Design Project Report - Mechanical...
Transcript of Design Project Report - Mechanical...
![Page 1: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/1.jpg)
Prepared By: Kushal Shah
Advisor: Professor John Hodgkinson
Graduate Advisor: Colin Alexander Sledge
12 June 2014
DESIGN PROJECT REPORT: Longitudinal and lateral-directional stability augmentation of
Boeing 747 for cruise flight condition.
![Page 2: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/2.jpg)
DESIGN PROJECT REPORT: 1
I. OBJECTIVE:.................................................................................................................................................................... 2
HISTORICAL PERSPECTIVE: ................................................................................................................................................................ 2
PURPOSE: ..................................................................................................................................................................................... 2
DESIGN SPECIFICATIONS .................................................................................................................................................................. 2
II. AIRCRAFT MODEL: ........................................................................................................................................................ 3
DESCRIPTION OF AIRCRAFT. .............................................................................................................................................................. 3
DESCRIPTION OF CHOSEN FLIGHT CONDITION: ..................................................................................................................................... 3
AIRCRAFT DATA AND DERIVATIVES DETAILS: ........................................................................................................................................ 3
LONGITUDINAL DYNAMICS โ ............................................................................................................................................................. 4
LATERAL-DIRECTION DYNAMICS โ ..................................................................................................................................................... 5
III. ASSESSMENT OF UNAUGMENTED DYNAMICS: ............................................................................................................. 6
LONGITUDINAL MODES ASSESSMENT ................................................................................................................................................. 6
LATERAL MODES ASSESSMENT .......................................................................................................................................................... 8
IV. STABILITY AUGMENTATION DESIGN: .......................................................................................................................... 10
LONGITUDINAL STABILITY AUGMENTATION ........................................................................................................................................ 10
LATERAL STABILITY AUGMENTATION: ............................................................................................................................................... 13
V. SIMULATION AND PERFORMANCE ASSESSMENT ........................................................................................................ 16
LONGITUDINAL MODES SIMULATION AND PERFORMANCE: ................................................................................................................... 16
LATERAL MODES SIMULATION AND PERFORMANCE:............................................................................................................................ 19
VI. REFERENCES: ............................................................................................................................................................... 21
APPENDIX: ........................................................................................................................................................................... 22
LONGITUDINAL MODE REQUIREMENTS: .............................................................................................................................. 23
SHORT PERIOD MODE REQUIREMENTS: ............................................................................................................................................ 23
PHUGOID MODE REQUIREMENTS: ................................................................................................................................................... 23
LATERAL MODE REQUIREMENTS:......................................................................................................................................... 24
ROLL MODE REQUIREMENTS: ......................................................................................................................................................... 24
SPIRAL MODE REQUIREMENTS: ....................................................................................................................................................... 24
DUTCH ROLL MODE REQUIREMENTS: ............................................................................................................................................... 24
![Page 3: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/3.jpg)
DESIGN PROJECT REPORT: 2
I. Objective:
Historical Perspective:
The emergence of fly-by-wire and digital control of an airplane has made understanding of flying
qualities and control & stability more crucial and essential. Many aircraft developments haven been
affected by pilot induced oscillations (PIOs) and other handling difficulties due to insufficient
understanding of flying qualities (Hodgkinson). This change in the industry provided the motivation for
this design study.
Purpose:
The objective of this design project is to design a longitudinal and lateral-directional stability
augmentation system for Boeing 747 for flight condition three (see section 2 for more details for this
aircraft and the flight condition). This augmentation will allow the pilot to reduce PIOs and fly the plane
safely and more comfortably. Furthermore, the secondary objective is to analyze the effect of gust on
the aircrafts stabilities.
Design Specifications
As had been mentioned, the primary purpose is to obtain โgoodโ flying qualities for Boeing 747
at flight condition three. The handling quality is characteristic of the combined performance of the pilot
and vehicle acting together as a system in support of an aircraft role (Hodgkinson). These flying handling
qualities are defined by Cooper โ Harper scale and they are in three different levels: Level 1, Level 2, and
Level 3.
In short, for this design study, the design specification is to have level 1 flying qualities for all the
longitudinal and lateral modes for given flight condition. How these qualities and their level 1
requirements are defined are summarized in the tables and charts in the appendix A.
![Page 4: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/4.jpg)
DESIGN PROJECT REPORT: 3
II. Aircraft Model:
Description of Aircraft.
