
3) to create the population of decisions and carry out
the necessary amount of the GA iterations above
them, using a code, given above.
The results of the GA solution at a choice of maximum
efficiency as a criterion of optimality are shown in Tab. 1.
III. CONCLUSIONS
Algorithm of the previous considered CP [5] in
comparison with the GA, allows to execute multi-criterion
optimization, that is its undoubted advantage. In addition,
the CP always gives only the synonymous best variant
among the existing ones. However, in the range of
varying of two variables ± 20 % from a base value (3976
combinations) the calculation time is approached up to 48
min.
Implementation of CA gives stunning results. At the
same varied variables and range of their change ± 100 %
(!) from a base value, the calculation time is only 40 sec!
However, GA, at least, in the present article task, does
not allow to execute optimization for a few criteria.
In the GA number of the varied variables and a range
of their change is not important from the point of view of
the productivity, because a set of the varied variables is
created dynamically, but not beforehand, as in the CP
method. In addition, all combinations of the variables and
values of objective function are realized in a binary form.
However, time of the GA work is very critical to the
number of the created populations and number of
iterations in the populations.
The choice of population’s number and iterations
realized by an experienced way increases until an
acceptable result will not be obtained. A result of the
optimization with the use of the GA will always be the
best for the chosen criterion, but there is not a guarantee,
that a better variant can exist. Actually, herein there is a
genetic selection logic - we get a result, approaching the
best among the random created populations. A most
reliable result depends on population’s number. Thus, the
degree of authenticity can be estimated by the variation of
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
the obtained results in repeated calculations at the same
population’s number.
Therefore, the GA productivity at effective variant
populations number is determined but not by the number
and range of the varied variables.
The result of the optimization is ambiguous and close to
the best. When production of approximate calculations in
maximum compressed terms is needed and quality of the
obtained results is written with a permissible error, then
using of the GA optimization will be the irreplaceable
instrument for designers.
69
REFERENCES
[1] Yemelyanov V.V., Kureychik V.V., Kureychik V.M. Theory and
practice of evolution modeling. М.: PHYSMATLIT, 2003. – 432 p.
[2] N. Zablodskiy, V. Pliugin, K. Buhr. CAD of electromechanic devices:
educational tutorial, part 2, 2013. - 330 p. (will be printed).
[3] http://evoj-frmw.appspot.com [ONLINE].
[4] N.K. Vereshchagin, A. Shen. Lectures on mathematical logic and
theory of algorithms. Beginning of sets theory. MCNMO, 2008. – 198p.
[5] N. Zablodskiy, V. Pliugin, K. Buhr, S. Khomitskiy. Asynchronous
motor optimal design with using of Cartesian product (will be printed).
REFERENCES ON RUSSIAN:
[1] Емельянов В.В., Курейчик В.В., Курейчик В.М. Теория и
практика эволюционного моделирования. М.: ФИЗМАТЛИТ, 2003.
– 432 с.
[4] Верещагин Н.К., Шень А. Лекции по математической логике и
теории алгоритмов. Начала теории множеств. МЦНМО, 2008. –
198с.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
70
New Ways of Controlling Dangerous Torsional
Vibration in Mechanical Systems
Jaroslav Homišin
Department of Design, Transportation and Logistics– Section of Design and Machine Elements
Technical University of Košice, Faculty of Mechanical Engineering, Letná 9, 040 01 Košice
Slovak Republic
e-mail: Jaroslav.Homisin@tuke.sk
Abstract — In general terms the mechanical systems (MS)
mean the systems of driving and driven machines arranged
to perform the required work. We divide them into MS operating with constant speed and MS working within a range
of speed. In terms of MS dynamics we understand the system of masses connected with flexible links between them,
i.e. systems that are able to oscillate. Especially piston machines bring heavy torsional excitation to the system, which
causes oscillation, vibration, and hence their noise. Based on
the results of our research, the torsional vibration control
as a source of given systems excitation, can be achieved by
application of pneumatic couplings, or pneumatic tuners of
torsional vibration. Existence of pneumatic couplings and
pneumatic tuners of torsional vibration developed by us,
creates the possibility of implementing new ways of torsional oscillating mechanical system tuning. Based on the
above, the aim of the article is to present new ways of dangerous torsional vibration control of mechanical systems by
application of pneumatic couplings and pneumatic tuners of
torsional vibration developed by us.
Keywords — torsion, oscillating mechanical system, pneumatic coupling, pneumatic tuner of torsional vibration, ways of
torsional vibration control
I. INTRODUCTION
Any MS, in terms of dynamics, we understand the system of masses connected with flexible links between
them, i.e. systems that are able to oscillate. Piston machines, either drivers or driven units, bring to the system
significant torsional vibration. This means that MS with
internal combustion engines, compressors and pumps can
be characterized as torsional oscillating mechanical systems (TOMS). In the range of operating speed there can
be a very intense resonance between the driver frequencies (reciprocating machines) and the natural frequencies
of the system. Consequently, there comes to an excessive
vibration and related excessive stress of the whole MS.
The excessive dynamic stress often causes malfunction of
various parts of the system, such as:
fatigue fractures of shafts,
gear failures,
deformation failures of flexible couplings and the like.
Therefore, it is necessary to control their dangerous torsional vibration.
Currently, the torsional vibration is reduced to a permissible degree by appropriate adjustment, respectively
tuning the system by application of an appropriately selected flexible coupling, based on a dynamic calculation.
Thus the principle of tuning is an appropriate adaptation
of the basic dynamic properties, particularly the dynamic
torsional stiffness of the flexible coupling to the system.
The general characteristics of flexible couplings include their dynamic torsional stiffness and damping coefficient. It should be noted that, they are affected by material (metal, rubber, plastics), shape, number and dimensions of the flexible elements. It follows that they depend
on various factors, which are divided into stable and unstable factors [1]. The shape, number, size and various
structural modifications of flexible elements can be added
to the group of stable factors, while the material of flexible members to the group of unstable factors, as a result of
fatigue and aging which are changing their original characteristics. By changing original properties there is a
change of coupling characteristics Mk = f(φ) (with respect
to initial characteristics), and thus a change of its basic
characteristics, which has a largely positive impact on the
magnitude of the dangerous torsional vibration of the mechanical system [1], [2], [3].
It should be emphasized that any linear or nonlinear
flexible coupling is only one characteristic, tightly coupled to the used flexible element. In the case of a linear
coupling it is only one characteristic of a constant dynamic torsional stiffness. Dynamic torsional stiffness of
the nonlinear coupling varies in some extent of its characteristics, obviously dependent on the working mode of the
system. Changing the characteristics of coupling due to
appropriate dynamic tuning of TOMS means the use of an
other coupling flexible element or other flexible shaft
coupling.
Influences such as: temperature of flexible coupling elements and the number of cycles causes that, by effect of
external forces any flexible member of the coupling is
exposed to fatigue and aging. Consequently, there is a
change of coupling characteristics, and thus a change of
its basic characteristic properties. This leads to the fact
that a suitably tuned TOMS becomes untuned. A flexible
coupling in this case does not act as a tuner, but rather as
a driver of torsional oscillations.
It should be noted that this method of tuning will be
suitable only in cases where no previously unforeseen
(random) effects occur during the operating mode, particularly in the turbo-machinery and reciprocating machinery [2], [3]. In case of random failure in an operating
mode of MS a very intense resonance of lower harmonic
excitation occurs, which is usually unexpected. Due to this
fact, an intense torsional excitation causes increased torsional vibration, mechanical vibration and hence a noisy
mechanism.
The torsional vibration control, based on the results of
our research, is achieved by use of a pneumatic flexible
coupling as well as by application of pneumatic flexible
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
coupling with auto-regulation – pneumatic tuner of torsional systems.
The aim of this scientific paper will be presentation of
the possibility to control dangerous torsional vibration of
mechanical systems by ways suggested by us, by tuning
and continuous tuning. Tuning of the given system is provided by application of the pneumatic flexible coupling
while continuous tuning (tuning during operation) of the
system will be ensured by application of the pneumatic
flexible coupling with auto-regulation – pneumatic tuner
of torsional vibration.
71
sure p of the gas media in it. Varying pressure will ensure
the fact that the coupling always works with different
characteristics (Fig. 2), which is defined by the formula
(1).
II. PROPOSED WAYS TO TORSIONAL VIBRATION CONTROL
OF MECHANICAL SYSTEMS
A change of the pneumatic couplings torsional stiffness
can be realized by changing the pressure of gaseous media, out of operation (Fig. 3) or during operation (Fig. 5)
of mechanical systems. This leads to two proposed ways
to the torsional vibration control of mechanical systems
[4]:
the torsional vibration control of mechanical systems
out of operation, ensuring the so called tuning of the
system [5], [6], [7], [8],
the torsional vibration control of mechanical systems
during operation, ensuring the so called continuous
tuning of the system [5], [9], [11].
Under the tuning of torsionally oscillating mechanical
systems with pneumatic coupling we understand the inflation space of the coupling compression suitable to pressure value of the gaseous medium out of the operation
system. The appropriate pressure value of the gaseous
medium, and hence the appropriate value of the dynamic
torsional stiffness of the coupling is based on the previously realized dynamic systems in terms of calculation of
the torsional dynamics. The mechanical systems run during their entire operation with such inflated pneumatic
coupling.
The principle of the torsional vibration control of the
mechanical system during its operation at steady state by
application of torsional oscillations pneumatic tuner [2],
[3] shows the adaptation of the basic dynamic properties,
particularly the dynamic torsional stiffness of the tuner to
the system dynamic . The basic principle of the pneumatic
tuner is the ability to auto-regulate the twist angle due to a
current change of the load torque on a predetermined constant angular value φK . This will ensure the autoregulation of gas pressure in the compression space of the
tuner, thus adapting it to the current value of the load
torque.
III. CHARACTERISTICS OF PNEUMATIC FLEXIBLE SHAFT
COUPLINGS
The differential pneumatic coupling (Fig.1) consists of
the driving part (1), driven part (2), between them there is
located the compression space filled with gaseous medium (air in our case). The compression space consists of
three circumferentially spaced and interconnected differential elements. Each differential element consists of a
compressed (3) and expanded pneumatic-flexible element (4).
Interconnection of differential elements is provided by
the interconnecting hose (5). The filling of compression
space of coupling through the valve (6) changes the pres-
Fig. 1. Differential pneumatic flexible shaft coupling.
To another characteristic other characteristic properties
always belong, thus still different torsional stiffness and
damping coefficient. Therefore, each pneumatic coupling
depending on the pressure is always defined by another
course of the torsional stiffness in Fig. 3, as described by
the formula (2).
(1)
M = a0 .ϕ + a3 .ϕ 3 ,
k = a0 +
3
a3 .ϕ 3 ,
4
(2)
where: ϕ – twist angle of the coupling,
a0, a3 – constants of the coupling characteristics.
The orsional stiffness, as the main component in the
field of the mechanical system tuning has a decisive influence on the natural frequency of the system
Ω0 = k / I red ,
(3)
where: Ired – reduced mass moment of inertia of the mechanical system.
It therefore follows the basic principle of the mechanical system tuning by pneumatic couplings. Its basic principle is to customize the natural frequency of the system
Ω0 to the angular excitation frequency ω, so that in the
range of the system working mode there is no resonance
condition ω = Ω0 and hence dangerous torsional vibration.
The pneumatic tuner of the torsional vibrations (Fig. 4),
which basic principle results from the patent claims [9],
[11] is compared with the differential pneumatic coupling
on a common structural base. The main difference is that
it does not have the valve, but the controller (6) to ensure
a coupling constant twist angle φk. The basic principle of
the tuner is the ability to auto-regulate the twist angle due
to the torque current load change on a predetermined constant angular value φk. This will ensure auto-regulation of
the gaseous media pressure in the compression space of
the tuner, thus adapting it to the current value of load
torque.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
72
In Fig. 5 there are in general terms shown the traces of
the pneumatic tuner of the torsional vibrations and torsional stiffness depending on the load torque. To each
constant twist angle φk1, φk2, φk3 and φk4 based on a calculation one course of torsional stiffness labelled a, b, c, d is
given.
Fig. 2. Courses of static characteristics of the differential pneumatic
coupling a, b, c, d, e, f, g shown in the general version belong to pressure
of the gaseous medium at p = 100 ÷ 700 kPa with 100 kPa.
Obr. 5. Courses of the pneumatic tuner of the torsional vibration and
torsional stiffness depending on the load torque M.
IV. THE RESULTS OF THE INVESTIGATION OF THE PROPER
Fig. 3. Courses of the torsional stiffness k of the differential pneumatic
coupling a, b, c, d, e, f, g shown in the general version belong to pressure
of the gaseous medium at p = 100 ÷ 700 kPa with 100 kPa.
Fig. 4. Pneumatic tuner of the torsional vibration.
Auto regulation of he pressure of the gaseous media
has a direct effect on the characteristics change of the
pneumatic tuner (Fig. 2) Of course, for change of the torsional stiffness value (Fig. 5), as a result, we can tune the
natural frequency of the system.
TUNING AND CONTINUOUS TUNING OF THE TORSIONALLY
OSCILLATING MECHANICAL SYSTEM
When investigating an appropriate tuning, or any continuous tuning of the torsionally oscillating mechanical
system we mostly start from the Campbell diagram showing the position of the critical speed nK (or the position of
the critical angular frequency ωK) depending on the rotational speed frequency N (or natural angular frequency
Ω0).
Magnitude of the torsional vibration for the tuning and
continuous tuning of the system is mostly presented by:
courses of the dynamic torque amplitude excited by the
torsional vibration in the mechanical system and hence
to the pneumatic coupling, depending on the speed.
A. Characteristics of the realized torsionally oscillating
mechanical system
The examination of an appropriate tuning and continuous tuning was performed on a realized torsionally oscillating mechanical system (Fig. 6). The realized system is
composed of the driving part (1), pneumatic flexible shaft
coupling (3) and driven part (2). The driving part, formed
by a DC electric motor type SM 160 L with a power of
16 kW and an auxiliary thyristor controller of the rotational frequency (4) type IRO with the possibility of continuous control from n = 0 ÷ 2000 min-1, using a pneumatic coupling that drives the exciter of the torsional vibrations represented by the three-cylinder compressor type
3–JSK–S. To increase the impact of the compressor tor-
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
sional excitation to the mechanical system we used a
compressor without a flywheel.
The load of the torsional oscillating mechanical system
by the compressor will be adjusted (regulated) with throttle valve (6) integrated into the output pipe of the compressor.
The analyzis the pneumatic coupling load at work of
the mechanical system in the steady state will be investigated by a schematic model of the realized torsional oscillating mechanical system (Fig. 7).
7
6
73
I 1 .ϕ&&1 + b.(ϕ& 1 − ϕ& 2 ) + k .(ϕ 1 − ϕ 2 ) = M i . sin .(i.ω .t + γ i ),
I 2 .ϕ&&2 − b.(ϕ& 1 − ϕ& 2 ) − k.(ϕ 1 − ϕ 2 ) = 0.
n
M d = ∑Mi.
i =1
4
I2
.θ . sin[(i.ω .t + γ i ) + β i + ϑi ]
I1 + I 2
(6)
while dynamic coefficients θ , ξ and phase angles βi , υi
are described by formulas
θ=
3
(5)
ξ=
 i.ω
1 + 
 Ω0
  i.ω
1 − 
  Ω 0



