835
1 INTRODUCTION
The problems of differential and integral calculus of
non-integer orders – commonly known as fractional
calculus – have been known since the days of famous
mathematicians Gottfried Wilhelm Leibniz (1646-
1716) and Guillaume François Antoine de l'Hospital
(1661-1704) [1–3, 10, 14–16]. However, until today, a
description of the dynamic properties of an object by
means of fractional calculus has not been used due to
the barriers resulting from the lack of appropriate
calculation methods and possibilities of verifying
them (related, among others, to the limited calculating
potential of earlier computers). Nowadays, technical
and calculating possibilities cause that former
limitations have disappeared and the said problems
can now be solved [8, 11]. There are more and more
publications dealing with the issue of differential
equations of non-integer orders. Majority of them,
however, deal with theoretical aspects of the problem.
The dynamic development of research in recent
years on the use of fractional calculus for the analysis
of dynamic systems has prompted the authors to use
it in the analysis and modelling of pneumatic systems
that have been described so far with "classical"
mathematical analysis [4–13]. The authors of the paper
have developed a method for describing the dynamic
properties of pneumatic systems, based on fractional
calculus which allows to analyse the properties of a
wide range of pneumatic systems of any order.
The simulation tests of the membrane pressure
transducer, presented in the paper [13], were
performed with the use of classical differential
calculus and fractional calculus. In the construction of
the mathematical model of the analysed dynamical
system, the Riemann-Liouville definition of the differ-
integral of non-integer order was used.
For simulation studies MATLAB were used [5, 6, 8,
9]. In the laboratory tests, which were the verification
of the simulation tests of the membrane pressure
Analysis of Dynamic Characteristics of Selected
Pneumatic Systems with Fractional Calculus. Simulation
and Laboratory Research
M. Luft
1
, K. Krzysztoszek
1
, D. Pietruszczak
1
& A. Nowocień
2
1
Kazimierz Pulaski University of Technology and Humanities in Radom, Radom, Poland
2
Complex of Electronic Schools in Radom, Radom, Poland
ABSTRACT: The section of the paper on simulation studies presents the application of fractional calculus to
describe the dynamics of pneumatic systems. In the construction of mathematical models of the analysed
dynamic systems, the Riemann-Liouville definition of differ-integral of non-integer order was used. For the
analysed model, transfer function of integer and non-integer order was determined. Functions describing
characteristics in time and frequency domains were determined, whereas the characteristics of the analysed
systems were obtained by means of computer simulation. MATLAB were used for the simulation research. The
section of the paper on laboratory research presents the results of the laboratory tests of the injection system of
the internal combustion engine with special attention to the verification of simulated tests of selected pneumatic
systems described with the use of fractional calculus.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 15
Number 4
December 2021
DOI: 10.12716/1001.15.04.16
836
transducer, the following assumptions were made: the
analysed pneumatic systems were modelled as a
linear system; the pneumatic system was described
with a transfer function characterizing the dynamics
of this system and the components contained therein,
assuming constant physical parameters and omission
of aging of its components; an assessment was
accepted of the dynamic properties of the pneumatic
systems in terms of amplitude and phase; pneumatic
systems with a pressure of up to 1MPa were analysed
while operating in the frequency range up to 500Hz;
with the variability of the thermodynamic parameters
of the air as a working medium, it can be treated as an
ideal gas; in the analysis of the pneumatic systems, an
adiabatic process was assumed whereas the pressure
distribution in the whole volume of the measuring
chamber was homogeneous.
2 MEMBRANE PRESSURE TRANSDUCER
Simulation tests of the membrane pressure transducer
were performed using a classical and fractional
differential calculus. The tested transducer was made
from a pressure chamber and an inlet pipe, which
supplied a working medium (air). To determine how
the connection of the intake pipe to the transducer
chamber affected its dynamic properties, the acoustic
system shown schematically in Figure 1 is considered.
