Vibration characteristic analysis and optimization of rotor hub for helicopters based on genetic algorithm

4.1. Mathematical model of optimization for the rotor hub

The key to optimization design of the bearing-less rotor hub is the flexible beam. Aero-elastic cutting was carried out for the flexible beam. Therefore, the requirement of the dynamic characteristic for the bearing-less rotor hub could be satisfied. The flexible beam structure was shown in Fig. 6(a), and its structural section was shown in Fig. 6(b).

Fig. 6Diagram of the flexible beam

a) Diagram of the flexible beam structure

b) Diagram of the typical section for the flexible beam

The flexible beam was constituted of five parts. Part I was the root joint section. Part II was the transition section. Part III was the torsional deformation section which bears the torsional deformation motion of the blade. Part IV was transmission section of the torsional deformation. Part V was joint section of the blades. The ply angle of the composite material for the flexible beam at the minimum section III was 0⁰. Optimization of the section shape was the key during optimization. The ply angles of the flexible beam at other four parts were random. Aero-elastic cutting was realized by the optimization of the ply angles. The aero-elastic cutting for the entire flexible beam could be completed by optimizing the five sections of the flexible beam. In this way, the dynamic characteristics of the entire rotor blade could meet the design requirements. Multi-objective optimization for reduction vibration of the rotor hub could be realized. The mathematical model was as follows:

The objective function in Eq. (5) was divided into two types as follows.

1) The horizontal force $F_Y$ and the vertical force $F_Z$ of the rotor hub were the minimized vibration loads. They were calculated by the following equation:

where subscripts $R$ and $H$ stood for a rotary coordinate system (non-transformed blade) and a fixed coordinate system, respectively. $T_C^T$ was a transfer matrix from the non-transformed blade to the fixed coordinate system.

2) $W$was the minimized blade mass. The blade mass included both structural and non-structural masses. They were calculated by the following equation:

In the equation, ${\rho }_i$ was the linear density of the $i$ unit and $L_i$ was the length of the $i$ unit.

The constraint function in Eq. (5) was explained in details as follows.

1) ${\alpha }_k$ was the modal stability coefficient of the $k$ order:

${\lambda }_k^R$, ${\lambda }_k^I$ were the real part and imaginary part of the $k$ order modal characteristic, respectively. $T$ was the number of circles.

2) $g_A$ was the rotary inertia of the blades:

$m_i$ was the dimensionless unit mass of the $i$ unit for the blades. $x_i$ was the distance from the left end point of the $i$ unit to the blade root. $R_A$ was the minimum rotational inertia ratio. The minimum rotational inertia ratio was the ratio of the rotational inertia of the blades in the calculation model to the reference rotational inertia.

3) $\sigma$ was the dynamic reaction at the blade end:

where, $F_R$ was the centrifugal force which was generated from blade motions. $A_R$ was the sectional area with maximum stress level, and ${\sigma }_0$ was the reference stress level of the blades.

4) In Eq. (5), ${\omega }_{Fk}$ was the $k$ order flapping frequency. ${\omega }_{Lk}$ was the $k$ order lead-lag frequency. ${\omega }_{\theta k}$ was the $k$ order pitch frequency. The low right subscripts $L$ and $U$ stood for the change scope of lower limit and upper limit for the constraint function. In order to avoid resonance and reduce vibration loads which would be transmitted to the rotor hub due to the lead-lag and flapping motions of the blades in a helicopter, the base-order inherent frequency should be kept within a certain scope as follows:

4.2. The optimization of the rotor hub based on GA

Genetic algorithm was used in the paper for optimization design of the rotor hub. Its necessity can be explained from the following aspects.

