Flight test 1 through 3 were conducted to investigate power required, power available and neutral point determination for the purpose of compiling a comprehensive performance summary table for the Kopp BD-4. The summary table contains both operational and design performance specifications useful to both the engineer and pilot alike. In this report results from flight test 4, phugoid mode determination, will be discussed and the summary table expanded to include those results.

The longitudinal phugoid mode is physically described as a damped vertical oscillation of the aircraft’s c.g. trajectory about the horizontal flight path with the velocity decreasing (or increasing) during an altitude increase (or decrease).[1] Mechanically this mode results in an exchange of potential energy with kinetic energy including subsequent loss of energy due to aerodynamic drag, which provides damping to the system. Generally, an aircrafts phugoid mode even when unstable does not greatly affect a pilots ability to safely handle the aircraft as the frequency is typically much lower than the frequency of pilot corrections. However, during IMC flight where pilot workload is increased, due to navigational procedures (chart reading, added communications, etc), an unstable phugoid will require the pilot to diligently monitor the aircrafts altitude and airspeed providing less hands free time for chart and communications work. This is especially true in single place aircraft where a single pilot must handle all the duties of flight. A total of six data collection runs were conducted in the Kopp BD-4. Three runs each were conducted at airspeeds of 100 and 140 mph (V_ias). Both stick free and stick fixed response were recorded during this test. Additionally, the prototype data acquisition system G.A.I.M.S. (general aviation inertial measurement system) was utilized to capture the phugoid mode by recording variations in normal acceleration during a single stick free maneuver. The data collected by hand will be used to help validate the general accuracy of the G.A.I.M.S. unit.

Part I – Flight Test

General Information

This test was conducted in the Kopp BD-4 on 8 Sept, 2000 departing from Monterey Peninsula Airport (MRY) at 10:00 am. Conditions at take-off were:

Wind: 310/8

Alt: 29.96

Sky Clear

Rwy: 28R

Crew and altitude assignments were as follows:

Table 1 Crew and altitude assignments

Crew	Altitude	Gross Weight (approx)
LT Ken Kopp / LT Jeff D’Latri	5500 ft	2025 lbs

The test area was restricted to Salinas Valley from Salinas to 15 miles South East of King City. Crew coordination and a thorough test procedures briefing preceded each flight. Data collection sheets were developed, printed and discussed in detail prior to flight as well. Specific responsibilities were delegated as follows:

Table 2 Flight Responsibilities

Responsibility	Pilot at the Controls	Pilot Not at the Controls
Flight Safety	Primary	Secondary
Airwork	Primary
Test Procedure		Primary
Data Recording		Primary
Communications	Primary	Secondary
Navigation	Secondary	Primary
Visual Lookout	Secondary	Primary
Emergencies	Primary	Secondary

ATC flight following was utilized to the maximum extent possible to aid in collision

avoidance. King City and Salinas Muni were designated primary diverts in the event an emergency due to mechanical failure or weather occurred.

Flight Test Technique

The aircraft was trimmed in level flight at 5,500 ft (indicated) at an initial airspeed of 100 mph (V_ias). The aircraft was slowly decelerated by smooth application of back stick (yoke) against trim until an airspeed change of –20 mph was attained. The yoke was released and the aircraft allowed to fly hands free until all oscillations had ceased. Upon yoke release a stop watch was started. Altitude, airspeed and time were recorded at the highest and lowest altitude of each cycle. This process is identical for the stick fixed case except that the yoke was held in place by the pilot throughout the maneuver. The entire procedure was repeated at an airspeed of 140 mph. An extra run was conducted for each airspeed where the initial input was an acceleration vice a deceleration. Additionally, during the 100 mph stick free (decel) run the GAIMS unit was started at exactly the moment the yoke was released and data recorded automatically throughout the maneuver. This data was serially transmitted to a host laptop computer connected to the unit and stored in a file for future analysis.

Part II – Data Reduction

All recorded data was entered into an excel spreadsheet for data analysis. As was done in previous flight test all airspeeds and altitude measurements are corrected for static position errors according to the following relationships.

Data for the first 100 mph stick free run was collected and reduced in the excel spreadsheet table shown below.

