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AERSP 420 – Principles of Flight Test

Final Report - #3

Brian Harrell Linda John

December 20, 2013

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ABSTRACT The Piper Arrow III-28R-201 belongs to a family of light aircraft design for flight

training, air taxi, and personal use. This report analyzes the performance of the Arrow III in a test

flight program, specifically during the phugoid maneuver. Furthermore, this report compares

experimental data results to values derived from a verified analytical simulation via three

different methods: a discrete time simulation of the phugoid maneuver as well as an energy

relation simulation and an eigenvalue simulation.

The discrete time step portion of the phugoid test resulted in a period of about 22

seconds and the phugoid frequency to be about 0.286 (1/seconds). The time to half amplitude for

the phugoid mode was found to be about 60 seconds based on the longitudinal velocity plot. The

short period was determined to be 1.3 seconds, resulting in a short period frequency of 4.83

(1/seconds) and a time to half amplitude for the short period mode of 4 seconds. An eigenvalue

simulation show the frequency of oscillation for the phugoid mode was 0.2933 (1/seconds).

This results in a period of 21.42 seconds and a time to half amplitude of about 31 seconds for the

phugoid mode. The short period frequency is 3.9512 (1/seconds) based on the eigenvalues

which leads to a short period of 1.590 seconds and a time to half amplitude of about 0.3 seconds.

An energy relation simulating the phugoid maneuver that the phugoid period is around

13 seconds, resulting in a frequency of 0.4833 (1/seconds). There is no damping in the energy

relation equations and thus do not have a time to half amplitude. The only major difference

between these result determined for a flight speed of 80 knots and a similar analysis ran for 100

knots was a increased amplitude and period of oscillation for both modes.

In comparison to the experimental results for the phugoid maneuver at both 80 knots and

100 knots, it was calculated that the phugoid period is about 24 seconds for 80 knots and about

26 seconds for 100 knots resulting in phugoid mode frequencies of 0.2618 (1/seconds) at 80

knots and 0.2417 (1/seconds) at 100 knots. The times to half amplitude for the phugoid modes

were 62 seconds at 80 knots and 75 seconds at 100 knots. As one can see, the results of the

analytical simulation were reasonably close to the experimental data result. It is important to note

that experimental data could only be taken for the phugoid mode and not for the short period

mode.

Overall, the three methods yielded slightly different results. The eigenvalue method and

the discrete time simulation were found to be the most accurate methods. The inaccuracy in the

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energy relation method was due to the fact that it did not account for the inescapable damping

that occurs while performing the phugoid maneuver. Other potential sources of error could be

inconstancies in data collection and initial headings, which could be rectified by better

synchronization between crew members in video recording or by redoing the test entirely.

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INTRODUCTION The following describes the purpose of the flight test experiments, a breakdown of the

test aircraft, and the scope and methodology of the tests performed for analysis.

Purpose The purpose of the combined analysis-flight test program is to:

• Collect and reduce flight test data for understanding the performance of a Piper Arrow

III-A-28R-201 during the phugoid maneuver and compare the experimental data results

to values determined from analytical analysis of a discrete time simulation of the phugoid

maneuver as well as an energy relation simulation and an eigenvalue simulation.

Description of Test Airplane

The Arrow III is a single engine, retractable landing gear, all metal airplane frequently

used for air taxi, flight training and personal use. It has seating for up to four people, a 200 pound

luggage compartment, and a maximum takeoff weight of 2750 pounds. The aircraft is not

configured for stunt maneuvers since its structure is not designed for aerobatic loads. The

fuselage is a semi-monologue structure with a conventionally designed, semi-tapered wing,

which employs a NACA 652-415 airfoil section. The four-positioning wing flaps are

mechanically controlled by a handle located between the front seats. When fully retracted, the

right flap locks into place to provide a step for cabin entry. A vertical stabilizer, all-movable

horizontal stabilator and a rudder make up the empennage.

