*Flight Dynamics* takes a new approach to the science and mathematics of aircraft flight, unifying principles of aeronautics with contemporary systems analysis. While presenting traditional material that is critical to understanding aircraft motions, it does so in the context of modern computational tools and multivariable methods. Robert Stengel devotes particular attention to models and techniques that are appropriate for analysis, simulation, evaluation of flying qualities, and control system design. He establishes bridges to classical analysis and results, and explores new territory that was treated only inferentially in earlier books. This book combines a highly accessible style of presentation with contents that will appeal to graduate students and to professionals already familiar with basic flight dynamics.

Dynamic analysis has changed dramatically in recent decades, with the introduction of powerful personal computers and scientific programming languages. Analysis programs have become so pervasive that it can be assumed that all students and practicing engineers working on aircraft flight dynamics have access to them. Therefore, this book presents the principles, derivations, and equations of flight dynamics with frequent reference to MATLAB functions and examples.

By using common notation and not assuming a strong background in aeronautics, *Flight Dynamics* will engage a wide variety of readers. Introductions to aerodynamics, propulsion, structures, flying qualities, flight control, and the atmospheric and gravitational environment accompany the development of the aircraft's dynamic equations.

Robert F. Stengel is Professor and former Associate Dean of Engineering and Applied Science at Princeton University, where he also directs the Program on Robotics and Intelligent Systems. He is the author of *Optimal Control and Estimation* (Dover, 1994). He was a principal designer of the Apollo Lunar Module manual control logic.

Princeton University Press | Cloth | November 2004 | $99.50 / £ 65.00 | ISBN: 0-691-11407-2 | 845 pages.

"A monumental piece of work. Its comprehensive treatment of flight dynamics makes it the broadest in its class and constitutes a major contribution to the aerospace community. Destined for students' shelves as well as mine, it will also be valuable as the methodological companion to the aircraft designer, flight test engineer, and pilot."--Eric Feron, Massachusetts Institute of Technology (currently, Georgia Institute of Technology)

"This book is definitely a significant contribution to the field. It is more comprehensive than any other work on flight dynamics I have seen; it includes newer concepts, such as neural nets and wind shear effects, some of these reflecting the author's own research; and it gives a very broad view of flight dynamics. Not only is it a fine textbook on flight dynamics, but it is so thorough and so well written that it will undoubtedly catch the attention of practicing engineers and airplane enthusiasts."--Haim Baruh, Rutgers University

Review by John Hodgkinson, AeroArts LLC, for the *AIAA Journal*

Review by Eric Feron, Georgia Institute of Technology, for the *IEEE Control Systems Magazine*

Review by John Valasek, Texas A&M University, for the *Journal of Guidance, Control, and Dynamics*

- Preface
- 1. Introduction
- Elements of the Airplane
- Airframe Components
- Propulsion Systems

- Representative Flight Vehicles
- Light General Aviation Aircraft
- Variable-Stability Research Aircraft
- Sailplane
- Business Jet Aircraft
- Turboprop Commuter Aircraft
- Small Commercial Transport Aircraft
- Large Commercial Transport Aircraft
- Supersonic Transport Aircraft
- Fighter/Attack Aircraft
- Bomber Aircraft
- Space Shuttle
- Uninhabited Air Vehicle

- The Mechanics of Flight
- References for Chapter 1

- Elements of the Airplane

- 2. Exploring the Flight Envelope
- The Earth's Atmosphere
- Pressure, Density, and the Speed of Sound
- Viscosity, Humidity, and Rain
- Wind Fields and Atmospheric Turbulence

- Kinematic Equations
- Translational Position and Velocity
- Angular Orientation and Rate
- Airflow Angles
- Summary of Axis Systems and Transformations

- Forces and Moments
- Alternative Axis Systems
- Aerodynamic Forces and Moments

- Static Aerodynamic Coefficients
- Lift
- Drag
- Pitching Moment
- Side Force
- Yawing Moment
- Rolling Moment
- Ground Effect

- Thrusting Characteristics of Aircraft Powerplants
- Propellers
- Reciprocating Engines
- Turboprop, Turbofan, and Turbojet Engines
- Ramjet and Scramjet Engines