Figure 1. Pressure chamber with inlet pipe: r, l - tube
dimensions, p0 - input pressure (force ), p - pressure in
transducer chamber
The relationship that binds the output signal p(t)
(pressure inside the chamber) to the signal p0(t)
(pressure at the open end) and referring to the RLC
electrical circuits can be represented as:
( ) ( )
( )
2
0
2
11
()
p
p p p p p
R
d p t dp t
p t p t
dt L dt L C L C
+ + =
(1)
Constants occurring in expression (1) can be
represented as:
25
2
p
V
C Ns m
c
−
=
(2a)
51
2
4
3
p
l
L m N
r
−
=
(2b)
5
2
8
p
l
R Nsm
r
−
=
(2c)
where:
ρ [kgm
-3
] - gas density;
η [kgm
-1
s
-1
] - dynamic viscosity;
Cp [Ns
2
m
-5
] - pneumatic capacity (gas compressibility);
p(t) [Pa] - pressure in transducer chamber;
p0(t) [Pa] - input pressure;
V [m
3
] - transducer chamber volume;
Lp [m
3
N
-1
] - pneumatic induction (gas inertia);
Rp [Nsm
-5
] - flow resistance;
c [ms
-1
] - speed of sound in the gas;
r, l [m] - dimensions of the inlet pipe.
By specifying the frequency ω0 and damping ratio
ξ as:
22
0
13
4
pp
c
lV
LC
==
(3a)
2
0
2 2 2
3
3
22
22
p p p p
p
lV
R C R C
lV
L r c r c
= = = =
(3b)
where: ξ <1.
Expression (1) finally assumes the form:
( ) ( )
( ) ( )
2
22
0 0 0 0
2
2
d p t dp t
p t p t
dt dt
+ + =
(4)
Equation (4) is a mathematical description of the
dynamics of the analysed pneumatic system, using
classical differential calculus (of integer orders). The
impulse response of the analysed pneumatic system is
given by:
( )
0
2
0
0
2
sin 1
1
t
g t e t
−
=−
−
(5)
The step response of the system is expressed by:
( )
(
)
0
2
0
2
1 sin 1
1
t
e
h t t
−
= − − +
−
(6)
where:
2
1
arctg
−
=
(7)
Given that the derivatives of integer orders in the
fractional calculus are a special case of fractional
derivatives, equation (4) can be written as:
( ) ( ) ( ) ( )
2 2 2
0 0 0 0 0 0
2
RL v RL v
tt
D p t D p t p t p t
+ + =
(8)
where v>0.
In order to determine the pressure in the
transducer chamber, the definition of Riemann-
Liouville differ-integral of non-integer order was
used, defined by a following formula (9):
837
( )
( )
( ) ( )
1
1
t
k
k
RL
at
k
a
d
D f t t f d
k dt
−−
=−
−
(9)
where:
α – the order of integration within bounds (a, t) of the
function f(t), k-1≤a≤k and α R+ and Γ - Euler's
gamma function.
The Laplace transform for a fractional derivative
defined by Riemann-Liouville takes the form:
( ) ( ) ( )
1
1
00
0
0
j
R L k R L k
tt
k
L D f t s F s s D f
−
− − − −
=
=−
(10)
where j-1≤α≤j N.
The practical application of the formula
determining the Laplace transform of a Riemann-
Liouville fractional derivative faces some difficulties
related to the lack of physical interpretation of the
initial values of successive derivatives of fractional
orders. Assuming zero initial conditions, the
difficulties associated with their physical
interpretation will be eliminated.
Using the Laplace transform to equation (8), for
zero initial conditions, we obtain:
( ) ( ) ( ) ( )
2 2 2
0 0 0 0
2
vv
s p s s p s p s p s
+ + =
(11)
Thus the transfer function of non-integer order of
the analysed pneumatic system is obtained:
( )
( )
2
0
22
00
2
v
vv
Gs
ss
=
++
(12)
Transfer function denominator of non-integer
order has two complex roots as the system damping is
ξ < 1.
