DESIGN PROJECT REPORT: Longitudinal and lateral-directional stability augmentation of Boeing 747 for cruise flight condition.

Transcription

1 DESIGN PROJECT REPORT: Longitudinal and lateral-directional stability augmentation of Boeing 747 for cruise flight condition. Prepared By: Kushal Shah Advisor: Professor John Hodgkinson Graduate Advisor: Colin Alexander Sledge 12 June 214

2 I. OBJECTIVE:... 2 HISTORICAL PERSPECTIVE:... 2 PURPOSE:... 2 DESIGN SPECIFICATIONS... 2 II. AIRCRAFT MODEL:... 3 DESCRIPTION OF AIRCRAFT 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:... 1 LONGITUDINAL STABILITY AUGMENTATION... 1 LATERAL STABILITY AUGMENTATION: V. SIMULATION AND PERFORMANCE ASSESSMENT LONGITUDINAL MODES SIMULATION AND PERFORMANCE: LATERAL MODES SIMULATION AND PERFORMANCE: VI. REFERENCES: APPENDIX: LONGITUDINAL MODE REQUIREMENTS: SHORT PERIOD MODE REQUIREMENTS: PHUGOID MODE REQUIREMENTS: LATERAL MODE REQUIREMENTS: ROLL MODE REQUIREMENTS: SPIRAL MODE REQUIREMENTS: DUTCH ROLL MODE REQUIREMENTS: DESIGN PROJECT REPORT: 1

3 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. DESIGN PROJECT REPORT: 2

4 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 4 passengers. Its first flight was in Its length is 231ft 1in, wingspan is 211ft 55in, and height is 63ft 8 in. Its cruise speed is Mach.85 (567 mph) and it cost approximately $25 million. Its maximum range is 726 nautical miles. Seating capacity is more than 366 with a seat arrangement in economy class and a 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 4, ft. and Mach number of.9 for take of weight if 636,636 lbs. This flight condition is the cruise condition of an airplane. Aircraft Data and Derivatives Details: DESIGN PROJECT REPORT: 3

5 Longitudinal Dynamics Linearized State Space Aircraft Dynamics Matrix Model: (Input Elevator) X = AX + Bu X u u X α gcos(θ 1 ) X Z α u Z α U 1 + Z q gsin(θ u δe 1) Z α δe [ q ] = U 1 U 1 U 1 U 1 [ ] + θ M u M α Mq + Mα. q U 1 δ e θ M δe [ 1 ] [ ] u u α α.211 [ q ] = [ ] [ ] + [ ] δ q e θ 1 θ Output State Space Model: For short period mode, pitch rate (q), angle of attack (α) and normal load factor in Z (n z ) 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. q α n z = Z u θ g [ u ] [ q α n z = θ [ u ] [ Y = CX + Du 1 1 Z α g 1 1 ] Z q g sin(θ 1) ] u α [ ] + q θ u α [ ] + q θ δ e [ ] δ e [ ] Summary: A Long B Long C Long D Long X u X α gcos(θ 1 ) X δe Z u Z α U 1 + Z q gsin(θ 1) Z δe U 1 U 1 U 1 U 1 U 1 M u M α Mq + Mα. M δe [ 1 ] [ ] [ ] [ ] [ 1 1 Z u g Z α g 1 1 ] Z q g sin(θ 1) [ ] [ 1 1 ] [ ] DESIGN PROJECT REPORT: 4

6 Lateral-Direction Dynamics Linearized State Space Matrix Model: (Inputs Rudder, & Aileron): Y β Y p x = Ax + Bu Y r β 1 g U p 1 U 1 U 1 U1 cos(θ1) Y β δa Y δr p U 1 U 1 = L r β L p L r [ ] + r L δa L δr [ Δδa Δδr ] [ ϕ ] N β N p N r ϕ N δa N δr [ 1 ] [ ] β β p p = [ ] [ ] + [ r r [ ϕ ] 1 ϕ Output State Space Model: ] [ Δδa Δδr ] 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 ( n y ) can be used to close the loop and obtain required handling qualities. Summary: Y β Y p Y = CX + Du p 1 r 1 β β 1 p = Y n β y U1 [ ] + [ Δδa r cos(θ1) Δδr ] g g ϕ [ ϕ ] [ ] [ 1 ] p 1 β r 1 β p = 1 [ ] + [ Δδa n r y Δδr ] [ ϕ ] [ ϕ 1] [ ] A Lat B Lat C Lat D Lat Y r 1 g U 1 U 1 U 1 U1 cos(θ1) Y δa Y δr U 1 U 1 L β L p L r L δa L δr N β N p N r N δa N δr [ 1 ] [ ] [ ] [ ] Y β U1 g g cos(θ1) [ 1 ] [ [ ] [ ] 1] DESIGN PROJECT REPORT: 5

