Paweł Lindstedt, Karol Golak
Premises of Parametrical Assessment of Turbojet Engine in Flight Regulation Condition During Ground Test
PREMISES OF PARAMETRICAL ASSESSMENT OF TURBOJET ENGINE
IN FLIGHT REGULATION CONDITION DURING GROUND TEST
Paweł LINDSTEDT*, Karol GOLAK*
*Faculty
of Mechanical Engineering,
Engineering Bialystok University of Technology, ul. Wiejska 45 C, 15-351
351 Bialystok,
Bialystok Poland
[email protected], [email protected]
Abstract: The article presents the theoretical bases of new
ew parametrical method of turbojet engine technical condition assessment. In this
method, engine technical condition is described by one (in other methods four are used) comprehensive model (binding engine input
i
– signals p2 and mp and engine output - n and p4 signals) with unique feature, that engine operation quality during ground
groun tests will provide necessary data on its performance in flight. The changes occurring in turbojet engine during its exploitation will be measurable by comparison
of standard model with parameters obtained from experiment (ground test).
Key words: Regulation, Computer Simulation, Turbine Jet Engine, Parametric Diagnostics, Ground Tests
1. INTRODUCTION
Proper regulation of turbine turbojet engines as well as other
objects is a necessary condition for safe usage admission. CurCu
rently, in process of engine performance signals courses and their
quality indicators values are researched in precisely determined
moments during ground tests. Such method of engine perforperfo
mance assessment is unreliable due to differences between
environment (temperature,
erature, pressure) influencing engine during
ground tests and in flight as well as impossibility to imitate noises,
usually unknown, affecting engine in flight during ground tests.
This may cause a situation where proper regulation during ground
tests may not provide sufficient utilitarian value for engine in flight.
Hence the necessity of finding new researching method allowing
engine performance determined during ground tests to provide
data on its performance in flight. One of such methods is comprecompr
hensive
ve (simultaneous analysis of four basic signals resulting from
engine operation), parametrical (engine performance is described
by 32 parameters) method of turbojet engine regulation condition
assessment.
2. THEORETICAL
HEORETICAL BASIS OF PARAMETRICAL ASSESSMENT
OF AIRCRAFT
RCRAFT ENGINE REGULATION CONDITION
DURING GROUND TESTS REFLECTING ITS STATE
IN FLIGHT
be solved by transforming signals into system parameters such
as amplification coefficients, time constants. Obtained parameters
allow to assess the value of other, unknown parameters that occur
in flight.
Simplified diagram of engine rotational speed regulation
regu
system is presented on Fig. 1.
Fig. 1. Simplified diagram of aircraft engine regulation system: GS- engine
transfer function, GR – regulator trans function, w – input function,
u – signal of influence of regulator onto object,
object z – interference,
y – applied signal (e.g. rotational speed),
speed) x – object incentive
signal, e – deviation signal
In order to assess the engine operation,
o
transfer functions
of closed-loop
loop system for an input function HW (1) (ground tests)
and of closed-loop
loop system for interference HZ (2) in flight tests
(Pełczewski, 1980; Piety, 1998):
Currently, during aircraft regulation condition assessment,
quality indicators of engine signals courses determined during
ground tests are of major significance. Howeverr these are often
inadequate to in flight indicators due to noise and environment
changes. Hence the need occurred to supplement the quality
indicators of signals courses determined during ground tests with
additional parameter – regulation potential, obtained
obta
from equation of state binding system operation qualityy and its
it technical
condition. (Balicki and Szczeciński, 2001; Gosiewski and Paszkowski, 1995; Lindstedt 2002, 2009). Noticeably, this problem may
44
ೄ ೃ
ೄ ೃ
ೄ
ೄ ೃ
(1)
(2)
Noticeably system ground test transfer function may be multimult
plied by controller transfer function reciprocal GR of given test,
and thus, by transfer functions determined during ground tests,
obtain the transfer function describing engine in flight.
