# Calculation of helicopter maneuverability in forward flight

## Transcript Of Calculation of helicopter maneuverability in forward flight

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

Calculation of helicopter maneuverability in forward flight based on energy method

Abstract

Nanjian Zhuang, Jinwu Xiang, Zhangping Luo*, Yiru Ren

School of Aeronautic Science and Engineering, Beihang University, Haidian Dist, Beijing, China Received 1 March 2014, www.tsi.lv

A new method for calculating helicopter maneuverability in forward flight is proposed. Empirical equations for evaluating rotor required power are employed. Using energy method, an algorithm to calculate the available overloads, rate of climb and flight trajectory is given. The maneuver performance of AH-1G helicopter is investigated and three kinds of maneuvers including level acceleration, deceleration turning and turning climb followed by accelerating climb are calculated and analysed. Numerical results indicate that the method is effective and feasible, even for three dimensional maneuvering problems. In addition, the method can be applied to predict flight trajectory during forward flight.

Keywords: Helicopter, Maneuverability, Energy Method

1 Introduction

With the development of aviation technology, plenty of new roles and missions are assigned for helicopter. Sometimes helicopters were required for rescues and aid in an unknown, dynamic and potentially hostile environment. In order to finish the specified task successfully, helicopters should equip collision avoidance system, which demands real-time and accurate helicopter maneuverability and flight trajectories [1].

Due to the complex helicopter aerodynamics, the maneuvering process need much more time based on the existing method. To obtain the designed performance, energy method is used by a lot of researchers to investigate two-dimensional (2-D) helicopter maneuver flight problems. Basic studies about helicopter required power were conducted in detail [2-4]. Helicopter performances were investigated using energy balance method [5, 6]. Some maneuver characteristics were studied by Xu [7] and Mikhailov [8-10]. Aerobatic maneuvers were analysed in detail based on mathematical description by Cao [11] and Hu [12, 13].

However, there are few researches on helicopter 3-D maneuver using energy method. In this paper, an algorithm based on energy method is described. Using the data of AH-1G helicopter, three kinds of maneuvers are analysed, including level acceleration, decelerating turn, turning climb followed by accelerating climb.

2 Mathematical model

The energy state of a helicopter can be written as:

E 1 mV 2 mgh 1 I 2 ,

(1)

2

2

where m is mass of helicopter, I is total rotor inertia, is rotor rotational speed. By taking the partial derivative with respect to time of equation 1, the energy rate is expressed as:

dE P mV dV mg dh . (2)

dt

dt

dt

2.1 REQUIRED POWER

The rotor power required in forward flight is given by the sum of parasite power, induced power, rotor blade profile power, compressibility power, stall power and climb power [14].

Preq,rotor 0.5 f V 3 TVi 0.125 bcR(1 4.6 2 )(R)3

(3) 3 R5M 3[0.0033 M (0.022 0.11M )] 1.5 (tc tcdiv )1.5 mgVh ,

where f is equivalent flat-plate drag area, is air

density, T is rotor thrust, Vi is induced velocity, is coefficient of blade drag, b is number of blades, c is blade chord, R is rotor radius, is advance ratio, M is the amount by which advancing blade tip Mach number exceeds drag divergent Mach number, is a

constant coefficient, tc is thrust coefficient, tcdiv is

*Corresponding author e-mail: [email protected]

50

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

thrust coefficient at which stall power occurs and Vh is x Vl cos

vertical velocity. The total power required is obtained y V sin

(10)

by rotor power and overall efficiency factor ( ) and

l

z V

h

HPreq,total HPreq,rotor .

(4) 3 Calculation algorithm

2.2 PERFORMANCE

Changes in horizontal velocity ( Vl ) for energy rate ( P1 ) is determined form the following:

Figure 1 presents the algorithm of calculating the maneuverability and flight trajectory. The required data for this algorithm are properties of helicopters, engine output power and control laws.

dVl 1P1 , dt mVl

(5)

Start

1 where 1

0.8

, when P1 0 , when P1 0

The relationship between normal load factor ( nn )

and the turn rate ( ) is given by:

gnn .

(6)

Vl

Energy rate ( P2 ) influences vertical velocity as follows:

V 2P2 ,

h m dVh mg

(7)

dt

1 where 2

0.8

, when P2 0 , when P2 0

2.3 ROTOR THRUST LIMITS

The maximum thrust of main rotor is restricted by the available power and maximum thrust coefficient.