3 IMPULSE RESPONSE TO THE MEMBRANE
PRESSURE TRANSDUCER
By transforming expression (12), we obtain
G
(v)
(s)=p(s)/p0(s) such that [12, 13]:
( )
( )
2
0
2
2
0
0
2
0
1
2
1
2
v
vv
vv
Gs
ss
ss
=
+
+
+
(13)
Using the properties of the geometric series, we
obtain:
( )
( )
( )
( )
2
2
0
0
2
2
0
0
0
2
2
k
v
k
vv
vv
k
Gs
ss
ss
=
−
=
+
+
(14)
Conducting elementary transformations, we
obtain:
( )
( )
( )
( )
( )
( )
2
2
2
0
0
1
0
0
!
!
2
k
v v vk
v
k
v
k
ks
Gs
k
s
−+
+
=
−
=
−−
(15)
Using the formula:
( )
( )
( )
1
,
1
!
m
m
m
ms
L t E at
sa
−
+−
+
=
−
(16)
we obtain:
( )
( )
( )
( )
( )
( )
( )
1
22
00
21
,2 0
0
2
!
vv
k
vk v vk k v
v v vk
k
g t L G s
t E t
k
−
+ + −
+
=
==
−
=−
(17)
where:
( )
( )
( )
,
0
!
!
n
k
n
nk
t
E
n n k
=
+
=
+ +
(18)
The simulation of the pressure impulse response in
the transducer chamber required a program to be
written in the MATLAB environment. The program
for the given parameters and derivative orders
calculates the function values and draws out their
impulse response. We present, for comparison, the
graphs of the function obtained for the classic solution
(v=1) and for several fractional orders.
Figure 2. Impulse response of a pneumatic transducer
described with an integer and non-integer order: F0,5 for
v=0.5, F0,7 for v=0.7, F0,9 for v=0.9, F1,0 for v=1, C2 - classical
model (integer order).
In Figure 2, the impulse response of the analysed
pneumatic transducer was determined by simulating
equation (17) for the selected parameter values: F0,5 for
v=0.5, F0,7 for v=0.7, F0,9 for v=0.9 and F1,0 for v=1.
The impulse response (characteristics C2 in the
above figure) was also presented, by simulating a
computer equation (5) which is a mathematical model
of the analysed pneumatic system, with the use of a
classical differential calculus (described by ordinary
differential equation). It is worth noting that while
reducing the row, it reduces the response time, which
is desirable in measuring transducers.
838
4 STEP RESPONSE OF THE MEMBRANE
PRESSURE TRANSDUCER
The step response of the tested transducer is defined
by the relationship:
( )
( )
( )
2
0
22
00
21
0
2
2
0
0
2
0
2
1
2
1
2
v
vv
vv
vv
Hs
s s s
s
ss
ss
−
==
++
=
+
+
+
(19)
Using the properties of geometric series, we obtain
( )
( )
( )
( )
2
21
0
0
2
2
0
0
0
2
2
k
v
k
vv
vv
k
s
Hs
ss
ss
−
=
−
=
+
+
(20)
Conducting elementary transformations, we
obtain:
( )
( )
( )
( )
( )
( )
2
21
2
0
0
1
0
!
!
2
k
v v vk
v
k
v
k
ks
Hs
k
s
− + +
+
=
−
=
−−
(21)
Using the formula (16), we obtain:
( )
( )
( )
( )
22
00
2 1 1
,2 1 0
0
2
!
k
v
vk v vk k v
x v vk
k
h s t E t
k
+ + + −
++
=
−
=−
(22)
in which
( )
,
k
E
is the Mittag-Leffler function defined
by the equation (18).
Figure 3. The step response of the pneumatic system: F0,5 for
v=0.5, F0,7 for v=0.7, F0,9 for v=0.9, F1,0 for v=1, C2 - classical
model (integer order).
Running a simulation of a pneumatic transducer, a
unit step signal was applied and the received step
response is shown in Figure 3.