Table 3 100 mph Stick Free (initial decel)

100 mph Stick Free with initial decel of 20 mph							Oat	63 F
Trim Conditions		Vcas	Vcas
Hi	Hc	mph	mph			sigma			Alt	A/S
5500	5507	100	102.03	Cas	Hi	std	Hpc	Hc	Amp	Amp
	Time	A/S	delt Vpc	mph	ft		ft	ft	ft	mph	e time
	0	80	6.36	86.36	5600	0.8528	27.35	5627	120	-16	0
	12.18	110	0.58	110.58	5450	0.8481	3.49	5453	-54	9	12.18
	28.42	85	4.99	89.99	5560	0.8523	22.85	5583	76	-12	16.24
	45.85	105	1.27	106.27	5480	0.8492	7.21	5487	-20	4	17.43
	54.68	90	3.85	93.85	5500	0.8517	18.69	5519	12	-8	8.83
	64	103	1.56	104.56	5500	0.8502	8.70	5509	2	3	9.32
	84.8	98	2.35	100.35	5520	0.8512	12.46	5532	25	-2	20.8

The amplitude of both airspeed and altitude response was determined by subtracting the trim condition from the raw data. This procedure is more valid for airspeed as it tended to oscillate consistently about the trim airspeed of 100 mph, whereas altitude tended to oscillate about an slightly increasing value as the maneuver progressed. A plot of the phugoid response is shown in the figure below.

Figure 1 100 mph Stick free phugoid

PERIOD

The GAIMS data was extracted and plotted in MATLAB and is shown in the figure below.

The period of oscillation is determined by subtracting the times of two successive peaks (or valleys). Because the data gathered by hand is based on pilot judgment as to when the aircraft reached a peak (or valley) the hand calculated period was determined by averaging the periods of successive peaks. GAIMS data is collected automatically and therefore the period was determined from the first two peaks directly. These results are shown in the table below.

Table 4 Phugoid Period

100 mph Stick Free Phugoid Period
	Hand collected Data	GAIMS Data
Period (sec)	28.27 sec	28.29 sec

The damping ratio ζ is determined using the log decrement procedure according to the following relationship:

, where Y_A and Y_B are the amplitudes of successive peaks. Solving algebraically for ζ results in:

The result of this calculation for both the hand and GAIMS collected data is shown in the table below.

Table 5 Phugoid Damping Ratio ζ

100 mph Stick Free Phugoid Damping Ratio ζ
	Hand collected Data	GAIMS Data
ζ damping ratio	.1148	.0986

Using both values of ζ and T (period) calculated above, the damped and undamped natural frequencies are determined by:

where ω_d is the damped natural frequency and

, where ω_n is the undamped natural frequency and is given in the table below along with ω_d.

Table 6 Natural Frequencies

100 mph Stick Free Phugoid Natural Frequencies
	Manual	GAIMS
Damped	.2217	.2219
Undamped	.2236	.22307

These values can be used to model the phugoid response as a second order system with a transfer function of the following form:

, substituting the calculated values of ω_n and ζ for both the manually collected and GAIMS collected data results in the following transfer functions.

Table 7 100 mph Phugoid Transfer Function

100 mph Stick Free Phugoid Transfer Functions
Manual	GAIMS

Using MATLAB to apply a step input to both transfer functions and plotting the result along with a plot of raw airspeed data shows good correspondence between the model and reality as shown in the figure below.

Figure 2 Model Response

Identical procedures were carried out for the remaining runs. Plots of amplitude vs. time for each are shown in the figures below. Data for each run is included in the appendix.

Figure 3 100mph Free (accel)

Figure 4 100 mph Stick Fixed

Figure 5 140 mph Stick Free (accel)

Figure 6 140 mph Stick Fixed

Part III - Conclusions

Using the same methods of data reduction as the 100 mph stick free run results in values of T (period), ζ , ω_d and ω_n for each case as shown in the table below.

Table 8 Phugoid Response Table

Phugoid Response Summary Table
			Wd		Wn	Initial Alt	Final Alt	Vias final
Vias	Mode	T	(r/sec)	Zeta	(r/sec)	ft	ft	mph
100	Free	28.27	0.2222	0.1148	0.2236	5500	5532	98
100	Fixed	28.19	0.2228	0.0723	0.2234	5550	5562	99
100	Free (accel)	27.68	0.2269	0.1320	0.2289	5550	5509	102
140	Free	33.13	0.1896	0.0979	0.1905	5500	5544	140
140	Fixed	33.59	0.1870	0.1169	0.1883	5500	5734	140
140	Free (accel)	32.82	0.1913	0.1160	0.1926	5500	5484	140

GAIMS Data		T	Wd	Zeta	Wn
100	Free	28.29	0.22199	0.0986	0.2231

The results are very consistent between similar runs. The correlation between the period calculated by GAIMS and the manual data run are excellent with only a .02 second difference. The period obviously grew longer with an increase in airspeed with the initial acceleration runs showing slightly reduced periods than the deceleration runs of similar trim airspeed. Differences between stick free and fixed are negligible. Values of ζ show slight variation between runs, which is attributed to the sensitivity of the natural log and square terms in the equation used to solve for ζ. Stick fixed runs have a tendency to result in final altitudes slightly higher than the stick free case, although trim airspeed was attained in all cases within 2 mph cas. Although the flight control system is not frictionless the relatively large trim tab area generates a significant restoring moment anytime the stab is displaced from its trim position. This is a noteworthy feature as it allows for trimming the airplane to a constant speed and altitude an easy task. The airplane exhibits stable phugoid response and returns to its trim state in under two minutes with little deviation in altitude and almost no deviation in airspeed a good quality for IMC flight.

