Design Project Report - Mechanical...

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.

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 – ............................................................................................................................................................. 4

LATERAL-DIRECTION DYNAMICS – ..................................................................................................................................................... 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


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 “good” 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 – 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.


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 3–4–3 seat arrangement in economy class and a

2–3–2 arrangement in first class on the main deck. The upper deck has a 3–3 seat arrangement in

economy class and a 2–2 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:


Longitudinal Dynamics –

Linearized State Space Aircraft Dynamics Matrix Model: (Input – Elevator) 𝑋 = 𝐴𝑋 + 𝐵𝑢

[

𝑢α��

θ

] =

[ Xu Xα 0 −gcos(θ1)

Zu

U1

Zα

U1

U1 + Zq

U1

−𝑔sin(θ1)

U1

Mu Mα Mq + 𝑀𝛼.

00 0 1 0 ]

[

𝑢α𝑞θ

] +

[ 𝑋𝛿𝑒

𝑍𝛿𝑒

𝑈1

𝑀𝛿𝑒

0 ]

∗ ∆𝛿𝑒

[

𝑢α��

θ

] = [

−0.0218 1.2227 0 −32.1850−0.0001 −0.3892 1 −0−0.0001 −1.6165 −0.5463 0

0 0 1 0

] [

𝑢α𝑞θ

] + [

0−0.0211−1.2124

0

] ∆𝛿𝑒

Output State Space Model:

For short period mode, pitch rate (q), angle of attack (α) and normal load factor in Z (𝑛𝑧) can be used

to close the loop and obtain required handling qualities.

For phugoid mode, pitch angle (θ), and forward speed (𝑢) can be used to close the loop and obtain

required handling qualities.

𝑌 = 𝐶𝑋 + 𝐷𝑢

[ 𝑞α𝑛𝑧

θ𝑢 ]

=

[

0 0 1 00 1 0 0

−Zu

𝑔−

Zα

𝑔−

Zq

𝑔sin(θ1)

0 0 0 11 0 0 0 ]

[

𝑢α𝑞θ

] +

[ 00000]

∆𝛿𝑒

[ 𝑞α𝑛𝑧

θ𝑢 ]

=

[

0 0 1 00 1 0 0

0.0018 10.5330 0 00 0 0 11 0 0 0]

[

𝑢α𝑞θ

] +

[ 00000]

∆𝛿𝑒

Summary:

𝑨𝑳𝒐𝒏𝒈 𝑩𝑳𝒐𝒏𝒈 𝑪𝑳𝒐𝒏𝒈 𝑫𝑳𝒐𝒏𝒈

[ Xu Xα 0 −gcos(θ1)

Zu

U1

Zα

U1

U1 + Zq

U1

−𝑔sin(θ1)

U1

Mu Mα Mq + 𝑀𝛼.

00 0 1 0 ]

[ 𝑋𝛿𝑒

𝑍𝛿𝑒

𝑈1

𝑀𝛿𝑒

0 ]

[

0 0 1 00 1 0 0

−Zu

𝑔−

Zα

𝑔−

Zq

𝑔sin(θ1)

0 0 0 11 0 0 0 ]

[ 00000]

[

−0.0218 1.2227 0 −32.1850−0.0001 −0.3892 1 −0−0.0001 −1.6165 −0.5463 0

0 0 1 0

] [

0−0.0211−1.2124

0

]

[

0 0 1 00 1 0 0

0.0018 10.5330 0 00 0 0 11 0 0 0]

[ 00000]


Lateral-Direction Dynamics –

Linearized State Space Matrix Model: (Inputs –Rudder, & Aileron):

𝑥 = 𝐴𝑥 + 𝐵𝑢

[ ∆��∆p∆��∆ϕ]

=

[ 𝑌𝛽

𝑈1

Yp

U1

𝑌𝑟

𝑈1

− 1 −g

U1∗ cos(𝜃1)

L𝛽 Lp Lr 0

N𝛽 Np Nr 0

0 1 0 0 ]

[

∆𝛽∆p∆𝑟∆ϕ

] +

[ 𝑌𝛿𝑎

𝑈1

𝑌𝛿𝑟

𝑈1

𝐿𝛿𝑎 𝐿𝛿𝑟

𝑁𝛿𝑎 𝑁𝛿𝑟

0 0 ]

[ΔδaΔδr

]

[ ∆��∆p∆��∆ϕ]

= [

−0.0640 0 −1 0.0370−1.2555 −0.4758 0.2974 01.0143 0.0109 −0.1793 0

0 1 0 0

] [


] + [

0 0.00430.1850 0.2974

−0.0135 −0.45890 0

] [ΔδaΔδr

]

Output State Space Model:

For roll mode, roll rate(∆p) can be used to close the loop and obtain required handling qualities.