The model described by equation (17) and (22)
correctly reproduces the amplitude of the input signal
as the classical model - the graphs coincide (graph F1,0
- the parameter v=1 coincides with C2 – the classical
model). This confirms the correctness of the method
and that the differential calculus with derivatives of
integer orders is a special case of fractional calculus.
The step response (Figure 3) shows that regardless of
the differential order, the amplitude of the signal is
constant. The smaller the order of the derivative leads
to the faster the reaction of the system to the unit step.
5 FREQUENCY RESPONSE OF THE MEMBRANE
PRESSURE TRANSDUCER
In order to determine the relationships describing the
frequency response, the spectral transfer function of
the tested transducer was determined [6, 9].
Substituting (23):
2
cos sin
22
j
s j e j
= = = +
(23)
to the formula (12), the spectral transfer function of
the transducer is obtained:
( )
( )
( ) ( )
2
0
2
2
00
2
v
vv
Gj
jj
=
++
(24a)
( )
( )
( ) ( )
2
0
22
00
cos sin 2 cos sin
22
v
vv
Gj
v j v v j v
=
=
+ + + +
(24b)
By making elementary transformations, the real
and imaginary part of the spectral transfer function
was calculated:
( )
( )
( )
( )
( )
( )
v v v
G j P jQ
=+
(25)
where:
( )
( )
( )
( ) ( )
2 2 3 4
0 0 0
22
2 2 2
0 0 0
cos 2 cos
2
cos 2 cos sin 2 sin
22
v
vv
v v v v
P
v
v
vv
vv
=
++
=
+ + + +
(26a)
( )
( )
( )
( ) ( )
2 2 3
00
22
2 2 2
0 0 0
sin 2 sin
2
cos 2 cos sin 2 sin
22
v
vv
v v v v
Q
v
v
vv
vv
=
+
=
+ + + +
(26b)
Knowing the real and imaginary part of the
spectral transfer function of the transducer, one can
determine the equation describing the logarithmic
amplitude characteristic:
( )
( )
( )
( )
( )
( )
22
20log
v v v
L P Q
=+
(27)
and the equation describing the logarithmic phase
step:
839
( )
( )
( )
( )
( )
( )
( )
( )
2
0
22
00
sin 2 sin
2
cos 2 cos
2
v
v
v
vv
vv
Q
arctg
P
v
v
arctg
v
v
==
+
=−
++
(28)
In order to verify the relationships describing
logarithmic steps of amplitude (27) and phase (28) of
the tested transducer, the pneumatic pressure
transducer was modelled in the MATLAB
environment, described by means of ordinary
differential equation and differential equation with
derivatives of non-integer order. Describing the
transducer with the use of a differential equation of
non-integer orders, the parameter v=1 was assumed
and the obtained logarithmic amplitude and phase
steps were compared to the logarithmic amplitude
and phase steps obtained from the transducer
description by means of the ordinary differential
equation.
The transfer function of the pneumatic pressure
transducer, calculated from the ordinary differential
equation, has the form:
( )
( )
( )
2
0
22
0 0 0
2
ps
Gs
p s s s
==
++
(29)
By performing the simulation of equation (29)
which presents the dynamics of the phenomena
occurring in the analysed pneumatic system, in the
MATLAB programming environment, the frequency
response presented in Figure 4 was obtained:
Figure 4. Logarithmic frequency response of the transducer
described by the ordinary differential equation
When simulating equations (27) and (28) in the
MATLAB environment which describe a pneumatic
pressure transmitter by means of a differential
equation of non-integer order, assuming a coefficient
v=1 for damping ξ=0.8, the response shown in Figure
5 was obtained:
Figure 5. Logarithmic frequency response of a pneumatic
transducer described by means of a differential equation
with non-integer order for v=1 (equation 27 and 28)
Logarithmic frequency response of amplitude and
phase presented by the simulation of ordinary
differential equation (Figure 4), coincide with
frequency response obtained by the simulation of the
equations describing logarithmic response of
amplitude (27) and phase (28), obtained from the
equation of the transducer described with the help of
non-integer orders (figure 5) for the parameter v=1.