Boeing 747, also known as Jumbo Jet or Queen of the Skies, is a wide-body double decker
commercial airliner and cargo transport aircraft. This is a heavy commercial transport aircraft, also
known as class III aircraft. The Boeing 747 is two-aisle airliner with four wing-mounted engines. It can
carry 400 passengers. Its first flight was in
1969. Its length is 231ft 10in, wingspan is 211ft
55in, and height is 63ft 8 in. Its cruise speed is
Mach 0.85 (567 mph) and it cost approximately
$250 million. Its maximum range is 7260
nautical miles. Seating capacity is more than 366 with a 3โ4โ3 seat arrangement in economy class and a
2โ3โ2 arrangement in first class on the main deck. The upper deck has a 3โ3 seat arrangement in
economy class and a 2โ2 arrangement in first class.
Description of Chosen Flight Condition:
For this study the flight condition that was chosen was the flight altitude of 40,000 ft. and Mach
number of 0.900 for take of weight if 636,636 lbs. This flight condition is the cruise condition of an
airplane.
Aircraft Data and Derivatives Details:
![Page 5: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/5.jpg)
DESIGN PROJECT REPORT: 4
Longitudinal Dynamics โ
Linearized State Space Aircraft Dynamics Matrix Model: (Input โ Elevator) ๐ = ๐ด๐ + ๐ต๐ข
[
๐ขฮฑ๏ฟฝ๏ฟฝ
ฮธ
] =
[ Xu Xฮฑ 0 โgcos(ฮธ1)
Zu
U1
Zฮฑ
U1
U1 + Zq
U1
โ๐sin(ฮธ1)
U1
Mu Mฮฑ Mq + ๐๐ผ.
00 0 1 0 ]
[
๐ขฮฑ๐ฮธ
] +
[ ๐๐ฟ๐
๐๐ฟ๐
๐1
๐๐ฟ๐
0 ]
โ โ๐ฟ๐
[
๐ขฮฑ๏ฟฝ๏ฟฝ
ฮธ
] = [
โ0.0218 1.2227 0 โ32.1850โ0.0001 โ0.3892 1 โ0โ0.0001 โ1.6165 โ0.5463 0
0 0 1 0
] [
๐ขฮฑ๐ฮธ
] + [
0โ0.0211โ1.2124
0
] โ๐ฟ๐
Output State Space Model:
For short period mode, pitch rate (q), angle of attack (ฮฑ) and normal load factor in Z (๐๐ง) can be used
to close the loop and obtain required handling qualities.
For phugoid mode, pitch angle (ฮธ), and forward speed (๐ข) can be used to close the loop and obtain
required handling qualities.
๐ = ๐ถ๐ + ๐ท๐ข
[ ๐ฮฑ๐๐ง
ฮธ๐ข ]
=
[
0 0 1 00 1 0 0
โZu
๐โ
Zฮฑ
๐โ
Zq
๐sin(ฮธ1)
0 0 0 11 0 0 0 ]
[
๐ขฮฑ๐ฮธ
] +
[ 00000]
โ๐ฟ๐
[ ๐ฮฑ๐๐ง
ฮธ๐ข ]
=
[
0 0 1 00 1 0 0
0.0018 10.5330 0 00 0 0 11 0 0 0]
[
๐ขฮฑ๐ฮธ
] +
[ 00000]
โ๐ฟ๐
Summary:
๐จ๐ณ๐๐๐ ๐ฉ๐ณ๐๐๐ ๐ช๐ณ๐๐๐ ๐ซ๐ณ๐๐๐
[ Xu Xฮฑ 0 โgcos(ฮธ1)
Zu
U1
Zฮฑ
U1
U1 + Zq
U1
โ๐sin(ฮธ1)
U1
Mu Mฮฑ Mq + ๐๐ผ.
