A NEW DISCOVERY: NONLINEAR INSTABILITY LEADING TO LOSS OF CONTROL - - PowerPoint PPT Presentation

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A NEW DISCOVERY: NONLINEAR INSTABILITY LEADING TO LOSS OF CONTROL - - PowerPoint PPT Presentation

Jul 19, 2023 415 likes β€’579 views

A NEW DISCOVERY: NONLINEAR INSTABILITY LEADING TO LOSS OF CONTROL OF AIRCRAFT AIAA Houston Section Annual T echnical Symposium June 1, 2019 Steve S. Tang Faiteve Inc. Neglected elephants in the equations The

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A NEW DISCOVERY: NONLINEAR INSTABILITY LEADING TO LOSS OF CONTROL OF AIRCRAFT AIAA Houston Section Annual T echnical Symposium June 1, 2019 Steve S. Tang Faiteve Inc.

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Β§ The original governing equations for aircraft roll (πœ’), pitch (πœ„), and yaw (πœ”) in the principal axis frame (𝐽& < 𝐽( < 𝐽)): 𝐽& ̈ πœ’ + 𝑐- Μ‡ πœ’ + 𝑙-πœ’ = 𝐽( βˆ’ 𝐽) Μ‡ πœ„ Μ‡ πœ” + 𝑁3(𝑒) 𝐽( ̈ πœ„ + 𝑐6 Μ‡ πœ„ + 𝑙6πœ„ = 𝐽) βˆ’ 𝐽& Μ‡ πœ’ Μ‡ πœ” + 𝑁7 𝑒 𝐽) ̈ πœ” + 𝑐8 Μ‡ πœ” + 𝑙8πœ” = 𝐽& βˆ’ 𝐽( Μ‡ πœ’ Μ‡ πœ„ + 𝑁9 𝑒 Β§ The linearized equations: 𝐽3 ̈ πœ’ + 𝑐- Μ‡ πœ’ + 𝑙-πœ’ = 𝑁3(𝑒) 𝐽7 ̈ πœ„ + 𝑐6 Μ‡ πœ„ + 𝑙6πœ„ = 𝑁7 𝑒 𝐽9 ̈ πœ” + 𝑐8 Μ‡ πœ” + 𝑙8πœ” = 𝑁9 𝑒 Β§ 𝐽7 βˆ’ 𝐽9 Μ‡ πœ„ Μ‡ πœ”, 𝐽9 βˆ’ 𝐽3 Μ‡ πœ’ Μ‡ πœ”, 𝐽3 βˆ’ 𝐽7 Μ‡ πœ’ Μ‡ πœ„ - the inertial moments, similar to Coriolis force for hurricanes, should not be neglected

Neglected elephants in the equations

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Β§ As nonlinear harmonic oscillation system: 𝐽3 ̈ πœ’ + 𝑐- Μ‡ πœ’ + 𝑙-πœ’ = 𝐽7 βˆ’ 𝐽9 Μ‡ πœ„ Μ‡ πœ” + 𝑁3(𝑒) 𝐽7 ̈ πœ„ + 𝑐6 Μ‡ πœ„ + 𝑙6πœ„ = 𝐽9 βˆ’ 𝐽3 Μ‡ πœ’ Μ‡ πœ” + 𝑁7 𝑒 𝐽9 ̈ πœ” + 𝑐8 Μ‡ πœ” + 𝑙8πœ” = 𝐽3 βˆ’ 𝐽7 Μ‡ πœ’ Μ‡ πœ„ + 𝑁9 𝑒

The elephants causing resonances

What matters the most is the exciting frequencies of these terms, not the amplitudes

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Β§The most common maneuver in takeoff, cruise leveling and landing approach Β§The original governing equations: 𝐽3 ̈ πœ’ + 𝑐- Μ‡ πœ’ + 𝑙-πœ’ = 𝐽7 βˆ’ 𝐽9 Μ‡ πœ„ Μ‡ πœ” 𝐽7 ̈ πœ„ + 𝑐6 Μ‡ πœ„ + 𝑙6πœ„ = 𝐽9 βˆ’ 𝐽3 Μ‡ πœ’ Μ‡ πœ” + 𝑁6-𝑑𝑝𝑑(πœ•6-𝑒 + 𝛽6-) 𝐽9 ̈ πœ” + 𝑐8 Μ‡ πœ” + 𝑙8πœ” = 𝐽3 βˆ’ 𝐽7 Μ‡ πœ’ Μ‡ πœ„ Β§The current longitudinal dynamics (pitch control only): 𝐽7 ̈ πœ„ + 𝑐6 Μ‡ πœ„ + 𝑙6πœ„ = 𝑁6-𝑑𝑝𝑑(πœ•6-𝑒 + 𝛽6-)