In order to obtain a Bode plot, the equations (27)
and (28) were simulated by writing an appropriate
program in the MATLAB environment. Written in the
MATLAB environment, the program allows analysing
the transducer for different orders of derivatives, with
any step, because the order was given as a parameter.
The simulation results for the selected values of
parameter v are shown in Figure 6 and Figure 7.
Figure 6. Logarithmic amplitude response of a pneumatic
transducer described by means of differential equation with
fractional derivatives of non-integer orders in the range (0.8-
1.2)
840
Figure 7. Logarithmic phase response of a pneumatic
transducer described by means of differential equation with
fractional derivatives of non-integer orders in the range (0.8-
1.2)
The analysis of the responses shows that for the
parameter v<1, the logarithmic amplitude responses
(Figure 6) are monotonically decreasing functions. For
the parameter v>1, the logarithmic amplitude
responses have a maximum depending on the order of
the differential. The maximum is achieved with
resonant frequency ωR=110rad/s.
Increasing the order of derivative, the frequency
responses acquire the character of a second-order
oscillatory element, and while decreasing the order of
the derivative, the responses acquire the character of
the first order inertial element.
Decreasing the order of the derivative causes the
transducer to become more linear, which allows the
scope of work to be increased.
Increasing the parameter v above one results in
resonance, although it should not be visible in the
response, because the simulation was carried out for
the damping ξ=0.8. The model then does not reflect
the real system.
6 LABORATORY TESTS OF THE PRESSURE
TRANSDUCER
In order to identify the dynamics of the pressure
transducer, the measuring system was constructed as
it is shown in Figure 8.
The AVL single-cylinder automatic ignition engine
was used for testing [6, 13]. It is an internal
combustion engine with a capacity of 511cm
3
, cylinder
diameter 85.01mm and a piston stroke of 90mm. The
measurements were made in the inlet air system to the
engine. The air supply was provided by an additional
system consisting of a rotary screw compressor.
Thanks to this system it was possible to regulate the
air pressure in the intake system.
Figure 8. View of measuring station: 1 - measuring chamber,
2 - intake manifold, 3-input pressure transducer, 4 - output
pressure transducer
In the air intake system leading into the measuring
chamber, the first pressure transducer was installed.
The second transducer was installed inside the
measuring chamber, at the outlet of the air into the
combustion chamber. Kulite pressure transducers,
type ETL-189- 190M-10 BARA were used. Two
identical pressure transducers were used in the
system.
The tests were performed with Concerto and Puma
software, whose interfaces are shown in Figure 9.
The presented measuring system allows studying
the dynamic properties of the pressure transducer.
The studies refer to the time and frequency analysis of
the investigated pressure transducer described with
integer and non-integer order.
Figure 9. Interfaces of the programs for motor control and
recording fast-changing parameters: 1 - Concerto program
window for fast variable parameters recording (monitor 1),
2 - Puma program window for controlling and archiving
engine parameters (monitor 2), 3 - Puma program window
for controlling and archiving engine parameters (monitor 3)
In the measuring system, computers with Concerto
and Puma software were used. The Concerto program
allows recording fast-changing system parameters
and recording them in time and numeric format. The
Puma program was used to control the engine. The air
supplied into the measuring system is provided by a
rotary screw compressor. The engine draws air into
the combustion chamber from the measuring system
841
by opening the valve located at the exit of the
measuring chamber. The valve opens and closes every
two turns of the engine crankshaft. Cyclical opening
of the valve caused the same effect as supplying the
system with a pneumatic rectangular signal generator,
which allowed experimental evaluation of the step
response of the measuring system. The step response
of the system was determined using the Concerto
program, and its graphic representation is shown in
Figure 10.
Figure 10. The step response of the measuring system in the
measuring chamber and inlet tube obtained experimentally.