The Arrow III incorporates a Lycoming 10-360-C 1C6 four-cylinder engine rated at 200

horsepower at 2700 rpm. The aircraft is equipped with McCauley 90DHA-16 propeller, which is

a constant speed, controllable pitch propeller with a maximum diameter of 74 inches. The

propeller control is located on the power quadrant between the throttle and mixture controls.

Engine controls consist of a throttle control, propeller control and a mixture control lever. The

throttle lever is used to adjust the manifold pressure. The propeller control lever is used to adjust

the propeller speed from high to low rpm. The mixture control lever is used to adjust the air to

fuel ratio. The horizontal stabilizer features a trim tab mounted on the trailing edge that provides

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trim control and pitch control forces. The rudder is of conventional design and includes a rudder

trim as well. Fuel is contained in two 38.5 U.S. Gallon tanks, one in each wing. Of the total 77

gallons, only 72 gallons are usable. The aircraft also has a system that supplies both pitot and

static pressure for the airspeed indicator and altimeter. Pitot pressure is picked up by the probe

on the bottom of the left wing. The Arrow III uses a traditional flight control configuration.

Scope of Test An actual takeoff weight was determined to be 2400 pounds and the altimeter was set at

29.92. This weight includes the empty weight of the aircraft, the combined weight of the

passengers and pilot, and the weight of the fuel. At the time of takeoff the fuel level in the

aircraft was at 23 gallons. Completing the phugoid test at different airspeeds had a combined

duration of approximately one hour and were filmed for later analysis. The phugoid test occurred

at altitude around 3700 feet. The outside air temperature during the time of the flight test was 28

degrees Fahrenheit. All tests were completed with the both the gear up and flaps up and within

the limitations of the Pilot’s Operating Handbook. The tables below provide important aircraft

parameters.

Table 1. Operating Limitations and Weights

2700 RPM Max Power

200 hp.

Max Takeoff Weight 2750 lbs.

Table 2. Important Physical Parameters of the Piper Arrow III

Name Abbreviation Value

Wing Planform Area S 170 ft^2

Wing Span b 35.417 ft

Aspect Ratio AR 7.3786

Moment of Inertia Iy 1249 ft*slugs

Mass m 74.60 slugs

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Method of Test

Phugoid Test

Following the completion of all necessary pre-flight checks and procedures, team

members board the aircraft and prepare for takeoff. At this point, outside air temperature and

initial fuel level are recorded. Once airborne, the aircraft is to climb to a stable cruise altitude

(between 3000-4000 feet) in order to preform the phugoid test. The aircraft is flying at a constant

speed of 80 knots with the gear up and the flaps up. Once in the correct configuration, the aircraft

is then to climb until reaching an airspeed of 60 knots. At the peak of the climb, a hand held

stopwatch is started to time and video recording of the dashboard begins. Throughout the

phugoid maneuver mark is called every 2-3 seconds and at all peaks and troughs of the

oscillation. The follow data is collected at each mark:

• Pressure Altitude

• Time

• Indicated Ground Speed

• Fuel Level

Once the data collection is complete and the aircraft has returned to stable flight, the flight test is

repeated again for a flight speed of 100 knots.

Instrumentation

Table 3. Relevant test parameters and the instruments used to measure them Parameter Instrument

Airspeed On-board ASI

Altitude On-board Altimeter

Ground-Speed On-board GPS

Time iPhone App

Fuel Levels On-board fuel indicator

Outside Air Temperature On-board temperature gauge

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2.0 ANALYSIS The following sections describe the theory behind the phugoid test as well as the method

used to implement and analyze the phugoid data.

2.1 Theory

Phugoid Test

The primary theory behind the phugoid test has to do with the longitudinal dynamics and

stability of the aircraft. In order to understand the phugoid motion, we must first examine the

body axis of the aircraft and assign variables to the velocities, forces and moments. Figure 1

show the body axis of the aircraft with forces and moments labeled. Additionally, table 4 shows

the variable names along with a short description of the variable.

