Helicopter Rotor Balance Adjustment Using GRNN Neural Network and Genetic Algorithm

Hongmei Liu, Yunlong Cai, Chen Lu, Jiahui Luan Department of System Engineering of Engineering Technology

Beihang University Haidian, Beijing, China, 100083

liuhongmeigg@ tom.com

Abstract

Considering the drawbacks of traditional

adjustment method without calculating possible nonlinear between rotor adjustments and fuselage vibration signals, a new rotor adjustment method based on general regression neural network (GRNN) and genetic algorithm is presented. GRNN network is employed to model the relationship of the rotor adjustments and fuselage vibrations, whose input parameters are rotor adjustment parameters and whose outputs are acceleration measurements along the three axes of rotor shaft and the fuselage. With helicopter vibration as objective function, genetic algorithm (GA) was used to make a global optimization to find the suitable rotor adjustments corresponding to the minimum vibrations. Flight test results indicate that proposed rotor adjustment method can minimize fuselage vibration at fundamental rotor frequency along the three axes, only in one or two adjustment flights. Moreover the neural networks are easily updated if new data becomes available thus allowing the system to evolve and mature in the course of its use. 1. Introduction

The large fuselage vibration not only makes crew and passenger fatigue, but also makes high-cycle fatigue of different components to decrease the reliability and to increase the maintenance costs. Consequently, the reduction of helicopter vibrations has been a concern of investigation [1]. The purpose of helicopter rotor balance adjustment is to provide a low vibratory level in 1 per rev along the 3 aircraft axes, to ensure optimized crew and passenger comfort.

The measured vibration data is used to calculate the rotor adjustments that are necessary to reduce the vibration magnitudes to below manufacturer’s

prescribed limits. There is a complicated nonlinear relationship between rotor adjustment parameters and fuselage vibration, but the adjustments are traditionally calculated with the use of a linear model, in which the helicopter vibration response to the rotor adjustments is represented by a set of linear coefficients. The linear model is a simplification of the actual rotor system as it is not capable to account for possible nonlinear interactions. Usually, the accuracy with which the linear coefficients may be determined is not better than about 20% and in many cases it is much worse [2]. Thus, in practice, the rotor smoothing is completed in several test flights, with consecutive sets of adjustments converging on an acceptable low vibration state of the aircraft.

Neural network has a strong nonlinear mapping ability. It is easy for neural network to realize the non-parameter mapping between the vibration measurements and the rotor adjustments [3, 4]. The mapping is extracted from empirical data in the neural network training process. Compared with RBF neural network, GRNN neural network have many advantages: faster convergence speed, no local minimum, less training samples and stronger robustness…etc. Consequently, the GRNN neural network can promptly identify the rotor system model, and may be easily updated (retrained) to include new data thus allowing the system to evolve and mature during the course of its use.

For above reasons, a new rotor adjustment method based on GRNN neural network is proposed in this paper, which combines GRNN neural network with genetic algorithm to optimize and adjust rotor adjustments parameters according to acceleration measurements along the three axes of rotor shaft and the fuselage.

Global Congress on Intelligent Systems

978-0-7695-3571-5/09 $25.00 © 2009 IEEE

DOI 10.1109/GCIS.2009.105

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2. Mathematical model of adjustment parameter and fuselage vibration

In order to bring 1 per rev vibration within specification, three types of adjustments can be made to the rotor system: pitch control rod, trim tab, and balance weight. Pitch control rods can be extended or contracted by a certain number of notches to alter the pitch of the rotor blades; positive push rod adjustments indicate extension. Trim tabs, which are adjustable surfaces on the trailing edge of the rotor blades, affect the aerodynamic pitch moment of the blade and consequently the overall 1 per rev vibration characteristics of the rotor. Tab adjustments are measured in thousandths of an inch, with positive and negative changes representing upward and downward tabbing, respectively. Finally, balance weights can be either added to or removed from the rotor hub to tune vibrations through changes in the center of gravity of the rotor. Balance weights are measured in ounces with positive adjustment representing the addition of weight [1].