∙
ೃ
(3)
This gives base for assessment
assessm
of regulation conditions
of turbine turbojet engine in flight based on its ground tests
(Lindstedt 2002, 2009).
acta mechanica et automatica, vol.6 no.1 (2012)
3. THEORETICAL FUNDAMENTALS FOR JOINT
CONSIDERATION OF ENGINE REGULATION CONDITION
ASSESSMENT MODELS
Four basic signals n – rotational speed, p2 – pressure behind
the compressor, mp- mass intensity of fuel flow, p4 – pressure
in engine nozzle, are considered in process of engine regulation
condition assessment (Fig. 2) (Lindstedt, 2009; Staniszewski,
1980; Szczeciński, 1965; Szevjakow, 1970).
‫ܩ‬௞௢௠௣௟௘௞௦ (‫= )ݏ‬
∆
∆
∆ ∆
(13)
Using inverse Laplace transform following is determined:
(Osiowski, 1981; Szabatin, 2000).
݃௞௢௠௣௟௘௞௦ ሺ‫ݐ‬ሻ ∗ ∆‫݌‬ସ ∗ ∆‫݌‬ଶ = ∆݊ ∗ ∆݉௣
(14)
As seen from dependences (13), (14), one comprehensive
engine model exists that corresponding to 4 classical models
applied hitherto in engine regulation condition assessment process. This model is a transfer function (13) or dependence of
courses n and mp tangle (14). Tangle model (14) is difficult to
solve. Model (13) is more suitable for further analysis. In case of
adopting model in form of transfer function (13), transition can be
made from space of variable s to space of frequency ω, hence
obtaining ability to analyze signals basing on power densities
and cross power densities for signals recorded during engine test.
Transfer function Gkompleks(jω) argument may be determined
from dependence (13):
‫ܩ݃ݎܣ‬௞௢௠௣௟௘௞௦ ሺ݆߱ሻ = ∆߮௡௣ ௣௠ = ∆߮௡௣ ସ − ∆߮௣௠ (15a)
Fig. 2. Engine regulation diagram (where W – intake, S – compressor,
KS – combustion chamber, T – turbine, D – nozzle, outlet,
1,2,3,4,5 – characteristic sections)
‫ܩ݃ݎܣ‬௞௢௠௣௟௘௞௦ ሺ݆߱ሻ = ‫݃ݎܣ‬
(15b)
Each relation between main signal, described by following
transfer functions, are researched in order to assess engine performance (Balicki, Szczeciński, 2001, Lindstedt, 2002):
∆௡
‫ܩ‬ଵ௠ =
‫ܩ‬ଵ௣ =
∆௡
(5)
∆௣
‫ܩ‬ଶ௠ =
‫ܩ‬ଶ௣ =
(4)
∆௠
∆௣
(6)
∆௠
∆௣
(7)
∆௣
Assumingly, model in form of four transfer functions might
be reduced to one comprehensive model with desired feature that
allows engine performance determined during ground tests
to provide data on its quality in flight.
After removing output signals ∆n and ∆p4 from equations
(4)÷(7), the following is obtained:
∆௣
‫ܩ‬ଵ௠ ଵ௣ =
∆௠
‫ܩ‬ଶ௠ ଶ௣ =
∆௠
(8)
∆௣
(9)
Subsequently, input signals ∆mp and ∆p2 are removed and the
following is obtained from equations (4)÷(7) as well:
∆௡
‫ܩ‬ଵ௡ଵ௣ =
∆௣
‫ܩ‬ଶ௡ଶ௣ =
∆௣
(10)
∆௡
(11)
In the end, model is created in form of quotient of relations
of output signals transform to relation of input signals transform:
‫ܩ‬௞௢௠௣௟௘௞௦ ሺ‫ݏ‬ሻ =
ீ
ீ =
ீ
ீ (12)
Subsequently, transfer function Gkompleks(jω) modulus square
may be determined:
|‫ܩ‬௞௢௠௣௟௘௞௦ ሺ݆߱ሻ|ଶ =
=
஺
஺ (16)
where: S – power spectral density or cross power spectral density,
A2(ω) – amplification square, φ(ω) – phase shift.