Figure 1. Aircraft body axis with labeled forces and moments

Table 4. Aircraft body axis parameters and short descriptions for each

Parameter Description u Aircraft velocity in the longitudinal direction v Aircraft velocity in the lateral direction w Aircraft velocity in the vertical direction p Aircraft body rate around the longitudinal axis q Aircraft body rate around the longitudinal axis r Aircraft body rate around the lateral axis X Aircraft body force in the vertical direction Y Aircraft body force in the lateral direction Z Aircraft body force in the vertical direction L Aircraft body moment around the longitudinal axis M Aircraft body moment around the lateral axis N Aircraft body moment around the vertical axis

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When considering the phugoid motion used in this experiment, we can simplify our analysis by

neglecting the parameters that do not affect the longitudinal motion of the aircraft. Because

phugoid motion is essentially an oscillation along the vertical body axis of the aircraft, we can

neglect the lateral body force and velocity as well as the moments and rates around the

longitudinal and vertical axis. This leaves us with only the parameters:

• u • w • X • Z • q • M

We then determined the equations of motion for the phugoid motion according to the equations,

€

U =

€

U0 + u (1)

€

W =

€

W0 + w (2)

€

X =

€

X0 + (ΔxG + Δx) = m ˙ u (3)

€

Z =

€

Z0 + (ΔzG + Δz) = m( ˙ w −U0q) (4)

€

Q =

€

Q0 + q (5)

€

M =

€

0 + ΔM = Iy ˙ q (6) We also know that Eq. (1) – (6) are dependent on four parameters:

• Longitudinal velocity, u

• Angle of attack, α

• Pitch attitude, θ

• Pitch rate, q

Using these parameters as well as the dimensional stability derivates for the Arrow III, we can

write the equations of motion for the phugoid motion according to,

€

˙ u =

€

Xuu + Xαα − gθ (7)

€

˙ α =

€

Zu

(u0 − Z ˙ α )u +

Zα(u0 − Z ˙ α )

α +Zq + u0

(u0 − Z ˙ α )q (8)

€

˙ θ =

€

q (9)

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€

˙ q =

€

Mαα + M ˙ α ˙ α + Mqq (10) Furthermore, these equations can be written in matrix form according to,

€

˙ u ˙ α

˙ q ˙ θ

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

€

A[ ]

uα

qθ

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

(11)

€

[! ˙ X ] =

€

A[ ]! X [ ] (12)

where,

€

[A] =

€

Xu Xα −g 0Zu

(u0 − Z ˙ α )Zα

(u0 − Z ˙ α )0

Zq + u0

(u0 − Z ˙ α )0 0 0 1

M ˙ α Zu

(u0 − Z ˙ α )Mα +

M ˙ α Zα(u0 − Z ˙ α )

0 Mq +M ˙ α (Zq + u0)

(u0 − Z ˙ α )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(13)

where, g, is the acceleration due to gravity. The stability derivatives used in the analysis are

shown below in table 5. The values for each stability derivative were given in class.

Table 5. Stability derivatives and their corresponding values. Given in class

Stability Derivative Value

€

Xu -5.33/m

€

Xα -2051/m

€

Xθ -2400/m

€

Zu -32.72/m

€

Zα -19149/m

€

Z ˙ α -73.4/m

€

Zq -2.655/m

€

Mα -21662/Iy

€

M ˙ α -892.4/Iy

€

Mq -2405/Iy Using the A matrix from Eq. (11), we can set up a time simulation in which we calculate each

successive X vector using discrete time steps and the equation,

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€

Xk+1 =

€

(AXk )Δt + Xk (14) which yields an X matrix containing four separate vectors, representing longitudinal velocity, u,

angle of attack, α, pitch attitude, θ, and pitch rate, q.

Additionally, equations for altitude versus time for the phugoid maneuver can be

generated using the eigenvalues of the A matrix. Calculating the eigenvalues of the A matrix

yields two sets complex conjugates in which the real parts represent the damping coefficients of

the short period and phugoid mode, and the imaginary parts represent the frequencies of

oscillation for the short period and phugoid mode. The altitude versus time equations for the

short period and phugoid mode are given by,

€

h =

€

γeDt cos(ωt + β) (15) where h represents the height relative to the starting point, t represents time, D is the damping

coefficient, ω is the frequency of oscillation and β and γ are coefficients used to scale the plots in

order to match the experimental data. These three separate methods will then be used to

compare with the experimental results from the phugoid test.