7 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 The Phugoid is a long-period, low frequency mode in which speed and Phugoid Mode (P) altitude are interchanged. The resulting oscillations are in pitch, speed, altitude, and flight path, while the angle of attack remains roughly constant. The short period is relatively rapid mode that governs the transient changes Short Period Mode (SP) 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) λ 1,2 (p ) =.976 ±.328j ξ p =.285 ω p =.342 Short Period Mode (4D) λ 1,2 (SP) =.469 ± 1.27j ξ sp =.347 ω sp = 1.35 Unaugmented Flying Handling Qualities: Parameter Unaugmented Value Unaugmented Handling Quality Short Period Mode λ 1,2 (sp).976 ±.328j N/A ξ sp.347 Level 2+3 ω sp( Rad s ) 1.35 Level 2+3 Phugoid Mode λ 1,2 (p).976 ±.328j N/A ξ p.285 Level 1 ω p ω sp =.25 <.1 Level 3 DESIGN PROJECT REPORT: 6

8 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 Phugoid Mode Augmented (Target value) Augmented Handling Quality λ 1,2 (sp).976 ±.328j N/A N/A N/A ξ sp.347 Level 2+3 ξ sp =.45 ±.5 Level 1 ω sp( Rad s ) 1.35 Level 2+3 Short Period Mode ω sp( Rad s ) = 3 ±.5 Level 1 λ 1,2 (p).976 ±.328j N/A N/A N/A ξ p.285 Level 1 ξ p =.6 ±.1 Level 1 ω p ω sp =.25 <.1 Level 3 ω p ω sp >.12 ±.1 Level 1 DESIGN PROJECT REPORT: 7

9 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 Roll Mode (R) Description Roll subsidence mode is simply the damping of rolling motion. Spiral Mode is a slow recovery or divergence from a bank angle disturbance. The Spiral Mode (SM) military specification contains a requirement which prevents too - rapid divergence. Dutch roll mode is the lateral-directional short period oscillatory mode. It Dutch Roll Mode (DR) 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) λ 1 (R) =.51 ξ R = 1 ω R =.51 T R = Spiral Mode (4D) λ 1 (SM) =.537 ξ SM = 1 ω SM =.537 T SM = 186 Dutch Roll (4D) λ 1,2 (DR) =.17 ± 1.1j ξ DR =.16 ω DR = 1.2 T DR = 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. DESIGN PROJECT REPORT: 8

10 Criteria Parameter Unaugmented Value Unaugmented Handling Quality Roll Mode (R) - λ 1 (R).51 N/A - ξ R ω R( Rad s ).51 Level T R (s) Level 3 Spiral Mode (SM) - λ 1,2 (SM).537 N/A - ξ SM ω SM.537 Level 3 5 T SM (s) 186 Level 1 Dutch Roll Mode (DR) - λ 1,2 (DR).17 ± 1.1j N/A 2 ξ DR.16 Level 2 3 ω DR 1.2 Level 1 4 ξ DR ω DR.18 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 Roll Mode (R) Unaugmented Unaugmented Augmented Augmented Value Handling Quality (Target value) Handling Quality - λ 1 (R).51 N/A N/A N/A - ξ R 1-1 (for stability) - - ω R( Rad s ).51 Level ±.25 Level 1 1 T R (s) Level 3.8 ±.1 Level 1 Spiral Mode (SM) - λ 1,2 (SM).537 N/A N/A N/A - ξ SM 1-1 (for stability) - - ω SM.537 Level 3.4 ±.5 Level 1 5 T SM (s) 186 Level 1 25 ± 3 Level 1 Dutch Roll Mode (DR) - λ 1,2 (DR).17 ± 1.1j N/A N/A N/A 2 ξ DR.16 Level 2.3 ±.5 Level 1 3 ω DR 1.2 Level 1 4 ± 1 Level 1 4 ξ DR ω DR.18 Level ±.55 Level 1 DESIGN PROJECT REPORT: 9

11 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) DESIGN PROJECT REPORT: 1

12 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): DESIGN PROJECT REPORT: 11