00 0 1 0 ]
[ ๐๐ฟ๐
๐๐ฟ๐
๐1
๐๐ฟ๐
0 ]
[
0 0 1 00 1 0 0
โZu
๐โ
Zฮฑ
๐โ
Zq
๐sin(ฮธ1)
0 0 0 11 0 0 0 ]
[ 00000]
[
โ0.0218 1.2227 0 โ32.1850โ0.0001 โ0.3892 1 โ0โ0.0001 โ1.6165 โ0.5463 0
0 0 1 0
] [
0โ0.0211โ1.2124
0
]
[
0 0 1 00 1 0 0
0.0018 10.5330 0 00 0 0 11 0 0 0]
[ 00000]
![Page 6: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/6.jpg)
DESIGN PROJECT REPORT: 5
Lateral-Direction Dynamics โ
Linearized State Space Matrix Model: (Inputs โRudder, & Aileron):
๐ฅ = ๐ด๐ฅ + ๐ต๐ข
[ โ๏ฟฝ๏ฟฝโpโ๏ฟฝ๏ฟฝโฯ]
=
[ ๐๐ฝ
๐1
Yp
U1
๐๐
๐1
โ 1 โg
U1โ cos(๐1)
L๐ฝ Lp Lr 0
N๐ฝ Np Nr 0
0 1 0 0 ]
[
โ๐ฝโpโ๐โฯ
] +
[ ๐๐ฟ๐
๐1
๐๐ฟ๐
๐1
๐ฟ๐ฟ๐ ๐ฟ๐ฟ๐
๐๐ฟ๐ ๐๐ฟ๐
0 0 ]
[ฮฮดaฮฮดr
]
[ โ๏ฟฝ๏ฟฝโpโ๏ฟฝ๏ฟฝโฯ]
= [
โ0.0640 0 โ1 0.0370โ1.2555 โ0.4758 0.2974 01.0143 0.0109 โ0.1793 0
0 1 0 0
] [
โ๐ฝโpโ๐โฯ
] + [
0 0.00430.1850 0.2974
โ0.0135 โ0.45890 0
] [ฮฮดaฮฮดr
]
Output State Space Model:
For roll mode, roll rate(โp) can be used to close the loop and obtain required handling qualities.
For spiral mode, bank angle(โฯ) can be used to close the loop and obtain required handling qualities.
For Dutch roll mode, yaw rate (โr), sideslip angle (โ๐ฝ) and normal load factor (โn๐ฆ) can be used to close
the loop and obtain required handling qualities.
๐ = ๐ถ๐ + ๐ท๐ข
[ โ๐โrโ๐ฝโ๐๐ฆ
โฯ ]
=
[ 0 1 0 00 0 1 01 0 0 0๐๐ฝ
๐0 โ
๐1
๐cos(๐1)
0 0 0 1 ]
[
โ๐ฝโpโ๐โฯ
] +
[ 0 00 00 00 00 0]
[ฮฮดaฮฮดr
]
[ โ๐โrโ๐ฝโ๐๐ฆ
โฯ ]
=
[
0 1 0 00 0 1 01 0 0 0
โ1.73 0 โ27.06 10 0 0 1]
[
โ๐ฝโpโ๐โฯ
] +
[ 0 00 00 00 00 0]
[ฮฮดaฮฮดr
]
Summary:
๐จ๐ณ๐๐ ๐ฉ๐ณ๐๐ ๐ช๐ณ๐๐ ๐ซ๐ณ๐๐
[ ๐๐ฝ
๐1
Yp
U1
๐๐
๐1
โ 1 โg
U1โ cos(๐1)
L๐ฝ Lp Lr 0
N๐ฝ Np Nr 0
0 1 0 0 ]
[ ๐๐ฟ๐
๐1
๐๐ฟ๐
๐1
๐ฟ๐ฟ๐ ๐ฟ๐ฟ๐
๐๐ฟ๐ ๐๐ฟ๐
0 0 ]
[ 0 1 0 00 0 1 01 0 0 0๐๐ฝ
๐0 โ
๐1
๐cos(๐1)
0 0 0 1 ]
[ 0 00 00 00 00 0]
[
โ0.0640 0 โ1 0.0370โ1.2555 โ0.4758 0.2974 01.0143 0.0109 โ0.1793 0
0 1 0 0
] [
0 0.00430.1850 0.2974
โ0.0135 โ0.45890 0
]
[
0 1 0 0
0 0 1 0
1 0 0 0
โ1.73 0 โ27.06 1
0 0 0 1]
[ 0 00 00 00 00 0]
![Page 7: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/7.jpg)
DESIGN PROJECT REPORT: 6
III. Assessment of Unaugmented Dynamics:
Longitudinal Modes Assessment
There are two modes that are associated with longitudinal mode: 1) Phugoid Mode and 2) Short
period mode. Their description, associated Eigen values and handling qualities are presented below.