- Steady Flight Performance
- Straight-and-Level Flight
- Steady Flight Envelope
- Cruising Range
- Gliding Flight
- Climbing Flight
- Maneuvering Envelope
- Steady Turning Flight

- References for Chapter 2

- The Earth's Atmosphere

- 3. The Dynamics of Aircraft Motion
- Momentum and Energy
- Translational Momentum, Work, Energy, and Power
- Energy-Changing Maneuvers
- Angular Momentum and Energy

- Dynamic Equations for a Flat Earth
- Rigid-Body Dynamic Equations
- Scalar Equations for a Symmetric Aircraft
- Alternative Frames of Reference
- Inertial Reference Frames
- Body-Axis Reference Frames
- Velocity- and Wind-Axis Reference Frames
- Air-Mass-Relative Reference Frame
- Direction Cosines and Quaternions
- Acceleration Sensed at an Arbitrary Point

- Dynamic Equations for a Round, Rotating Earth
- Geometry and Gravity Field of the Earth
- Rigid-Body Dynamic Equations

- Aerodynamic Effects of Rotational and Unsteady Motion
- Pitch-Rate Effects
- Angle-of-Attack-Rate Effects
- Yaw-Rate Effects
- Roll-Rate Effects
- Effects of Wind Shear and Wake Vortices

- Aerodynamic Effects of Control
- Elevators, Stabilators, Elevons, and Canards
- Rudders
- Ailerons
- Spoilers and Flaps
- Other Control Devices
- Isolated Control Surfaces
- Trailing-Edge Flaps

- Solution of Nonlinear Differential Equations
- Numerical Algorithms for Integration
- Equations of Motion
- Representation of Data
- Trimmed Solution of the Equations of Motion

- References for Chapter 3

- Momentum and Energy

- 4. Methods of Analysis and Design
- Local Linearization of Differential Equations
- Stability and Control Derivatives
- Incorporating Unsteady Aerodynamic Effects
- Symmetric Aircraft in Wings-Level Flight
- Longitudinal Equations of Motion
- Lateral-Directional Equations of Motion
- Stability-Axis Equations of Motion

- Solution of Linear Differential Equations
- Numerical Integration and State Transition
- Static and Quasistatic Equilibrium Response to Inputs
- Initial Response to Control Inputs
- Controllability and Observability of Motions
- Truncation and Residualization

- Stability and Modes of Motion
- Stability of Transient Response
- Fourier and Laplace Transforms
- Modes of Aircraft Motion
- Phase Plane

- Frequency-Domain Analysis
- Transfer Functions and Frequency Response
- Nyquist Plot and Nichols Chart
- Root Locus

- Dealing with Uncertainty
- Random Variables and Processes
- Dynamic Response to Random Inputs and Initial Conditions
- Effects of System Parameter Variations
- System Survey
- Monte Carlo Evaluation
- Stochastic Root Locus

- Linear Aeroelasticity
- Stress, Strain, and Material Properties
- Monocoque and Semi-Monocoque Structures
- Force and Moments on a Simple Beam
- Static Deflection of a Simple Beam under Load
- Vibrations of a Simple Beam
- Coupled Vibrations of an Elastically Restrained Rigid Airfoil
- Vibrations of a Complex Structure
- The Four-Block Structure
- Fuel Slosh

- Introduction to Flying Qualities and Flight Control Systems
- Cognitive/Biological Models and Piloting Action
- Aircraft Flying Qualities
- Linear-Quadratic Regulator
- Steady-State Response to Command Input
- Implicit Model-Following and Integral Compensation
- Optimal State Estimation
- Linear-Quadratic-Gaussian Regulator
- Design for Stochastic Robustness

- References for Chapter 4

- Local Linearization of Differential Equations

- 5. Longitudinal Motions
- Longitudinal Equations of Motion
- Reduced-Order Models of Long-Period Modes
- Second-Order Phugoid-Mode Approximation
- Effects of Compressibility
- Effects of Altitude Variation
- Effects of Wind Shear

- Reduced-Order Model of the Short-Period Mode
- Second-Order Approximation
- Effects of Compressibility and High Angle of Attack

- Coupled Phugoid/Short-Period Dynamics
- Residualized Phugoid Mode
- Fourth-Order Model
- Longitudinal Flying Qualities