T

tc 0.5bc2 R5 tcmax ,

(8)

P1 P2 (Preq,total mgVh ) Pava ,

(9)

where tcmax and Pava is maximum thrust coefficient and available power.

2.4 KINEMATIC EQUATIONS

The kinematic equations can be written as follows:

Initial Data

Properties of Helicopter

Current Engine Power

Control Laws

Calculating Required Power

Excess Power

Climb of Rate and Overloads

Yes Beyond Limits

No

Flight Kinematic Equation

Time Out

No

Yes

End

FIGURE 1 Calculation algorithm for maneuverability

4 Examples and results

A program is completed in MATLAB to calculate AH1G helicopter maneuverability. The properties of AH-1G helicopter [15], which are used in program are presented in Table 1. Three kinds of maneuvers are analysed as follows.

51

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

TABLE 1 Properties of AH-1G helicopter

Parameter (unit)

m (kg)

R (m)

b (-) c (m)

Value 3400 6.71

2 0.69

Parameter (unit)

(s-1) f (m2)

(-) (-)

Value 34 1.82

736 0.0075

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

350 340 330 320 310

Power(kW)

4.1 LEVEL ACCELERATION

Level acceleration is a very necessary and integral component of the maneuver capability of a helicopter. It is very important to predict acceleration in danger situations. In the present example, the engine is at the maximum power output condition and the initial velocity is 25.7 m/s. Figure 2 shows the computed trajectory of the helicopter in 10 seconds. Figure 3 and Figure 4 demonstrate the time histories of the helicopter speed and power for the acceleration maneuver. It can be seen from the figures that the excess power is maximum at the velocity about 32 m/s.

Y(m)

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

0

50

100

150

200

250

300

350

400

X(m)

FIGURE 2 Acceleration trajectory

50

45

40

300

290

280

270

260

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 4 Time history of excess power for acceleration maneuver

4.2 DECELERATING TURN

The pilots often choose decelerate at constant altitude in order to supply more power for a 180 degree turn. Figure 5 presents an example of a decelerating turn at 1.72-g overload.

0

-50

-100

Y(m)

-150

-200

-250

-300 0

50

100

150

200

250

X(m)

FIGURE 5 Turning trajectory

The time histories of the helicopter speed and power in this maneuver are given in Figure 6 and Figure 7. In a decelerating turn, the helicopter must maintain engine idle speed [16]. Therefore, the value of excess power is always below zero, shown in Figure 7. The maximum excess power is at the velocity about 37 m/s.

Velocity(m/s )

35

30

25

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 3 Time history of velocity for acceleration maneuver

4.3 TURNING CLIMB FOLLOWED BY ACCELERATING CLIMB

A complex maneuver is presented in which a turning climb is following by an accelerating climb. The rate of climb is 5 m/s in whole maneuver. The helicopter maintains engine idle speed in climbing turn and the engine work at the maximum power output condition in accelerating climb.

52

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

60

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

50

Velocity(m/s )

Velocity(m/s )

55 45

50 40

45 35

40

30 35

Power(kW)

30

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 6 Time history of velocity for turning maneuver

-400

-420

-440

-460

-480

-500

-520

-540

-560

-580

-600

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 7 Time history of power for turning maneuver

There is a transitional period between these two conditions. Figure 8 shows the flight trajectory of this maneuver. Figure 9 demonstrates the time history of the helicopter speed. It can be seen from figure 10 that there is a transitional period about 3 seconds.

Z(m )

60

40

20

0 150

100

50

X(m)

00

-300 -250 -200 -150 -100 -50

Y(m)

FIGURE 8 3-D flight trajectory

Power(kW)

25

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 9Time history of velocity for 3-D maneuver

200 100

0

-100

-200

-300

-400

-500

-600

-700

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 10 Time history of power for 3-D maneuver

5 Conclusions

This paper proposed a fast and accurate approach to evaluate helicopter maneuverability in forward flight. A three dimensional complex maneuver of AH-1G helicopter is presented, and some important characteristics including maneuver time, trajectory, excess power and velocity are obtained. The results indicate that this method is feasible and effective for helicopters during forward flight. In order to get a fast calculation algorithm, some empirical studies which are only appropriate for forward flight situation are employed in mathematical model. Future research would focus on the rapid method for evaluating maneuverability of the helicopter in hover or vortex ring state.