Modes Description:
Mode Description
Phugoid Mode (P)
The Phugoid is a long-period, low frequency mode in which speed and
altitude are interchanged. The resulting oscillations are in pitch, speed,
altitude, and flight path, while the angle of attack remains roughly constant.
Short Period Mode (SP)
The short period is relatively rapid mode that governs the transient changes
in angle of attack, pitch, flight path and normal load factor that occur
following rapid control or gust inputs. Forward speed stays constant
Eigen Values Description: o Using the Matlab Disp (A) Following Results were obtained. (This performs Det|A - ๐๐ผ| )
Mode Eigenvalues Damping Frequency (rad/s)
Phugoid Mode (4D) ๐1,2 (๐ ) = โ0.00976 ยฑ 0.0328๐ ๐๐ = 0.285 ๐๐ = 0.0342
Short Period Mode (4D) ๐1,2 (๐๐) = โ0.469 ยฑ 1.27๐ ๐๐ ๐ = 0.347 ๐๐ ๐ = 1.35
Unaugmented Flying Handling Qualities:
Parameter Unaugmented Value Unaugmented Handling Quality
Short Period Mode
๐1,2 (๐ ๐) โ0.00976 ยฑ 0.0328๐ N/A
๐๐ ๐ 0.347 Level 2+3
๐๐ ๐(๐ ๐๐
๐ ) 1.35 Level 2+3
Phugoid Mode
๐1,2 (๐) โ0.00976 ยฑ 0.0328๐ N/A
๐๐ 0.285 Level 1 ๐๐
๐๐ ๐
0.0342
1.35= 0.025 < 0.1 Level 3
![Page 8: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/8.jpg)
DESIGN PROJECT REPORT: 7
o This table above show that most of the parameters have unaugmented values that are level 2 or 3.
Short period damping and frequency is level 2 and level 3 so it needs augmentation to achieve
level 1 handling qualities. Furthermore, for Phugoid mode, the frequency is low and needs to be
higher to meet the level 1 flying qualities. These assessments were made using the requirements
provided in the appendix.
Targeted Augmented Flying Handling Qualities:
o The following table describes what the values for different parameters should be to achieve level 1
flying qualities. These values were chosen so they meet the level 1 requirements. The
augmentation is well described in section IV of this report.
Parameter Unaugmented Value Unaugmented
Handling Quality Augmented (Target
value)
Augmented Handling Quality
Phugoid Mode
๐1,2 (๐ ๐) โ0.00976 ยฑ 0.0328๐ N/A N/A N/A
๐๐ ๐ 0.347 Level 2+3 ๐๐ ๐ = 0.45 ยฑ .05 Level 1
๐๐ ๐(๐ ๐๐๐ )
1.35 Level 2+3 ๐๐ ๐(๐ ๐๐๐ )
= 3 ยฑ 0.5 Level 1
Short Period Mode
๐1,2 (๐) โ0.00976 ยฑ 0.0328๐ N/A N/A N/A
๐๐ 0.285 Level 1 ๐๐ = 0.06 ยฑ .01 Level 1
๐๐
๐๐ ๐
0.0342
1.35= 0.025 < 0.1 Level 3
๐๐
๐๐ ๐> 0.12 ยฑ 0.01 Level 1
![Page 9: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/9.jpg)
DESIGN PROJECT REPORT: 8
Lateral Modes Assessment
There are two modes that are associated with longitudinal mode: 1) Phugoid Mode and 2) Short
period mode. Their description, associated Eigen values and handling qualities are presented below.
Modes Description: Mode Description
Roll Mode (R) Roll subsidence mode is simply the damping of rolling motion.
Spiral Mode (SM)
Spiral Mode is a slow recovery or divergence from a bank angle disturbance. The
military specification contains a requirement which prevents too - rapid
divergence.
Dutch Roll Mode (DR)
Dutch roll mode is the lateral-directional short period oscillatory mode. It
generally occurs at frequencies similar to those of the longitudinal short period
mode.
