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University of Sydney Aerospace Engineering

AERO3560 FLIGHT MECHANICS 1 Lecture Notes

P. W. Gibbens

13 March 2014

University of Sydney

Aerospace Engineering

AERO3560

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AERO3560 FLIGHT MECHANICS 1 GENERAL Lecturer: Peter Gibbens, Rm N124 Unit of Study Resources: See the on-line Flight Mechanics T&L Resource Centre

Tutor: See References: Nelson, Flight Stability and Automatic Control, 2nd Edn, 1998, McGraw-Hill McCormick, Aerodynamics, Aeronautics and Flight Mechanics, 2nd Edn, 1995, Wiley Etkin, Dynamics of Atmospheric Flight, 1972 Roskam, Airplane Flight Dynamics and Automatic Flight Controls AIAA ANSI R-004-1992, Recommended Practice – Atmospheric and Space Flight Vehicle Coordinate Systems

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GOALS AND EVALUATION Goals To develop a concise and clear understanding of ∙ The nature of aircraft equilibrium and stability ∙ The determination of equilibrium ∙ Dynamic stability and modes of motion ∙ Handling qualities To develop generic engineering skills in ∙ Analysis ∙ Design ∙ System structure, breakdown, specification and integration, ∙ Project orientated approach and management principles. To develop a professional engineering ethos and approach to problems.

Performance Evaluation Evaluation of student performance in this course will be guided by the following principles: ∙ Demonstrated ability to provide sufficiently accurate solution to problems, ∙ Selection and demonstrated application of techniques appropriate to the problem, ∙ Demonstrated understanding of the basic concepts taught in the course via verbal and written communication, ∙ The demonstration of generic skills such as analytic capability, design strategy, problem breakdown and priority allocation, ability to interact and work in a team, ∙ Methodical approach to problem formulation and solution ∙ Resourcefulness in obtaining and using reference material, tools, tutorial assistance, and participation in class and group work.

Assessment and Assignments Course assessment will be based on tutorial assignments (50%) and a final examination (50%). There will be four (4) tutorial assignments. These will be due and marked according to the schedule available on the CUSP assessment page: . Note: A pass in the exam is required to pass the course Mark allocation will be on the basis of the quality of problem solutions and the manner in which they are obtained, judged according to the criteria set on CUSP.

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COURSE OUTLINE ∙ General equations of aircraft motion ∙ Linearised equations of aircraft motion ∙ Static lateral-directional characteristics and stability ∙ Static lateral-directional equilibrium ∙ Dynamic lateral-directional characteristics and equilibrium ∙ Lateral-directional control forces ∙ Linear perturbation equations of motion ∙ Modes of dynamic aircraft motion ∙ Dynamic stability This course deals with the static longitudinal lateral-directional equilibrium and stability of fixed-wing aircraft. It follows on by considering longitudinal and lateraldirectional dynamic stability. Beginning from a general discussion of stability, the conditions required for aircraft longitudinal stability are given. As opposed to aircraft performance (see other notes in Flight Mechanics series) which deals primarily with force balance and the ramifications thereof, stability deals with the moment balance that keeps the forces in equilibrium, and in particular, the nature of the net moments that arise if the aircraft is disturbed from the equilibrium. Equilibrium is investigated through the effects of the longitudinal controls. Various control mechanisms and their modes of operation are discussed. The mechanics of these controls are developed. This leads to an investigation of the effects of controls on equilibrium conditions, and the nature and magnitude of control forces required to keep the aircraft in balance when it is flying in an untrimmed (off-equilibrium) condition. The effect on stability of freeing the controls is presented. This leads to an analysis of longitudinal stability in manoeuvring flight (longitudinal manoeuvring in the vertical plane), this is compared to stability in steady-level flight. These equilibrium concepts are then built upon by considering equilibrium and stability for lateral-directional flight (in the horizontal plane). Finally the general case of aircraft dynamic stability is considered. Natural modes on motion are studied and linked to analyses of their dynamic properties. The relationship of these dynamic descriptions of aircraft motion are related the concept of aircraft handling qualities, and assessments of acceptable handling qualities.

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Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2

Assessment and Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Aerospace Engineering

FLIGHT MECHANICS 1 Contents GOALS AND EVALUATION

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COURSE OUTLINE

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1 INTRODUCTION

7

1.1 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Longitudinal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Lateral-directional Motion . . . . . . . . . . . . . . . . . . . . . . .

8 8 8

1.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 10

1.4 Def initions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 STATIC LONGITUDINAL STABILITY 2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Wing Contribution to Static Longitudinal Stability . . . . . . . . . . . . . 2.3 Horizontal Tail Contribution to Static Longitudinal Stability . . . . . . . 2.4 Fuselage Contribution to Static Longitudinal Stability . . . . . . . . . . .