2.2 Implementation

Phugoid Test

The phugoid analysis was performed using a custom computer code implemented in

MATLAB. The code begins with the declaration of several constants for the aircraft as well as

the stability derivatives. In the next step, the A matrix is calculated, to be used throughout the

remainder of the code.

In the first major section of the code, a discrete time simulation is performed using a for

loop and Eq.(14). Before the iteration occurs, the initial X matrix is declared. Ideally this matrix

would be a matrix of zeros, except that in order for the simulation to correctly evaluate the

longitudinal dynamics of the phugoid motion, one of the values in the X matrix is set to one.

Finally, after the X matrix is calculated, it is separated into four separate arrays representing the

longitudinal velocity, angle of attack, pitch attitude and pitch rate.

In the second major section of the code, height versus time plots are generated for the

phugoid maneuver using the eigenvalues of the A matrix. This section of code begins with a

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command to find the eigenvalues of the A matrix and store them in a separate vector. Next, the

code takes the real and imaginary parts of each separate eigenvalue and stores them in variables

corresponding to the variable that they represent (decay rate and frequency for short period and

phugoid modes). Finally, the values are put into equations that describe the height of both the

phugoid and short period modes versus time.

In the third section of the code, a for loop is used to fill two vectors representing height

versus time and vertical acceleration versus time. These equations were given in class and use

the initial values of the flight test along with several other parameters. This section of the

analysis is the least accurate because of its simplicity.

Finally, in the last section of the code, the experimental data for indicated airspeed and

altitude (comparable to height) are plotted from a Microsoft Excel spreadsheet containing out

flight test data.

2.2.1 Verification Approach The phugoid code was verified by using a substitute A matrix that had been used for

calculations in another class. Several discrete time steps were calculated and checked against the

results against several hand calculations. The resulting plots using this substitute matrix should

be oscillations that damp out over time, with the exception of the plot for the energy relation.

This plot has no damping and will therefore oscillate at constant amplitude. Additionally, we

expect to see similar plots for the discrete time simulation and the eigenvalue method because

the two methods both utilize the A matrix heavily.

Additionally, after running the code with the substitute A matrix, we also expect to obtain

two pairs of complex conjugate eigenvalues.

2.2.2 Verification Results

After running the phugoid test code with the substitute matrix, we found that the code

worked and yielded plots that matched what we had expected to see. Figure 1 shows plots of the

discrete time simulation using the substitute matrix. We can see that the altitude oscillates over

time and we can also see the angle of attack oscillating rapidly at the beginning of the simulation.

This is indicative of the short period mode.

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Figure 2. Plot of the discrete time simulation using the substitute matrix. Note that it matches what was expected, verifying the code.

Additionally, we can see that the eigenvector plots also match what was expected. Figure

3 shows the eigenvalue plots for the substitute matrix. We can see that both the phugoid mode

and the short period mode behave as we expected.

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Figure 3. Plot of altitude versus time using eigenvalues and the substitute matrix. Note that the plots match

what we expected, verifying the code.

Finally, the code was verified by simply checking to make sure that it yielded complex

conjugate eigenvalues when using the substitute matrix (which we already knew had complex

conjugate pairs for eigenvalues). Figure 4 shows the result of the eigenvalue calculation for the

substitute matrix, A.

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Figure 4. Shows the resulting eigenvalues and the original matrix used in verifying the code.

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3.0 RESULTS & DISCUSSION

Phugoid Test

The discrete time step portion of the phugoid test yielded very good results. First, run at

80 knots, figure 5 shows the longitudinal velocity, angle of attack, pitch attitude, and pitch rate

versus time. The plot of longitudinal velocity very clearly models the phugoid maneuver

because it shows a slight short period oscillation at the beginning but settles into a larger phugoid

mode oscillation for the remainder of the test. The angle of attack plot very clearly shows the

short period mode and its frequency and damping. Observing these plots, we found phugoid

period to be about 22 seconds and the phugoid frequency to be about 0.286 (1/seconds).