The comparison of manually collected data to the electronically collected data from the G.A.I.M.S. unit shows good correlation, however; results from this test cannot be used to conclusively validate either method. Particularly because the only sensor collecting data was the vertical accelerometer which is also sensitive to roll angles thus corrupting the phugoid data anytime a slight roll was initiated. To accurately depict the true phugoid response using the GAIMS unit requires correlation of accelerations in all three axis to offset the effect of lateral directional motion during the maneuver. This will be the subject of future flight test and GAIMS validation.

Further phugoid testing should be done at various c.g. locations to investigate the longitudinal stability with both extreme forward and aft c.g. configurations. The Kopp BD-4 Performance Summary table is included below.

Table 9 Kopp BD-4 Performance Summary Table

Altitude / Weight	Max C_l/C_d	Min Thrust Required
3000 ft / 1950 lbs	8.8235	217.87 lbs
7500 ft / 2130 lbs	9.0329	229.63 lbs
Parameters	Drag Polar	Power Curve
C_do	0.0440	0.0425
e	0.7031	0.6507
Altitude	Minimum Thrust Horsepower Required
3500 ft 1950 lbs	52.33 HP
7500 ft 2130 lbs	60.59 HP
Standardized	59.16 HP
V_{x (ias)}	75 mph
V_{y (ias)}	90 mph
R/C_max S.L max GW	799 fpm
AOC_max S.L. max GW	5.71°
Service Ceiling @ max GW	10,600 ft
Absolute Ceiling @ max GW	11,400 ft
Longitudinal Stability
Stick Free / Fixed Neutral Point	36.5% (mac)
Forward cg static margin	16.5% (mac)
Aft cg static margin	5.25% (mac)
Phugoid Period (100 mph)	28.29 seconds
Phugoid Period (140 mph)	33.2 seconds
ζ damping ratio	.11

Appendix

Table 10 Data

Stick Free with initial accel of 20 mph					100 mph	Free
Trim Conditions
Hi	Hc	Vias mph	Vcas mph						Altitude	A/S
5500	5507	100	102.03	Cas	Hi	sigma std	delt Hpc	Hc	Amplitude	Amplitude
OAT	Time	A/S (mph)	delt Vpc	mph	ft		ft	ft	ft	mph	e time
63	0	120	-0.59	119.41	5440	0.8528	-3.8246	5436	-72.04	17.38	0
	13.75	85	4.99	89.99	5620	0.8481	22.958	5643	134.74	-12.04	13.75
	25.83	112	0.33	112.33	5460	0.8523	1.9916	5462	-46.23	10.30	12.08
	40.83	93	3.25	96.25	5580	0.8492	16.351	5596	88.13	-5.78	15
	52.82	107	0.99	107.99	5480	0.8517	5.699	5486	-22.52	5.96	11.99
	68.58	98	2.35	100.35	5540	0.8502	12.471	5552	44.25	-1.68	15.76
	79.45	105	1.27	106.27	5500	0.8512	7.1929	5507	-1.03	4.24	10.87
	96.6	100	2.03	102.03	5505	0.8511	10.943	5516	7.72	0.00	17.15
	110.7	102	1.71	103.71	5500	0.8512	9.437	5509	1.22	1.68	14.1

Stick Fixed with initial decel of 20 mph					100 mph	Fixed
Trim Conditions
Hi	Hc	Vias mph	Vcas mph						Altituded	A/S
5500	5507	100	102.03	Cas	Hi	sigma std	delt Hpc	Hc	Amplitude	Amplitude
OAT	Time	A/S (mph)	delt Vpc	mph	ft		ft	ft	ft	mph	e time
63	0	80	6.36	86.36	5660	0.8471	27.534	5688	181	-16	0
	12.8	115	-0.03	114.97	5530	0.8505	-0.2133	5530	23	13	12.8
	30.19	90	3.85	93.85	5650	0.8474	18.784	5669	162	-8	17.39
	43.34	108	0.85	108.85	5550	0.8499	4.9662	5555	48	7	13.15
	56.42	92	3.45	95.45	5600	0.8487	17.151	5617	110	-7	13.08
	67.3	104	1.41	105.41	5560	0.8497	7.954	5568	61	3	10.88
	84.37	97	2.52	99.52	5580	0.8492	13.249	5593	86	-3	17.07
	98.65	99	2.19	101.19	5550	0.8499	11.715	5562	55	-1	14.28