Acknowledgments

This study has been supported by the State Key Basic Research Program of China (2011CB707002).

53

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

References

[1] Meng S, Xiang J, Luo Z 2013 Navigation of micro aerial vehicle in unknown environments 25th Chinese Control and Decision Conference: Guiyang 322-7

[2] Li J, Gao Z 2009 A simplified algorithm of rotor power and pull to determine helicopter performance Journal of Air Force Engineering University (Natural Science Edition) 10(4) 9-12 (in Chinese)

[3] Zhang Y 2003 Research of computing method for helicopter required power Helicopter Technique 22(1) 1-5 (in Chinese)

[4] The General Editorial of Aircraft Design Manual 2005 The 19th volume of aircraft design manual Aviation Industry Press: Beijing (in Chinese)

[5] Wells C D, Wood T L 1973 Journal of the American Helicopter Society 18(1) 10-22

[6] Kiwan A R 1994 Helicopter performance evaluation (HELPE) computer model AD Report AD-A284 319/1

[7] Xu X, Wang W 1994 Study on Z-9 helicopter turning maneuver characteristics by energy method Journal of Nanjing University of Aeronautics and Astronautics 26(1) 121-6 (in Chinese)

[8] Mikhailov S A, Onushkin A Y 2007 Power balance method in calculation of helicopter maneuverability taking into account specific operational conditions. Russian Aeronautics 50(2) 121-8

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

[9] Mikhailov S A, Onushkin Y P, Safonov A A, Kochish S I 2009 Numerical simulation of helicopter maneuverability in aerobatics research Russian Aeronautics 52(2) 176-83

[10] Mikhailov S A, Onushkin A Y, Onushkin Y P, Safonov A A, Kochish S I 2009 Investigation of helicopter maneuverability by the power balance method Russian Aeronautics 52(3) 296-301

[11] Cao Y, Zhang G, Su Y 2004 Mathematical modeling of helicopter aerobatic maneuvers Aircraft Engineering and Aerospace Technology 76(2) 170-78

[12] Hu H, Zhang X 2012 Simulation method of helicopter maneuver flight based on mathematical description Fire Control and Command Control 37 99-101 (in Chinese)

[13] Hu H, Wu S, Shao H 2011 Mathematical description and simulation of helicopter maneuver flight System Simulation Technology 7(4) 305-10 (in Chinese)

[14] Johnson W 1994 Helicopter theory Dover Publications: New York [15] Wood T L, Livingston C L 1971 An energy method for prediction

of helicopter maneuverability AD Report, ADA021266 [16] Gao Z, Sun C 1997 Probing of helicopter maneuverability Journal

of Nanjing University of Aeronautics and Astronautics 29(6) 66673 (in Chinese)

Authors

Nanjian Zhuang, born in June, 1987, Hunan, China

Current position, grades: Ph.D. student at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree, Beihang University, 2008 Scientific interest: helicopter flight dynamic, structural dynamics. Publications: 2 papers

Jinwu Xiang, born in February, 1964, Hunan, China

Current position, grades: Professor at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree (Nanjing University of Aeronautics and Astronautics, 1980), Master degree (Northwestern Polytechnical University, 1990), PhD (Nanjing University of Aeronautics and Astronautics, 1993). Scientific interest: aircraft design, aeroelasticity and structural dynamics, etc. Publications: 161 papers, 12 patents

Zhangping Luo, born in December, 1968, Guizhou, China

Current position, grades: associate professor at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree (Beihang University, 1991), Ph. D. (Beihang University, 2004). Scientific interest: structural dynamics, impact dynamics and crashworthiness, aircraft design. Publications: 16 papers

Yiru Ren, born in January, 1983, Hunan, China

Current position, grades: post-doctor at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree (Harbin Engineering University, 2006), Ph. D. (Beihang University, 2011). Scientific interest: structural dynamics, impact dynamics and crashworthiness, aircraft design. Publications: 10 papers

54

Mathematical and Computer Modelling

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

Calculation of helicopter maneuverability in forward flight based on energy method