The aim of investigating the mass imbalance and aerodynamic imbalance is to find the effect of tab, pitch link and weight adjustment on rotor vibration and tracking, and to research their different adjustment effect. In this paper neural network is applied to establish the mapping relationship between rotor adjustments and fuselage, and genetic algorithm is used for optimizing the mapping relationship to obtain the rotor adjustment parameters. 2.1. Description of the rotor adjustment system

According to the rotor imbalance mechanism, extra vibration at the fundamental (1/rev) rotor frequency will occur when mass imbalance and aerodynamic imbalance exists. Therefore, mathematical model is established directed toward the fundamen-tal (1/rev) rotor frequency. For each aircraft, the vibration levels in the cabin can be written as:

( )h ja iHγ αΔ = (1)

The adjustment parameters are noted jiα where i

represents the index blade（ 1,2...i N= ，the blades are indexed 1 to N, where N is the number of blades of the rotor) and j represents the parameter type (weight, pitch rod, tab). The Fast Fourier Transform coefficient increments of the vibrations in the cabin are noted

haγΔ where a represents the accelerometer index and

h represents the harmonic. Where H is the transfer

function between the adjustment parameters and the vibration levels.

Obviously, Fuselage vibration information haγΔ and

adjustment parameters jiα have a complicated

nonlinear mapping relationship. Consequently, neural network is adopted to describe precisely transfer function between one harmonic vibration of fuselage and adjustment parameters. The helicopter vibration levels can be predicted according to adjustment parameters input to neural network. But the predicted results are only mapping value, not optimal adjustment value. Therefore, optimization method is applied to obtain optimal blade adjustment parameters.

Due to genetic algorithm is an effective global search method and independent of gradient information, it is especially applicable to complicated nonlinear problem which traditional search method is difficult to solve. Therefore genetic algorithm is adopted to optimize the GRNN neural network model, and to obtain the optimal blade adjustments corresponding to the minimum vibrations.

The schematic of the proposed method is shown in Fig.1. In this method, a neural network model is used as the basis of search for the appropriate blade modifications by use of genetic algorithm, According to vibration measures of rotor shaft and fuselage, 1/rev vibration signal is extracted, and its magnitude and phase relative to the synchronous signal are also calculated. And its magnitude and phase are input into neural network to obtain proper blade modifications, which are used for balancing rotor to make the 1/rev vibration within the specification.

Once a set of blade modifications is found within the feasible region of this model, the modifications are applied to the aircraft and its vibration is measured during a proceeding test flight. The neural network model is then adapted to better fit the measured vibration from the aircraft.

Geneticalgorithm

Estimated vibrationChange of vibration

Measured vibrationHelicopter rotor

Optimizationsearch

Adjustmentparameters

Neural network

Optimizedadjustments

Figure 1. The strategy of the proposed method

2.2. GRNN network model of rotor system

2.2.1. GRNN neural network model. Due to training samples originate from flight test, data size is

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limited. Identification precision of GRNN is far higher than that of BP network and RBF network in lack of training samples, so the GRNN is chosen to model the rotor. As shown in Fig.2, Adjustment parameters , 1,2, 4, 1,2,3j

i i jα = = are inputs (Where

i represents the index blade, and j represents the

parameter type, 1-weight, 2-pitch rod, 3-tab). Outputs haγΔ （where a represents the accelerometer index

and h represents the harmonic）are sine or cosine component increment of vibration in the cabin along three axes (Considering the vibration data with magnitude and phase relative to the synchronous signal, for convenience, transform vibration data into two-dimension cartesian coordinate, then two components respectively represent the increment of the vibration’s sine or cosine component.). , 1, 2,if i m= are

activation functions of the hidden layer. , 1, 2,iw i m= are weights of output layer.