- Control Mechanisms, Stick-Free Stability, and Trim
- Elevator Control Mechanism
- Short-Period/Control-Mechanism Coupling
- Control Force for Trimmed Flight
- Elevator Angle and Stick Force per g
- "Tail-Wags-Dog" Effect

- Longitudinal Aeroelastic Effects
- Truncated and Residualized Elastic-Body Models
- Coupling of the Short Period with a Single Elastic Mode

- References for Chapter 5

- 6. Lateral-Directional Motions
- Lateral-Directional Equations of Motion
- Reduced-order Model of the Dutch Roll Mode
- Reduced-order Model of Roll and Spiral Modes
- Coupled Lateral-Directional Dynamics
- A Truncated Dutch roll/Roll Model
- Residualized Lateral-Directional Models
- Fourth-Order Model
- Lateral-Directional Flying Qualities

- Control Mechanisms, Nonlinearity, and Time Delay
- Rudder Control Mechanism
- Dutch roll/Rudder Coupling
- Quasi-linear Representation of Nonlinearity
- Quasi-linear Root Locus Analysis
- Roll-Spiral/Aileron Coupling
- Spoiler Nonlinearity and Time Delay

- Lateral-Directional Aeroelastic Effects
- Equilibrium Response to Control
- Eigenvalues and Root Locus Analysis of Parameter Variations
- Response to Initial Conditions and Step Control Inputs

- References for Chapter 6

- 7. Coupled Longitudinal and Lateral-Directional Motions
- Small-amplitude Motions
- Effects of Rotating Machinery
- Asymmetric Inertial and Aerodynamic Properties
- Asymmetric Flight Condition and Constant Angular Rate
- Coupling Controls

- Inertial Coupling of Pitch and Yaw Motions
- Fifth-Order Model of Coupled Dynamics
- Truncated and Residualized Fourth-Order Models
- Response to Controls During Steady Rolling

- Multiple Equilibrium Points
- Second-order Examples of Multiple Equilibria
- Effects of Cross-Coupling and Control on Rolling Equilibrium

- Flight at High Angle of Attack
- High-Angle-of-Attack Aerodynamics and Control Effects
- Fully Developed Spins
- Simulated Motions of a Business Jet Aircraft
- Stability of High-Angle-of-Attack Maneuvers
- Pilot-Aircraft Interactions
- Gain-Scheduled Stability and Command Augmentation
- Adaptive Neural Network Control
- Robust Nonlinear-Inverse-Dynamic Control

- References for Chapter 7

- Small-amplitude Motions

- Epilogue
- Appendices
- Constants, Units, and Conversion Factors
- Mathematical Model and Six-Degree-of-Freedom Simulation of a
Business Jet Aircraft
- Main Program for Analysis and Simulation (FLIGHT)
- Low-Angle-of-Attack, Mach-Dependent Model (LoAeroModel)
- High-Angle-of-Attack, Low-Subsonic Model (HiAeroModel)
- Supporting Functions
- Equations of Motion (EoM)
- Cost Function for Aerodynamic Trim (TrimCost)
- Direction Cosine Matrix (DCM)
- Linear System Matrices (LinModel)
- Wind Field (WindField)
- Atmospheric State (Atmos)

- Linear System Survey
- Main Program for Analysis and Simulation (SURVEY)
- Supporting Functions
- Reduced-Order Models (LonLatDir)
- Transient Response (Trans)
- Static Response (Static)
- Controllability and Observability (ConObs)
- Natural Frequency (NatFreq)
- Stability and Modes of Motion (StabMode)

- Paper Airplane Program
- Bibliography of NASA Reports Related to Aircraft Configuration Aerodynamics
### MATLAB Code for Analysis and Simulation