A pitch control case

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Β§ By the current practice, the aircraft response is (caterpillar mode)

πœ’ = 0 πœ„ = πœ„@(𝑁6-) cos πœ•6-𝑒 + 𝛽6- + 𝜊6- πœ” = 0 Β§ This pitch-only response is not always stable. Under certain conditions, the following roll, pitch and yaw bifurcation solutions (butterfly mode) exist. πœ’ 𝑒 = βˆ‘FG-

H πœ’F 𝑒 = βˆ‘FG- H 𝐡-F cos πœ•-F𝑒 + 𝛾-F

πœ„ 𝑒 = βˆ‘KG-

H πœ„ K 𝑒 = βˆ‘KG- H 𝐡6K cos πœ•6K𝑒 + 𝛾6K

πœ” 𝑒 = βˆ‘LG-

H πœ”L 𝑒 = βˆ‘LG- H 𝐡8L cos πœ•8L𝑒 + 𝛾8L

Solutions and bifurcations

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Β§ Solve the amplitudes:

𝐡-- = 2 πœ•-- 𝐽7𝐽9π‘Ž6-π‘Ž8- (𝐽9 βˆ’ 𝐽3)(𝐽7 βˆ’ 𝐽3) 𝑁6- 2𝐽7π‘Ž6- 𝐽9 βˆ’ 𝐽7 𝐽7 βˆ’ 𝐽3 𝐽3𝐽9π‘Ž--π‘Ž8-

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βˆ’ 1

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𝐡6-= 2 πœ•6- 𝐽3𝐽9π‘Ž--π‘Ž8- (𝐽9 βˆ’ 𝐽7)(𝐽7 βˆ’ 𝐽3)

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𝐡8- = 2 πœ•8- 𝐽3𝐽7π‘Ž6-π‘Ž-- (𝐽9 βˆ’ 𝐽7)(𝐽9 βˆ’ 𝐽3) 𝑁6- 2𝐽7π‘Ž6- 𝐽9 βˆ’ 𝐽7 𝐽7 βˆ’ 𝐽3 𝐽3𝐽9π‘Ž--π‘Ž8-

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βˆ’ 1

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Roll impedance: π‘Ž-- = (πœ•--

6 βˆ’ πœ•-@ 6 )6+( QRSRR TU

)6 /πœ•-- Yaw impedance: π‘Ž8- = (πœ•8-

6 βˆ’ πœ•8@ 6 )6+( QVSVR TW

)6/πœ•8-

Amplitudes of the 1st modes

Pitch amplitude cutoff from 𝑁6- Pitch moment 𝑁6- driving roll and yaw

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Β§ Pitch unstable if 𝑁6- > 2𝐽7π‘Ž6-

TUTWYRRYVR (TWZT[)(T[ZTU)

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, pitch moment threshold Pitch amplitude > 𝐡\]^ ≑

6 S`R TUTWYRRYVR (TWZT[)(T[ZTU)

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, pitch threshold Β§ Pitch stable if

𝑁6- ≀ 2𝐽7π‘Ž6-

TUTWYRRYVR (TWZT[)(T[ZTU)

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Β§ Proved in the book, if the first modes πœ’-, πœ„-, πœ”- exist, πœ’6, πœ„6, πœ”6; πœ’8, πœ„8, πœ”8, … exist.

Pitch conditional instability criterion

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Β§ A commercial aircraft model

was used to do the numerical simulations. Β§ Pitch instability threshold in Scenario 1 Β§ Pitch instability threshold in Scenario 2

Pitch instability threshold verification

Pitch frequency (rad/s) Pitch threshold (deg)

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10 20 30 40 50 60 70 80 Analytical Numerical

Pitch frequency (rad/s) Pitch threshold (deg)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 20 40 60 80 100 120 Analytical Numerical

Scenario 1 Scenario 2 The worst case: πœ•6- = πœ•-@ + πœ•8@ The worst case: πœ•6- = πœ•-@ βˆ’ πœ•8@

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Β§ The pitch critical frequency:

Β§ Pitch amplitude threshold Β§ Two dangerous situations,

  • 1. At stall,

Roll damping; 𝑐- β†’ 0, then 𝐡\]^ β†’ 0

  • 2. Yaw damper malfunction, turned off or hardover

Yaw damping: 𝑐8 β†’ 0, then 𝐡\]^ β†’ 0

The worst case – a resonance mode

𝐡\]^ ≑ 2 πœ•-@ + πœ•8@ 𝑐-𝑐8 (𝐽9 βˆ’ 𝐽7)(𝐽7 βˆ’ 𝐽3)

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πœ•6- = πœ•efFgFehL ≑ πœ•-@ + πœ•8@

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Β§ At the pitch

critical frequency πœ•efFgFehL, aircraft is prone to nonlinear pitch instability before stall during takeoff

The danger during takeoff

caterpillar sketch

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Β§ A pitch control producing

18Β° pitch oscillation at πœ•6- = 5.2 (𝑆𝑏𝑒/𝑑) Β§The pitch amplitude threshold: 𝐡\]^ = 12Β° Β§ Pitch unstable: pitch 18Β° > 𝐡\]^ = 12Β° Β§The final steady state amplitudes depend on the external pitch control amplitude

Loss of control of aircraft

Time (sec) Roll (deg) 20 40 60 80 100

  • 80
  • 40

40 80 Time (sec) Pitch (deg) 20 40 60 80 100

  • 50
  • 25

25 50 Time (sec) Yaw (deg) 20 40 60 80 100

  • 50
  • 25

25 50

β€œUncommanded” roll 55Β° β€œUncommanded” pitch 37Β° β€œUncommanded” yaw 36Β°

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Nonlinear pitch instability leading to Loss of control of aircraft on B737-236 Advanced G-BGJI

B737-236 mode change from caterpillar to butterfly happened 1.5 min before autopilot disconnect

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Nonlinear pitch instability leading to Loss of control

  • f aircraft on Ethiopian Airline Flight 302 B737 Max

Pitch β‰ˆ 18Β° > Boeing recommended pitch 15βˆ’16Β° Uncommanded roll began 4 seconds earlier AOA β‰ˆ 15Β°, critical AOA β†’ 𝑐- β†’ 0, 𝐡\]^ β†’ 0, π’’π’‹π’–π’…π’Š π’π’‘π’π’Žπ’‹π’π’‡π’ƒπ’”π’Žπ’› π’—π’π’•π’–π’ƒπ’„π’Žπ’‡ Stick shaker began 4 seconds later

Flight 302 bifurcated from caterpillar to butterfly mode 4 seconds before stick shaker activated

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Β§ Northwest Flight 255 MD DC-9-82 crashed in1987

Β§ Delta Airlines Flight 1141 B727-232 crashed in1988 Β§ USAir Flight 405 Fokker F-28 crashed in1992 Β§ American Airline Flight 587 A300-605R crashed in 2001 Β§ PT. Mandala Airlines Flight 091 B737-200 crashed in 2005 Β§ Air Transat A310-308 C-GPAT upset in 2008 Β§ Spainair Flight 5022 MD DC-9-82 crashed in 2008 Β§ Gulfstream GVI (G650) N652GD test flight crashed in 2011 Β§ Ethiopian Airlines Flight 302 B737 Max crashed in 2019

The list of nonlinear pitch instability leading to loss of control of aircraft during takeoff

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Β§ Aircraft model with restoring and damping for roll and yaw

πœ•-@ = 2𝜌, π‘ˆ

  • @ = 1 𝑑𝑓𝑑

πœ•8@ = 𝜌, π‘ˆ8@ = 2 𝑑𝑓𝑑

Β§ The dangerous pitch frequencies

Scenario 1: πœ•6- = πœ•-@ + πœ•8@ = 3𝜌 π‘ˆ6- = 0. 7𝑑, 𝐡\]^Z- = 6

8‰ QRQV (TWZT[)(T[ZTU)

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= 𝐡\]^Z6/3 Scenario 2: πœ•6- = πœ•-@ βˆ’ πœ•8@ = 𝜌 π‘ˆ6- = 2 𝑑, 𝐡\]^Z6 = 2 𝜌 𝑐-𝑐8 (𝐽9 βˆ’ 𝐽7)(𝐽7 βˆ’ 𝐽3)

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§ Experimental observation: 𝐡\]^Z6 = 3𝐡\]^Z-

Nonlinear pitch instability demonstration

(www.youtube.com/watch?v=gG2-mu6I11A)