Eigen Values Description:
o Using the Matlab Disp (A) Following Results were obtained. (This performs Det|A - ๐๐ผ| )
Mode Eigenvalues Damping Frequency (rad/s) Time Constant (s)
Roll Mode (4D) ๐1 (๐ ) = โ0.510 ๐๐ = 1 ๐๐ = 0.510 ๐๐ = 1.967
Spiral Mode (4D) ๐1 (๐๐) = 0.00537 ๐๐๐ = โ1 ๐๐๐ = 0.00537 ๐๐๐ = 186
Dutch Roll (4D) ๐1,2 (๐ท๐ ) = โ0.107 ยฑ 1.01๐ ๐๐ท๐ = 0.106 ๐๐ท๐ = 1.02 ๐๐ท๐ = 9.24
Unaugmented Flying Handling Qualities:
o This table below show that most of the parameters have unaugmented values that are level 2 or 3.
Dutch roll frequency is level 1 but however, it is very close to minimum value of level 1, thus also
need good augmentation to ensure stability. In addition, spiral mode is unstable mode because it
has a positive Eigen value and thus, it also need augmentation to ensure stability. For the roll mode,
time constant is very slow and needs to be fast and responsive, therefore, this mode also needs a
augmentation to bring the time constant lower for faster level 1 response.
![Page 10: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/10.jpg)
DESIGN PROJECT REPORT: 9
Criteria Parameter Unaugmented Value Unaugmented Handling
Quality
Roll Mode (R)
- ๐1 (๐ ) โ0.510 N/A
- ๐๐ 1 -
- ๐๐ (๐ ๐๐
๐ ) 0.510 Level 2+3
1 ๐๐ (s) 1.967 Level 3
Spiral Mode (SM)
- ๐1,2 (๐๐) 0.00537 N/A
- ๐๐๐ โ1 -
- ๐๐๐ 0.00537 Level 3
5 ๐๐๐(s) 186 Level 1
Dutch Roll Mode (DR)
- ๐1,2 (๐ท๐ ) โ0.107 ยฑ 1.01๐ N/A
2 ๐๐ท๐ 0.106 Level 2
3 ๐๐ท๐ 1.02 Level 1
4 ๐๐ท๐ โ ๐๐ท๐ 0.108 Level 2
Targeted Augmented Flying Handling Qualities:
o The following table describes what the values for different parameters should be to achieve level 1
flying qualities. These values were chosen so they meet the level 1 requirements as shown in the
tables and charts in the appendix.
Criteria Parameter Unaugmented
Value Unaugmented
Handling Quality Augmented
(Target value) Augmented
Handling Quality
Roll Mode (R)
- ๐1 (๐ ) โ0.510 N/A N/A N/A
- ๐๐ 1 - 1 (for stability) -
- ๐๐ (๐ ๐๐
๐ ) 0.510 Level 2+3 1.25 ยฑ 0.025 Level 1
1 ๐๐ (s) 1.967 Level 3 0.8 ยฑ .01 Level 1
Spiral Mode (SM)
- ๐1,2 (๐๐) 0.00537 N/A N/A N/A
- ๐๐๐ โ1 - 1 (for stability) -
- ๐๐๐ 0.00537 Level 3 0.04 ยฑ 0.005 Level 1
5 ๐๐๐(s) 186 Level 1 25 ยฑ 3 Level 1
Dutch Roll Mode (DR)
- ๐1,2 (๐ท๐ ) โ0.107 ยฑ 1.01๐
N/A N/A N/A
2 ๐๐ท๐ 0.106 Level 2 0.3 ยฑ .05 Level 1
3 ๐๐ท๐ 1.02 Level 1 4 ยฑ 1 Level 1
4 ๐๐ท๐ โ ๐๐ท๐ 0.108 Level 2 1.2 ยฑ .55 Level 1
![Page 11: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/11.jpg)
DESIGN PROJECT REPORT: 10
IV. Stability Augmentation Design:
In this section, steps of augmentation is described. This augmentation will correct the
deficiencies that are needed to improve the handling qualities of the aircraft and meet the level 1
requirements.
Following steps were taken, in order to augment the system.
1) Assessment of Unaugmented results. (Done in previous section)
2) Choose target values for augmentation to achieve level 1 flying qualities. (Done in previous section)
3) Analyze the transfer functions and analyze their root loci individually and create a tentative table.
4) Using the summarized tentative table, choose the root loci to close and tune the gains to achieve level 1.
5) Generate A_Augumented Matrix
6) Analyze and verify the augmented Eigen values and flying qualities.