Abstract

Nanjian Zhuang, Jinwu Xiang, Zhangping Luo*, Yiru Ren

School of Aeronautic Science and Engineering, Beihang University, Haidian Dist, Beijing, China Received 1 March 2014, www.tsi.lv

A new method for calculating helicopter maneuverability in forward flight is proposed. Empirical equations for evaluating rotor required power are employed. Using energy method, an algorithm to calculate the available overloads, rate of climb and flight trajectory is given. The maneuver performance of AH-1G helicopter is investigated and three kinds of maneuvers including level acceleration, deceleration turning and turning climb followed by accelerating climb are calculated and analysed. Numerical results indicate that the method is effective and feasible, even for three dimensional maneuvering problems. In addition, the method can be applied to predict flight trajectory during forward flight.

Keywords: Helicopter, Maneuverability, Energy Method

1 Introduction

With the development of aviation technology, plenty of new roles and missions are assigned for helicopter. Sometimes helicopters were required for rescues and aid in an unknown, dynamic and potentially hostile environment. In order to finish the specified task successfully, helicopters should equip collision avoidance system, which demands real-time and accurate helicopter maneuverability and flight trajectories [1].

Due to the complex helicopter aerodynamics, the maneuvering process need much more time based on the existing method. To obtain the designed performance, energy method is used by a lot of researchers to investigate two-dimensional (2-D) helicopter maneuver flight problems. Basic studies about helicopter required power were conducted in detail [2-4]. Helicopter performances were investigated using energy balance method [5, 6]. Some maneuver characteristics were studied by Xu [7] and Mikhailov [8-10]. Aerobatic maneuvers were analysed in detail based on mathematical description by Cao [11] and Hu [12, 13].

However, there are few researches on helicopter 3-D maneuver using energy method. In this paper, an algorithm based on energy method is described. Using the data of AH-1G helicopter, three kinds of maneuvers are analysed, including level acceleration, decelerating turn, turning climb followed by accelerating climb.

2 Mathematical model

The energy state of a helicopter can be written as:

E 1 mV 2 mgh 1 I 2 ,

(1)

2

2

where m is mass of helicopter, I is total rotor inertia, is rotor rotational speed. By taking the partial derivative with respect to time of equation 1, the energy rate is expressed as:

dE P mV dV mg dh . (2)

dt

dt

dt

2.1 REQUIRED POWER

The rotor power required in forward flight is given by the sum of parasite power, induced power, rotor blade profile power, compressibility power, stall power and climb power [14].

Preq,rotor 0.5 f V 3 TVi 0.125 bcR(1 4.6 2 )(R)3

(3) 3 R5M 3[0.0033 M (0.022 0.11M )] 1.5 (tc tcdiv )1.5 mgVh ,

where f is equivalent flat-plate drag area, is air

density, T is rotor thrust, Vi is induced velocity, is coefficient of blade drag, b is number of blades, c is blade chord, R is rotor radius, is advance ratio, M is the amount by which advancing blade tip Mach number exceeds drag divergent Mach number, is a

constant coefficient, tc is thrust coefficient, tcdiv is

*Corresponding author e-mail: [email protected]

50

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

thrust coefficient at which stall power occurs and Vh is x Vl cos

vertical velocity. The total power required is obtained y V sin

(10)

by rotor power and overall efficiency factor ( ) and

l

z V

h

HPreq,total HPreq,rotor .

(4) 3 Calculation algorithm

2.2 PERFORMANCE

Changes in horizontal velocity ( Vl ) for energy rate ( P1 ) is determined form the following:

Figure 1 presents the algorithm of calculating the maneuverability and flight trajectory. The required data for this algorithm are properties of helicopters, engine output power and control laws.

dVl 1P1 , dt mVl

(5)

Start

1 where 1

0.8

, when P1 0 , when P1 0

The relationship between normal load factor ( nn )

and the turn rate ( ) is given by:

gnn .

(6)

Vl

Energy rate ( P2 ) influences vertical velocity as follows:

V 2P2 ,

h m dVh mg

(7)

dt

1 where 2

0.8

, when P2 0 , when P2 0

2.3 ROTOR THRUST LIMITS

The maximum thrust of main rotor is restricted by the available power and maximum thrust coefficient.

T

tc 0.5bc2 R5 tcmax ,

(8)

P1 P2 (Preq,total mgVh ) Pava ,

(9)

where tcmax and Pava is maximum thrust coefficient and available power.