#### Nonlinear Six-Degree-of-Freedom Aircraft Simulation

#### Linear System Survey

#### Paper Airplane Simulation

#### Performance of a Business Jet Aircraft

### Errata

- page xvi, URL in third line of last paragraph should be http://www.stengel.mycpanel.princeton.edu. (RS, 3/11/09)
- page 8, "... currently owned and operated by Flight Level Engineering in cooperation with Embry-Riddle Aeronautical University." reference: Croft, J., "Feeling FBW," Aviation Week & Space Technology, May 15-28, 2017, pp. 44-46.
- page 56, Change second sentence to read "With zero sideslip angle, the vertical flight-path angle is ..." (AV, 2/26/08)
- page 58, MATLAB output: xid = 2.8624. (AV, 2/26/08)
- page 62, line 9: rho(z) should be divided by rho-sub-SL. (DM, 11/4/04)
- page 63, eq. 2.3-14b: multiply C-sub-D-sub-alpha by alpha. (IT, 1/21/05)
- page 63, eq. 2.3-16: C-sub-L-sub-alpha^2 should be (C-sub-L-sub-alpha)^2. (IT, 1/21/05)
- page 72, line 6, "eq. 2.4-7" should be "eq. 2.4-8". (ME, 10/20/10)
- page 88, Figure 2.4-9 caption: "movement" should be "moment". (IT, 1/21/05)
- page 90, line 11, change "equation 2.3-20" to "equation 2.3-21". (AJL, 10/11/14)
- page 90, line 14, change "equation 2.2-33" to "equation 2.2-32". (AJL, 10/11/14)
- page 92, Insert C-sub-m-sub-alpha after "forces" on line 6 and remove from line 7. (DL, 7/12/05)
- page 97, 2 lines below eq. 2.4-62: Change S-4 to S-2. (RS, 10/4/10)
- page 109, line 24: co-axial counter-rotating propellers are sometimes called "contra-rotating" propellers. (RK, 2/2/11)
- page 117, eq. 2.5-37: change rho^2 to rho V^2. (DL, 9/23/05)
- page 136, Figure 2.6-5: Horizontal axis should be labeled "Equivalent Airspeed, m/s". (BK, 10/26/16)
- page 179, eq. 3.2-93a and 3.2-93b: add prime to v on left side of equation, following eq. 3.2-91. (DL, 12/12/06)
- page 184-187: Replace discussion of quaternions with material at Quaternions.
- page 187, Below eq. 3.2-121, reference should be made to eq. 3.2-36 rather than 3.2-35. (DL, 5/11/07)
- page 188, eq. 3.2-127: third term on right should be "- (r^2 v+ p^2) delta-y". (JG, 9/2011)
- page 193, line 13: "... and delta g-sub-E contains ..." (DL, 11/10/04)
- page 196, line 9: "eq. 3.2-111 that ..." (DL, 11/10/04)
- page 200, eq. 3.4-1: dot missing over the final "q" in the equation (RS, 10/12/06)
- page 232, eq. 3.5-18: first line, remove term between first and second "=" signs (RS, 3/8/13)
- page 260, line 4: replace "fmins" by "fminsearch" (RS, 11/15/12)
- page 281, "sensitivity". (RS, 11/14/06)
- page 284, line 3: Should be (eq. 4.1-58). (RK, 2/2/11)
- page 291, eq. 4.1-107: F-sub-LD(5,3) should be tan (theta-sub-o); F-sub-LD(6,3) should be sec (theta-sub-o). (RS, 10/24/08)
- page 294, eq. 4.1-125: F-sub-LD(5,3) should be tan (gamma-sub-o); F-sub-LD(6,3) should be sec (gamma-sub-o). (RS, 10/24/08)
- page 348, Figure 4.4-3: caption should be "(a) Amplitude ratio. (b) Phase angle." (RS, 12/1/06)
- page 354, Figure 4.4-7: Labeling of omega-sub-n-sub-SP and z-sub-2 is reversed. Dashed vertical line for z-sub-2 missing on phase angle plot. (RS, 1/19/05)
- page 390, eq. 4.6-2: This is a single integral over z, not a double integral over x and z. Ix = integral(zmin, zmax, x(z)z^2 dz). (RS, 2/14/08)
- page 422, eq. 4.7-2: fraction should be "-(s - 2/tau)/(s + 2/tau). (RS, 9/16/05)
- page 475, eq. 5.2-67: C-sub-L-sub-V should be C-sub-D-sub-V. (RS, 11/14/06)