The obtained graph is a step response of a typical
oscillating element with frequency ω0 and damping
ratio ξ. In order to identify the dynamic properties of
the tested pneumatic system it is convenient to
determine its transfer function.
Knowing the dependency (30):
( )
( )
( )
2
0
22
0 0 0
2
ps
Gs
p s s s
==
++
(30)
and the parameters ω0 and ξ (31a 31b) obtained from
Figure 10:
0.042
=
(31a)
0
84.684
rad
s
=
(32b)
allows determining the transfer function of non-
integer order of the analysed pneumatic system. The
pulsatance and the damping ratio of the tested system
can be determined directly from the obtained step
response. The transfer function of non-integer order of
the tested system presents the following dependence
(32):
( )
( )
2
7171.38
7.1135 7171.38
v
vv
Gs
ss
=
++
(32)
Using the dependence (33):
( )
( )
( )
( )
22
00
2 1 1
,2 1 0
0
2
!
v
k
vk v vk k v
v v vk
k
hs
t E t
k
+ + + −
++
=
=
−
=−
(33)
we obtain a relationship describing the step response
of non-integer order of the analysed pneumatic
system:
( )
( )
( )
( )
22
,2 1
0
7171.38 7171.38
7.1135
!
v
k
vk v k v
v v vk
k
hs
t E t
k
+
++
=
=
−
=−
(34)
The equation (34) was simulated with the
MATLAB software and as a result the step response of
the non-integer order of the tested pneumatic system
inside the measuring chamber was obtained, the
graph of which is shown in Figure 11.
Figure 11. The step response of the measuring system in the
transducer measuring chamber
The step response shown in Figure 11, resulting
from the simulation of equation (35), for the
parameter v=1 (F_1) coincides with the graph
determined by means of the differential equation of
integer order (C_1) and the graph of the step response
obtained experimentally. This means that the model
has been correctly designated. Decreasing the order of
the derivative causes a decrease in the amplitude of
the step response.
( )
( )
( )
( )
( )
( )
v v v
G j P jQ
=+
(35)
The spectral transfer function of non-integer order
of the tested transducer was obtained by using the
experimentally determined values (31a) and (31b) in
equations (36a) and (36b), which is the real and
imaginary part of the spectral transfer function (35).
( )
( )
( )
( ) ( )
2 2 3 4
0 0 0
22
2 2 2
0 0 0
cos 2 cos
2
cos 2 cos sin 2 sin
22
v
vv
v v v v
P
v
v
vv
vv
=
++
=
+ + + +
(36a)
( )
( )
( )
( ) ( )
2 2 3
00
22
2 2 2
0 0 0
sin 2 sin
2
cos 2 cos sin 2 sin
22
v
vv
v v v v
Q
v
v
vv
vv
=
+
=
+ + + +
(36b)
The equation describing the logarithmic amplitude
characteristic can be determined from the equation
(27):
842
( )
( )
( )
( )
( )
( )
22
20log
v v v
L P Q
=+
(37)
The equation describing the logarithmic phase
characteristic is given by the dependence (38):
( )
( )
( )
( )
2
2
7171.38 sin 51013.295 sin
2
7171.38 cos 51013.295 cos 51428689.039
2
vv
v
vv
v
v
arctg
v
v
+
=−
++
(38)
After simulating equation (37), with the use of
equation (36a), a logarithmic amplitude response of
the non-integer order of the tested measurement
system was obtained for different values of the
parameter v, which is shown in Figure 12.
Logarithmic phase response (Figure 12) was obtained
by simulating equation (36b). Against the response
obtained by means of simulation of the mathematical
model in which the actual parameters of the tested
transducer were applied, the transducer response
obtained experimentally was presented.
Figure 12. Logarithmic amplitude response determined
experimentally and theoretically for different values of the
parameter v: red – laboratory results.
Figure 13. Logarithmic phase response determined
experimentally and for different values of the parameter v:
red – laboratory results.