2



2
2
 2χ
.
 Ω0
  i.ω
 + 
  Ω 0






2
2
 2χ
.
 Ω0



1
  i.ω
1 − 
  Ω 0



2
2
  i.ω  2  2 χ
 .
 + 
  Ω 0   Ω 0
,
(7)
,
(8)
2



2
Where for the damping coefficient 2.χ, natural angular
frequency of the system Ω0 and separation margin η it is
applied
5
2
1 1
2.χ = b. + ,
 I1 I 2 
1
1 1
Ω 0 = k . +  ,
 I1 I 2 
η=
i.ω i.n
= . (9)
Ω0 N
B. Results of a proper tuning of the torsionally
oscillating mechanical system
The Campbell diagram according Fig. 8 describes the
tuning of the realized mechanical system by the tangential
pneumatic coupling type name 4–1/70–T–C in the speed
range n = 0 ÷ 2000 min-1. Operating mode of the system is
defined in the speed range n = 750 ÷ 1500 min-1.
Fig. 6. Realized torsional oscillating mechanical system.
Fig. 7. Schematic model of the realized torsional oscillating mechanical
system.
When calculating the loads for the steady-state mechanical system within its operational mode, we suppose
that the mechanical system is rotating at an angular speed,
which varies in a wide range. On mass (1) with the mass
moment
of
inertia
I1
a
load
torque
acts
.
From
the
above
it
is
clear
that
M N + ∑ M i . sin(i.ω .t + γ i )
pneumatic coupling and thus the whole torsional oscillating mechanical system is loaded by both unvariable with
time medium torque MN in steady state and excitation of
harmonic components Mi. On this basis an additional
component of dynamic torque Md is introduced in the
pneumatic coupling. Thus pneumatic coupling will be in
this case loaded with load torque MS that causes the maximum twist angle φS:
MS = MN + Md,
(4)
The magnitude of the additional dynamic torque calculated from the equations of motion (5) can be described by
equation (6).
Fig.8. The Campbell diagram of the mechanical system with the applied
tangential pneumatic coupling at the constant pressure p = 100 ÷
700kPa.
The diagram shows the position of the critical speed,
depending on the natural speed frequencies. Those given
pneumatic couplings are nearly linear, natural speed frequencies are shown by the horizontal straight lines a, b, c,
d, e, f, g for the entire range of the gaseous medium pressure p = 100 ÷ 700 kPa. Based on the diagram it is possi-
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
ble to say that the pneumatic coupling is capable to operate at all pressures of the gaseous medium (p = 100 ÷
700 kPa).
On the other hand, in terms of the dynamic tuning we
conclude that the pneumatic coupling is suitable for the
given system in the pressure range p = 200 to 600 kPa.
This is due to the fact that at the beginning of the operation mode at p = 100 kPa there is a resonance with the first
harmonic component of the load torque speed n =
820 min-1, while at p = 700 kPa a resonance occurs also
with the first harmonic component but at the speed n =
1480 min-1.
Based on the above we shall in following focus on the
dynamic tuning characteristics of the realized system by
the tangential pneumatic coupling at the pressure range
p = 200 to 600 kPa.
Based on the figure of the Campbell diagram it is possible to say that a given coupling extends from the range
of operating speed harmonic series i = 2 ÷ 12 The critical
speed due to the main harmonic component (i = 3) at pressures p = 200, 300, 400, 500 and 600 kPa appears at the
speed nK = 330, 360, 405, 440 and 460 min-1. These values indicate that the realized mechanical system is welltuned with regard to the operating mode beginning. This
is confirmed by the separation margin η = i. n / N, which
for the investigated pressures has relatively high values of
η = 2,3 to 1,6 for the investigated pressures. At the same
time we can see in the figure that the harmonic series i = 1
extends in the range of the gaseous medium pressure p =
200 to 600 kPa into operating speed range (OSR). It follows that when using the pneumatic coupling there is a
resonance with a harmonic component at these pressures.
Specifically, for the pressure p = 200, 300, 400, 500 and
600 kPa resonances occur at speeds nK = 980, 1090, 1220,
1330 and 1430 min-1.
The tuning system realized by the pneumatic coupling
for one disabled cylinder with regard to the main (i = 3)
and secondary (i = 2, 1) harmonics within the operating
speed (n = 750 ÷ 1500min-1) is characterized in Fig. 9.
Fig. 9. The dependence of the dynamic component of the torque Md at
speeds in the range n = 0 ÷ 2000min-1 of the mechanical system on the
tangential pneumatic coupling application with constant pressure at
p = 100 ÷ 700 kPa.
74
The overall analysis shows that differential pneumatic
coupling can be applied in the torsional oscillating mechanical system with a range of speed only in a fault-free
case of work of the piston device. In case of faults caused
mainly by the piston device (unbalanced excitation of engine cylinders, one disabled cylinder) it is to use the linear coupling in mechanical systems with a range of unsuitable speed . The reason of this is that, in this case,
particularly lower harmonics cause increased amplitude of
the mechanical load across the system (Fig. 9).
The results indicate that the linear differential pneumatic coupling would be particularly suitable for the mechanical system operating with constant operating speed.
C. The results of the continuous tuning of the torsionally
oscillating mechanical system
The result of continuous tuning of the system realized
by a pneumatic tuner of the torsional oscillation type 4–
1/70–T–C is presented by the Campbell diagram in Fig.
10. In the figure there are represented eight waveforms of
the natural speed frequencies marked a, b, c, d, e, f, g, h,
corresponding to a constant twist angle of the pneumatic
tuner φK = 0,5 °; 1 °; 1,5 °; 2 °; 2,5 °; 3 °; 3,5 °; 4 ° and are
characterized by a broken line.
Based on the Campbell diagram it is possible to say that
critical speeds by φK = 0,5 ° and 1 ° from the main harmonic i = 3 of the load torque are in a sufficient distance
with regard to the start of the operating mode (n =
750 min-1) by the lowest pressure p = 100 kPa and also by
the highest pressure p = 700 kPa. This fact is confirmed
by the separation margin, which for that case has values in
the range η100 = 2,3 and η700 = 1,51. At the same time we
can see that within the operating speed range of the system, particularly for the speed n = 1480 min-1, a resonance is at the harmonic component series of i = 1 by φK
= 0,5° and 1° of the pneumatic tuner with the maximum
pressure value of the gaseous medium p = 700 kPa. Based
on the above it can be concluded that the constant twist
angles of the pneumatic tuner φK = 0,5 ° and 1 ° are not
suitable for the realized system.
By the minimum pressure value of the gaseous medium
of the pneumatic tuner p = 100 kPa with φK = 1,5 ° a
resonance occurs at the harmonic component i = 1 at the
operating speed n = 850 min-1 . With rising pressure up to
the maximum value no resonance is at the harmonic
component i = 1. For example at the maximum pressure
the separation margin for i = 1 has a value η = 1,34. It
indicates that the pneumatic coupling with φK = 1,5 ° is
appropriate for the realized system except the beginning
of the operating mode.
When using the pneumatic tuner with constant angles
φK = 2 °; 2,5 °; 3 °; 3,5 ° and 4 ° no resonance is within
the operating speed range of the realized system from any
harmonic components of the load torque.
The results of the torsional vibration magnitude of the
realized mechanical systems in the case of a disabled cylinder are shown in Fig.11. They are characterized by
courses of the dynamic torque amplitudes Md depending
on the operating speed in the range n = 0 ÷ 2000min-1.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
Fig. 10. The Campbell diagram of the realized mechanical system by use
of the pneumatic tuner of the torsional vibration type 4–1/70–T–C.
It results from the overall analysis in Fig. 11 that the
lowest dynamic loads of the mechanical system in the
range of the operating speed n = 750 ÷ 1500min-1 , are
obtained at constant twist angles of the pneumatic tuner
of torsional vibration φK = 2°; 2,5°; 3°; 3,5° and 4°. It is
caused by the fact that the pneumatic tuner in that case
acts as a highly flexible pneumatic coupling, thus coupling
with a relatively low torsional stiffness.
The results indicate that the pneumatic tuner of torsional vibration will be especially suitable for mechanical
systems working within a range of operating speed.
75
characteristic. The change of the flexible coupling characteristics, due to appropriate adaptation of its dynamic
properties to the system dynamics means to use a different element of the flexible coupling or using a different
flexible shaft coupling. In any case, it is not possible to
forget the fatigue and aging of flexible materials, which
finally have a major impact on the initial dynamic properties. Thus the unsteadiness of flexible coupling dynamic
properties caused by aging and fatigue of their flexible
elements and as well as the frequent failure rate of some
other elements of the system causes the detuning of the
tuned torsional oscillating mechanical system. In this case
its tuning element, the flexible coupling, has no possibility
to remove or reduce the increasing dangerous torsional
vibration.
Taking into account the given facts we propose to use
the pneumatic flexible shaft couplings developed by us in
order to reduce dangerous torsional vibration by optimal
tuning or rather optimal continuous tuning of torsional
oscillating mechanical systems. Based on the presented
results it is possible to say that presented differential
pneumatic coupling, as well as the pneumatic tuner of the
torsional vibration, fulfil all the requirements for their
application in torsional oscillating mechanical systems.
Based on the detailed analysis of the realized mechanical
system we can say that linear pneumatic couplings are
especially suitable for mechanical systems operating with
constant operating speed. On the other hand, the pneumatic tuners of torsional vibrations will fulfil all the requirements of mechanical systems within a range of operating speeds.
ACKNOWLEDGMENT
This paper was written in the framework of Grant Project VEGA: „ 1/0688/12 – Research and application of
universal regulation system in order to master the source
of mechanical systems excitation”.
REFERENCES
Fig.11. Courses of the dynamic torque amplitudes Md depending on the
operating speed in the range n = 0 ÷ 2000min-1 for the realized mechanical system with application of the pneumatic tuner of torsional vibration
V. CONCLUSION
Based on presented results we can say that negative impact of the dangerous torsional vibration is possible to
reduce by application of classical flexible couplings. On
this occasion it is necessary to note, that each linear or
nonlinear presently used flexible coupling has only one
[1] J. Homišin a kol., “Súčasné trendy optimalizácie strojov a zariadení”, C–Press Košice, 2006, ISBN 80-7099-834-2.
[2] J, Böhmer, “Einsatz elastischer Vulkan-Kupplungen mit linearer
und progressiver Drehfeder-charakteristik”, MTZ, 44/5, 1983.
[3] V. Zoul, V, “Torzní vibrace v pohonech a způsob jejich snižování”,
Praha, ČSVTS 1984.
[4] J. Homišin, J., “Methods of tuning torsionally oscillating mechanical systems using pneumatic tuners of torsional oscillations”,
Transactions of the TU of Košice, 3/4, England, 1993, pp. 415
[5] J. Homišin, J., “Mechanická sústava optimálne vyladená pneumatickou spojkou”, UV SR/5274/2009.
[6] J. Homišin, “Plynulo riadená mechanická sústava”. UV SR/ 5275/
2009.
[7] J. Homišin, “Pneumatická pružná hriadeľová spojka”. Patent č.
222411/86.
[8] J. Homišin, “Pneumatická pružná hriadeľová spojka s diferenčnými
členmi”. UV SR/5278/2009.
[9] J. Homišin, “Regulačný systém pre zabezpečenie plynulej zmeny
charakteristiky pneumatických spojok”. P ČSSR/259225/87.
[10] J. Homišin, “Regulačný systém pre realizáciu plynulého ladenia
mechanickej sústavy”. P SR/276927/92.
[11] J. Homišin, “Pneumatická pružná hriadeľová spojka so schopnosťou autoregulácie”. P ČSSR 278025/95.
[12] J. Homišin, M. Jurčo, “Aplication of differential pneumatic dutches
voith and without autoregulation in torsionally oscillating me-
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chanical systems”. The shock and vibration digest, 29/3, 1997,
USA, pp.44, (80%/20%).
[13] J. Homisin, M. Jurčo, “Application of differential pneumatic clutch
with an additional regulating system”. The Shock and Vibration
Digest, USA, 30/6, 1998, pp.490, (80%/20%).
[14] J. Homišin, “Dostrajanie ukadów mechanicznych drgajacych skretnie przy pomocy sprzegie pneumatycznych. Kompendium
wyników pracy naukowo-badawczych autora”. Bielsko-Biała, ATH,
2008, [106 p]. ISBN 978-83-60714-55-3.
[15] R. Grega, “Prezentácia výsledkov dynamickej torznej tuhosti
pneumatickej pružnej spojky s autoreguláciou na základe experimentálnych meraní”. Acta Mechanica Slovaca, 2/2002, ročník 6,
s. 29 – 34.
[16] P. Kaššay, M. Urbanský, “Úvod do problematiky prechodových
dejov v torzne mechanických sústavách”. Zborník 51. MVK
KČSaM, 2010. s. 130 – 136.
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[17] P. Kaššay, R. Grega, J. Krajňák, “Determination of objective function for extremal control of torsional oscillating mechanical system“. Transactions of the Universities of Košice. Nr. 3, 2009, pp.
17–20. ISSN 1335-2334.
[18] P. Kaššay, “Effect of Pneumatic Flexible Shaft Coupling on the
Size of Torsional Vibration“. 2 Międzynarodowa Konferencja Studentów oraz Młodych Naukowców. Bielsko-Biała, Wydawnictwo
naukowe Akademii techniczno-humanistycznejj, 2012 pp. 99‒104.
ISBN 978-83-63713-23-2.
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42 (3/4), 2000, pp. 127‒135, Zagreb, Croatia.
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s. 75–78.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
77
Experience with Torque Measurement
on Rotating Shaft
Zdeněk Folta1), Milena Hrudičková 2)
1)
VŠB-Technical University of Ostrava/Faculty of Mechanical Engineering, Ostrava, Czech Republic, e-mail:
zdenek.folta@vsb.cz
2)
VŠB-Technical University of Ostrava/Faculty of Mechanical Engineering, Ostrava, Czech Republic, e-mail:
milena.hrudickova@vsb.cz
Abstract — This article deals with problem of torque
measurement on the turning shaft and the signal
transmission to the measuring equipment. There is
described the experience with contact transmission and both
non-contact and high frequency transmission (with ESA
Messtechnik equipment). Also the examples of the practical
realization are presented.
Keywords — Torque measurement, telemetry, strain gauge
measurement, experiment
I. INTRODUCTION
Torque measurement on rotating shaft by the method of
strain gauge measurements, especially with the existing
equipment, brings usually two problems:
- the first problem is the installation of the strain gauge to
places giving representative data. These are usually
areas without mounting, the groove for tongue area
and similar discontinuities – simply such location on
the shaft, where the relative deformation is in
accordance with the torque value (under known
formulas);
- the second problem is the transmission of the measured
signal from the rotating shaft to the recording
apparatus.
Our article deals with these issues.
II. CONTACT TRANSMISSION
The contact transmission by means of rings and brushes
used to be the only option for the signal transmission in
the past. There were usually the direct signals from strain
gauges transmitted, that is why the apparatus had to be
reliable and with only very low transition resistance
values. Despite the use of high quality materials –
especially precious metals – such devices were not only
liable to failure but also unfit for measurements on the
existing equipment.
Contact transmission of signal is still used at present –
the Hottinger sensor is one of the examples (see Fig. 1 and
Fig. 2).
The torque measurement by means of the existing
equipment is very often required in practice. For this
purpose one very simple, cheap and at the same time
reliable signal transmission method was developed by
professor Dejl and his colleagues – the signal transmission
by means of rings and „wire“ brushes. There are rings
with peripheral groove mounted on the insulation lining.
The signal is then transmitted by a wire encircling the
rings which is tighten by means of springs or coils.
Fig. 1: Hottinger torque sensor
Fig. 2: Hottinger torque sensor - detail of the signal transmission
This system was used successfully in many applications
– for example for signal transmission of strain gauge data
of the industrial automatic washer drum and shaft (Fig. 3
and Fig 4), or articulated cardan shafts of a rolling mill
(Fig. 5 and Fig 6).
Fig. 3. Wired signal transmission on industrial automatic washer drum
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
78
gauge measurement on the automatic washer drum serves
as an example (Fig. 7 and Fig. 8).
Fig. 7. Some strain gauges placed on the automatic washer drum
Fig. 4. Wired and wireless transmission
on shaft of an industrial automatic washer drum
Fig. 8. Detail of strain gauges
Fig. 5. Torque measurements on the cardan shaft
of rolling mill vertical cylinders
III. TELEMETRY WITH DISC AERIAL
Modern electronics has brought a miniaturization and
also possibilities of non-contact signal transmission