10 13 13 14 17 19

2.5 Power Contribution to Static Longitudinal Stability . . . . . . . . . . . . . 20 2.6 Total Aircraft Static Longitudinal Stability Characteristics . . . . . . . . 21 2.7 Control Power and Moment Equilibrium . . . . . . . . . . . . . . . . . . . 2.8 Static Test for Neutral Point Location . . . . . . . . . . . . . . . . . . . . . 3 LIFT COEFFICIENT AND STABILITY

25 27 29

3.1 Effect of Static Stability on Lift Coefficient . . . . . . . . . . . . . . . . . . 29 3.2 Trimmed Lift Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 STABILISERS AND ELEVATORS

31

4.1 Horizontal Stabiliser Authority Limits . . . . . . . . . . . . . . . . . . . . . 31 4.2 Elevators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.1 Elevator Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 REVERSIBLE CONTROL SYSTEMS 35 5.1 Control Surface Hinge Moment . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Longitudinal Stability with Reversible Controls Free . . . . . . . . . . . . 37 5.3 Lift Coefficient with Reversible Controls Free . . . . . . . . . . . . . . . . 5.4 Control Forces for Reversible Control Systems . . . . . . . . . . . . . . .

38 38

5.5 Static Test for Control Force Neutral Point Location . . . . . . . . . . . .

42

6 STATIC LONGITUDINAL STABILITY IN STEADY MANOEUVRES

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6.1 Longitudinal Stability in Steady Pull-ups . . . . . . . . . . . . . . . . . . . 44 6.2 Longitudinal Stability in Steady Turns . . . . . . . . . . . . . . . . . . . . 47 6.3 Reversible Control System Forces in Steady Manoeuvres . . . . . . . . . 48 6.4 Static and Manoeuvring Longitudinal Stability: Summary . . . . . . . . . 7 GENERAL EQUATIONS OF AIRCRAFT MOTION 7.1 Axis Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 51 51

7.1.1 General Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 51 7.2 General equations of Aircraft Dynamic Motion . . . . . . . . . . . . . . . 58 7.3 Orientation and Axes Transformations . . . . . . . . . . . . . . . . . . . . 61 7.3.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.3.2 Direction Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.3.3 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.4 Aerodynamic Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 LINEARISED EQUATIONS OF AIRCRAFT MOTION 8.1 Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72

8.2 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.3 Linearised Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 73 8.4 Linearised Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . 76 9 STATIC LATERAL-DIRECTIONAL STABILITY 9.1 Static Stability Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Lateral-Directional Control Power . . . . . . . . . . . . . . . . . . . . . . .

79 79 79

9.3 Lateral Stability Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.3.1 Sideforce due to Sideslip . . . . . . . . . . . . . . . . . . . . . . . . 79 9.3.2 Directional (Weathercock) Stability . . . . . . . . . . . . . . . . . . 79 9.3.3 Dihedral Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4 Lateral Control Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4.1 Sideforce due to Rudder . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4.2 Directional Control Power . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4.3 Roll due to Rudder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4.4 Lateral Control Power . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4.5 Adverse Yaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 10 STATIC LATERAL-DIRECTIONAL EQUILIBRIUM - STEADY HEADING SIDESLIP 79 10.1 Sidelip Angle in a Steady Heading Sideslip . . . . . . . . . . . . . . . . . . 10.2 Bank Angle in a Steady Heading Sideslip . . . . . . . . . . . . . . . . . . .

79 79

10.3 Rudder Deflection in a Steady Heading Sideslip . . . . . . . . . . . . . . . 79 10.4 Aileron Deflection in a Steady Heading Sideslip . . . . . . . . . . . . . . . 79 10.5 Rudder Authority and Large Yawing Moments . . . . . . . . . . . . . . . . 79 10.6 Equilibrium with Zero Bank Angle

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11 STEADY LATERAL-DIRECTIONAL RATE DERIVATIVES

79 79

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11.1 Sideforce due to Yaw Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 11.2 Yaw Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 11.3 Rolling Moment due to Yaw Rate . . . . . . . . . . . . . . . . . . . . . . . . 79 11.4 Sideforce due to Roll Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 11.5 Yaw Moment due to Roll Rate . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Roll Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 STEADY TURNING FLIGHT

79 79 79

12.1 Balanced Zero-Sideslip Turn . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Aileron Only Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Spiral Mode Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 79

12.4 Steady Rolling Flight

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

13 LATERAL-DIRECTIONAL CONTROL FORCES 13.1 Rudder Pedal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79

13.2 Aileron Control Forces

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14 LINEAR PERTURBATION EQUATIONS OF MOTION 14.1 Laplace Transforms of Equations of Motion . . . . . . . . . . . . . . . . . 14.2 Time Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Non-dimensional Forms of Equations . . . . . . . . . . . . . . . . . . . . .