For spiral mode, bank angle(∆ϕ) can be used to close the loop and obtain required handling qualities.

For Dutch roll mode, yaw rate (∆r), sideslip angle (∆𝛽) and normal load factor (∆n𝑦) can be used to close

the loop and obtain required handling qualities.

𝑌 = 𝐶𝑋 + 𝐷𝑢

[ ∆𝑝∆r∆𝛽∆𝑛𝑦

∆ϕ ]

=

[ 0 1 0 00 0 1 01 0 0 0𝑌𝛽

𝑔0 −

𝑈1

𝑔cos(𝜃1)

0 0 0 1 ]

[


] +

[ 0 00 00 00 00 0]

[ΔδaΔδr

]

[ ∆𝑝∆r∆𝛽∆𝑛𝑦

∆ϕ ]

=

[

0 1 0 00 0 1 01 0 0 0

−1.73 0 −27.06 10 0 0 1]

[


] +

[ 0 00 00 00 00 0]

[ΔδaΔδr

]

Summary:

𝑨𝑳𝒂𝒕 𝑩𝑳𝒂𝒕 𝑪𝑳𝒂𝒕 𝑫𝑳𝒂𝒕

[ 𝑌𝛽

𝑈1

Yp

U1

𝑌𝑟

𝑈1

− 1 −g

U1∗ cos(𝜃1)

L𝛽 Lp Lr 0

N𝛽 Np Nr 0

0 1 0 0 ]

[ 𝑌𝛿𝑎

𝑈1

𝑌𝛿𝑟

𝑈1

𝐿𝛿𝑎 𝐿𝛿𝑟

𝑁𝛿𝑎 𝑁𝛿𝑟

0 0 ]

[ 0 1 0 00 0 1 01 0 0 0𝑌𝛽

𝑔0 −

𝑈1

𝑔cos(𝜃1)

0 0 0 1 ]

[ 0 00 00 00 00 0]

[

−0.0640 0 −1 0.0370−1.2555 −0.4758 0.2974 01.0143 0.0109 −0.1793 0

0 1 0 0

] [

0 0.00430.1850 0.2974

−0.0135 −0.45890 0

]

[

0 1 0 0

0 0 1 0

1 0 0 0

−1.73 0 −27.06 1

0 0 0 1]

[ 0 00 00 00 00 0]


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 - 𝜆𝐼| )

Mode Eigenvalues Damping Frequency (rad/s)

Phugoid Mode (4D) 𝜆1,2 (𝑝 ) = −0.00976 ± 0.0328𝑗 𝜉𝑝 = 0.285 𝜔𝑝 = 0.0342

Short Period Mode (4D) 𝜆1,2 (𝑆𝑃) = −0.469 ± 1.27𝑗 𝜉𝑠𝑝 = 0.347 𝜔𝑠𝑝 = 1.35

Unaugmented Flying Handling Qualities:

Parameter Unaugmented Value Unaugmented Handling Quality

Short Period Mode

𝜆1,2 (𝑠𝑝) −0.00976 ± 0.0328𝑗 N/A

𝜉𝑠𝑝 0.347 Level 2+3

𝜔𝑠𝑝(𝑅𝑎𝑑

𝑠) 1.35 Level 2+3

Phugoid Mode

𝜆1,2 (𝑝) −0.00976 ± 0.0328𝑗 N/A

𝜉𝑝 0.285 Level 1 𝜔𝑝

𝜔𝑠𝑝

0.0342

1.35= 0.025 < 0.1 Level 3


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

𝜆1,2 (𝑠𝑝) −0.00976 ± 0.0328𝑗 N/A N/A N/A

𝜉𝑠𝑝 0.347 Level 2+3 𝜉𝑠𝑝 = 0.45 ± .05 Level 1

𝜔𝑠𝑝(𝑅𝑎𝑑𝑠 )