telemetric systems. The telemetric system by ESA
Messtechnik GmbH, Munich is an example. Its principle
is described in Fig. 9, one of the practical realizations
(experimental stand – Faculty of Mechanical Engineering,
STU Bratislava) is in Fig. 10. The strain gauges are
connected in the module with strain gauge amplifier,
bridge charger and transmitting block. By means of a
fixed aerial and a rotational aerial system both the power
supply of strain gauge module and the signal transmission
from the strain gauge bridge to the evaluation unit are
secured.
Fig. 6. Torque measurements on the cardan shaft - detail
The application of this system depends on certain
experience; it is necessary to set the angle of wire
encircling, the tension of springs and also the dimensions
and material for the wire and the groove in a appropriate
ratio in accordance with the shaft diameter and its
peripheral velocity.
By means of this equipment it is possible to transmit
not only the torque measuring strain gauge signal, but also
signals from the strain gauge measuring any other
quantities or from quite different sensors. The strain
Fig. 9: Telemetry block diagram
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
Fig. 10: Torque sensoring on the experimental stand
79
Above described system is relatively simple as for
installation and usage; it is necessary to have space for
placing of the rotational and fixed aerials. Maximum
possible diameter of the rotational aerial depends on the
power supply unit output. The output 1 W is enough for
the diameter of 400 mm, the output 3 W is suitable for the
diameter of 750 mm.
Despite some problems with sufficient space this
equipment was installed into the gearbox of fork-lift truck
Jungheinrich Hamburg (Fig. 13) and performed
measurements during test operation with the option of
switching between strain gauges for the tensile-pressure
strength and strain gauges for torque measurements on
the conical gear pinion (Fig. 14).
The example of such telemetry usage in practice may be
torque measurement of a conveyor belt drive. The space
for the sensor locating was very limited and confined (see
Fig. 11) and besides this fact it was necessary to place the
strain gauges near the shaft mounting, which required
additional specification of the scale between the
measuring voltage and torque by means of FEM (Fig. 12).
Fig. 13: Location of aerials and strain gauges
in the gearbox of fork-lift truck
Fig. 11: Torque measurement of a conveyor belt drive
Fig. 14: Location of the amplifier and connecting bar for strain gauges
Fig. 12: Calculating of conversion scale
„measuring voltage – torque“ by means of FEM
This equipment can be used (after applying some
suitable insulation measures) even in environment with
running water – we have tested it during measurements on
a running mill in Budapest (Fig. 15).
The conditions of cylinders fixing unfortunately caused
such shifts of the cardan shaft both in horizontal and
vertical directions that the fixed aerial could not be
installed properly – so the data obtained were only
informative.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
Fig. 15: Telemetry on the cardan shaft of a rolling mill in Budapest
80
V. TELEMETRY WITH HIGH FREQUENCY TRANSMISSION
This telemetry is used in more difficult conditions. The
transmission is realized by means of high frequency of
433 MHz. The strain gauge (or other sensor) signal is
filtered for values over 1 kHz (antialiasing) and digitized
by the 12-bit converter. The digital signal is then
transmitted at a distance of minimum 50 m (in free space),
then decoded – and the final output is an analogue signal
again.
The block diagram of this telemetry is in Fig. 18, its
realization in Fig. 19. This system is now being tested and
will be used for measurements on an experimental car in
Josef Božek Competence Centre for Automotive Industry.
IV. TELEMETRY WITH AERIAL RIGHT ON THE SHAFT
If there is not enough space in the vicinity of the shaft,
it is possible to install the rotational aerial right on the
shaft. In our case this solution was used during
measurements of torque on Škoda Fabia half-axles.
The rotational aerial is installed right on the shaft (see
Fig. 16 and Fig. 17)) and the fixed aerial is in this case
closer to the shaft.
Fig. 16. Torque sensoring on a car half-axle
Fig. 18: Block diagram of the high frequency telemetry
Fig. 17. Torque sensoring on a car half-axle
The used aerial incorporates also the revolution sensor.
At each shaft revolution there is a short rectangular pulse
on the output connector. Then it is possible to calculate
the current revolutions from the pulse frequency – and in
this case also the vehicle velocity (speed).
Fig. 19. High frequency telemetric system real appearance
The device can be also used for the signal scanning
from moving objects. An example is the measurement of
the go-karts frame loading, which realized a student of our
university as part of his thesis.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
81
ACKNOWLEDGMENT
This research has been realized using the support of
Technological Agency, Czech Republic, programme
Centres of Competence, project # TE01020020 Josef
Božek Competence Centre for Automotive Industry. This
support is gratefully acknowledged.
REFERENCES
Fig. 20 –Telemetry on the go-car frame
Fig. 21 – Receiving point
CONCLUSION
Despite this article looks rather as an advertising leaflet
for ESA Messtechnik products, its aim is to inform
colleagues in our branch. Especially those of them who
need to measure the torque or other tensile strength of a
rotating equipment – we would like to inform them of
some options we have experience with.
While the strain gauge system is based on the
evaluation of the voltage on the Wheatstone bridge
measurement diagonal, it is possible to connect any other
sensor (which output is voltage) to the amplifier - and then
transmit this signal to the evaluation unit. For example
temperature, force, vibrations and other quantities can be
transmitted in this way.
[1] Dejl, Z., Folta, Z., Zieschang, T. Zkrácené životnostní zkoušky
kuželočelní převodovky vysokozdvižného vozíku Retrak.
(Shortened life tests bevel gear forklift reach truck Retrak) .In
Proceedings konference Trwalošč elementow i wezlow
konstrukczyjnych maszyn górniczych. Gliwice-Ustoň: TEMAG,
2004. ISBN 83-917265-4-1.
[2] Folta, Z., Polák, J., Nečas, J., Vrána, V. Ověření účinnosti pohonu
pásového dopravníku. (Verification of the effectiveness of belt
conveyor drive) Liberec: TU Liberec, 2005. International
conference of Departments of Machine Parts and Mechanisms
anthology 2005.
[3] Dejl, Z., Folta, Z. Měření zátěžného spektra automobilové
převodovky. (Measurement of the automotive gearbox loading
spektra.) Bratislava: STU Bratislava, 2007.
International
conference of Departments of Machine Parts and Mechanisms
anthology 2007. ISBN 978-80-227-2708-2.
[4] Folta, Z., Hrudičková, M. Zpracování zátěžných spekter převodovky
osobního automobilu. (Processing of loading spectra of a car
gearbox.) Košice: TU v Košiciach, 2010. International conference
of Departments of Machine Parts and Mechanisms 2010. ISBN
978-80-970-294-1-8.
[5] Dejl, Z., Němček, M.: Research results in the field of automotive
transmissions on dept. of Machine Parts and Mechanisms VŠB –
Technical University of Ostrava. In: Proceedings of abstracts of
Conference „Mechanical Engineering 2006, Bratislava, Slovak
University of Technology, p. 46, ISBN 80-227-2513-7.
[6] ESA Messtechnik GmbH Munich, Technical documentation.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
82
Performance of Steady-State Voltage Stability
Analysis in MATLAB Environment
Jan Veleba 1)
1)
University of West Bohemia in Pilsen, Regional Innovation Centre of Electrical Engineering, Pilsen, Czech Republic,
e-mail: jveleba@rice.zcu.cz
Abstract — In recent years, electric power systems have
been often operated close to their working limits due to
increased power consumptions and
installation of
renewable power sources. This situation poses a serious
threat to stable network operation and control. Therefore,
voltage stability is currently one of key topics worldwide for
preventing related black-out scenarios. In this paper,
modelling and simulations of steady-state stability problems
in MATLAB environment are performed using authordeveloped computational tool implementing
both
conventional and more advanced numerical approaches.
Their performance is compared with the Simulink-based
library Power System Analysis Toolbox (PSAT) in terms of
solution accuracy, CPU time and possible limitations.
Keywords — Steady-state voltage stability, continuation load
flow analysis, predictor-corrector method, voltage stability
margin, voltage-power sensitivity, Power System Analysis
Toolbox
I. INTRODUCTION
Steady-state voltage stability is defined as the capability
of the system to withstand a small disturbance (e.g. fault
occurrence, small change in parameters, topology
modification, etc.) without abandoning a stable operating
point [1]-[5]. Voltage stability problems are generally
bound with long "electrical" distances between reactive
power sources and loads, low source voltages, severe
changes in the system topology and low level of var
compensation. However, this does not strictly mean that
voltage instability is directly connected only with low
voltage scenarios. Voltage collapse can arise even during
normal operating conditions (e.g. for voltages above
nominal values). Moreover, variety of practical situations
can eventually lead to voltage collapse, e.g. tripping of a
parallel connected line during the fault, reaching the var
limit of a generator or a synchronous condenser, restoring
low supply voltage in induction motors after the fault. All
these cause the reduction of delivered reactive power for
supporting bus voltages followed by increases of branch
currents and further voltage drops to even lower reactive
power flow or line tripping until the voltage collapse
occurs. This entire process may occur in a rather large
time frame from seconds to tens of minutes.
To prevent voltage collapse scenarios, several types of
compensation devices are massively used worldwide both shunt capacitors/inductors, series capacitors, SVCs,
synchronous condensers, STATCOMs, etc. To reduce
voltage profiles (in case of low demand), var
consumptions must be increased by switching in shunt
reactors, disconnecting cable lines (if possible), reducing
voltage-independent MVAr output from generators and
synchronous condensers, etc. To increase bus voltages,
opposite corrective actions are to be taken. These include
reconfigurations (connecting parallel lines / transformers /
cables), power transfer limitations and activations of new
generating units at most critical network areas.
Furthermore, the voltage load shedding of low-priority
loads (usually by 5, 10 or 20 % in total) is usually realized
at subtransmission substations using undervoltage relays.
These relays work similarly as on-load tap-changing
(OLTC) transformers. They are activated by long-term
voltage dips (in region between 0.8 and 0.9 pu) and as the
result, they trip the load feeders - typically in steps of 1 to
2 % of the load at any given time (with time delays of 1-2
minutes after the voltage dip). The larger voltage dip, the
faster and larger response of the relay [2].
Low voltage profiles are usually averted by actions of
OLTC transformers. However, each tap position
corresponds to an increase of the load which eventually
leads to higher branch losses and further voltage drops
[1]-[2],[4]. Therefore, OLTC transformers should be
blocked during low voltage stability scenarios. Negative
effects of OLTC actions during low voltage conditions are
presented in many studies with voltage stability margin
calculation from synchrophasor measurements [6]-[7].
This paper is organized as follows. Chapters II and III
describe conventional Cycled Newton-Raphson (N-R) and
more robust Continuation Load Flow (CLF) methods for
the voltage stability analysis, respectively. Independent
tool - Power System Analysis Toolbox (PSAT) - is briefly
introduced in Chapter IV. In Chapter V, key properties of
both of the author-developed codes are discussed. Chapters
VI and VII show the results of individual approaches when
solving voltage stability of a broad variety of test power
systems. Chapter VIII closes the paper with some
concluding remarks and the evaluation of each technique
applied.
II. CONVENTIONAL NUMERICAL CALCULATION OF THE
VOLTAGE STABILITY PROBLEM
When increasing the loading (or loadability factor λ) of
the system, its bus voltages slowly decrease due to the
lack of reactive power. At the critical point (called
singular or bifurcation), characterized by maximum
loadability factor λmax and critical bus voltages, the system
starts to be unstable and voltage collapse appears (system
black-out). From this point on, only lower loading with
low voltage values leads to the solution. The dependence
between bus voltage magnitudes and λ is graphically
represented by the V-P curve (also referred to as the nose
curve). Unfortunately, the current (so-called base-case)
position of the system operating point on the V-P curve is
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
not known along with its distance from the voltage
collapse (so-called voltage stability margin). Thus,
location of the singular point must be found during the
voltage stability analysis.
Note: Values of λmax and critical voltages are rather
theoretical since they do not reflect voltage/flow limits of
network buses/branches. When incorporating these
practical restrictions, the real maximum loadability λmax*
can be found (i.e. the maximum value for keeping all
network buses and branch loadings within limits).
Traditional approach for finding the maximum system
loadability is to apply the standard N-R method [8] for the
base-case load flow calculation (i.e. for λ = 1.0). When
obtaining current position on the V-P curve, network
loading (i.e. loads/generations in selected network buses)
is increased in defined manner by a certain step and the
load flow is computed repetitively along with a new
position on the V-P curve. This process continues in an
infinite loop until the singular point is reached. However,
total number of iterations in each V-P step is gradually
increasing so that when close to the singular point, the NR method fails to converge, i.e. no solution is provided.
This relates to the fact, that Jacobian J becomes singular
(i.e. det J ≈ 0) and its inverse matrix cannot be computed
for successful numerical convergence.
For speeding up the calculation, a variable step change
is applied. Usually, a single default step value is used.
When obtaining the divergence of the N-R method, the
step size is simply divided by two and the calculation for
the current V-P point is repeated until the convergence is
achieved. When the current step size value reaches the
pre-set minimum value, the calculation is stopped. Despite
of the relatively simple procedure, the Cycled N-R method
enables the completion of the stable V-P part only. The
unstable part including the singular point cannot be
examined. Also, high CPU requirements prevent this
method from being employed for larger power systems.
In this paper, the Cycled N-R algorithm was developed
and further tested on wide range of test power systems.
83
for the selected CP in each network bus k at the current
point on the V-P curve. Remaining elements in (1) are the
newly computed Jacobian J and step size σ of the CP.
The tangent predictor is relatively slow, anyway shows
good behaviour especially in steep parts of the V-P curve.
Unlike the tangent predictor, secant predictor is simpler,
computationally faster and behaves well in flat parts of the
V-P curve. In steep parts (i.e. close to the singular point
and at sharp corners when a generator exceeds its var
limit) it computes new predictions too far from the exact
solution. This may eventually lead to serious convergence
problems in the next corrector step. Therefore, the tangent
predictor is more recommended to be applied.
The corrector is a standard N-R algorithm for correcting
state variables from the predictor step to satisfy load flow
equations. Due to one extra parameter λ, additional
condition (2) must be included for keeping the value of
the CP constant in the current corrector step. This
condition makes the final set of equations non-singular
even at the bifurcation point.
λ
x k − x kpredicted = 0 , x = 
V
if CP is λ
(2)
if CP is V
Difference between both types of predictors and the
entire process of the predictor/corrector algorithm is
graphically demonstrated in Fig. 1. Horizontal/vertical
corrections are performed with respect to the chosen CP
type.
III. CONTINUATION LOAD FLOW ANALYSIS
The CLF analysis [1],[9] suitably modifies conventional
load flow equations to become stable also in the
bifurcation point and therefore being capable of drawing
both upper/lower parts of the V-P curve. It uses a two-step
predictor/corrector algorithm along with the new unknown
state variable called continuation parameter (CP).
The predictor (1) is a tangent extrapolation of the
current operation point estimating approximate position of
the new point on the V-P curve.
−1
θ 
V 
 