79 79 79 79 79

14.4 Dimensional Derivatives as Functions of the Non-dimensional Aerodynamic Coefficients a 14.5 Power Related Stability and Control Derivatives . . . . . . . . . . . . . . . 79 15 APPLICATION OF EQUATIONS OF MOTION TO TYPICAL AIRCRAFT

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15.1 Longitudinal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 15.1.1Phugoid Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 15.1.2Short-period Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 15.1.3Effects of Trim Speed Change . . . . . . . . . . . . . . . . . . . . . 79 15.2 Lateral-Directional Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 15.2.1Roll Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2Spiral Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3Dutch Roll Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 79

15.3 Bounds on Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 15.3.1Routh’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 15.3.2Stability Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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INTRODUCTION

Flight Mechanics incorporates the study of; ∙ static stability, ∙ dynamic stability, ∙ the response of an aircraft to control inputs, ∙ controllability, and ∙ closed loop flight control. These fields are critical to aircraft design, development, handling and operations. The Mechanics of fixed wing aircraft, are covered in this school according to the following breakdown; ∙ Longitudinal static stability and equilibrium, Lateral static stability and equilibrium. Longitudinal and lateral dynamic stability - AERO3560 Flight Mechanics 1 ∙ Longitudinal and lateral dynamic stability and response to control inputs and wind gusts. Introduction to closed loop control - AERO4560 Flight Mechanics 2 ∙ Digital flight control and advanced topics in flight dynamic system identification - AERO4590 Advanced Flight Mechanics and Digital Flight Control (elective). NOTE: The study of dynamic stability and response, and closed loop control will also make extensive use of the dynamics material covered in AMME2500, AMME3500. Flight dynamics, flight mechanics, stability and control analysis determines how an aircraft’s flight path and attitude will vary in response to inputs, and how the mass, moments of inertia and geometry effect the stability of the aircraft and its response to control inputs. These may include pilot control inputs, automatic flight control system inputs, weight and inertia distribution changes and aerodynamic disturbances such as gusts. This course is intended to give a substantial grounding in aircraft flight statics and dynamics. It begins with a comprehensive analysis of stability, specifically with respect to the stability of flight. This is done initially with respect to what is called longitudinal flight, meaning flight in a vertical plane. It will then expand to consider the stability of lateral-directional flight. This considers flight motion in the horizontal plane. To make this move we will need to define the the axis systems used to define aircraft motion. The general nonlinear equations of aircraft motion arise from the inter-relation of these axis systems, these will be developed from the general equations of linear and angular motion. In the analysis of aircraft dynamics and stability it is often more convenient to use a linearised version of the aircraft equations of motion, accordingly these will be developed from the general equations of motion. The forces and moments which act upon an aircraft determine its motion and are effectively inputs to the equations of motion but are dependent on the variables, herein referred to as state variables, which are determined by the equations of motion. Mechanics of Flight 1 introduced the forces and moments which affect the longitudinal motion of an aircraft. In this course we introduce and characterise the forces and moments which affect the lateral and directional motions of the aircraft. The static lateral-directional equilibrium and stability are then investigated, leading to the study of lateral-directional control forces and dynamic motion.

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Using the concept of small amplitude motion about a trim point, the linear perturbation form of the equations of motion is developed in order to study the natural dynamic modes and dynamic stability.

1.1 Degrees of Freedom The general motion of an aircraft in 3-D space is said to have 6 Degrees of Freedom. This means it can translate in 3 dimensions, effectively having components of linear motion along three pre-defined orthogonal axes (3DOF). It can also rotate with components of rotation about each of the same axes (the other 3DOF). In the study of aircraft motion, it turns out that in most cases these 6 Degrees of Freedom can be separated into two groups that are largely independent (uncoupled). The groups describe the aircraft longitudinal and lateral-directional motions respectively. 1.1.1 Longitudinal Motion The longitudinal motion of an aircraft effectively describes its motion in a vertical plane. 1.1.2 Lateral-directional Motion

1.2

Stability

Stability describes the response of a system parameter following an external disturbance of that parameter. A system may be positively stable, neutrally stable, or unstable (or metastable, but this is not a common state for the simple linear aircraft undergoing small disturbances). The classic illustration of the three cases consists of a ball on surfaces of various shapes.

Static Stability Refers to the gradient of a parameter immediately after a small disturbance. If the parameter tends towards the undisturbed state the system is statically stable. For example, consider 𝛼 (angle of attack) being perturbed by a small gust, causing a change Δ𝛼

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Static stability is a description of the immediate post- disturbance gradient of the parameter response. Dynamic Stability Refers to the medium to long term post-disturbance response of a disturbed parameter (of the order of 1 sec to 1 min). For the typical rigid-body aircraft with simple control systems and small-disturbance linear aerodynamics, long term responses are frequently oscillatory in nature and well approximated by combinations of first and second order systems. One common method of describing such second order responses, widely used in airworthiness regulatory documentation, is in terms of an oscillatory frequency, either undamped (𝜔𝑛 ) or damped (𝜔𝑑 ) and a viscous damping ratio (𝜁). There are several likely combinations of static and dynamic stability, which may be viewed in terms of their eff...