Considering helicopter with the four blades, each blade has three adjustment parameters (weight, pitch rod and tab); therefore, the number of total adjustment parameters is twelve. So the number of neurons in input layer is determined as twelve, and the number of neurons in output layer is determined as one. Output neuron is a vibration FFT sine or cosine component increment of a certain accelerometer. Consequently, total 48 neural networks are established (considering four flight regimes, six accelerometer, each has x and y components).the structure of each neural network is shown Fig.2.

ijw

kjv 2

1α

22α

1f

mf

11cγ

12α

31α

11α

iii

34α

iii

Figure 2. GRNN neural network model of rotor system

2.2.2. Learning of neural network. For GRNN, once learning samples are determined, network structure and connection weights between neurons are also determined. Network training in fact is the process of determining smoothness factor σ . Different from traditional error back propagation algorithm, there is no need for general regression neural network to adjust connection weights between neurons in training

process. The optimal regression results can be obtained by changing smoothness factor and adjusting transfer function between units.

To carry out the learning, it is necessary to collect representative learning couples of the application field. These couples consist of an input vector , 1,2, 4, 1,2,3j

i i jα = = (adjustment

parameters) and an output vector haγΔ (vibration

levels). These vectors satisfy the relation: ( )h j

a iGRNNγ αΔ = , where H is the transfer function of the system to be modeled.

The learning couples are given by an intentional misadjustment of the rotor by measuring the vibration levels associated. In order to reduce the number of learning flights some hypotheses are necessary:

(a) Superposition: the effect of each adjustment parameter is independent.

(b) Isotropy: for each blade, each parameter type has the same effect on the vibration levels with a different phase (geometrical angle between the blades).

Only four flights are carried out for the learning: a reference flight, a flight with a weight misadjustment, a pitch rod misadjustment flight and a tab misadjustment flight. Each flight has four specific configurations during which vibration measurements are taken. These configurations are typical of the aircraft use: Ground, Hover, 100 kts speed and MCP.

After having completed the flights, the whole learning couples are generated with the flight measurements. 3. Optimization rotor adjustment based on AGA and GRNN

With the vibration in the cabin as objective function

and blade adjustments as optimization variables, genetic algorithm (GA) was used to make a global optimization to find the suitable rotor adjustments corresponding to the minimum vibrations by use of the neural network. 3.1. Optimization objective function

Considering the blade adjustments shouldn’t too large when the vibration in the cabin is minimized, optimization objective function is determined according to cabin’s vibration and blade adjustments. Therefore the research for the adjustment parameters is carried out by minimizing the below objective function:

4 32 2

1 1 1

1 1( ) ( ) ( )2 2

mj h h j

i a a i ija i j

F αα γ γ β γ α η= = =

= Δ + +∑ ∑∑ (3)

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where hαγΔ =GRNN( j

iα ) is the FFT increment of the

a th acceleration sensor signal, jiα is the adjustment

parameter to be determined and hαγ is the vibration

level in the cabin before adjustment, γ is a relative coefficient, aβ is importance coefficient of the a th vibration sensor signal, ijη is importance coefficient of i th blade’s j th adjustment parameter. 3.2. Constraint condition

1) Vibration along the three axes in the cabin and shaft should be within prescribed limits, that is to

say, haGRNN( )+ j

i Kα γ <

2) Blade adjustment parameters (pitch rod, weight and tab) should also be within specification:

jia a b< <

3.3. Transformation of the objective function

Due to genetic algorithm can not directly deal with constraint condition, penalty function is adopted to transform the constraint problem into unconstrained problem. objective function can be rewritten as：

{ }42 2

1( , ) ( ) max 0, ( )j j j

i i k ik

P a r F a r g a=

⎡ ⎤= + −⎣ ⎦∑ (4)

Where ( ) 0.2 H( )j j hk i ig a a αγ= − − （ k=1,2…12 ） ，

( )j jk i ig a a a= − （ k=13,14,…,24 ）

( )j jk i ig a b a= − （ k=25,26,…,36 ） ； r is the

penalty factor and is used for regulating the penalty function to make optimized solve entirely satisfy the constraint condition, here 1.0r = . 3.4. Optimization operation of genetic algorithm

Flow chart of optimization process is shown in Fig.3. The AGA optimization procedure for rotor adjustment is as follow: a) encode variable initial space; b)set population size and produce initial population; c)calculate the individual objective function value, and fitness transformation; d)selection, crossover and mutation operation; e)Check if new population satisfy the Terminate rule, if satisfy, optimum solve is obtained, otherwise, turn to other steps.