1.35 Level 2+3 𝜔𝑠𝑝(𝑅𝑎𝑑𝑠 )

= 3 ± 0.5 Level 1

Short Period Mode

𝜆1,2 (𝑝) −0.00976 ± 0.0328𝑗 N/A N/A N/A

𝜉𝑝 0.285 Level 1 𝜉𝑝 = 0.06 ± .01 Level 1

𝜔𝑝

𝜔𝑠𝑝

0.0342

1.35= 0.025 < 0.1 Level 3

𝜔𝑝

𝜔𝑠𝑝> 0.12 ± 0.01 Level 1


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 - 𝜆𝐼| )

Mode Eigenvalues Damping Frequency (rad/s) Time Constant (s)

Roll Mode (4D) 𝜆1 (𝑅) = −0.510 𝜉𝑅 = 1 𝜔𝑅 = 0.510 𝑇𝑅 = 1.967

Spiral Mode (4D) 𝜆1 (𝑆𝑀) = 0.00537 𝜉𝑆𝑀 = −1 𝜔𝑆𝑀 = 0.00537 𝑇𝑆𝑀 = 186

Dutch Roll (4D) 𝜆1,2 (𝐷𝑅) = −0.107 ± 1.01𝑗 𝜉𝐷𝑅 = 0.106 𝜔𝐷𝑅 = 1.02 𝑇𝐷𝑅 = 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.


Criteria Parameter Unaugmented Value Unaugmented Handling

Quality

Roll Mode (R)

- 𝜆1 (𝑅) −0.510 N/A

- 𝜉𝑅 1 -

- 𝜔𝑅(𝑅𝑎𝑑

𝑠) 0.510 Level 2+3

1 𝑇𝑅(s) 1.967 Level 3

Spiral Mode (SM)

- 𝜆1,2 (𝑆𝑀) 0.00537 N/A

- 𝜉𝑆𝑀 −1 -

- 𝜔𝑆𝑀 0.00537 Level 3

5 𝑇𝑆𝑀(s) 186 Level 1


- 𝜆1,2 (𝐷𝑅) −0.107 ± 1.01𝑗 N/A

2 𝜉𝐷𝑅 0.106 Level 2

3 𝜔𝐷𝑅 1.02 Level 1

4 𝜉𝐷𝑅 ∗ 𝜔𝐷𝑅 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)

- 𝜆1 (𝑅) −0.510 N/A N/A N/A

- 𝜉𝑅 1 - 1 (for stability) -

- 𝜔𝑅(𝑅𝑎𝑑

𝑠) 0.510 Level 2+3 1.25 ± 0.025 Level 1

1 𝑇𝑅(s) 1.967 Level 3 0.8 ± .01 Level 1

Spiral Mode (SM)

- 𝜆1,2 (𝑆𝑀) 0.00537 N/A N/A N/A

- 𝜉𝑆𝑀 −1 - 1 (for stability) -

- 𝜔𝑆𝑀 0.00537 Level 3 0.04 ± 0.005 Level 1

5 𝑇𝑆𝑀(s) 186 Level 1 25 ± 3 Level 1


- 𝜆1,2 (𝐷𝑅) −0.107 ± 1.01𝑗

N/A N/A N/A

2 𝜉𝐷𝑅 0.106 Level 2 0.3 ± .05 Level 1

3 𝜔𝐷𝑅 1.02 Level 1 4 ± 1 Level 1

4 𝜉𝐷𝑅 ∗ 𝜔𝐷𝑅 0.108 Level 2 1.2 ± .55 Level 1


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 (Δq) to Elevator Input (Δδe):

2. Angle of Attack (Δ α) Response to Elevator Input (Δδe)


3. Load Response (Δnz) to Elevator Input (Δδe):

4. Pitch Angle (Δ θ) Response to Elevator Input (Δδe):

5. Forward Speed (Δu) Response to Elevator Input (Δδe):


Tentative Summary Table:

Used for Case Gain (Sign) Tentative Gain 𝝃𝒔𝒑(𝟒𝑫) 𝝎𝒔𝒑(𝟒𝑫) 𝝃𝒑(𝟒𝑫) 𝝎𝒑(𝟒𝑫)