13 Tentative Summary Table: Used for Case Gain (Sign) Tentative Gain ξ sp(4d) ω sp(4d) ξ p(4d) ω p(4d) SP (damping) (Δq) to (Δδe) SP (Frequency) (Δ α) to (Δδe) SP (Δnz) to (Δδe) + P (Frequency) (Δθ) to (Δδe): P (Damping) (Δu) to (Δδe) 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: K Long = [ ] Step 5) Generate A_Augumented Matrix A Augumented = A Long(Unaugumented) + B Long K Long C Long A Augumented = [ ] Damp(Aaug) Table below Step 6) Analyze and verify the augmented Eigen values and flying qualities. Parameter Unaugmented Value Short Period Mode Unaugmented Augmented (Target Actual Augmented Augmented Handling Quality value) Values Handling Quality λ 1,2 (sp).976 ±.328j N/A N/A 1.49 ± 2.89j N/A ξ sp.347 Level 2+3 ξ sp =.45 ±.5 ξ sp =.459 Level 1 ω sp( Rad 1.35 Level 2+3 ω sp( = 3 ±.5 ω s ) Rad sp( = 3.25 s ) Rad s ) Level 1 Phugoid Mode λ 1,2 (p).976 ±.328j N/A N/A.252 ±.314j N/A ξ p.285 Level 1 ξ p =.6 ±.1 ξ p =.626 Level 1 ω p ω sp =.25 <.1 Level 3 ω p ω sp >.12 ±.1 ω p ω sp = =.124 Level 1 o The table above shows that all the longitudinal mode parameters meet level 1 handling qualities. DESIGN PROJECT REPORT: 12

14 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) DESIGN PROJECT REPORT: 13

15 4. Normal Load Factor ( Ny) Response to Aileron Input (Δδa) 5. Roll Angle ( ϕ) Response to Aileron Input (Δδa) Tentative Summary Table: Used for Roll Mode (Frequency) Dutch Roll Damping Dutch Roll Frequency Dutch roll damping Case Tentative Gain (Sign) Tentative Gain ω R ω DR ξ DR ω SM ( p) to (Δδa) / ( r) to (Δδr) ( β) to (Δδr) + 36 ( ny) to (Δδa): +.45 SM Frequency ( ϕ) to (Δδa) +.1 / DESIGN PROJECT REPORT: 14

16 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: Step 5) Generate A_Augumented Matrix A Augumented = [ K = [ ] A Augumented = A Lat(Unaugumented) + B Lat K Lat C Lat ] Damp(Aaug) Results below are shown 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) λ 1 (R).51 N/A -1.25± N/A ξ R 1-1 (for stability) 1 - ω R( Rad s ).51 Level ± Level 1 T R (s) Level 3.8 ±.1.8 Level 1 Dutch Roll Mode (DR) λ 1,2 (DR).17 ± 1.1j N/A N/A 1.17 ± 3.67j N/A ξ DR.16 Level 2.3 ±.5.34 Level 1 ω DR 1.2 Level 1 4 ± Level 1 ξ DR ω DR.18 Level ± Level 1 Spiral Mode (SM) λ 1,2 (SM).537 N/A.4 ± N/A ξ SM 1-1 (for stability) 1 - ω SM.537 Level 3.4 ± Level 1 T SM (s) 186 Level 1 25 ± Level 1 o The table above shows that all the lateral modes parameters meet level 1 handling qualities. DESIGN PROJECT REPORT: 15

17 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. DESIGN PROJECT REPORT: 16

18 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. DESIGN PROJECT REPORT: 17

19 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: 1 seconds and 3 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. DESIGN PROJECT REPORT: 18

20 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. DESIGN PROJECT REPORT: 19

21 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. DESIGN PROJECT REPORT: 2

22 VI. References: [1] Hodgkinson, J. Aircraft Handling Qualities. AIAA Education Series [2] Cook, Michael V. Flight Dynamics Principles. Boston, 213. DESIGN PROJECT REPORT: 21

23 Appendix: DESIGN PROJECT REPORT: 22

24 Longitudinal Mode Requirements: Short Period Mode Requirements: Figure 1: Typical Short-Period Mode Frequency Requirements Phugoid Mode Requirements: Table 1: Short-Period Mode Damping Requirements Table 2: Phugoid Damping Ratio Requirements DESIGN PROJECT REPORT: 23

25 Lateral Mode Requirements: Roll Mode Requirements: Spiral Mode Requirements: Table 3: Roll Subsidence mode time constant Dutch Roll Mode Requirements: Table 4: Spiral Mode time constant Table 5: Dutch Roll frequency and damping DESIGN PROJECT REPORT: 24