 λ 
predicted
M


0 

J M K   
θ 0 

M
   (1)
= V0  + σ 
M

 0 
 λ0 
L
L
L
L

 1

  
ek
Vector K contains base-case power generations and
loads. Variables θ0, V0, λ0 define the system state from the
previous corrector step. The vector ek is filled with zeros
and certain modifications (see [1],[9]) are implemented
Fig. 1. Predictor/Corrector Mechanism for the CLF Analysis [10].
As the CP, state variable with the highest rate of change
must be chosen (i.e. λ and V in flat and steep parts of the
V-P curve, respectively). When the process starts
diverging, parameter σ must be halved or parameter CP
must be switched from λ to V.
The step size should be carefully increased to speed-up
the calculation when far from the singular point or
decreased to avoid convergence problems when close to
the peak. The step size modification based on the current
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
position on the V-P curve (i.e. as a function of the line
slope for previous two corrected points on the V-P curve)
is recommended in [11]. This approach belongs to socalled rule-based or adaptive step size control algorithms.
In [10], several voltage stability margin indices
(VSMIi, VSMIik) are presented along with relative var
reserve coefficient and voltage-load sensitivity factors
(VSFi) for comprehensive voltage stability analysis and
location of weak or sensitive system buses/branches/areas.
In these regions, preventive or remedial actions should be
taken. Procedures for allocating individual compensation
devices and possible effects are also discussed.
The CLF analysis still remains very popular for highspeed solving of voltage stability studies. Due to its
reliable numerical behaviour, it is often included into the
N-R method providing stable solutions even for illconditioned load flow cases. Moreover, it is applied in
foreign control centres for N-1 on-/off-line contingency
studies with frequencies of 5 and 60 minutes [2],
respectively.
IV. POWER SYSTEM ANALYSIS TOOLBOX (PSAT)
PSAT [12] is a Simulink-based open-source library for
electric power system analyses and simulations. It is
distributed via the General Public License (GPL), its
download and use is free of charge. However, there is no
warranty that the Toolbox will provide correct and
accurate results. All corrections and possible repairs or
improvements are to be done on the customer side.
It contains the tools for Power Flow (busbars, lines,
two-/three-winding transformers, slack bus(es), shunt
admittances, etc.), CLF and OPF data (power
supply/demand bids and limits, generator power reserves
and ramping data), Small Signal Stability Analysis and
Time Domain Simulations. Moreover, line faults and
breakers, various load types, machines, controls, OLTC
transformers, FACTS and other can be also modelled.
User defined device models can be added as well.
All studies must be formulated for one-line network
diagram only - either in input data *.m file in required
format or in graphical *.mdl file, where the schema is
manually drawn. For the former option, input data
conversions from and to various common formats (PSS/E,
DIgSILENT, IEEE cdf, NEPLAN, PowerWorld and more
others) are available.
When compared to another MATLAB-based opensource tool MATPOWER [13], PSAT is more efficient
and highly advanced by providing more analyses, problem
variations, possible outputs and other useful features in its
user-friendly graphical interface. MATPOWER does not
support most of advanced network devices, entirely omits
CLF analysis and has no graphical user interface or
graphical network construction ability. Also, it does not
consider var limits in PV buses. Incorrect interpretation of
reactive power branch losses can be also observed.
V. PROPERTIES OF AUTHOR-DEVELOPED CODES IN
MATLAB ENVIRONMENT
Both Cycled N-R and CLF procedures were developed
in MATLAB environment for providing fundamental
examination of medium-sized and larger power systems in
terms of steady-state voltage stability. Several key aspects
of these codes are discussed below.
84
1] Predictor: Despite of computationally more complex
algorithm, the tangent predictor was used for finding
reliable estimations of new V-P points especially around
the singular point. It is applied in CLF algorithm only.
2] Corrector: First, a corrector step is used at the start of
the CLF program to find the base-case point for further
calculations. Due to possible weak numerical stability at
this point (for badly-scaled power systems), the One-Shot
Fast-Decoupled (OSFD) procedure is implemented to the
standard N-R method for providing more stable solutions
and thus preventing numerical divergence. Moreover,
voltage truncation (SUT algorithm) is also included into
the state update process at every N-R's iteration. Both of
these stability approaches were introduced in [14] and
further tuned and tested in [15]. Both were also applied to
the Cycled N-R algorithm to increase the loading range,
for which the stable load flow solutions can be obtained
(i.e. closer proximity to the singular point can be reached).
3] Step size: Largest-load PQ network bus is chosen for
computing the angle α between the horizontal and the line
interconnecting two adjacent V-P points. Based on this,
the step size evaluation function (3) is applied - see Fig. 2.
σL