SP (damping) (Δq) to (Δδe) + 0.265 ↑↑↑ ↑ ↑↑ ↓

SP (Frequency) (Δ α) to (Δδe) + 7.11 ↑↑ ↑↑↑ ↓ ↑

SP (Δnz) to (Δδe) + 0 ↑↑ ↑↑↑ ↓ ↑

P (Frequency) (Δθ) to (Δδe): + 0.431 ↓↓ ↑↑ ↑↑ ↑↑

P (Damping) (Δu) to (Δδe) + 0.000126 ↓ ↑↑↑ ↑↑ ↓↓

Step 4) choose the root loci to close and tune the gains to achieve level 1

It was decided to close the loop for (Δq) to (Δδe) to adjust the damping of short period, (Δnz) to (Δδe) to

adjust short period frequency, (Δθ) to (Δδe) to adjust Phugoid frequency and (Δu) to (Δδe) to adjust

Phugoid damping. (Δ α) to (Δδe) was not used for augmentation as normal load factor provided similar

root locus plot and in reality the angle of attack (Δ α) sensor some time doesn’t provide better data for

augmentation. Therefore it’s better to use sensor for normal load factor.

After tuning a bit, following K matrix was chosen:

𝐾𝐿𝑜𝑛𝑔 = [2 0 0.5 2.5 −0.12]

Step 5) Generate A_Augumented Matrix

𝐴𝐴𝑢𝑔𝑢𝑚𝑒𝑛𝑡𝑒𝑑 = 𝐴𝐿𝑜𝑛𝑔(𝑈𝑛𝑎𝑢𝑔𝑢𝑚𝑒𝑛𝑡𝑒𝑑) + 𝐵𝐿𝑜𝑛𝑔𝐾𝐿𝑜𝑛𝑔𝐶𝐿𝑜𝑛𝑔

𝐴𝐴𝑢𝑔𝑢𝑚𝑒𝑛𝑡𝑒𝑑 = [

−0.0218 1.2227 0 −32.1850

0.0024 −0.5001 0.9579 −0.0526

−0.1443 −8.0016 −2.9711 −3.0310

0 0 1 0

] → 𝐷𝑎𝑚𝑝(𝐴𝑎𝑢𝑔) → 𝑇𝑎𝑏𝑙𝑒 𝑏𝑒𝑙𝑜𝑤

Step 6) Analyze and verify the augmented Eigen values and flying qualities.



value) Actual Augmented

Values Augmented

Handling Quality

Short Period Mode

𝜆1,2 (𝑠𝑝) −0.00976 ± 0.0328𝑗 N/A N/A −1.49 ± 2.89𝑗 N/A

𝜉𝑠𝑝 0.347 Level 2+3 𝜉𝑠𝑝 = 0.45 ± .05 𝜉𝑠𝑝 = 0.459 Level 1

𝜔𝑠𝑝(𝑅𝑎𝑑

𝑠) 1.35 Level 2+3 𝜔𝑠𝑝

(𝑅𝑎𝑑

𝑠)= 3 ± 0.5 𝜔𝑠𝑝

(𝑅𝑎𝑑

𝑠)= 3.25 Level 1

Phugoid Mode

𝜆1,2 (𝑝) −0.00976 ± 0.0328𝑗 N/A N/A −0.252 ± 0.314𝑗 N/A

𝜉𝑝 0.285 Level 1 𝜉𝑝 = 0.06 ± .01 𝜉𝑝 = 0.0626 Level 1 𝜔𝑝

𝜔𝑠𝑝

0.0342

1.35= 0.025 < 0.1 Level 3

𝜔𝑝

𝜔𝑠𝑝> 0.12 ± 0.01

𝜔𝑝

𝜔𝑠𝑝=

0.403

3.25= 0.124 Level 1

o The table above shows that all the longitudinal mode parameters meet level 1 handling qualities.