Longitudinal Stability Augmentation
Step 3) Analyze the transfer functions and analyze their root loci
1. Pitch Rate Response (ฮq) to Elevator Input (ฮฮดe):
2. Angle of Attack (ฮ ฮฑ) Response to Elevator Input (ฮฮดe)
![Page 12: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/12.jpg)
DESIGN PROJECT REPORT: 11
3. Load Response (ฮnz) to Elevator Input (ฮฮดe):
4. Pitch Angle (ฮ ฮธ) Response to Elevator Input (ฮฮดe):
5. Forward Speed (ฮu) Response to Elevator Input (ฮฮดe):
![Page 13: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/13.jpg)
DESIGN PROJECT REPORT: 12
Tentative Summary Table:
Used for Case Gain (Sign) Tentative Gain ๐๐๐(๐๐ซ) ๐๐๐(๐๐ซ) ๐๐(๐๐ซ) ๐๐(๐๐ซ)
SP (damping) (ฮq) to (ฮฮดe) + 0.265 โโโ โ โโ โ
SP (Frequency) (ฮ ฮฑ) to (ฮฮดe) + 7.11 โโ โโโ โ โ
SP (ฮnz) to (ฮฮดe) + 0 โโ โโโ โ โ
P (Frequency) (ฮฮธ) to (ฮฮดe): + 0.431 โโ โโ โโ โโ
P (Damping) (ฮu) to (ฮฮดe) + 0.000126 โ โโโ โโ โโ
Step 4) choose the root loci to close and tune the gains to achieve level 1
It was decided to close the loop for (ฮq) to (ฮฮดe) to adjust the damping of short period, (ฮnz) to (ฮฮดe) to
adjust short period frequency, (ฮฮธ) to (ฮฮดe) to adjust Phugoid frequency and (ฮu) to (ฮฮดe) to adjust
Phugoid damping. (ฮ ฮฑ) to (ฮฮดe) was not used for augmentation as normal load factor provided similar
root locus plot and in reality the angle of attack (ฮ ฮฑ) sensor some time doesnโt provide better data for
augmentation. Therefore itโs better to use sensor for normal load factor.
After tuning a bit, following K matrix was chosen:
๐พ๐ฟ๐๐๐ = [2 0 0.5 2.5 โ0.12]
Step 5) Generate A_Augumented Matrix
๐ด๐ด๐ข๐๐ข๐๐๐๐ก๐๐ = ๐ด๐ฟ๐๐๐(๐๐๐๐ข๐๐ข๐๐๐๐ก๐๐) + ๐ต๐ฟ๐๐๐๐พ๐ฟ๐๐๐๐ถ๐ฟ๐๐๐
๐ด๐ด๐ข๐๐ข๐๐๐๐ก๐๐ = [
โ0.0218 1.2227 0 โ32.1850
0.0024 โ0.5001 0.9579 โ0.0526
โ0.1443 โ8.0016 โ2.9711 โ3.0310
0 0 1 0
] โ ๐ท๐๐๐(๐ด๐๐ข๐) โ ๐๐๐๐๐ ๐๐๐๐๐ค
Step 6) Analyze and verify the augmented Eigen values and flying qualities.
Parameter Unaugmented Value Unaugmented
Handling Quality Augmented (Target
value) Actual Augmented
Values Augmented
Handling Quality
Short Period Mode
๐1,2 (๐ ๐) โ0.00976 ยฑ 0.0328๐ N/A N/A โ1.49 ยฑ 2.89๐ N/A
๐๐ ๐ 0.347 Level 2+3 ๐๐ ๐ = 0.45 ยฑ .05 ๐๐ ๐ = 0.459 Level 1
๐๐ ๐(๐ ๐๐
๐ ) 1.35 Level 2+3 ๐๐ ๐
(๐ ๐๐
๐ )= 3 ยฑ 0.5 ๐๐ ๐
(๐ ๐๐
๐ )= 3.25 Level 1
Phugoid Mode
๐1,2 (๐) โ0.00976 ยฑ 0.0328๐ N/A N/A โ0.252 ยฑ 0.314๐ N/A
๐๐ 0.285 Level 1 ๐๐ = 0.06 ยฑ .01 ๐๐ = 0.0626 Level 1 ๐๐
๐๐ ๐
0.0342
1.35= 0.025 < 0.1 Level 3
๐๐
๐๐ ๐> 0.12 ยฑ 0.01
๐๐
๐๐ ๐=
0.403
3.25= 0.124 Level 1
o The table above shows that all the longitudinal mode parameters meet level 1 handling qualities.