σ=
σU
 A / sin 2 α + B

for
α ≥ π/ 8
for
α ≤ π/ 32
otherwise
(3)
The upper and lower step limit constants σU and σL
define the step size for the flat part of the V-P curve and
for close vicinity to the singular point, respectively.
Fig. 2. Step size evaluation function [10].
For the Cycled N-R algorithm, this is a rather too
complex concept of the step size control. Therefore, only a
single step size is chosen at the start and a simple stepcutting technique (dividing by 2) is applied in case of
divergence.
4] Ending criterion: Only stable part of the V-P curve
(incl. exact singular point calculation) is computed by the
CLF code. Thus, if the computed value of λ begins to
decrease, the process is stopped. For the Cycled N-R code,
the calculation is terminated when the step size falls below
a certain small value (e.g. 1×10-8). For each load flow
case, maximum number of iterations and permitted
tolerance for convergence is set to 20 and 1×10-8,
respectively.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
5] Calculation speed and accuracy: For excessively
accurate voltage stability solutions, values of σU and σL
2.5×10-2 and 6.25×10-4 are used in the CLF algorithm.
Rather compromise values of 5×10-2 and 1×10-2 are also
used to obtain fast and fairly accurate solutions for any of
tested power systems. For the Cycled N-R algorithm,
initial step size of 2.5×10-2 is chosen as sufficient.
6] Code versatility: Both Cycled N-R and CLF
procedures are programmed so that the user directly
specifies an arbitrary group of network buses for an
load/generation increase. From this set of buses, only
those non-slack buses with non-zero active power
loads/generations are activated for the analysis. In all
studies performed, a load/generation increase in the entire
network (i.e. all network buses selected) is considered.
Two scenarios can be activated by the user. a) L
scenario increases both P/Q loads in selected PQ/PV buses
with a constant power factor (i.e. with identical increase
rate). b) L+G scenario increases both P/Q loads in selected
PQ/PV buses and P generations in selected PV buses (with
identical increase rate).
7] Var limits: In both approaches, bus-type switching
logics are applied to iteratively computed reactive powers
QGi in PV buses when exceeding the var limit (4) or to
relevant bus voltages when returning the vars back inside
the permitted var region (5). Symbol (p) denotes the current
iteration number.
if QGi( p ) > QGi max
Q
QGi( p ) =  Gi max
( p)
QGi min if QGi < QGi min
Vi = Vi sp
(4)
QGi( p ) = QGi max AND Vi > Vi sp 