Initial variable space encoding

Set population sizeand produce initial

population

Calculate fitness by GRNN

Adaptive adjustment of crossover rateand mutation rate

Selection, crossover and mutation

Terminal condition

Optimal adjustment parameter

Yes

No

Figure 3. Flow chart of optimization process

4. Flight test 4.1. Test planning

The learning and optimization flights were accomplished on a 2T helicopter. Four flights are necessary for the learning phase:①Reference flight: rotor well tuned by a traditional method.②Flight with a weight misadjustment: +0.5Kg on blade 1.③Flight with a pitch rod misadjustment: +18 notches on blade 2.④Flight with a tab misadjustment: +12° on blade 4.

Each flight has four specific configurations during which vibration measurements are taken. These configurations are typical of the aircraft use: Ground, Hover, 100 kts speed and MCP. Between the learning phase and the optimization phase, the rotor is intentionally misadjusted. Thus the aircraft has a high vibration level in the cabin. This level will be computed by the rotor adjustment system to find the applied misadjustment.

Six accelerometers were located in the cabin: 1) Pilot sit: Xpil and Ypil 2) Copilot sit: Zcop 3) Under the rotor shaft: Xsha,Ysha and Zsha

4.2. Learning

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Table 1. Flight test data

Flight regime Flight sortie

Ground（acceleration/mg） Hover（acceleration/mg）

Xsha Ysha Zsha Xpil Ypil Zcop Xsha Ysha Zsha Xpil Ypil Zcop

Weight misadjustment 2.73 60.1 6.21 41.3 22.6 26.9 0.63 0.74 1.82 0.93 1.27 1.08

Pitch rod misadjustment 1.56 31.7 2.98 23.4 16.3 11.9 2.76 0.94 6.55 3.02 9.01 7.98

Tab misadjustment 1.23 9.78 2.34 10.3 5.78 3.46 0.98 1.98 1.01 2.67 4.98 2.46

Flight regime

Flight sortie

100kts（acceleration/mg） MCP（acceleration/mg）

Xsha Ysha Zsha Xpil Ypil Zcop Xsha Ysha Zsha Xpil Ypil Zcop

Weight misadjustment 4.17 4.22 0.97 2.03 8.53 8.32 3.62 3.47 4.34 2.32 2.87 5.42

Pitch rod misadjustment 3.02 3.37 16.1 1.03 14.5 15.1 2.03 3.45 5.06 2.17 4.02 2.97

Tab misadjustment 1.23 1.33 5.84 1.03 8.65 8.06 1.46 4.56 3.38 1.09 7.52 7.13

4.3. Optimization result analysis

In order to test the validity of proposed method, the rotor was intentionally misadjusted with the following parameters as shown in table2.

Table 2. Misadjusted parameter of rotor

Mass (g) Pitch Rod (notches)

Tabs (°)

Blade 1 300 0 0 Blade 2 0 0 8 Blade 3 0 10 0 Blade 4 0 0 0

And flight test is carried out in the imbalance

condition, during which vibration measurements are taken. Then adjustment parameters are optimized according to objective function established above by the adaptive genetic algorithm. Select population size is 100, and iteration number is n=500. The optimization process of genetic algorithm is shown in Fig.4.

0 100 200 300 400 50018

20

22

24

26

28

generation

fitne

ss

Figure 4. Optimization process of genetic algorithm

It is can be seen that objective function find the global optimum solve until 450 generations. Genetic algorithm needs more the number of generations when searching for the minimum vibration. This is likely due to genetic algorithm has strong global search ability, but has weaker local search ability. The parameters suggested by genetic algorithm to correct the vibration levels are shown in table 3. These parameters are almost identical to the parameters applied before the optimization.