Signals power S spectral density functions is determined basing on their correlation functions with Fourier transform applied.
Therefore, when courses n(t), p4(t), p2(t) and mp(t) are known,
determination of their correlation and cross correlation functions
and, subsequently, power spectral densities and cross power
spectral densities should prove no difficulty. In the end, transfer
function Gkompleks(jω) and, then, signals amplification square
| Gkompleks(jω) |2 might be determined. Similarly, basing on cross
power spectral density, phase shift ∆߮௡௣௣ ௠ is determined
(‫ܣ‬ଶ௡௡௣௣ and ∆߮௡௣௣௠ being values physically interpretable).
(Osiowski, 1981; Szabatin, 2000).
4. COMPREHENSIVE, PARAMETRICAL ANALYSIS
OF ENGINE REGULATION CONDITION
BASING ON ENGINE EXPLOITATION RESEARCH
Recorded courses of input and output signals of turbojet engine are shown in Fig. 3. and Fig. 4. (Pawlak et al., 1996).
Additionally, assumption is made that DProb course corresponds with signal mp course, signal P4 with signal p4, signal N
with signal n and P2 with signal p2.
Ranges for determination of amplification value
| Gkompleks(jω) |2, as well as phase shift ∆߮௡௣௣ ௠ were determined dividing signal N onto sections as seen in Fig. 5.
and Tab. 1.
Taking dependences (10) and (12) into consideration, the following is obtained:
45
Paweł Lindstedt, Karol Golak
Premises of Parametrical Assessment of Turbojet Engine in Flight Regulation Condition During Ground Test
autocorrelation and cross-correlation
correlation functions, bilateral Fourier
transform was used. Subsequently
uently engine models in form
of amplification | Gkompleks(jω) |2 and phase shift ∆రమು during ground test were determined in general form of:
| | ೔ఴ ఴ ೔ళ ళ ೔ల ల ೔ఱ ఱ ೔ర ర ೔య య ೔మ మ ೔భ ೔బ
య
మ
೔ఴ ఴ ೔ళ ళ ೔ల ల ೔ఱ ఱ ೔ర ర ೔య
೔ ೔మ ೔భ (19)
∆రమು | ೔ఴ ఴ ೔ళ ళ ೔ల ల ೔ఱ ఱ ೔ర ర ೔య య ೔మ మ ೔భ ೔బ
ర
య
మ
೔ఴ ఴ ೔ళ ళ ೔ల ల ೔ఱ ఱ ೔ర
ర ೔య ೔మ ೔భ (20)
Fig. 3. Courses of normalized engine input signals
(signal observation time 350 – 500 [s])
Fig. 5. Courses of normalized engine output signals
(signal observation time 350 – 500 [s])
Changes occurring in engine during its exploitation may be
determined
termined by determining percentage values of each δ parameter (24) deviation from approximate µ (21) for each signal course
types (Fig. 4.) and comparing them to variability coefficient v (22)
presented as a percentage and calculated for standard deviations
σ(23), 2σ and 3σ.
Fig. 4. Courses of normalized engine output signals
(signal observation time 350 – 500 [s])
Tab. 1. Signal n ranges for beginning and end of signal course types
Signal n range for the
beginning of signal
<0,0.33>
<0,0.33>
(0.33,0.67>
(0.33,0.67>
(0.67,1>
(0.67,1>
Signal n range
for the end of signal
(0.33,0.67>
(0.67,1>
<0,0.33>
(0.67,1>
<0,0.33>
(0.33,0.67>
Signal type
1
2
3
4
5
6
(21)
೗
೗
∙ 100%
೔೗ ೗
೗
∙ 100%
(22)
(23)
(24)
where: x – parameter a, b, c or d; l – parameter number.
Results are presented as percentage of regulation potential for each parameter.
Recorded characteristics of basic signals n(t),, p4(t), p2(t) i mp(t)
were divided onto sections according
ng to assumptions presented in
table 1. Hanning window was put on each of obtained sections.
