9. Math Modeling and Flight Simulation5_9

8/12/2019 9. Math Modeling and Flight Simulation5_9

1/31

9. MATH MODELING AND

FLIGHT SIMULATION


2/31

GOALS OF FLIGHT SIMULATION

Predict mathematically:

1. Equilibrium flight condition (Trim)

2. Flight mode stability of the aircraft (Stab

Analysis),

3. Time history response of the aircraft due to

pilot inputs, external disturbances.(Maneuver)


3/31

NON REAL TIME GLOBAL MODELS

BASED ON KNOWN AVAILABLE PHYSICS

PRIMARY USES:

COMPONENT DESIGN (Rotor, fuselage, etc)

INTER-RELATIONSHIP AMONG COMPONENTS

DEVELOPMENT OF MORE ADVANCED MODLES LIMITED BY

ACCURATE KNOWLEDGE OF PHYSICS

DEVELOPMENT OF DIGITIAL COMPUTERS

ONE ANALYSIS/PROGRAM APPLICABLE TO SEVERAL

AIRCRAFT MODELS


4/31

REAL TIME MODELS

INDIVIDUAL MODEL FOR EACH AIRCRAFT MODEL

MODEL LOOSELY BASED IN PHYSICS BUT TWEAKED

TO AIRCRAFT FLIES LIKE THE REAL AIRCRAFT TO

TEST PILOTS

PRIMARY USES: PILOT FAMILIARZATION, PILOT

TRAING, CONTROL SYSTEM DESIGN AND

EVALUATION, ACCIDENT INVESTIGATION

MAY NOT BE USEFUL FOR DETAILED DESIGN OF

MECHANICAL COMPONENTS


5/31


6/31

MODES OF OPERATION

TRIM

STAB (Linearized Model)

TIME HISTORY


7/31

TRIM

Required Inputs:

1. Complete structural and aerodynamic description of

aircraft using either measured aerodynamic data or analytic functions.

2. Complete description of Control System

3. Flight Condition description (gross weight, cg, speed,atmospheric properties)

Typical Results:

1. Attitude of aircraft (yaw, pitch, roll), Rotor flapping angles

(fore/aft and lateral flapping) for 2 rotors

2. Position of pilots controls

3. Steady state performance (HP, fuel flow,etc) and

Steady state oscillatory rotor and fuselage loads


8/31

TRIM MATHEMATICS

Trimmed Helicopter flight condition is described by 11

independent variables: 3 fuselage angles(yaw, pitch,

roll); 4 pilot controls (collective, f/a cyclic, lateral cyclic,

and pedals) and 4 rigid blade flapping angles (f/a and

lateral) for 2 independent rotors.

These 11 independent variables must satisfy 10 static

equilibrium equations ( 3 forces and 3 moments at and

about cg, fore/aft and lateral flapping moments for 2independent rotors


9/31

TRIM SOLUTION TECHNIQUE

1. Assume fixed value of 1 flight condition

independent variable (usually fuselage

pitch or roll angle) to obtain 10

nonlinear algebraic equations in 10unknowns.

2. Guess values for remaining 10

independent variables and iterate on

them until all 10 static equilibriumequations are satisfied


10/31

STAB ANALYSIS

GOAL: Produce a linearized mathematical model

that is valid over a small region about the equilibrium

flight condition obtained in the Trim process.

METHOD: Calculate the changes in all fuselage

and rotor forces and moments due to small

perturbations in the flight variables- displacements,

velocities, and control inputs.

Produces 10x10 mass, damping, and stiffness

matrices; and a 10x4 control effectiveness matrix


11/31

STAB RESULTS

Stick fixed aircraft stability obtained from Eigen value

solution. System damped natural frequencies and

mode shapes. Time to half (stable roots) or time to

double amplitude (unstable roots)

Single input-Single output transfer functions to show

aircraft response to control inputs. Starting point in

designing automatic flight control system.M

Many System requirements are described in terms

resulting for the Stab Results.


12/31

TIME HISTORY SOLUTION

TASK: Determine time history response of aircraft, performance,

rotor loads, fuselage vibrations, etc following Pilot control input,

change is aircraft configuration change is atmosphere.