![Page 14: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/14.jpg)
DESIGN PROJECT REPORT: 13
Lateral Stability Augmentation:
Step 3) Analyze the transfer functions and analyze their root loci
1. Roll Rate (โp) Response to Aileron Input (ฮฮดa)
2. Yaw Rate (โ๐) Response to Rudder Input (ฮฮดr)
3. Slide Slip Angle (โ๐ฝ) Response to Rudder Input (ฮฮดr)
![Page 15: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/15.jpg)
DESIGN PROJECT REPORT: 14
4. Normal Load Factor (โNy) Response to Aileron Input (ฮฮดa)
5. Roll Angle (โฯ) Response to Aileron Input (ฮฮดa)
Tentative Summary Table:
Used for Case Tentative Gain
(Sign) Tentative Gain ๐๐น ๐๐ซ๐น ๐๐ซ๐น ๐๐บ๐ด
Roll Mode (Frequency)
(โp) to (ฮฮดa) + 3.94 โโ โ โ / โ โ
Dutch Roll Damping
(โ๐) to (ฮฮดr) - 2.23 โ โโ โโโ โโ
Dutch Roll Frequency
(โ๐ฝ) to (ฮฮดr) + 36 โ โโโ โโโ โ
Dutch roll damping
(โny) to (ฮฮดa): + 0.045 โ โโ โโโ โ
SM Frequency (โฯ) to (ฮฮดa) + 0.01 โโ โโ โ / โ โโ
![Page 16: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/16.jpg)
DESIGN PROJECT REPORT: 15
Step 4) choose the root loci to close and tune the gains to achieve level 1
It was decided to close the loop for (ฮp) to (ฮฮดa) to adjust the frequency of roll mode, ((โฯ)) to (ฮฮดa) to
adjust spiral mode frequency and damping, (โ๐) to (ฮฮดr) to adjust Dutch roll damping and (ฮ๐ฝ) to (ฮฮดr) to
adjust Dutch roll frequency. (โny) to (ฮฮดa) was not used for anything in this report; however it can be used in
other planes and flight conditions to adjust Dutch roll damping.
After tuning a bit, following K matrix was chosen:
๐พ = [โ4.5 0 0 0 โ0.3
0 4.25 โ30 0 0]
Step 5) Generate A_Augumented Matrix
๐ด๐ด๐ข๐๐ข๐๐๐๐ก๐๐ = ๐ด๐ฟ๐๐ก(๐๐๐๐ข๐๐ข๐๐๐๐ก๐๐) + ๐ต๐ฟ๐๐ก๐พ๐ฟ๐๐ก๐ถ๐ฟ๐๐ก
๐ด๐ด๐ข๐๐ข๐๐๐๐ก๐๐ = [
โ0.1921 0 โ0.9819 0.0370
โ10.1775 โ1.3083 1.5614 โ0.0555
14.7813 0.0716 โ2.1296 0.0040
0 1 0 0
] โ ๐ท๐๐๐(๐ด๐๐ข๐) โ ๐ ๐๐ ๐ข๐๐ก๐ ๐๐๐๐๐ค ๐๐๐ ๐ โ๐๐ค๐.
Step 6) Analyze and verify the augmented Eigen values and flying qualities.
Parameter Unaugmented Value Unaugmented
Handling Quality Augmented (Target
value)
Actual Augmented
Values
Augmented Handling Quality
Roll Mode (R)
๐1 (๐ ) โ0.510 N/A -1.25ยฑ0.025 -1.25 N/A
๐๐ 1 - 1 (for stability) 1 -
๐๐ (๐ ๐๐
๐ ) 0.510 Level 2+3 1.25 ยฑ 0.025 1.25 Level 1
๐๐ (s) 1.967 Level 3 0.8 ยฑ .01 0.8 Level 1
Dutch Roll Mode (DR)
๐1,2 (๐ท๐ ) โ0.107 ยฑ 1.01๐ N/A N/A โ1.17 ยฑ 3.67๐ N/A
๐๐ท๐ 0.106 Level 2 0.3 ยฑ .05 0.304 Level 1
๐๐ท๐ 1.02 Level 1 4 ยฑ 1 3.85 Level 1
๐๐ท๐ โ ๐๐ท๐ 0.108 Level 2 1.2 ยฑ .55 1.17 Level 1
Spiral Mode (SM)
๐1,2 (๐๐) 0.00537 N/A โ0.04 ยฑ 0.005 -0.0395 N/A
๐๐๐ โ1 - 1 (for stability) 1 -
๐๐๐ 0.00537 Level 3 0.04 ยฑ 0.005 0.0395 Level 1
๐๐๐(s) 186 Level 1 25 ยฑ 3 25.31 Level 1
o The table above shows that all the lateral modes parameters meet level 1 handling qualities.