For obtained signal courses autocorrelations and cross correlacorrel
tions of signals n and p4 as well as p2 and mp were calculated.
Obtained charts of autocorrelations and cross-correlations
correlations were
approximated with precision of R2>0.995
995 (described by determinadetermin
tion coefficient) using 4 degree polynomials in general form of:
(17)
(18)
In order to determine function spectral power from obtained
46
∑೙
೔సభ ೔೗
೗ ೗
೗
∙ 100%
(25)
Results of undertaken research in form of regulation potential
of parameters from 5 tests for each of six signal types are prepr
sented in Tab. 2. for amplification as well as in Tab. 3. for phase
shift.
Engine condition is described by 34 parameters with specific
value. For various courses, different configurations and parameter
values are obtained. During consecutive tests with identical propr
gram, parameters values should remain unchanged. Regulation
changes applied during engine ground test, expressed as change
of regulator transfer function reciprocal 1/GR may be introduced
into model and ultimately allow determination of engine parameparam
ters in flight.
acta mechanica et automatica, vol.6 no.1 (2012)
Tab. 2. A2 model parameters
type
1
2
3
4
5
6
nr
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
a0
43
128
86
271
-29
297
62
40
77
24
110
-14
24
276
103
299
42
38
58
64
152
154
149
-100
146
57
45
124
282
-8
a1
158
59
114
-64
234
-97
153
152
121
171
85
222
174
-72
90
-98
150
173
126
149
48
46
51
300
54
181
137
103
-92
172
a2
40
160
85
253
-38
297
27
60
79
37
117
-27
29
268
114
291
64
11
100
35
152
154
149
-100
146
0
76
77
292
55
a3
166
9
118
-33
240
-93
195
117
126
154
86
227
169
-70
87
-68
110
212
59
187
49
46
51
300
55
205
119
137
-91
130
a4
16
249
71
180
-15
286
-3
113
52
52
69
25
38
297
72
188
146
-36
205
-4
151
155
149
-100
145
30
60
49
299
62
a5
195
-67
145
187
40
-84
169
73
183
159
178
-76
51
171
176
191
-13
201
-28
149
151
150
150
-100
149
-90
178
88
158
165
a6
104
50
95
-27
277
234
186
99
-28
9
57
261
157
-31
56
-57
125
186
33
214
49
47
51
300
53
259
71
144
-46
73
a7
18
250
69
179
-16
284
-6
117
49
56
65
27
43
298
67
173
150
-34
216
-4
151
153
149
-100
147
19
69
44
296
71
a8
247
-43
149
29
118
-94
149
104
173
168
199
-71
79
90
202
-15
54
260
34
168
49
48
51
300
52
-81
172
63
171
175
b1
2
0
63
245
190
193
223
9
109
-34
60
249
182
-21
30
202
18
141
-51
191
130
96
157
-87
204
255
45
177
-14
37
b2
184
220
129
-37
4
45
-43
200
72
226
138
-47
17
225
167
3
184
55
251
7
62
81
45
295
16
-53
154
20
215
163
b3
46
-50
85
219
201
64
263
0
164
10
71
240
186
-37
40
188
14
153
-51
196
144
134
153
-99
168
246
48
188
-20
39
b4
81
291
86
34
8
261
-15
152
10
93
86
7
5
278
123
37
190
25
249
-1
53
57
48
300
42
47
116
-49
253
133
b5
218
-74
150
64
143
-92
148
98
179
167
205
-69
88
77
199
39
19
244
1
196
149
149
150
-100
151
-84
168
71
175
170
b6
-59
139
42
233
145
276
136
49
51
-12
37
270
159
12
22
242
34
96
-43
171
50
48
50
300
52
272
37
154
5
32
b7
82
292
82
35
9
276
0
134
15
75
79
15
14
287
106
35
184
25
253
3
151
152
149
-100
148
20
130
-40
245
144
b8
237
-55
156
41
121
-94
147
105
176
167
200
-72
83
89
200
2
44
258
19
177
49
48
51
300
52
-81
167
64
181
169
a3