Additionally, we found the time to half amplitude for the phugoid mode to be about 60 seconds

based on the longitudinal velocity plot. In addition, we used the angle of attack plot to find the

short period of 1.3 seconds, resulting in a short period frequency of 4.83 (1/seconds). Finally,

we found the time to half amplitude for the short period mode to be about 4 seconds.

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Figure 5. Discrete Time Simulation Plots for 80 knots.

Additionally, we found that the discrete time simulation results yielded a similar result at

100 knots. The major differences were the increased amplitude and period of oscillation for both

the short period mode and the phugoid mode. Figure 6 shows the plots of longitudinal velocity,

angle of attack, pitch attitude and pitch rate versus time for the discrete time simulation. Based

on the longitudinal velocity plot, we found the period of oscillation to be 23 seconds, resulting in

a frequency of 0.273 (1/seconds). Additionally, we found the time to half amplitude for the

phugoid motion to be about 65 seconds. We also found the short period to be about 1.4 seconds

with a frequency of about 4.488 (1/seconds) based on the angle of attack plot. The time to half

amplitude for the short period mode is approximately13 seconds based on figure 6.

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Figure 6. Discrete Time Simulation Plots for 80 knots.

In the next section of the phugoid simulation code, eigenvalues were used to generate

plots of altitude versus time for both short period and phugoid modes. Figure 7 shows altitude

versus time for both the short period and phugoid mode at 80 knots. The frequency of oscillation

for the phugoid mode was determined by the eigenvalues in the computer code to be 0.2933

(1/seconds). This results in a period of 21.42 seconds and a time to half amplitude of about 31

seconds for the phugoid mode. Additionally, we can determine that the short period frequency is

3.9512 (1/seconds) based on the eigenvalues. This gives us a short period of 1.590 seconds and a

time to half amplitude of about 0.3 seconds for the short period.

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Figure 7. Eigenvalue method plots for altitude versus time

Additionally, we can use a similar method at 100 knots. The eigenvalues resulted in a

phugoid frequency of 0.266 (1/seconds) and a short period frequency of 3.968 (1/seconds).

These frequency values lead to periods of 23.567 seconds for the phugoid mode and 1.583

seconds for the short period mode. Finally, based on the plot in figure 8, we can determine the

times to half amplitude to be about 31 seconds for the phugoid mode and about 0.5 seconds for

the short period mode.

Figure 8. Eigenvalue method plots for altitude versus time

In the next step we used an energy relation to simulate the phugoid maneuver and analyze

the altitude and vertical acceleration against time. Figure 9 shows plots of altitude versus time

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and vertical acceleration versus time for the phugoid mode at 80 knots. Based on figure 9, we

can see that the phugoid period is approximately 13 seconds, resulting in a frequency of 0.4833

(1/seconds). There is no damping in the energy relation equations therefore they do not have a

time to half amplitude.

Figure 9. Energy Relation plots for altitude versus time and vertical acceleration versus time

The process was repeated for an initial velocity of 100 knots and the resulting energy

relation plots are shown in figure 10. We can see that the period of oscillation is about 23

seconds based on the plot. From this value we determined the frequency of the phugoid

oscillation to be about 0.2732 (1/seconds). There is no damping in the energy relation equations

therefore they do not have a time to half amplitude. Additionally, the energy relation does not

account for the short period mode and therefore does not yield any data for it.

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Figure 9. Energy Relation plots for altitude versus time and vertical acceleration versus time

Finally, the experimental results for the phugoid maneuver at both 80 knots and 100

knots are shown in figure 11. Based on the plots we determined the phugoid period to be about

24 seconds for 80 knots and about 26 seconds for 100 knots resulting in phugoid mode

frequencies of 0.2618 (1/seconds) at 80 knots and 0.2417 (1/seconds) at 100 knots. Additionally,

the times to half amplitude for the phugoid modes were 62 seconds at 80 knots and 75 seconds at

100 knots.

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Figure 11. Experimental plots of altitude and airspeed versus time at 80 and 100 knots.