Lateral Stability Augmentation:

Step 3) Analyze the transfer functions and analyze their root loci

1. Roll Rate (∆p) Response to Aileron Input (Δδa)

2. Yaw Rate (∆𝑟) Response to Rudder Input (Δδr)

3. Slide Slip Angle (∆𝛽) Response to Rudder Input (Δδr)


4. Normal Load Factor (∆Ny) Response to Aileron Input (Δδa)

5. Roll Angle (∆ϕ) Response to Aileron Input (Δδa)

Tentative Summary Table:

Used for Case Tentative Gain

(Sign) Tentative Gain 𝝎𝑹 𝝎𝑫𝑹 𝝃𝑫𝑹 𝝎𝑺𝑴

Roll Mode (Frequency)

(∆p) to (Δδa) + 3.94 ↑↑ ↓ ↑ / ↓ ↑

Dutch Roll Damping

(∆𝑟) to (Δδr) - 2.23 ↓ ↓↓ ↑↑↑ ↑↑

Dutch Roll Frequency

(∆𝛽) to (Δδr) + 36 ↓ ↑↑↑ ↑↑↑ ↓

Dutch roll damping

(∆ny) to (Δδa): + 0.045 ↑ ↓↓ ↑↑↑ ↓

SM Frequency (∆ϕ) to (Δδa) + 0.01 ↓↓ ↑↑ ↑ / ↓ ↑↑


Step 4) choose the root loci to close and tune the gains to achieve level 1

It was decided to close the loop for (Δp) to (Δδa) to adjust the frequency of roll mode, ((∆ϕ)) to (Δδa) to

adjust spiral mode frequency and damping, (∆𝑟) to (Δδr) to adjust Dutch roll damping and (Δ𝛽) to (Δδr) to

adjust Dutch roll frequency. (∆ny) to (Δδa) 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:

𝐾 = [−4.5 0 0 0 −0.3

0 4.25 −30 0 0]

Step 5) Generate A_Augumented Matrix

𝐴𝐴𝑢𝑔𝑢𝑚𝑒𝑛𝑡𝑒𝑑 = 𝐴𝐿𝑎𝑡(𝑈𝑛𝑎𝑢𝑔𝑢𝑚𝑒𝑛𝑡𝑒𝑑) + 𝐵𝐿𝑎𝑡𝐾𝐿𝑎𝑡𝐶𝐿𝑎𝑡

𝐴𝐴𝑢𝑔𝑢𝑚𝑒𝑛𝑡𝑒𝑑 = [

−0.1921 0 −0.9819 0.0370

−10.1775 −1.3083 1.5614 −0.0555

14.7813 0.0716 −2.1296 0.0040

0 1 0 0

] → 𝐷𝑎𝑚𝑝(𝐴𝑎𝑢𝑔) → 𝑅𝑒𝑠𝑢𝑙𝑡𝑠 𝑏𝑒𝑙𝑜𝑤 𝑎𝑟𝑒 𝑠ℎ𝑜𝑤𝑛.

Step 6) Analyze and verify the augmented Eigen values and flying qualities.



value)

Actual Augmented

Values

Augmented Handling Quality

Roll Mode (R)

𝜆1 (𝑅) −0.510 N/A -1.25±0.025 -1.25 N/A

𝜉𝑅 1 - 1 (for stability) 1 -

𝜔𝑅(𝑅𝑎𝑑

𝑠 ) 0.510 Level 2+3 1.25 ± 0.025 1.25 Level 1

𝑇𝑅(s) 1.967 Level 3 0.8 ± .01 0.8 Level 1


𝜆1,2 (𝐷𝑅) −0.107 ± 1.01𝑗 N/A N/A −1.17 ± 3.67𝑗 N/A

𝜉𝐷𝑅 0.106 Level 2 0.3 ± .05 0.304 Level 1

𝜔𝐷𝑅 1.02 Level 1 4 ± 1 3.85 Level 1

𝜉𝐷𝑅 ∗ 𝜔𝐷𝑅 0.108 Level 2 1.2 ± .55 1.17 Level 1

Spiral Mode (SM)

𝜆1,2 (𝑆𝑀) 0.00537 N/A −0.04 ± 0.005 -0.0395 N/A

𝜉𝑆𝑀 −1 - 1 (for stability) 1 -

𝜔𝑆𝑀 0.00537 Level 3 0.04 ± 0.005 0.0395 Level 1

𝑇𝑆𝑀(s) 186 Level 1 25 ± 3 25.31 Level 1

o The table above shows that all the lateral modes parameters meet level 1 handling qualities.


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.


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.


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.


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.


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.


VI. References:

[1] Hodgkinson, J. Aircraft Handling Qualities. AIAA Education Series. 1999

[2] Cook, Michael V. Flight Dynamics Principles. Boston, 2013.


Appendix:


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


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

Design Project Report - Mechanical...

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