188
-4
101
-17
233
-100
161
142
147
150
94
237
144
-71
96
-44
84
219
36
204
144
127
155
-98
173
213
118
150
-85
105
a4
18
247
51
190
-6
300
40
61
51
49
64
20
47
297
72
169
150
-31
219
-7
56
90
46
294
14
14
60
46
295
84
a5
172
-29
207
-13
163
-100
153
142
152
153
180
-93
103
141
169
91
32
251
-36
162
52
47
47
300
53
-79
174
59
178
169
a6
70
67
-35
270
128
262
172
19
28
19
58
280
119
-18
61
114
33
177
-52
228
148
139
153
-100
159
259
67
154
-36
56
a7
25
269
46
158
2
300
44
58
49
49
62
24
49
298
68
121
162
-23
242
-2
86
90
85
155
83
0
72
43
289
96
a8
179
-5
190
-39
175
-100
151
145
151
152
191
-81
128
75
187
-44
90
248
43
164
111
109
113
53
114
-76
170
52
178
175
b1
89
-26
71
282
85
257
174
-7
56
20
69
248
176
-32
39
241
22
127
-43
153
43
11
56
292
97
237
87
179
-47
44
b2
99
249
115
-66
102
111
-84
215
102
155
125
-42
20
240
157
-39
179
70
244
46
156
180
146
-96
114
-33
110
18
250
156
b3
125
-84
115
221
122
-87
209
100
147
131
89
222
192
-57
55
233
19
138
-46
156
45
30
52
298
75
219
104
188
-59
48
b4
34
292
30
105
39
299
33
68
49
51
52
87
-15
285
91
-10
184
38
251
36
153
161
149
-99
137
145
-12
10
265
91
b5
179
-18
193
-26
173
-100
152
143
152
154
192
-81
136
71
183
22
40
261
4
172
49
48
50
300
54
-78
175
56
176
170
b6
39
102
21
293
45
276
142
5
56
21
41
284
129
17
30
263
34
95
-32
139
150
145
151
-100
154
263
51
164
-11
33
b7
39
296
36
88
41
300
44
59
48
49
47
92
-2
290
74
-4
176
36
258
34
86
88
85
158
83
130
8
-11
269
104
b8
179
-12
193
-32
172
-100
151
145
152
152
190
-82
129
77
186
-14
75
255
14
169
112
110
113
50
115
-76
166
52
187
170
Tab. 3. Regulation potential for ∆ϕ model
type
1
2
3
4
5
6
nr
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
a0
27
95
111
279
-12
300
54
47
54
46
100
-15
45
283
87
300
54
42
43
61
60
69
43
298
29
64
34
93
293
16
a1
178
84
85
-69
222
-100
150
151
146
153
94
228
150
-76
104
-98
134
174
132
157
141
130
156
-98
171
181
141
129
-97
145
a2
15
152
115
250
-32
300
44
53
54
48
109
-37
54
271
103
284
85
4
108
19
58
71
45
298
29
-5
75
58
291
81
47
Paweł Lindstedt, Karol Golak
Premises of Parametrical Assessment of Turbojet Engine in Flight Regulation Condition During Ground Test
REFERENCES
5. SUMMARY
Comprehensive model for turbojet engine regulation condition
assessment was executed. This model allows calculating amplification |Gkompleks(jω)|2 and phase Shift ∆߮௡௣௣௠ , that may be
physically interpreted. Engine condition is described by 34 parameters of specific value, assuming various configurations for different ground tests signal courses. Obtained parameters present
regulation condition of turbojet engine. Changes in engine occurring during its exploitation are expressed as parameters
|Gkompleks(jω)|2 and ∆߮௡௣௣௠ changes and by changes of regulator adjustment. Parameters of ground model and regulations
may be the basis to determine engine in flight model according
to dependence HZ=HW·1/GR.
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This work was realized within the Dean’s Projects: Paweł Lindstedt
W/WM/4/2009, Karol Golak W/WM/9/2011 in Faculty of Mechanical
Engineering on Bialystok University of Technology.
.
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