Overall, the data gathered during the phugoid test was reasonably close to what we

determined during out analysis. The tabulated results along with reported error is listed in tables

6 and 7. Note that the experimental data could only be taken for the phugoid mode and not for

the short period mode.

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Table 6. Tabulated data for the phugoid and short period modes for each separate simulation method, at 80

knots

Table 7. Tabulated data for the phugoid and short period modes for each separate simulation method, at 100 knots

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4.0 CONCLUSIONS The main focus of this flight test was to collect data during a phugoid maneuver and

compare it to a somputer simulation of the maneuver implemented in MATLAB. Data for

indicated airspeed and pressure altitude were collected during the flight test and plotted against

time. In addition, using physical parameters for the Piper Arrow III, we were able to construct a

discrete time simulation of the phugoid maneuver as well as an energy relation simulation and an

eigenvalue simulation. All of these methods yielded slightly different results with the eigenvalue

method and the discrete time simulation proving to be the most accurate methods. The primary

issue with the energy relation method was that it did not account for damping which inevitably

happens during the course of the phugoid maneuver, otherwise the aircraft would not be

dynamically stable.

A few potential sources of error during the phugoid test include:

• Inconsistent starting points for data collection. Resulting in an unclear zero time

point.

• Inconsistent heading which would lead to some transverse velocity, effecting the

overall dynamic response of the aircraft

These sources of error could be mitigated by adding several improvements including:

• Calling out mark and start during the test to insure the video recordings are

synced with one another.

• Video monitoring of the GPS heading and the option to repeat the flight test of

too much transverse motion is experienced during the test.

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5.0 REFERENCES Arrow PA-28R-201 Pilot's Operating Handbook. The New Piper Aircraft Inc., Publications

Department. Rev 24, Oct. 24, 2011. McCormick, Barnes W. AIAA (2011), Introduction to Aeronautics and Flight Testing. 312-314.

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6.0 APPENDIX APPENDIX A: Test Card:

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APPENDIX B: Phugoid Test Code % AERSP 420 -- Brian Harrell % Flight Test #3 -- Phugoid Code % % The following code models a Phugoid Maneuver using four different % methods. 1) Discrete Time Simulation 2) Eigenvector method using the % equation, Gamma*e^-rt * cos(wt + beta) where r is the dacay rate, w is % the frequency, and Gamma and Beta are coefficients used to match the % plots with the other methods. 3) Energy method 4) Experimental Data % plots g = 32.17; %standard gravity (slugs/ft^3) %%% Some Initial Values % Initial Velocity for the Phugoid Maneuver (this will be changed for each % test) u_0kts = 100; %knots u_0 = u_0kts*1.6878; %fps %%% The following Weight and Inertia values were given W = 2400; %lbs Iy = 1249; %slugs m = W/g; %%% Some stability derivatives for the Piper Archer III % These values were given in class Xu = -5.33/m; Xalpha = -2051/m; Xtheta = -2400/m; Zu = -32.72/m; Zalpha = -19149/m; Zalpha_dot = -73.4/m; Zq = -2.655/m; Malpha = -21662/Iy; Malpha_dot = -892.4/Iy; Mq = -2405/Iy; %%%%% A MATRIX %%%%% denom = (u_0-Zalpha_dot); A = [ Xu Xalpha -g 0; ... Zu/denom Zalpha/denom 0 (Zq+u_0)/denom; ... 0 0 0 1; ... (Malpha_dot*Zu)/denom Malpha+(Malpha_dot*Zalpha)/denom 0 Mq+(Malpha_dot*(Zq+u_0))/denom]; % I USED THE MATRIX BELOW TO TEST THE ITERATION METHOD USED IN THE CODE AND % COMPARED TO SOME HAND CALCULATIONS. THIS MATRIX WAS USED IN ANOTHER CLASS % AND THE ANSWER IS ALREADY KNOWN %A = [ -0.0025 0.0781 -61.3333 -31.9437;... % -0.0689 -0.4395 603.0773 -3.8698;... % 0.0002 -0.0017 -0.4211 0;... % 0 0 1.0000 0]; %%%%%%% DISCRETE TIME SIMULATION %%%%%%%