2.4 KINEMATIC EQUATIONS

The kinematic equations can be written as follows:

Initial Data

Properties of Helicopter

Current Engine Power

Control Laws

Calculating Required Power

Excess Power

Climb of Rate and Overloads

Yes Beyond Limits

No

Flight Kinematic Equation

Time Out

No

Yes

End

FIGURE 1 Calculation algorithm for maneuverability

4 Examples and results

A program is completed in MATLAB to calculate AH1G helicopter maneuverability. The properties of AH-1G helicopter [15], which are used in program are presented in Table 1. Three kinds of maneuvers are analysed as follows.

51

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

TABLE 1 Properties of AH-1G helicopter

Parameter (unit)

m (kg)

R (m)

b (-) c (m)

Value 3400 6.71

2 0.69

Parameter (unit)

(s-1) f (m2)

(-) (-)

Value 34 1.82

736 0.0075

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

350 340 330 320 310

Power(kW)

4.1 LEVEL ACCELERATION

Level acceleration is a very necessary and integral component of the maneuver capability of a helicopter. It is very important to predict acceleration in danger situations. In the present example, the engine is at the maximum power output condition and the initial velocity is 25.7 m/s. Figure 2 shows the computed trajectory of the helicopter in 10 seconds. Figure 3 and Figure 4 demonstrate the time histories of the helicopter speed and power for the acceleration maneuver. It can be seen from the figures that the excess power is maximum at the velocity about 32 m/s.

Y(m)

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

0

50

100

150

200

250

300

350

400

X(m)

FIGURE 2 Acceleration trajectory

50

45

40

300

290

280

270

260

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 4 Time history of excess power for acceleration maneuver

4.2 DECELERATING TURN

The pilots often choose decelerate at constant altitude in order to supply more power for a 180 degree turn. Figure 5 presents an example of a decelerating turn at 1.72-g overload.

0

-50

-100

Y(m)

-150

-200

-250

-300 0

50

100

150

200

250

X(m)

FIGURE 5 Turning trajectory

The time histories of the helicopter speed and power in this maneuver are given in Figure 6 and Figure 7. In a decelerating turn, the helicopter must maintain engine idle speed [16]. Therefore, the value of excess power is always below zero, shown in Figure 7. The maximum excess power is at the velocity about 37 m/s.

Velocity(m/s )

35

30

25

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 3 Time history of velocity for acceleration maneuver

4.3 TURNING CLIMB FOLLOWED BY ACCELERATING CLIMB

A complex maneuver is presented in which a turning climb is following by an accelerating climb. The rate of climb is 5 m/s in whole maneuver. The helicopter maintains engine idle speed in climbing turn and the engine work at the maximum power output condition in accelerating climb.

52

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

60

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

50

Velocity(m/s )

Velocity(m/s )

55 45

50 40

45 35

40

30 35

Power(kW)

30

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 6 Time history of velocity for turning maneuver

-400

-420

-440

-460

-480

-500

-520

-540

-560

-580

-600

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 7 Time history of power for turning maneuver

There is a transitional period between these two conditions. Figure 8 shows the flight trajectory of this maneuver. Figure 9 demonstrates the time history of the helicopter speed. It can be seen from figure 10 that there is a transitional period about 3 seconds.

Z(m )

60

40

20

0 150

100

50

X(m)

00

-300 -250 -200 -150 -100 -50

Y(m)

FIGURE 8 3-D flight trajectory

Power(kW)

25

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 9Time history of velocity for 3-D maneuver

200 100

0

-100

-200

-300

-400

-500

-600

-700

0

1

2

3

4

5

6

7

8

9

10

Time(s)

FIGURE 10 Time history of power for 3-D maneuver

5 Conclusions

This paper proposed a fast and accurate approach to evaluate helicopter maneuverability in forward flight. A three dimensional complex maneuver of AH-1G helicopter is presented, and some important characteristics including maneuver time, trajectory, excess power and velocity are obtained. The results indicate that this method is feasible and effective for helicopters during forward flight. In order to get a fast calculation algorithm, some empirical studies which are only appropriate for forward flight situation are employed in mathematical model. Future research would focus on the rapid method for evaluating maneuverability of the helicopter in hover or vortex ring state.