if 
OR
 (5)
(
p
)
sp
Q = Q

Gi min AND Vi < Vi 
 Gi
The terms QGimax and QGimin are the upper and lower var
limits, the term Visp determines the specified value of the
voltage magnitude for each PV bus.
8] Code limitations: a) With increased loading,
lower/upper var limits in PV buses should not be fixed but
variable proportionally to the generated active power. In
both codes, constant var limits are used for more
pessimistic V/Q control. b) Only identical increase rate is
applied. However, implementing user-defined increase
rates for each load/generation would not pose any serious
problem.
9] Sparse programming: Sparsity techniques along with
smart vector/matrix programming are used in both Cycled
N-R and CLF codes to significantly decrease the CPU
time needed for each load flow case.
10] Outputs: Theoretical value of λmax and V-λ data
outputs for V-P curves are computed and stored or
graphically projected. Respective values of λ for switching
some of PV buses permanently to PQ are also recorded.
Voltage and power flow limits were not considered for the
evaluation of the real maximum loadability λmax*.
85
VI. TESTING OF CYCLED N-R AND CLF ALGORITHMS FOR
SOLVING VOLTAGE STABILITY LOAD FLOW PROBLEMS
Total number of 50 test power systems between 3 and
734 buses were analyzed using developed Cycled N-R and
CLF algorithms in the MATLAB environment. Identical
increase rate was applied to all network buses (before
filtering those with non-zero active power loads or
generations). For both L and L+G scenarios, only stable
part of the V-P curve was calculated with included var
limits. Settings of both codes are as introduced in Chapter
V, paragraphs 4] and 5]. In Tab. I., voltage stability
solutions of several test cases are shown. Presented results
contain the maximum loadability, numbers of stable V-P
points and CPU times in seconds needed.
For each case, the first two rows show the outputs of
the CLF code for excessive accuracy and compromise
accuracy, respectively. For comparison purposes, the third
row provides the results of the Cycled N-R code.
TABLE I.
VOLTAGE STABILITY SOLUTIONS USING CYCLED N-R AND CLF
ALGORITHMS - L AND L+G SCENARIOS
Scenario L
λmax [-]
pts
CPU [s]
1.302632 331
0.5616
IEEE9II 1.302632
27
0.1404
1.302632
23
0.4056
1.760331 658
1.2012
IEEE14 1.760331
87
0.2340
1.760331
43
0.5460
1.536905 854
1.9500
IEEE30 1.536905
88
0.2808
1.536905
37
0.6396
1.406778 891
2.9016
IEEE57 1.406778 229
0.6864
1.406778
27
0.8112
1.079959 1640 12.9169
IEEE162 1.079960 464
3.1044
1.079960
13
1.7628
1.024573 8457 103.8655
IEEE300 1.024573 529
7.0044
1.024573
16
2.4180
3.104162 139
4.5864
EPS734II 3.104083
46
1.8720
3.104162
96
8.2369
Case
Scenario L+G
λmax [-]
pts
CPU [s]
1.162053 215
0.3900
1.162052 24
0.1248
1.162053 20
0.4212
1.777995 506
0.9360
1.777995 59
0.2028
1.777995 45
0.6396
1.546751 726
1.6536
1.546752 124
0.4212
1.546751 37
0.6552
1.616845 399
1.3884
1.616845 57
0.2652
1.616845 37
0.8112
1.138996 1185 9.3913
1.138996 65
0.8112
1.138996 16
1.8408
1.058820 311
4.0092
1.058819 94
1.4508
1.058820 17
2.5584
3.104162 139
4.8360
3.104083 46
1.8408
3.104162 96
8.1745
As can be seen, exact solutions of maximum loadability
were obtained for both of tested methods and each of the
three accuracy settings. The first setting was definitely too
much focused on producing exact results. Therefore,
numbers of V-P points and CPU times were pushed often
above 200 and 1 second, respectively. When using fair
compromise setting, the maximum error for λmax from all
50 test power systems was only 0.0185 percent, while
numbers of points and CPU times were decreased on
average by 75.27 percent and 64.11 percent, respectively.
The Cycled N-R code obtains highly accurate results in
terms of solution accuracy. In majority of cases, it
provides even better solutions than CLF algorithm with
compromise accuracy. Surprisingly, it always computes
slightly higher maximum loadability values than by the
high-accurate CLF code. This seems to be one visible
drawback of the Cycled N-R method. Only low numbers
of V-P points are needed for reaching close proximity to
the singular point. These numbers are well comparable to
those needed for the compromise CLF code.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
Unfortunately, each divergence case (between 22 and 28)
significantly delays the entire computation process of the
Cycled N-R method. Therefore, the Cycled N-R code
suffers from being extremely time-dependent on
computing of each V-P point. When compared with the
compromise CLF code, the CPU time needed by the
Cycled N-R method is on average about 167 % higher.
Therefore, the compromise CLF code seems to be the
best method for providing fast and highly accurate voltage
stability solutions.
Stable V-P curves of the IEEE 30-bus power system
(L+G scenario) are computed by both Cycled N-R and
CLF methods and shown in Figs. 3 and 4, respectively.
For the CLF method, the V-P curves are extended to
demonstrate numerical stability of the CLF algorithm
around the singular point. Extension of V-P curves in the
unstable region is provided for 0.97×λmax < λ < λmax.
Voltage Magnitude [pu]
1
0.9
0.8
0.7
0.6
0.5
0.4
1
1.1
1.2
1.3
1.4
Loadability Factor λ [-]
1.5
Fig. 3. V-P curves for the IEEE 30-bus system (Cycled N-R method).
86
VII. TESTING OF PSAT FOR SOLVING VOLTAGE STABILITY
LOAD FLOW PROBLEMS
Before solving voltage stability in PSAT, the load flow
analysis of a system must be performed. Therefore, large
number of load flow studies is solved using PSAT to
detect any of its possible weaknesses. Results were
compared with the author-developed N-R code in
MATLAB.
Despite of unconstrained network size to be solved,
several limitations of PSAT were found during the testing
stage. 1] Inefficient PV-PQ bus type switching logic is
applied. Probably, reverse switching logic (5) is not used
and the need for convergence is requested to activate
forward switching logic (4). As a result, unnecessarily
more PV buses are being switched permanently to PQ.
Furthermore, the switching logic completely fails to
switch PV buses to PQ for larger systems with high
numbers of PV buses. 2] Nominal voltages must be
defined in the input data file or the error message
'Divergence - Singular Jacobian' is obtained during the
simulation. This seems to be entirely illogical since
nominal voltages should not be necessary for the 'in per
units defined' problem. 3] It seems that no advanced
stability techniques are applied for the N-R method in
PSAT because of severe numerical oscillations appearing
in several studies. 4] PSAT intentionally neglects
transformer susceptances and thus causes errors in final
load flow results. A column for shunt susceptances is
available for power lines only. For transformers, this
column is filled with zeros by default.
Under these limitations, load flow results show very
good congruity between the author-developed N-R
method and PSAT. Higher total numbers of iterations are
needed by PSAT due to missing stability technique(s).
Also, CPU times are higher in PSAT due to combining the
codes with other analyses and related tool features.
As an example, the load flow and voltage stability
analysis of the IEEE 14-bus system is done by PSAT
(Figs. 5-9).
Voltage Magnitude [pu]
1
0.9
0.8
0.7
0.6
0.5
0.4
1
1.1
1.2
1.3
1.4
Loadability Factor λ [-]
1.5
Fig. 4. Extended V-P curves for the IEEE 30-bus system (CLF method).
As Tab. I. indicates, applied version of the CLF method
is still not applicable for real-time voltage stability
monitoring, but it can be useful for off-line reliability,
evaluation or planning studies of even larger networks.
Fig. 5. GUI in PSAT for the load flow analysis of IEEE 14-bus system.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
87
Fig. 8. Settings for the CLF analysis when solving the IEEE 14-bus
power system.
Fig. 6. Final voltage magnitudes of IEEE 14-bus power system in PSAT.
The CLF algorithm in PSAT is defined so, that power
increases are realized by adding a power increment
(loadability factor multiplied by the increase rate) to the
base-case loading, i.e. initial λ is zero. In the authordeveloped Cycled N-R and CLF codes, power increases
are performed by multiplying the base-case loading with
λ. Therefore, the maximum loadability in PSAT must be
increased by unity when comparing both codes.
Fig. 7. Final voltage angles of IEEE 14-bus power system in PSAT.
For voltage stability studies, PSAT contains the
advanced CLF algorithm, which is combined with
contingency and OPF analyses. Load flow data are
extended by two matrices specifying the sets of PQ and
PV buses, where the loads and generations are to be
increased (different increase rates are possible). Thus,
various loading scenarios of the system can be modelled.
The CLF code is then started via a specialized window
(Fig. 8). Calculation can be adjusted by the user for better
computational performance - e.g. by setting a more
suitable step size, maximum number of V-P points or by
checking the option for controlling voltage, flow or var
limits. PSAT offers two CLF methods - perpendicular
intersection (PI, as in Fig. 1) and local parametrization
(LP). Three stopping criteria are available: The complete
Nose Curve (computing both stable/unstable parts of the
V-P curve), Stop at Bifurcation (when singular point
exceeded) and Stop at Limit (when voltage/flow/point
limit hit).
Fig. 9. Nose curves for all network buses of the IEEE 14-bus test system
in PSAT.
In Tab. II., voltage stability results for medium-sized
IEEE test systems are provided by the author-developed
Cycled N-R and compromise CLF codes. These outputs
are compared to those obtained by PSAT - see Tab. III. In
PSAT, both of the CLF modes were tested (i.e. PI with
step 0.025 and LP with default step 0.5).
TABLE II.
VOLTAGE STABILITY ANALYSIS OF MEDIUM-SIZED IEEE TEST SYSTEMS
(CYCLED N-R VS. COMPROMISE CLF)
L+G
Cycled N-R code
Compromise CLF code
Case
λmax [-]
pts
CPU [s]
λmax [-]
pts
CPU [s]
IEEE9
2.485393
74
0.5460
2.485382
84
0.2964
IEEE13
4.400579
148
0.6708
4.400577 112
0.3120
IEEE14
4.060253
137
0.8268
4.060252
92
0.3276
IEEE24
2.279398
61
0.6396
2.279398
58
0.2496
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
IEEE30
2.958815
88
0.8112
2.958814
57
0.2964
0.3432
IEEE35
2.888962
91
0.6864
2.888950 107
IEEE39
1.999203
57
0.7644
1.999202
30
0.2184
IEEE57
1.892091
47
0.8892
1.892089
92
0.4836
IEEE118
3.187128
100
1.6536
3.187128
66
0.5772
TABLE III.
VOLTAGE STABILITY ANALYSIS OF MEDIUM-SIZED IEEE TEST SYSTEMS
(PSAT - PI VS LP MODE)
L+G
PSAT - PI mode
88
ACKNOWLEDGMENT
This paper has been supported by the European
Regional Development Fund and the Ministry of
Education, Youth and Sports of the Czech Republic under
the Regional Innovation Centre for Electrical Engineering
(RICE), project No. CZ.1.05/2.1.00/03.0094. This work
has been also sponsored by Technology Agency of the
Czech Republic (TACR), project No. TA01020865 and
by student science project SGS-2012-047.
PSAT - LP mode
Case
λmax [-]
pts
CPU [s]
λmax [-]
IEEE9
2.481220
7
0.2093
2.482000
13
0.3241
IEEE13
4.390420
13
0.3292
4.399570
20
0.4832
IEEE14
4.060100
18
0.4098
4.059420
19
0.4939
IEEE24
2.277550
10
0.2600
2.278670
16
0.4313
IEEE30
2.958550
16
0.8761
2.958250
20
1.5023
IEEE35
2.872940
16
1.1242
2.878420
10
0.2940
IEEE39
1.999110
11
0.2932
1.997840
12
0.3692
pts CPU [s]
IEEE57
1.891920
12
0.9089
1.892090
26
3.9090
IEEE118
3.187100
613 19.1693
3.187120
82
19.7464
Theoretical values of λmax, numbers of stable V-P points
and CPU times in seconds are provided for comparison.
For all voltage stability studies in PSAT, identical power
increase rates were considered. Only the L+G scenario
was examined, logics for var limits were deactivated.
Both of PSAT modes showed only average accuracy
with satisfiable numbers of V-P points and rather lower
computational speed. The LP mode was computationally
more time-consuming, but needed lower numbers of V-P
points and usually provided more accurate results. The
compromise CLF code provided the best combination of
solution accuracy and CPU time requirements in each of
the cases. Although higher numbers of V-P points were
needed, CPU times were still significantly smaller than
those in PSAT due to optimized sparse programming
applied. Identical conclusions can be made when mutually
comparing CLF and Cycled N-R codes.
VIII. CONCLUSION
For solving voltage stability problems, both the Cycled
N-R and CLF codes were programmed and
comprehensively tested on a broad range of test power
systems in MATLAB environment. Various stability
techniques, step size approaches and numerical settings
were applied and used to upgrade their performance in
order to find the algorithm with fair compromise between
calculation speed and solution accuracy. The results were
compared with outputs obtained from PSAT. The studies
imply that the best technique (i.e. best combination of
precision level and CPU requirements) is the CLF
algorithm with compromise step size settings programmed
by Author in MATLAB. However, final technique can be
applicable in practice only for off-line planning and
development studies of electric power systems. For realtime evaluations of system's voltage stability, a more
robust algorithm with minimized numbers of stable V-P
points must be developed. Therefore, follow-up research
activities will be concentrated especially on this area of
interest.
REFERENCES
[1] P. Kundur, Power System Stability and Control, McGraw-Hill,
1994.
[2] C. Canizares, A.J. Conejo and A.G. Exposito, Electric Energy
Systems: Analysis and Operation, CRC Press, 2008.
[3] V. Ajjarapu, Computational Techniques for Voltage Stability
Assessment and Control, Springer, 2006.
[4] I. Dobson, T.V. Cutsem, C. Vournas, C.L. DeMarco, M.
Venkatasubramanian, T. Overbye and C.A. Canizares, “Voltage
Stability Assessment: Concepts, Practices and Tools - Chapter 2,”
Power System Stability Subcomittee Special Publication, IEEE
Power Engineering Society, 2002.
[5] J.H. Chow, F.F. Wu and J.A. Momoh, Applied Mathematics for
Restructured Electric Power Systems - Optimization, Control and
Computational Intelligence, Springer, 2005.
[6] I. Šmon, M. Pantoš and F. Gubina, “An improved voltage-collapse
protection algorithm based on local phasors,” Electric Power
Systems Research 78 (2008), pp. 434-440, 2008.
[7] Y. Gong and N. Schulz, “Synchrophasor-Based Real-Time Voltage
Stability Index,” Proceedings of PSCE conference, pp. 1029-1036,
2006.
[8] J.J. Grainger and W.D. Stevenson, Power System Analysis,
McGraw-Hill, 1994.
[9] M. Crow, Computational Methods for Electric Power Systems,
CRC Press, 2002.
[10] J. Veleba, “Application of Continuation Load Flow Analysis for
Voltage Collapse Prevention,” Journal Acta Technica, vol. 57, pp.
143-163, 2012.
[11] P. Zhu, “Performance Investigation of Voltage Stability Analysis
Methods,” PhD. thesis, Brunel University of West London, 2008.
[12] Homepage
of
PSAT.
[Online].
Available:
http://www3.uclm.es/profesorado/federico.milano/psat.htm.
[Accessed: 3 Mar. 2013].
[13] Homepage
of
MATPOWER.
[Online].
Available:
www.pserc.cornell.edu/ matpower/. [Accessed: 23 Dec. 2012].
[14] Z. Tate, “Initialization Schemes for Newton-Raphson Power Flow
Solvers,”
[Online].
Available:
http://grb.physics.unlv.edu/~zbb/files/upload/29UV3GPCVQWO6
ERVE9RABFQ5M.pdf. [Accessed: 15 Apr. 2010].
[15] J. Veleba, “Acceleration and Stability Techniques for Conventional
Numerical Methods in Load Flow Analysis,” Proceedings of ELEN
conference, pp. 1-10, 2010.
THE AUTHORS
Jan Veleba received his Master degree in Power
Engineering at the University of West Bohemia, Pilsen,
Czech Republic in 2008. Currently, he is the PhD. student
at the Department of Electrical Power Engineering and
Environmental Engineering at the University of West
Bohemia in Pilsen, Faculty of Electrical Engineering. His
main research activities concern load flow, voltage-power
control and voltage stability analyses of particularly larger
and more complex electric power systems.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
89
Cross slide mathematical model
for solving chatter
Jiří Ondrášek
VÚTS, a.s./Mechatronics Department, Svárovská 619, 460 01 Liberec XI, Czech Republic,
e-mail: jiri.ondrasek@vuts.cz
Abstract—The paper deals with the issues of creating a
mathematical model of self-excited chatter of the cross slide.
This type of oscillation occurs in systems where there is still
an internal source from which the system consumes energy
to maintain or even to increase the amplitude of oscillation.
This consumption is controlled by an oscillatory movement
of the system itself. Such an energy source greatly affects the
dynamic and stability properties of the system.
objects, in the following part of the text there are again
provided some basic relationships of continuous cutting
machining operation that are based on those assumptions:
1) Cutting force F is proportional to width b and depth
of cut yN in the direction of the normal to the machined
surface by the relation:
Keywords—chatter, cross slide, chip thickness, lobe diagram
2) Cutting force constant C0 is independent of speed.
3) Cutting edge geometry does not affect the direct
correlation between force F and chip depth yN in the
normal direction.
4) Angle β (an angle between cutting force F and the
axis perpendicular to the normal of the machined surface)
does not change with chip depth yN.
5) Friction forces between tool, workpiece and leaving
chip are neglected.
In Fig. 1, there is shown a block diagram of the
continuous machining process in which the dynamic
system can be described by the so-called oriented
dynamic compliance Gy(s). This is a transfer function
between the Laplace transforms of cutting force F(s) and
the movement of the tool group y(s):
I. INTRODUCTION
One of the main causes of generating self-excited
vibration in mechanical systems is dry friction between
two mutually moving parts that are directly related to
damping in the system. Chatter is undesirable and there
are efforts to avoid it either by increasing positive
damping or eliminating the causes of negative damping.
The frequency of steady chatter is close to the natural
frequency of a mechanical system.
This phenomenon often occurs during machining
operation when a part of the energy of cutting process
during cutting operation can change in the energy
oscillating the machine as a whole. Vibration then
manifests in a significant waviness of the cut surface and
is usually accompanied by noise. Generally, it is
theoretically possible to establish a certain range of cutting
conditions under which, when applying them, no chatter
arises. One of the means of such a designation is speed
stability diagram – lobe diagram, which expresses the
dependence of chip thickness on the workpiece speed. The
methodology of creating lobe diagrams is based on the
Laplace transform images of the cutting force and
movement of a tool group in the direction of material
removal.
The very issue of dealing with chatter in cutting
machining operation is creating the mathematical models
of the following physical objects: Model of the machining
process, Model of mechanical system inclusive flexible
links and real constraints, Drive model that presents a
model of electromotor itself and its control.
II. CHATTER
A mathematical process of machining process is
described in detail in publication [2], see Chapter 5, a
brief description of continuous machining is given in
article [1], as the case may be. In the case of creating a
mathematical model of the cross slide to investigate
chatter it is assumed the latter mentioned method of
machining. For this reason and for the reason of the
logical linking of mathematical models of particular
F = −C0byN .
G y (s) =
y(s)
,
F(s)
(1)
(2)
where s is the complex variable.
If depth of chip y0(t) < 0 is specified, cutting force F is
generated which will cause the movement of tool group
y(t) that will be superimposed to the specified depth, so
instantaneous chip depth yN(t) is given by an expression:
yN (t ) = y0 (t ) + y (t ).
(3)
The chip thickness is negative because chip cutting
occurs in the opposite direction of the y-axis orientation. If
there is a case of yN(t) > 0, the cutting edge got out from
engagement. When working with a fixed time link
between the cuts, the freshly machined surface will get
again into contact with the tool after a defined time with
the socalled transport time delay Td. This fact can be
expressed with a relation:
y N (t ) = y0 (t ) + y (t ) − y (t − Td ),
(4)
where y0(t) is the feed per one revolution, term y(t) is
valid for immediate waviness and term y(t-Td) applies for
waviness from the previous cut which will come under
the cutting edge of the tool over a time period Td.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
90
y
y(t-Td)
-Tds
F
+y(t1-Td)
–
y0
yN(t1)
β
y0(t1)
R
y(t)
–y(t1)
x
e
+
+
yN
−C 0 b
F
G y (s)
y
+
vobv = 2πRnZ
Fig. 1. Continuous machining and its block diagram.
3) From pairs f ↔ nZ and f ↔ bmez, there are assembled
pairs nZ ↔ bmez which are shown as a set of curves – lobs
for various values N again.
4) The individual „lobs“ may intersect each other and
the area of the stable widths is under their lower
envelope.
Waviness y(t-Td) from the previous cut together with an
immediate tool position y(t) always affects the variation of
chip depth yN(t), see Fig. 1, where the instantaneous chip
depth yN at time t = t1 is shown. The Laplace image of
equation (4) is:
(
y N (s) = y0 (s) + y(s) 1 - e
-Td s
).
(5)
III. EFFECT OF FEED DRIVES ON THE CHATTER
Using equations (1), (2) and (5), the closed loop transfer
as to Fig. 1 can be determined in the following form:
G Celk (s) =
−C 0 bG y (s)
y(s)
=
y 0 (s) 1 + C b G (s) 1 - e- Td s
0
y
(
)
.
The shape of speed diagram and defining the areas of
stability in the diagram depend, in addition to the
properties of the coupled mechanical system, also on the
characteristics of feed drives. In the same way as the
behavior and properties of a mechanical system, also the
behavior and properties of a control drive can be
considered. They are commonly expressed by the dynamic
flexibility of the control when it is again a transfer
function. The total dynamic compliance of the associated
dynamic system is then determined on the basis of the
causal interconnection of particular dynamic systems, i.e.,
feed drives and a mechanical system. This interconnection
is simple since the outputs of the one system are inputs of
the second system, as shown in Fig. 2 and Fig. 3.
In computer simulations it is assumed that the drive of
the feeds of machine tool will be implemented by a
3phase synchronous electromotor with permanent
magnets with which an exciting magnetic rotor flow is
produced. For the purposes of simulations, the model of
this servomotor was replaced by a simplified model which
is based on the description of a DC motor, see Fig. 2, in
which the block Mechanical system is a motor rotor with
mass moment of inertia J. The parameters of a single coil,
ie., inductance Ls, electrical resistance Rs and motor
voltage constant KE are substituted in this model. Only at
torque constant KT it is necessary to substitute the value:
(6)
To ensure the stability of the machining process it is
necessary so that chip width b does not exceed the limit
chip width bmez at which the amplitude characteristic of the
closed loop transfer (6) shall not exceed the unit level at
any angular frequency ω. This condition is expressed by
the relationship:
b < bmez = −
1
= funkce1(ω ) , (7)
2C0 Reneg Gy ( jω ) min
(
)
in which RenegGy represents negative values of the real
component of dynamic compliance Gy, see [2].
The stability lobe diagram expresses the dependence of
limit chip width bmez on the speed of workpiece nZ. The
speed equation, see [1], is given by the expression:

Re G y ( jω ) 
1
 = funkce 2(ω ) , (8)
f = nz  N + 1 − arctg

π
Im G y ( jω ) 

where f = ω/2π is the frequency of self-excited vibrations
on the limit of stability for each value of N individually.
Number N indicates the number of whole waves of the
workpiece surface ripple to be incurred over period Td =
1/nZ. The full version of the stability lobe diagram is
generated in practice as follows:
1) To the theoretically or experimentally determined
set of the values of dynamic compliance, there are
completed arranged pairs f ↔ bmez as to (7).
2) By means of (8), there are created pairs f ↔ nZ
repeatedly for values N = 0,1,2,… which may be further
displayed by a set of curves f = f(nZ).
U
+
–
1
Lss + R s
UE
I
K TCelk =
3
KT ,
2
(9)
whereby the common interaction of all three coils is taken
into account, see [4]. Furthermore, in the scheme, there
stands U for voltage, I electrical current, M
electromagnetic torque, ω rotor angular velocity and φ
rotor angular displacement. The above given parameters
of the electromotor were substituted in its simulation
model from the data sheet stated by the manufacturer.
KTCelk
M
Mechanical
system
KE
Fig. 2. A simplified block diagram of a synchronous motor.
φ
ω
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
+
y*
Ky
–
y
v*
+
91
Kv
–
Motion
function
Tiv s + 1 I*
Tiv s
+
–
v
Mechanical ω
system
KI
TiIs + 1
TiIs
I
U
Electromotor
M
Fig. 3. A cascade control circuit with a current speed and position feedback.
In the vector control of this type of electric motor, it is
almost exclusively used the cascade control circuit with
three hierarchically arranged feedbacks: current, speed
and position, see Fig. 3. Maintaining the required values
of position, revolutions and current is ensured by PID
linear controllers. Constant K is a proportional component
of the controller, constant Ti expresses the integral time
constant of the controller and Td is a derivative time
constant, υ = y, v, I. Index υ reflects competence of the
various constants to the position, velocity and current
feedback. Those constants in the control structure
according to Fig. 3 were debugged for the needs of the
simulation model by the Ziegler-Nichols method, see [5].
In the diagram in Fig. 3, y expresses the position of the
tool group, v its velocity, and * denotes the desired value
of a particular variable.
IV. MATHEMATICAL MODEL OF A MECHANICAL SYSTEM
A. Mathematical description of a coupled mech. system
To build equations of motion of a coupled mechanical
system with flexible links, it is usually based on the
Lagrange equations of mixed type which in matrix form
are as follows:
d  ∂Ek

dt  ∂q&
  ∂Ek
 − 
  ∂q
  ∂E p   ∂Rd
 + 
 + 
  ∂q   ∂q&
 ∂f V

 = Q + 

 ∂q

 λ , (10)

where Ek and Ep represent the kinetic and potential energy
of a mechanical system, Rd the so-called Rayleigh
dissipative function and vector Q represents the vector of
action generalized forces whose components correspond
to appropriate coordinates qj. To describe the coupled
mechanical system, there are used generally dependent
physical coordinates q of dimension r, which are coupled
by a system s of scalar constraint conditions:
Knowing kinetic Ek, potential Ep, and dissipative Rd
energies of the coupled mechanical system, it is possible
to establish equations of motion. Together with the second
time derivative of constraint conditions fV, a summary
record of these equations in matrix form can be included:
M

J
T
&&  p1 − Dq& − Kq 
− J  q
  = 
.
p2
0  λ  

(12)
in which:
J=
∂f V
T
∂q
(13)
expresses the Jacobi matrix of the system of coupling
equations. Symbols M, K and D are gradually mass,
stiffness and damping matrices of the mechanical system
as a whole. Vectors p1 and p2 are:
T
& q& + 1  ∂M q& 
p1 = Q − f g − M
2  ∂q 
∂
∂  ∂f V
p 2 = − T (Jq& )q& − 2 T 
∂q
∂q  ∂t
q& ,