SOLUTION: Use 4 cycle Runge_Kutta method to numericallyintegrate NDE nonlinear, coupled differential equations

NDE= Fuselage Rigid Body Degrees of Freedom 6

number of elastic fuselage modes from NASTRAN ??

number of elastic rotor modes from MYKLESSAD ???

5< NDE


13/31

ROTOR MODELS

1. Rotor Disk or Frisbee Model

2. Blade Element Model

3. Aeroelastic Model


14/31

ROTOR DISK or FRISBEE

Linear Model

Rotor Forces as function of (radius, tipspeed,

solidity, average slope of lift curve, collectivepitch, cyclic pitch, rotor inflow, twist)

Neglects: Stall, compressibility effects, mach

number effects, different airfoils, taper,

nonlinear twist, elastic blade deflections andvelocities

Produce quick Back of the Envelope results


15/31

]T+T+T+T[2

a

=C 423T201T

Thrust Coefficient from

Approximate Linearized Integration


16/31

BLADE ELEMENT MODEL

Blade Aerodynamic forces are calculated at 20

radial stations and 24 azimuth locations

Requires solution of Rigid Blade Flapping

Differential Equation

Includes nonlinear aerodynamic effects (stall,

compressibility, nonlinear twist, non uniform

airfoil radial distribution

Neglects effects of blade/fuselage Elastic

displacements and velocities


17/31

BLADE ELEMENT AERODYNAMIC

MODEL


18/31

BLADE ELEMENT AERODYNAMICS


19/31

BLADE FLAPPING DIFFERENTIAL

EQUATION


20/31

BLADE ELEMENT AERO FORCES

CL from Airfoil data tables

CD from Airfoil data tables


21/31

AIRFOIL LIFT AS A FUNCTION

OF MACH AND ALPHA

MACH NUMBERANGLE OF ATTACK 0.3 0.4 0.5 0.68 0.74 0.92 1

(DEGREES) -25 -0.892 -0.892 -0.892 -0.892 -0.892 -0.8922 -0.892

-14 -0.81 -0.81 -0.81 -0.783 -0.76 -0.69 -0.69

-12 -0.816 -0.816 -0.816 -0.786 -0.761 -0.59 -0.59

-10 -0.766 -0.766 -0.766 -0.758 -0.751 -0.48 -0.48

-8 -0.655 -0.67 -0.66 -0.68 -0.69 -0.39 -0.39

-6 -0.51 -0.54 -0.53 -0.625 -0.681 -0.35 -0.35

-4 -0.34 -0.35 -0.34 -0.44 -0.5 -0.3 -0.3

-3 -0.24 -0.25 -0.24 -0.313 -0.35 -0.2 -0.2

-2 -0.14 -0.15 -0.13 -0.161 -0.17 -0.1 -0.1

-1 -0.03 -0.03 0 0.006 0.02 0 00 0.09 0.095 0.12 0.163 0.2 0.1 0.1

1 0.21 0.22 0.24 0.3177 0.35 0.2 0.2

2 0.32 0.33 0.36 0.447 0.47 0.3 0.3

3 0.44 0.45 0.49 0.56 0.56 0.4 0.4

4 0.55 0.57 0.65 0.663 0.64 0.5 0.5

6 0.78 0.79 0.87 0.863 0.78 0.59 0.59

8 0.99 1.02 1.13 1.021 0.91 0.69 0.69

10 1.2 1.24 1.36 1.122 1.03 0.73 0.73

11 1.31 1.35 1.47 1.15 1.09 0.74 0.74

12 1.41 1.435 1.55 1.195 1.14 0.74 0.74

12.5 1.46 1.43 1.43 1.212 1.17 0.738 0.738

13 1.51 1.39 1.42 1.229 1.2 0.735 0.735

13.5 1.54 1.34 1.42 1.251 1.23 0.732 0.732

14 1.535 1.29 1.41 1.269 1.26 0.73 0.73

14.5 1.5 1.26 1.39 1.285 1.29 0.73 0.73

15 1.435 1.24 1.36 1.297 1.32 0.72 0.72

16 1.392 1.19 1.325 1.274 1.295 0.7 0.7

17 1.326 1.15 1.3 1.288 1.316 0.67 0.67

20 1.172 1.172 1.172 1.294 1.3 0.64 0.64

25 0.892 0.892 0.892 1.036 1.16 0.65 0.65


22/31

TYPICAL AIR FOIL LIFT DATA TABLE

-1.5

-1

-0.5

0

0.5

1

1.5

2

-30 -20 -10 0 10 20 30

Angle of Attack (degrees)