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t_step = 0.2; %seconds time = 0:t_step:90; %seconds % Initial X vector, this is ideally a vector of zeros, but we must perturb % one of the values in order to see the system response X_0 = [0; 1; 0; 0]; n = length(time); % Initialize the X-vector X = zeros(4,n); X(:,1) = X_0; % The below for loop steps through the time simulation, creating an % X-matrix where each of the rows represents a different state variable for i=1:n-1 X(:,i+1) = (A*X(:,i)).*t_step + X(:,i); end % Once the X-matrix is completed, the rows are seperated into the four % seperate state-variable vectors u = X(1,:); alpha = X(2,:); theta = X(3,:); q = X(4,:); %PLOTS figure(1) subplot(4,1,1) plot(time, u) title('Longitudinal Velocity vs. Time (DISCRETE TIME SIMULATION)') ylabel('Longitudinal Velocity, u (ft/sec)') xlabel('Time (sec)') subplot(4,1,2) plot(time, alpha) title('Angle of Attack vs. Time (DISCRETE TIME SIMULATION)') ylabel('Angle of Attack, alpha (radians)') xlabel('Time (sec)') subplot(4,1,3) plot(time, theta) title('Pitch Attitude vs. Time (DISCRETE TIME SIMULATION)') ylabel('Pitch Attitude, theta (radians)') xlabel('Time (sec)') subplot(4,1,4) plot(time, q) title('Pitch Rate vs. Time (DISCRETE TIME SIMULATION)') ylabel('Pitch Rate, q (radians/sec)') xlabel('Time (sec)') %%%%%%% USING EIGENVECTORS %%%%%%% E_Vect_of_A = eig(A); % The lines below take the real and imaginary parts of each eigenvector and % stores them in seperate variables to be used later decay_1 = real(E_Vect_of_A(1)); freq_1 = imag(E_Vect_of_A(1)); decay_2 = real(E_Vect_of_A(3)); freq_2 = imag(E_Vect_of_A(3)); % Adjust the below coefficients to get the plots to line up with the % discrete time simulation plots (they will be different for 80 vs 100 kts

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Gamma = 320; Beta = 0; h_0 = 3750; %Starting altitude for the maneuver % Each of the for loops below fill vectors representing the short period % and phugoid maneuvers (altitude versus time) for i=1:n eq_1(i) = Gamma*exp(decay_1*time(i))*cos(freq_1*time(i) + Beta) + h_0; end for i=1:n eq_2(i) = Gamma*exp(decay_2*time(i))*cos(freq_2*time(i) + Beta) + h_0; end %PLOTS figure(2) plot(time, eq_1, '--', time, eq_2) title('Altitude vs. Time (EIGENVECTOR RELATION)') legend('Short Period Mode', 'Phugoid Mode') ylabel('Altitude (feet)') xlabel('Time (sec)') %%%%%%% ENERGY RELATION %%%%%%% % The for loop below fills a vector representing the energy relation for % ALTITUDE vs TIME for the phugoid maneuver for i=1:n h(i) = sin(sqrt(2*(g/u_0)*(g/u_0))*time(i)); end % The for loop below fills a vector representing the energy relation for % VERTICAL ACCELERATION vs TIME for the phugoid maneuver for i=1:n h_doubledot(i) = -2*(g/u_0)*(g/u_0)*h(i); end %PLOTS figure(3) subplot(2,1,1) plot(time, h) title('Altitude vs. Time (ENERGY RELATION)') ylabel('Altitude (feet)') xlabel('Time (sec)') subplot(2,1,2) plot(time, h_doubledot) title('Vertical Acceleration vs. Time (ENERGY RELATION)') ylabel('Vertical Acceleration (feet/sec^2)') xlabel('Time (sec)') %%%%%%% EXPERIMENTAL DATA %%%%%%% % I HAVEN'T PLOTTED THIS AGAINST EVERYTHING YET, BUT THE EXCEL SPREADSHEETS % ARE BUILT AND CAN BE PLOTTED AT ANY TIME % Calls the file and sheet where the data is stored filename = 'Phugoid_Data.xlsx'; sheet = 1; % Assigns a range value to each range variable to be called later Range_time1 = 'B5:B34'; Range_time2 = 'F5:F43';