![Page 17: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/17.jpg)
DESIGN PROJECT REPORT: 16
V. Simulation and Performance Assessment
In this section of the report time histories illustrating the effect of the stability augmentation.
Blue lines in the plots represent augmented response and black lines represents unaugmented response.
In addition, in the pole-zero map (PZ map), red poles represent unaugmented system and green poles
represent augmented system.
Longitudinal Modes Simulation and Performance:
PZ Map:
Short Period Time Response: In the graph below it can be seen that augmented response always starts with negative value to
cancel the effect and achieve the final stability with faster response. Furthermore, it can be also seen
in the figure below that the augmented response does not have many oscillations and settles faster
when compared to unaugmented response.
![Page 18: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/18.jpg)
DESIGN PROJECT REPORT: 17
Phugoid Mode Time Response: In the graph below it can be seen that augmented response always starts with negative value to
cancel the effect and achieve the final stability with faster response. Furthermore, it can be also seen
in the figure below that the augmented response does not have many oscillations and settles faster
when compared to unaugmented response.
![Page 19: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/19.jpg)
DESIGN PROJECT REPORT: 18
Gust Time Response: Wind gust is a sudden, brief increase in speed of the wind. If a gust of wind strikes the aircraft
from the right it will be in a slip and the fin will get an angle of attack causing the aircraft to yaw until
the slip is eliminated. In this section angle of attack response is shown to gust input. Gust input
primarily affects only longitudinal dynamics. This response is showed in two time scales: 10 seconds
and 30 seconds. It can be seen in the figure below that the augmented response does not have many
oscillations and settles faster when compared to unaugmented response.
![Page 20: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/20.jpg)
DESIGN PROJECT REPORT: 19
Lateral Modes Simulation and Performance:
PZ Map:
Roll Mode Time Response: It can be seen in the figure below that the augmented response does not have many oscillations
and settles faster when compared to unaugmented response. It can be seen that in roll mode the
unaugmented response of yaw rate to rudder input and bank angle to aileron input is unstable and
does not achieve stability at all.
![Page 21: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/21.jpg)
DESIGN PROJECT REPORT: 20
Spiral Mode Time Response: It can be seen in the figure below that the augmented response does not have many oscillations
and settles faster when compared to unaugmented response. In this figure it can be seen that
unaugmented response of roll rate to aileron input is faster than augmented response.
Dutch Roll Mode Time Response: It can be seen in the figure below that the augmented response does not have many oscillations
and settles faster when compared to unaugmented response.
![Page 22: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/22.jpg)
DESIGN PROJECT REPORT: 21
VI. References:
[1] Hodgkinson, J. Aircraft Handling Qualities. AIAA Education Series. 1999
[2] Cook, Michael V. Flight Dynamics Principles. Boston, 2013.
![Page 23: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/23.jpg)
DESIGN PROJECT REPORT: 22
Appendix:
![Page 24: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/24.jpg)
DESIGN PROJECT REPORT: 23
Longitudinal Mode Requirements:
Short Period Mode Requirements:
Figure 1: Typical Short-Period Mode Frequency Requirements
Table 1: Short-Period Mode Damping Requirements
Phugoid Mode Requirements:
Table 2: Phugoid Damping Ratio Requirements
![Page 25: Design Project Report - Mechanical Engineeringkushalshahmae.weebly.com/.../designproject_final.pdfDESIGN PROJECT REPORT: 3 II. Aircraft Model: Description of Aircraft. Boeing 747,](https://reader035.fdocuments.net/reader035/viewer/2022070801/5f028d617e708231d404d4af/html5/thumbnails/25.jpg)
DESIGN PROJECT REPORT: 24
Lateral Mode Requirements:
Roll Mode Requirements:
Table 3: Roll Subsidence mode time constant
Spiral Mode Requirements:
Table 4: Spiral Mode time constant
Dutch Roll Mode Requirements:
Table 5: Dutch Roll frequency and damping