LiftC

oefficient(cl)

0.3

0.4

0.5

0.68

0.74

0.92

1


23/31

AEROELASTIC MODEL

REQUIRES AS INPUT FULLY COUPLED

(BEAM/CHORD/TORSION) BLADE MODE SHAPES

AND NATURAL FREQUIENCES)

SOLUTION REQUIRES NUMERICAL INTEGRATIONOF SEVERAL NON-LINEAR SECOND ORDER

COUPLED DIFFERENTIAL EQUATIONS WITH TIME

DEPENDENT COEFFICENTS

PRODUCES ROTOR SHEAR AND MOMENT

DISTRIBUTIONS INCLUDING ELASTIC EFFECTS


24/31

AIRPLANES VS. HELICOPTERS

AIRPLANES

6 DEGREES OF FREEDOM: 3 FORCES AND 3 MOMENTS

7 IND. VARIABLES: YAW PITCH ROLL ANGLES

THROTTLE F/A STICK LAT STICK PEDALS

HELICOPTERS

10 DEGREES OF FREEDOM: 3 FUSE FORCES AND 3 FUS MOMENTS

+ F/A AND LATERAL MOMENTS ON 2 ROTORS

11 IND. VARIABLES: FUSELAGE YAW PITCH AND ROLL ANGLES

COLLECTIVE F/A CYCLIC LAT CYCLIC PEDAL+ F/A AND LATERAL FLAPPING FOR 2 ROTORS


25/31

PROBLEMS ASSOCIATED WITH GLOBAL

SIMULATION PROGRAMS

COPTER

CAMRAD

FLIGHT LAB


26/31

VALIDITY OF IMBEDDED ANALYTICAL

MODELS

INDIVIDUAL COMPONENTS (Induced Velocity, Tip

Vortex, Mechanical Friction) cannot be verified as a

stand alone component

INNERACTION OF MAJOR COMPONENTS ARE

DIFFICULT, IF NOT IMPOSSIBLE, TO MODEL

(Operating environment of Tail Rotor)

MATH MODELS OF PILOT/HUMAN REACTION

ARE SUSPECT


27/31

VALIDITY OF INPUT DATA

STEADY/UNSTEADY AERODYNAMIC

COEFFICIENTS OVER TOTAL OPERATING REGION

(Mach, Angle of Attack)

FINITE ELEMENT MODEL/RESULTS FOR FIXED

COMPONENTS

FULLY COUPLED BEAM/CHORD/TORSION BLADE

NATURAL FREQUENCIES AND MODE SHAPES


28/31

USE OF OUTPUT DATA

MANAGING THE NUMBER OF OUTPUT DATA

INTERPRETATION OF RESULTS (Knowledge ofCoordinate System, Units, etc)


29/31

ADVANCES IN DIGITAL COMPUTERS

1972: IBM 360 MAIN FRAME COMPUTER

BLADE ELEMENT MODEL REQUIRED 15

MINUTES TO MODEL 4 SEC REAL TIME

RATIO: 1 SEC REAL TIME=225 SEC CPUTIME

2002: DESKTOP PC:

1 SEC REAL TIME=.2 SEC CPU TIME


30/31

DESK TOP SIMULATION PROGRAMS

MICROSOFT FLIGHT SIMULATOR 2002

X PLANE

FLY II

STRONG POINTS:

GRPHICS, EYE CANDY, NAVIGATION

AIDS, AIRPORTS, WEATHER

WEAK POINTS:

ACCURATE REPRESENTATION OF

VEHICLE DYNAMICS


31/31

ROTOR MODELING PROBLEMS

AERODYNAMIC MODELS HAVE DIFFICULTY WITH

1 UNSTEADY AERODYNAMICS

2 STALL/ REVERSED FLOW3 TIP VORTEX

4 ROTOR WAKE

INCREASING MODELING DETAIL DOES

NOT NECESSARILY PRODUCE BETTER

ANSWERS *** UNIFORM TEXTURE***

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9. Math Modeling and Flight Simulation5_9