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Range_time3KIAS = 'J5:J57'; Range_time3PA = 'L5:L57'; Range_time4KIAS = 'O5:O61'; Range_time4PA = 'Q5:Q55'; Range_PA1 = 'D5:D34'; Range_PA2 = 'H5:H43'; Range_PA3 = 'M5:M57'; Range_PA4 = 'R5:R55'; Range_KIAS1 = 'C5:C34'; Range_KIAS2 = 'G5:G43'; Range_KIAS3 = 'K5:K57'; Range_KIAS4 = 'P5:P61'; % Takes the data from the Excel sheet and stores it in the arrays below Exp_time1 = xlsread(filename, sheet, Range_time1); Exp_time2 = xlsread(filename, sheet, Range_time2); Exp_time3KIAS = xlsread(filename, sheet, Range_time3KIAS); Exp_time3PA = xlsread(filename, sheet, Range_time3PA); Exp_time4KIAS = xlsread(filename, sheet, Range_time4KIAS); Exp_time4PA = xlsread(filename, sheet, Range_time4PA); Exp_PA1 = xlsread(filename, sheet, Range_PA1); Exp_PA2 = xlsread(filename, sheet, Range_PA2); Exp_PA3 = xlsread(filename, sheet, Range_PA3); Exp_PA4 = xlsread(filename, sheet, Range_PA4); Exp_KIAS1 = xlsread(filename, sheet, Range_KIAS1); Exp_KIAS2 = xlsread(filename, sheet, Range_KIAS2); Exp_KIAS3 = xlsread(filename, sheet, Range_KIAS3); Exp_KIAS4 = xlsread(filename, sheet, Range_KIAS4); %PLOTS figure(4) subplot(4,1,1) plot(Exp_time1, Exp_PA1, Exp_time2, Exp_PA2) title('Altitude vs. Time (80 knots) (EXPERIMENTAL DATA)') xlabel('Time (sec)') ylabel('Altitude (feet)') legend('Test #1 (80kts)', 'Test #2 (80kts)') subplot(4,1,2) plot(Exp_time1, Exp_KIAS1, Exp_time2, Exp_KIAS2) title('Indicated Airspeed vs. Time (80 knots) (EXPERIMENTAL DATA)') xlabel('Time (sec)') ylabel('Indicated Airspeed (knots)') legend('Test #1 (80kts)', 'Test #2 (80kts)') subplot(4,1,3) plot(Exp_time3PA, Exp_PA3, Exp_time4PA, Exp_PA4) title('Altitude vs. Time (100 knots) (EXPERIMENTAL DATA)') xlabel('Time (sec)') ylabel('Altitude (feet)') legend('Test #3 (100kts)', 'Test #4 (100kts)') subplot(4,1,4) plot(Exp_time3KIAS, Exp_KIAS3, Exp_time4KIAS, Exp_KIAS4) title('Indicated Airspeed vs. Time (100 knots) (EXPERIMENTAL DATA)') xlabel('Time (sec)') ylabel('Indicated Airspeed (knots)') legend('Test #3 (100kts)', 'Test #4 (100kts)')

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APPENDIX C: Hazard Mitigation

Hazard Mitigation – Team 5

1. Risk: Phone camera battery dies/ doesn't work

Probability Rank: 1 Severity Rank: 3 Mitigation: Bring a spare phone or an actual digital camera

2. Risk: Motion sickness occurs during the phugoid maneuver

Probability Rank: 2 Severity Rank: 4 Mitigation: Bring a flight sickness bag and be prepared to repeat the procedure if necessary

3. Risk: Stall Test – The videos do not begin at the same time (lack of consistent zero point)

Probability Rank: 4 Severity Rank: 2 Mitigation: Have someone call out start and mark throughout the test in order to sync several videos