∂ 2f V
q& − 2 ,

∂t

(14)
where vector fg is the vector of gravitational forces.
Equations (12) compose a system (r+s) of algebraicdifferential equations for r unknown generally dependent
physical equations q and s unknown Lagrange multipliers
λ. These equations are currently assembled and solved by
computational mechanics.
Comprehensive
information
about
creating
mathematical models of multibody systems with flexible
bodies is cited e.g. in publication [3].
For i-th flexible (pliable) body, coordinate vector qi can be
written as:
B. Frequency response of the coupled mechanical system
Transfer functions between the acting forces and the
mass displacements of the controlled mechanical system
with discretely distributed mass and stiffness parameters
are generally determined by the relevant matrix elements:
qi = ri , pi , qei ,
G ( s ) = ( M s 2 + D s + K ) −1 .
in which ri is the vector of the coordinates that define the
location of the given body in a fixed coordinate system
Oxyz. Orientation of the body in the basic space is
determined with Euler parameters pi. The elastic
deformations of the body are expressed by the vector of
elastic (or standardized modal) coordinates qei.
Furthermore, in equations (10), vector λ of Lagrange
multipliers is found in the number of s that have a direct
connection with the reactionary forces in kinematic
constraints. The coupled mechanical system is
characterized by i = r – s degrees of freedom.
The method of determining the transfer function of the
mechanical system according to relation (15) is
applicable only for linear non-conservative mechanical
systems. In the case of mechanical systems in which there
occur nonlinearities due to passive resistances in
kinematic constraints for example, it is advisable to use a
method in which the system is excited by the swept
signal. It is one of the possible methods that allows the
study of nonlinearities in the time domain only. Getting a
frequency response of a nonlinear dynamic system is
possible only round about the operating point.
f (q, t ) = 0 .
V
(11)
(15)
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
H(jω)
u(t)
If any of the eigenvalues should have a positive real part,
it would be an unstable mechanical system. The
relationship between the imaginary part of the eigenvalue
βi and the natural frequency of the system Ω0i is
determined by the following conversion:
y(t)
Fig. 4. Time-invariant system block.
Consider a dynamic system described by the frequency
response:
H ( jω ) =
Y ( jω )
,
U ( jω )
(16)
where Y(jω) and U(jω) are the images of output and input
signal y(t) and y(t), see Fig. 4. Because it is a complex
function, it can be decomposed into the real and
imaginary parts:
H ( jω ) = V (ω ) + jW (ω ) ,
(17)
with which the amplitude part A(ω) and the phase part
φ(ω) of frequency response can be determined:
A(ω ) = H ( jω ) = V 2 (ω ) + W 2 (ω ) ,
 W (ω ) 
 .
 V (ω ) 
ϕ (ω ) = arg (H ( jω ) ) = arctan 
(18)
Alternatively:
Re{H ( jω )} = H ( jω ) cos ϕ (ω ),
Im{H ( jω )} = H ( jω ) sin ϕ (ω ) .
(19)
If a swept sine wave signal is fed to the system input:
u (t ) = A1 sin(πf 0 t 2 ),
(20)
then, a signal with variable frequency and amplitude will
be stabilized at the system output:
y (t ) = A2 (t ) sin(ω (t )t + ϕ 0 ).
(21)
Subsequently, both signals are converted with Fourier
transformation into the frequency domain and by their
dividing according to (16), it is set the appropriate
frequency response around the given operating point. In
other words, there is a linear approximation of the
appropriate transfer function.
C. The natural frequency of the coupled mech. system
If the coupled mechanical system is generally
expressed in a system of equations of motion in matrix
form:
&&(t ) + Dq& (t ) + Kq (t ) = f (t ),
Mq
(22)
it can be solved at f(t) = 0 the problem of eigenvalues,
which is given in n-dimensional space by the expression:
(
)
det λ2 M + λ D + K = 0 .
(23)
Roots λi of the characteristic equation (23) are the
eigenvalues of the system. Those can be complex coupled
in pairs:
λi = −α i + jβ i ,
λ∗i = −α i − jβ i ,
i = 1,2,..., m
or real:
λi = −α i ,
i = 2m + 1,2m + 2,..., n .
92
Ω 0i =
βi
.
2π
(24)
V. MATHEMATICAL MODEL OF A CROSS SLIDE
The aim of the calculations in the mathematical model
of the chatter of the cross-slide was to calculate the limit
chip width bmez and to create the stability lobe diagram in
the machining processes of grooving (machining in udirection) and longitudinal turning (machining in wdirection) while both operations do not occur
simultaneously. To establish them, it is necessary to
determine the appropriate dynamic compliance Gy(s), y =
u, w, in the given direction depending on the method of
machining.
As stated above, the problems of vibration during
machining can be divided into the description of three
basic objects:
1) The process of cutting itself.
2) The description of the mechanical system – machine
tool or machine group.
3) Description of the drive, i.e., the engine itself and its
control.
In the first case it was used the knowledge contained
primarily in [2], and which are briefly mentioned in the
previous Chapter II. Here are two parameters that, on the
basis of assumptions, were considered for constants that
had to be put into the resulting simulation model. These
are:
- β = 22.5°
cutting angle,
- C0 = 2·109 Nm-2
cutting force constant.
A cross slide mathematical model is generally
expressed in equation (12), see Fig. 5. These equations
were compiled based on the following assumptions.
This is a spatial system of 24 perfectly rigid bodies
(without considering the frame 1) and 4 flexible bodies –
a console of U and W axes and a slide of U and W axes.
Between the bodies there were defined kinematic pairs in
such a way so that the analyzed mechanical system is free
of redundant constraints.
W axis console seating to the frame, ball screw support,
ball screw nut and the ball screw itself of both axes were
considered as flexible. Their compliance was defined by
stiffness as to TABLE I. Stiffness of both ball screws is
given by the ball screw nut distance from the ball screw
support. These are the following:
lW = 140 mm.
lU = 125 mm,
Both ball screws are overhung.
TABLE I.
STIFFNESS OF FLEXIBLE ELEMENTS
Torsion stiffness of the coupling
Screw support stiffness
Nut axial stiffness
Screw longitudinal stiffness
Screw torsion stiffness
Unit
[Nmrad-1]
106 [Nm-1]
106 [Nm-1]
106 [Nm-1]
[Nmrad-1]
U-axis
825
450
410
688
17620
W-axis
825
450
410
614
15730
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
U-axis motor
93
Fu
U-axis slide
W-axis slide
W-axis motor
Fw
v
U-axis console
w
W-axis console
u
Fig. 5. Cross slide model.
The damping of these elements was selected
proportional to their stiffness in the simulation model,
i.e.:
dU,W = 2.5·10-6kU,W
in the case of longitudinal stiffness and
dtU,W = 2.5·10-3ktU,W
in the case of torsion stiffness.
The deformation field of the flexible bodies was
approximated with deformation modes, see [3]. It is a
linear combination of static wave shapes of the body that
respect the boundary conditions and the natural vibrations
of the given flexible body. The modes of individual
elements were established on the basis of the model of
the particular body, created by using the FEM. Through
those modes, there were determined further generalized
mass matrix M and stiffness matrix K of the given
flexible body whose dimension was acceptable for the
purposes of subsequent calculations already. Damping
matrix D of the flexible body was defined on the basis of
modal damping coefficients ci. That one was introduced
by means of damping ratio bri that expresses the ratio of ci
modal damping coefficient in relation to cicr critical
damping of the given mode of the appropriate pliable
body. The coefficients of proportional damping and
damping ratio were verified with respect to the
experimentally determined modal damping of the natural
wave shapes with the real mechanical system.
TABLE II.
PROPORTIONAL DAMPING OF THE OWN SHAPES OF PLIABLE BODIES
W axis console
W axis slide + U axis console
U axis slide
bri [%]
10
25
20
The dovetail slides of the cross slide were modeled
through the interaction of two bodies. It is ensured by the
action of coupling forces as linear dampers with bυ
damping coefficients. The origin of forces is spread over
6 points in each functional area of each from the dovetail
slides of U and W machining axes. This gives a total of
2x24 force relations between two pairs of bodies. Points
were distributed evenly in such a way so that the
appropriate pairs of points can lie always opposite each
other in the normal to the functional surfaces of the
groove.
Oil viscosity was estimated at 250 mPa·s and the
normal clearance on one side of the groove 10 µm, while
these values correspond to the specific damping
coefficient δ in a size of 5·1010 Nsm-3. Then, damping
coefficients bυ of one damper for each of the functional
surfaces of the dovetail slide of U and W machining axes
can be determined as follows:
bν =
1
δ lν dν ,
12
ν = Uh ,Us ,Wh ,Ws ,
(25)
in which U and W indices stand for belonging to U and
W axes and h and s indices express the horizontal and
inclined functional surface of the dovetail slide. Knowing
length lυ and width dυ of the contact surfaces of the
dovetail slide groove, damping coefficients bυ of linear
dampers were determined and which are given in
TABLE III.
TABLE III.
DAMPING COEFFICIENTS OF LINEAR DAMPERS
U
W
h
s
h
s
lυ [mm]
260
260
304
304
dυ [mm]
35.29
19.03
40.30
23.54
bυ·107 [Nsm-1]
3.8
2.1
5.1
3.0
The resulting simulation model was characterized by
108 DOF.
In the model of feed drive, there were substituted the
numerical values of the relevant parameters and which
are given in TABLE IV.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
TABLE IV.
DRIVE PARAMETERS
pp – Number of pole pairs
Rs – Electrical resistance of one phase
Ls – Inductance of one phase
KE – Motor voltage constant
KT – Motor torque constant
rI – Current controller proportional component
TI – Current controller time integration constant
rω – Speed controller proportional component
Tω – Speed controller time integration constant
rφ – Position regulator proportional component
Unit
[-]
[Ω]
[mH]
[Vrad]
[NmA-1]
[VA-1]
[s]
[Asrad-1]
[s]
[s-1]
Value
3
1.5
13.3
0.4444
0.4054
0.8
0.004
0.75
0.05
5
VI. SIMULATION RESULTS
At first, the mathematical model of the cross slide was
verified on the basis of measured natural frequencies on a
real machine installation. By comparing the values of
calculated and measured natural frequencies, a quite good
compliance between those variables is apparent, see
TABLE V. Since according to the measurement the third
mode dominates, this model was not further verified in
terms of damping due to modal damping 1 of natural
oscillation shape.
TABLE V.
NATURAL FREQUENCIES
Measured modal Calculated Calculated modal
damping [%]
f0 [Hz]
damping [%]
2.8
118.6
1.4
2.6
160.0
2.6
2.6
252.0
2.6
(
)
Fu , v , w = AF sin πf 0t , AF = 50 N , f 0 = 400Hz. (26)
2
To determine inertance, it is necessary to know the
progress of acceleration at the site. Then, both signals are
converted through Fourier transformation into the
frequency domain and with their division, the appropriate
course is determined. In Fig. 6 up to Fig. 8, there is
shown the course of real and imaginary components of
the inertances of the cross slide provided both by
measurements and calculations on a mathematical model
of this mechanical system further then.
By comparing the courses of inertances determined by
calculating in different directions with the measured ones,
a very similar character of the dynamic properties of the
mathematical model of the cross slide with a real object is
obvious. For its further refinement it is necessary to better
specify the submodel of damping caused mainly with
passive resistances in the dovetail slide, which has a
significant influence on the courses of inertances.
Measured real inertance component
Calculated real inertance component
ReInU [m/Ns2]
1
2
3
Measured
f0 [Hz]
118.0
157.0
260.0
In the next step, the mathematical model was verified
with experimentally determined transfer functions of a
real mechanical system. In this case, it is the course of
inertances in the given direction. As an inertation it is
called the transfer function between the Laplace images
of cutting force and the acceleration of a tool group.
In the point of acting of cutting forces, the frequency
variable course of harmonic force acts in the appropriate
direction according to the following equation:
f [Hz]
Measured imaginary inertance component
Calculated imaginary inertance component
ImInU [m/Ns2]
Shape
94
f [Hz]
Fig. 6. Real and imaginary inertance component in u – direction.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
95
ReInV [m/Ns2]
Measured real inertance component
Calculated real inertance component
f [Hz]
ImInV [m/Ns2]
Measured imaginary inertance component
Calculated imaginary inertance component
f [Hz]
ReInW [m/Ns2]
Fig. 7. Real and imaginary inertance component in v – direction.
Measured real inertance component
Calculated real inertance component
f [Hz]
ImInW [m/Ns2]
Measured imaginary inertance component
Calculated imaginary inertance component
f [Hz]
Fig. 8. Real and imaginary inertance component in w – direction.
96
Chip width b [mm]
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
bmezmin = 18.9 mm
Spindle speed nZ [rev/min]
Fig. 9. Lobe diagram in the machining process of grooving.
By comparing the course of inertances determined by
calculations in different directions with the measured
ones, a very similar character of the dynamic features of
the mathematical model of the cross slide with the real
object is obvious. For its further refinement it is
necessary to better specify the submodel of damping
caused mainly by passive resistances in the dovetail slide
that has a significant influence on the course of
inertances.
A sample of the speed stability diagram of grooving is
shown in Fig. 9. The diagram was designed in accordance
with the procedure referred to in Paragraph II. For its
formation, transfer functions determined on the basis of
calculations on a mathematical model of the cross slide
were used. The theoretical value of the chip minimum
limit width achieves the size bmezmin = 18.9 mm. The
stable area of cutting conditions is highlighted with
hatching.
VII. CONCLUSION
Creating a mathematical model of the cross slide as a
coupled mechanical system with flexible links and real
constraints are not an entirely trivial matter. Especially in
the case of the dynamic properties of the dovetail slide it
is important to realize that this is a non-linear dynamic
system. In order to take into account the dynamic
properties of the real system it is necessary to create a
relatively detailed computational model which leads to a
large number of degrees of freedom and often shows a
nonlinear character of dynamic behavior as well.
It was selected such a procedure of the creation of the
linked system of the cross slide when it is a composition
of abstract dynamic subsystems with causal orientation
input – output. This assembly is very simple because the
outputs of one model are the inputs of another model.
Such a model of the related mechanical system can be
solved in both time domain and frequency domain.
The course of transfer functions of the analyzed
mechanical system is highly dependent on the submodel
of damping - above all, on the damping in the dovetail
slides of the cross slide, i.e., on passive resistances. The
course of transfer functions is not significantly sensitive
to the size of the specific damping coefficient δ.
The shape of Lobe diagrams depends on:
Cutting force constant C0,
The direction of cutting force β,
The course of the oriented dynamic compliance
of the mechanical system.
To create Lobe diagrams, it is therefore necessary to
identify the internal damping of the system and the
damping due to the effect of passive resistances in the
given system.
ACKNOWLEDGMENT
This paper was created within the work on the 2A2TP1/038 – Project supported by the Ministry of Industry
and Trade - Ministerstvo průmyslu a obchodu.
REFERENCES
[1] J. Ondrášek, “Creating a mathematical model for solving chatter
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Media Dordrecht, 2012, ISBN 978-94-007-5124-8.
[2] P. Souček, A. Bubák, Vybrané statě z kmitání v pohonech
výrobních strojů, (in Czech), Česká technika – nakladatelství
ČVUT, 2008, ISBN 978-80-01-04048-5
[3] J. Slavík, V. Stejskal, V. Zeman, Základy dynamiky strojů, (in
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ČVUT, 1995, ISBN 80-01-01276-X.
Transactions on Electrical Engineering, Vol. 2 (2013), No. 3
Zablodskiy, N., Lettl, J., Pliugin, V., Buhr, K., Khomitskiy, S.: Induction Motor Design by Use of Genetic
Optimization Algorithms
The problem of the automated calculation and optimal design of an induction motor is presented. The problem of
optimization by use of genetic algorithms is set and solved. The analysis of the obtained results is executed.
Homišin, J.: New Ways of Controlling Dangerous Torsional Vibration in Mechanical Systems
In general terms the mechanical systems (MS) mean the systems of driving and driven machines arranged to perform the
required work. We divide them into MS operating with constant speed and MS working within a range of speed. In terms of
MS dynamics we understand the system of masses connected with flexible links between them, i.e. systems that are able to
oscillate. Especially piston machines bring heavy torsional excitation to the system, which causes oscillation, vibration, and
hence their noise. Based on the results of our research, the torsional vibration control as a source of given systems excitation,
can be achieved by application of pneumatic couplings, or pneumatic tuners of torsional vibration. Existence of pneumatic
couplings and pneumatic tuners of torsional vibration developed by us, creates the possibility of implementing new ways of
torsional oscillating mechanical system tuning. Based on the above, the aim of the article is to present new ways of
dangerous torsional vibration control of mechanical systems by application of pneumatic couplings and pneumatic tuners of
torsional vibration developed by us.
Folta, Z., Hrudičková, M.: Experience with Torque Measurement on Rotating Shaft
This article deals with problem of torque measurement on the turning shaft and the signal transmission to the measuring
equipment. There is described the experience with contact transmission and both non-contact and high frequency
transmission (with ESA Messtechnik equipment). Also the examples of the practical realization are presented.
Veleba, J..: Performance of Steady-State Voltage Stability Analysis in MATLAB Environment
In recent years, electric power systems have been often operated close to their working limits due to increased power
consumptions and installation of renewable power sources. This situation poses a serious threat to stable network operation
and control. Therefore, voltage stability is currently one of key topics worldwide for preventing related black-out scenarios.
In this paper, modelling and simulations of steady-state stability problems in MATLAB environment are performed using
author-developed computational tool implementing both conventional and more advanced numerical approaches. Their
performance is compared with the Simulink-based library Power System Analysis Toolbox (PSAT) in terms of solution
accuracy, CPU time and possible limitations.
Ondrášek, J.: Cross Slide Mathematical Model for Solving Chatter
The paper deals with the issues of creating a mathematical model of self-excited chatter of the cross slide. This type of
oscillation occurs in systems where there is still an internal source from which the system consumes energy to maintain or
even to increase the amplitude of oscillation. This consumption is controlled by an oscillatory movement of the system itself.
Such an energy source greatly affects the dynamic and stability properties of the system.
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TRANSACTIONS ON ELECTRICAL ENGINEERING VOL. 2, NO. 3 HAS BEEN PUBLISHED ON 30TH OF SEPTEMBER 2013