Acknowledgments

This study has been supported by the State Key Basic Research Program of China (2011CB707002).

53

Mathematical and Computer Modelling

COMPUTER MODELLING & NEW TECHNOLOGIES 2014 18(5) 50-54

References

[1] Meng S, Xiang J, Luo Z 2013 Navigation of micro aerial vehicle in unknown environments 25th Chinese Control and Decision Conference: Guiyang 322-7

[2] Li J, Gao Z 2009 A simplified algorithm of rotor power and pull to determine helicopter performance Journal of Air Force Engineering University (Natural Science Edition) 10(4) 9-12 (in Chinese)

[3] Zhang Y 2003 Research of computing method for helicopter required power Helicopter Technique 22(1) 1-5 (in Chinese)

[4] The General Editorial of Aircraft Design Manual 2005 The 19th volume of aircraft design manual Aviation Industry Press: Beijing (in Chinese)

[5] Wells C D, Wood T L 1973 Journal of the American Helicopter Society 18(1) 10-22

[6] Kiwan A R 1994 Helicopter performance evaluation (HELPE) computer model AD Report AD-A284 319/1

[7] Xu X, Wang W 1994 Study on Z-9 helicopter turning maneuver characteristics by energy method Journal of Nanjing University of Aeronautics and Astronautics 26(1) 121-6 (in Chinese)

[8] Mikhailov S A, Onushkin A Y 2007 Power balance method in calculation of helicopter maneuverability taking into account specific operational conditions. Russian Aeronautics 50(2) 121-8

Zhuang Nanjian, Xiang Jinwu, Luo Zhangping, Ren Yiru

[9] Mikhailov S A, Onushkin Y P, Safonov A A, Kochish S I 2009 Numerical simulation of helicopter maneuverability in aerobatics research Russian Aeronautics 52(2) 176-83

[10] Mikhailov S A, Onushkin A Y, Onushkin Y P, Safonov A A, Kochish S I 2009 Investigation of helicopter maneuverability by the power balance method Russian Aeronautics 52(3) 296-301

[11] Cao Y, Zhang G, Su Y 2004 Mathematical modeling of helicopter aerobatic maneuvers Aircraft Engineering and Aerospace Technology 76(2) 170-78

[12] Hu H, Zhang X 2012 Simulation method of helicopter maneuver flight based on mathematical description Fire Control and Command Control 37 99-101 (in Chinese)

[13] Hu H, Wu S, Shao H 2011 Mathematical description and simulation of helicopter maneuver flight System Simulation Technology 7(4) 305-10 (in Chinese)

[14] Johnson W 1994 Helicopter theory Dover Publications: New York [15] Wood T L, Livingston C L 1971 An energy method for prediction

of helicopter maneuverability AD Report, ADA021266 [16] Gao Z, Sun C 1997 Probing of helicopter maneuverability Journal

of Nanjing University of Aeronautics and Astronautics 29(6) 66673 (in Chinese)

Authors

Nanjian Zhuang, born in June, 1987, Hunan, China

Current position, grades: Ph.D. student at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree, Beihang University, 2008 Scientific interest: helicopter flight dynamic, structural dynamics. Publications: 2 papers

Jinwu Xiang, born in February, 1964, Hunan, China

Current position, grades: Professor at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree (Nanjing University of Aeronautics and Astronautics, 1980), Master degree (Northwestern Polytechnical University, 1990), PhD (Nanjing University of Aeronautics and Astronautics, 1993). Scientific interest: aircraft design, aeroelasticity and structural dynamics, etc. Publications: 161 papers, 12 patents

Zhangping Luo, born in December, 1968, Guizhou, China

Current position, grades: associate professor at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree (Beihang University, 1991), Ph. D. (Beihang University, 2004). Scientific interest: structural dynamics, impact dynamics and crashworthiness, aircraft design. Publications: 16 papers

Yiru Ren, born in January, 1983, Hunan, China

Current position, grades: post-doctor at School of Aeronautic Science and Engineering, Beihang University, China. University studies: Bachelor degree (Harbin Engineering University, 2006), Ph. D. (Beihang University, 2011). Scientific interest: structural dynamics, impact dynamics and crashworthiness, aircraft design. Publications: 10 papers

54

Mathematical and Computer Modelling