Table 3. Parameter optimization result corresponding to minimum vibration

Mass (g) Pitch Rod (notches)

Tabs (°)

Blade1 －342 0 －2 Blade 2 －54 2 6 Blade 3 0 11 －1 Blade 4 0 0 0 Vibration level before and after adjustment is

compared in Fig.5~Fig.8. Acceleration(mg)40

30

20

10

0Xsha Ysha Zsha Xpil Ypil Zcop

Figure 5. Vibration levels before & after tuning-

ground

105

Acceleration(mg)15

10

5

0Xsha Ysha Zsha Xpil Ypil Zcop

Figure 6. Vibration levels before & after tuning-

hove

Acceleration(mg)20

15

10

5

0Xsha Ysha Zsha Xpil Ypil Zcop

Figure 7. Vibration levels before & after tuning-

100 kts

Acceleration(mg)30

20

10

0Xsha Ysha Zsha Xpil Ypil Zcop

Figure 8. Vibration levels before & after tuning-MPC

It is can be seen that vibration level is largely

reduced after the rotor is corrected according to the optimization adjustments given by genetic algorithm. Obviously combining genetic algorithm and GRNN neural network is effective in rotor track and balance. 5. Conclusions

GRNN neural network is adapted to model rotor system, and genetic algorithm is used for searching for rotor adjustment parameters corresponding to minimum vibration in the cabin. Considering the drawbacks of premature convergence and the worse local search ability of SGA, an adaptive genetic algorithm is applied, which can automatically adjust genetic control parameters according to individual fitness and population dispersion. Flight test results indicate that:

(1) Rotor adjustment methodology proposed can minimize fuselage vibration (on first harmonics of the rotor frequency) in as many points as desirable, along the three axes, and in one or two adjustment flights only.

(2) The neural networks rotor adjustments system is easily updated if new data becomes available thus allowing the system to evolve and mature in the course of its use.

6. Acknowledgements

This project is supported by the China National Nature & Science foundation (Grant No.50705005) and National 863 program (Grant No.2007AA04Z431). The authors would like to acknowledge Chen Lu’s assistance during the evaluation phase of this research. 7. References [1] Sam Ventres, Richard E. Hayden. Rotor Tuning Using Vibration Data Only[J] Wang and I. Chapara, “Dynamics of helicopters with dissimilar blades”,American Helicopter Soci-ety 56th Forum, Virginia Beach, Virginia, May 2-4, 2000,623~629. [2] Dariusz Wroblewski, Paul Grabill, Neural Network System for Helicopter Rotor Smooth-ing, Intelligent Automation Corporation, San Diego,CA,2000.271~276 [3] Shengda Wang, Kourosh Danai, mark Wilson. Adaptive Method of Helicopter track and balance[J], Journal of Dynamic Systems, Measurement, and control.,vol.127, JUNE 2005 :275~282 [4] A.ROSE, R.BEN_ARI. Mathematical modeling of a helicopter rotor track and balance: theory [J]. Journal of sound and vibration, vol.200, No.5, 1997, 589~603 [5] R.BEN_ARI, A.ROSE. Mathematical modeling of a helicopter rotor track and balance: Results [J]. Journal of sound and vibration, vol.200, No.5,1997, 605-620 [6] ZHOUHao, ZHENGLi-gang, FAN Jian-ren…etc. Application of general regression neural network in prediction of coal ash fusion temperature[J]. Journal of Zhejiang Uni-versity Vol.38, No.11, Nov. 2004,1479~1481 [7] XIA Xiao-yan, MEEK L. Computer cutting pattern generation of membrane structures [J].International Journal of Space Structures, 2002, 15(2): 95~110 [8] Fan Hui, Li Weiji. Application of MATLAB Neural Network and Optimization Toolbox in Design Optimization[J].Computer engineering and application, 2006.16,187~189(in Chinese)

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