Stick Free with initial accel of 20 mph					140 mph	Free	Accel
Trim Conditions
Hi	Hc	Vias mph	Vcas mph						Altitude	A/S
5500	5484	140	137.9	Cas	Hi	sigma std	delt Hpc	Hc	Amplitude	Amplitude
OAT	Time	A/S (mph)	delt Vpc	mph	ft		ft	ft	ft	mph	e time
65	0	160	-2.46	157.54	5400	0.8538	-21.509	5378	-106	20	0
	16.91	129	-1.42	127.58	5600	0.8487	-9.9743	5590	106	-10	16.91
	34.87	150	-2.40	147.60	5420	0.8533	-19.58	5400	-84	10	17.96
	51.25	136	-1.90	134.10	5540	0.8502	-14.068	5526	42	-4	16.38
	67.56	146	-2.31	143.69	5460	0.8523	-18.406	5442	-42	6	16.31
	82.94	138	-2.01	135.99	5500	0.8512	-15.08	5485	1	-2	15.38
	100.22	143	-2.22	140.78	5460	0.8523	-17.3	5443	-41	3	17.28
	114.88	140	-2.10	137.90	5500	0.8512	-16.034	5484	0	0	14.66

Stick Fixed with initial decel of 20 mph					140 mph	Fixed	Decel
Trim Conditions
Hi	Hc	Vias mph	Vcas mph						Altitude	A/S
5500	5484	140	137.9	Cas	Hi	Sigma Std	Delt Hpc	Hc	Amplitude	Amplitude
OAT	Time	A/S (mph)	delt Vpc	mph	ft		ft	ft	ft	mph	e time
65	0	120	-0.59	119.41	5800	0.8435	-3.8667	5796	312	-18	0
	17.91	155	-2.45	152.55	5500	0.8512	-20.739	5479	-5	15	17.91
	33.76	128	-1.34	126.66	5780	0.844	-9.3823	5771	287	-11	15.85
	51	144	-2.26	141.74	5650	0.8474	-17.791	5632	148	4	17.24
	64.59	132	-1.64	130.36	5750	0.8448	-11.882	5738	254	-8	13.59
	83.66	140	-2.10	137.90	5720	0.8456	-16.142	5704	220	0	19.07
	100.34	138	-2.01	135.99	5780	0.844	-15.208	5765	281	-2	16.68
	117.56	140	-2.10	137.90	5750	0.8448	-16.157	5734	250	0	17.22

					140 mph	Free	Decel		Stick Free with initial decel of 20 mph
Trim Conditions
Hi	Hc	Vias mph	Vcas mph						Altitude	A/S
5500	5484	140	137.9	Cas	Hi	Sigma Std	Delt Hpc	Hc	Amplitude	Amplitude
OAT	Time	A/S (mph)	delt Vpc	mph	ft		ft	ft	ft	mph	e time
65	0	120	-0.59	119.41	5700	0.8461	-3.8549	5696	212	-18	0
	17.66	155	-2.45	152.55	5450	0.8525	-20.707	5429	-55	15	17.66
	32.55	130	-1.50	128.50	5650	0.8474	-10.622	5639	155	-9	14.89
	50.45	150	-2.40	147.60	5480	0.8517	-19.616	5460	-24	10	17.9
	64.95	135	-1.84	133.16	5580	0.8492	-13.543	5566	82	-5	14.5
	84.01	145	-2.29	142.71	5500	0.8512	-18.079	5482	-2	5	19.06
	100.1	138	-2.01	135.99	5550	0.8499	-15.102	5535	51	-2	16.09
	115.95	142	-2.19	139.81	5520	0.8507	-16.922	5503	19	2	15.85
	135.59	140	-2.10	137.90	5560	0.8497	-16.064	5544	60	0

About the author:

LT Kenneth G. Kopp is an active duty Naval Aviator flying SH-60B “Seahawk” helicopters. He has been selected to attend the Naval Test Pilot School in Patuxent River, MD and recently completed all academic requirements for a Masters of Science in Aeronautical Engineering from the Naval Post Graduate School in Monterey, California, which will be awarded upon successful completion of the Test Pilot curriculum. He is co-builder and owner of the Kopp BD-4. His FAA ratings included: Fixed/Rotary commercial, instrument and CFI. He is current in the following aircraft models: SH-60B, TC-12B, T-34, BD-4, C-172,

C-152.

[1] Schmidt, L.V., Introduction to Aircraft Flight Dynamics

List of Figures

List of Tables

Sky Clear

Crew

ATC flight following was utilized to the maximum extent possible to aid in collision

avoidance. King City and Salinas Muni were designated primary diverts in the event an emergency due to mechanical failure or weather occurred.

Altitude / Weight

Min Thrust Required

Parameters

Drag Polar

Power Curve

Cdo

0.0440

0.0425

e

0.7031

0.6507

C_do