- page 475, M in numerators of eq. 5.2-68 and 5.2-69 should be M^2. (RS, 12/6/08)
- page 475, eq. 5.2-69: Numerator should be (eta - ...) (RS, 12/6/08)
- page 476, Damping ratios in Table 5.2-3 should be adjusted accordingly. (RS, 12/6/08)
- page 476, eq. 5.2-70: (1 + eta) should be (1 - eta)^(1/2), where eta is less than or equal to 0. (RS, 12/6/08)
- page 481, eq. 5.2-91, change D-sub-L to D-sub-V. (LV, 10/28/14)
- page 482, eq. 5.2-97, change 2nd term to sqrt((g L-sub-V / V-sub-o) (1 - ((V-sub-o / g) (del V-sub-W / del z)))). (LV, 10/28/14)
- page 500, Fig. 5.3-8b, vertical axis: 1 should be 0.1 (RS, 10/23/08)
- page 522, Figure 5.4-5: caption should be "Frequency response of flight path angle (solid line), angle of attack (dashed line), and pitch angle (dashed-dotted line) to sinusoidal thrust input." (RS 12/5/06)
- page 528, Footnote 2: First-order time to double is -0.693 tau, where tau < 0. (RS, 8/16/10)
- page 547, eq. 5.5-51, change C-sub-L-sub-q-hat to C-sub-H-sub-q-hat. (LV, 10-28-14)
- page 569, Reference B-6: the date is 1904 rather than 1903. (RS, 6/19/09)
- page 570, Reference H-4: NACA Report 767. (RS, 6/19/09)
- page 570, Reference L-1: revised title,
*Aerodynamics: Constituting the First Volume of a Complete Work on Aerial Flight*. (RS, 6/19/09) - page 570, add Reference L-2: Lanchester, F. W.,
*Aerodonetics: Constituting the Second Volume of a Complete Work on Aerial Flight*, Arnold Constable, London, 1908. (RS, 6/19/09) - page 575, eq. 6.1-15: F-sub-LD(4,1) should be tan (gamma-sub-o). (RS, 10/24/08)
- page 580, eq. 6.2-13: "- s" should be "- (N-sub-r + Y-sub-beta / V-sub-zero)s" (RS 12/10/06)
- page 581, lines 6 and 7: re-phrase: "... supersonic speed. The vertical tail's center of pressure may move aft; however, ..." (RK, 12/6/10)
- page 590, eq. 6.3-19, change "+ L-sub-phi" to "- L-sub-phi". (LV, 10-28-14)
- page 597, Table 6.4-1 caption: change "5.4" to 6.4". (RS, 11/30/06)
- page 603, eq. 6.4-38: Add the 1st term of eq. 6.4-37 to the right side of eq. 6.4-38. The 2 equations take the general form, lambda-sub-1,2 = c1 +/- c2, and c1 is missing from eq. 6.4-38 (NK, 6/23/14)
- page 627, next to last line: should be |delta-phi / delta-beta|. (RS 12/13/06)
- page 683, eq. 7.1-6: F-sub-LD(4,3) should be tan (gamma-sub-o). (RS, 10/24/08)
- page 758, eq. 7.4-8, change "cos 2 alpha" to "sin 2 alpha". (LV, 10-28-14)
- page 816, line 9 of computer code: "CLa = 3.141592 * AR/(1 + sqrt(1 + (AR / 2)^2);" (RS, 8/23/05)
- page 834, "Deep (super) stall, 502, 682, 750, 762" (RS, 11/3/14)
- page 844, "Van der Pol oscillator, 734" (RS, 1/3/06)

#### Course and Lecture Slides

- Aircraft Flight Dynamics, MAE 331, Princeton University.
- Lecture Slides for Aircraft Flight Dynamics, MAE 331, Princeton University.

**http://www.stengel.mycpanel.princeton.edu/FlightDynamics.html**

*key words:*aircraft flight dynamics, flight mechanics, stability and control, short period, phugoid, Dutch roll, roll, spiral, aeronautical case histories, aircraft performance, longitudinal dynamics, lateral-directional dynamics, coupled motions, inertial coupling, high angle of attack, aeroelasticity, nonlinear equations of motion, flight simulation, flight vehicles, earth's atmosphere.

Last updated on November 11, 2018.

Copyright 2018 (c) by Robert F. Stengel. All rights reserved.