The test stand facilitates changing the rotation
speed of the rotor and positioning it so as to take
measurements. The speed change allows setting the
opening frequency of the valve and thus changing the
air pressure in the measuring chamber. The regulation
of the opening speed of the valve gives the same effect
as feeding the system with a pneumatic generator
with adjustable frequency, which facilitates obtaining
logarithmic frequency response.
Changes in the pressure values in the measuring
chamber were recorded for different rotor speed
values in the range (100rpm-3200rpm), which
corresponds to the frequency ω(104.717[rad/s]-
335.093[rad/s]).
The input signal is the pressure in the intake pipe
p0, the output signal is the pressure in the transducer
chamber (Figure 1). Logarithmic phase response was
determined by measuring the phase shift between the
output and input signal for each set motor shaft
speed.
The minimum rotation speed of the internal
combustion engine used in the tests is 1000rpm. This
limitation made it impossible to obtain experimentally
the full frequency response shown in Figures 12 and
13. The obtained responses were made from
frequency ω=104.717[rad/s] to ω=335.093 [rad/s].
By comparing the obtained responses with the
responses of non-integer order for the parameter v=1,
it can be stated that the tested transducer is of a
slightly smaller order than the second order
oscillating element. Experientially determined
frequency responses are included between the
simulated frequency response for parameter v=0.98
and v=1. This means that the transducer should be
modelled with the equations of non-integer order. It
can therefore be concluded that the description with
the classical method would be inaccurate.
The presented simulation studies were performed in the
MATLAB development environment (manufacturer: The
MathWorks). The authors of the paper declare that, using
the above mentioned trademark, they did so only with
reference to this publication and with such an intention
that it would be for the benefit of the trademark holders but
without the intention of infringing the trademark.
7 CONCLUSIONS
The pneumatic system, analysed in the paper in the
part on simulation, represents the second order
damping oscillator with damping ratio ξ<1, which
means that the characteristic equation of the model
does not have real solutions. Therefore, the authors
were required to develop the original method for
determining the relationships describing the time and
frequency responses for dynamic systems described
with fractional calculus. In the construction of the
mathematical model of the analyzed dynamic system,
the definition of Riemann-Liouville differ-integral (of
fractional order) was used.
The paper presents the results of the laboratory
tests of the pressure transducer, which was described
with a mathematical model. The analysis of the
dynamic properties of the model in terms of time and
frequency was conducted. The parameters of the
tested pressure transducer were determined
experimentally - the damping ratio ξ and the
frequency ω0, which were used to determine the
transfer function of integer and non-integer order.
Having the knowledge of transfer function, the step
response was determined as well as logarithmic
843
amplitude response and phase response of integer
and non-integer order. It was found out that the time
response obtained experimentally, coincides with the
response determined by the developed model of non-
integer order of the transducer for the parameter v=1.
This confirms the correctness of the designated model.
Frequency responses obtained experimentally
differ slightly from the responses obtained in
computer simulation. The logarithmic amplitude
response and phase response obtained experimentally
are of the order of the oscillating element for the
parameter v of the interval 0.98<v<1.
In the real system shown in the paper, the pressure
in the transducer chamber was measured at the outlet
of the air into the combustion chamber, where, at the
moment of aspiration of air through the engine, the air
reaches the speed of sound. Such conditions may
account for a slight decrease in the order of the
oscillating element under testing.
The analysis of the logarithmic amplitude response
of the models presented in the paper shows that the
local maximum present in these characteristics is
dependent on the order of the derivative and the
bigger the amplitude, the higher the order of the
derivative. For the parameter v=1 (classical model) the
amplitude reaches the maximum at the resonant
frequency for damping ξ<1. With the decreasing order
of the derivative, the increase in the resonant
frequency of the circuit can be observed.
Fractional calculus is particularly useful in
building dynamic models of mathematical systems
working in conditions that cannot be described with
differential equations of integer orders. This can be
deduced by analyzing systems such as the long
electric line of infinitely large length or the
supercapacitor of a few thousand Farads, which are
now also described with fractional calculus.
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