1) Multiple constraint variables were considered in the paper, and then the multi-objective optimization design was carried out to the rotor hub of a helicopter. An absolute optimal solution could hardly be found for the multi-objective optimization problem. In general, multiple Pareto optimal solutions are obtained and form an assembly. In addition, evolutionary operations are carried out to the whole group by genetic algorithm, and an assembly of individuals is processed. Because of such similarity, genetic algorithm becomes an effective method to obtain Pareto optimal solution of a multi-objective problem. It was widely used in the aerospace field. For example, the genetic algorithm was used by Liu to redesign aerodynamic appearance of a rotor blade, and experiments were conducted to the optimized structure [10]. As a result, effectiveness of genetic algorithm in multi-objective optimization of a helicopter was verified. Genetic algorithm was also used by Crossley to research the multi-objective optimization design problem with the minimum rotor mass and required power. Parameters like discrete element stiffness were taken as the design variables [11].

2) If the finite element method is directly used for calculation of the rotor hub, calculation time would be long and cost would be high due to the huge mesh quantity. In this paper, the finite element method was replaced by RBF (radial basis function) surrogate model during the numerical calculation of the rotor hub. In this way, calculation time was saved a lot in comparison with the finite element method. In addition, RBF surrogate model is characterized by whole and partial estimation. It is a surrogate model with good flexibility, simple structure, relatively less calculation amount and relatively high efficiency. The surrogate model cannot be integrated with the deterministic algorithm yet. In most of the current researches, the surrogate model is combined with genetic algorithm.

3) Research in this paper was carried out based on ISIGHT software. In this software, processes like genetic algorithm and surrogate model are integrated. Only some parameters in the software are required to be modified. Too much time needs not to be spent on programming and calculation. In this way, calculation efficiency is increased. Such software has been widely used.

Based on the mentioned analysis, genetic algorithm was used in the paper for structure optimization of the rotor hub. The basic flow was shown in Fig. 7.

Realization of the optimization design for the rotor hub based on genetic algorithm was as follows.

1) Chromosome coding. Before searching, the target function was encoded into a binary string with fixed length, and such string was a chromosome. Different search points in the search space were constituted of different combinations of these strings.

2) Generation of the initial population. Size of the initial population was set to be 30.

3) Fitness calculation. Convergence rate of genetic algorithm and whether an optimal solution can be found are directly influenced by fitness. In this paper, negatives of the target function values were taken as the fitness values.

4) Selection. Individuals which showed relatively stable fitness during evolution were selected.

5) Crossover. Some chromosomes among 30 individuals were exchanged under a probability. 30 new individuals were then generated. The crossover probability was set to be 0.9.

6) Mutation. Mutation was carried out to the selected 30 individuals according to the given probability. A new generation group was then formed. The mutation probability in this paper was 0.05.

7) Judgment. Whether the new generation group could satisfy constraint conditions was judged. If the conditions could be satisfied, the operation would be stopped, otherwise, step 3 would be continued.

Fig. 7Basic flow of genetic algorithm

Fitness was taken as an evaluation index for convergence of genetic algorithm. The change curve of fitness with the evolutionary generation number during optimization was shown in Fig. 8. According to this figure, the fitness has tended to be stabilized when the evolution developed to the 100th generation. As a result, the calculation results were convergent.

Fig. 8Change curve of fitness with the evolutionary generation number

During the optimization process, RBF surrogate model of the aero-elastic dynamic for the rotor hub was built in order to reduce optimization burdens. RBF surrogate model is a kind of function. In the function, Euclidean distances between the points to be measured and the sample points were taken as the variables. See Reference [12, 13] for the mathematical description of RBF surrogate model.

The principle of RBF surrogate model is simple. The solving flow of the RBF surrogate model is as follows.

1) Firstly, an optimal $P_m$ with the sample number of$m$was obtained by LHD method.

2) $L_p$ was the difference between an experience accumulated error distribution function and a uniform accumulated error distribution function. In other words, $L_p$ was used to describe heterogeneity of the experiment. Among the relevant descriptions about $L_p$, $L_2$ was widely used by designers, because it can be calculated and analyzed easily. Hickernell [14] put forward three kinds of descriptive forms for $L_2$. The center $L_2$ deviation was most popular and expressed as Eq. (12):

3) A group of test points $P_i$ was selected. A test point was selected from the remained points of each subspace to constitute a new group of test points $P_{i+1}$. This was repeated until all the test points were grouped.

4.3. Optimization results and discussion

TOPSIS method is a common method for the multi-objective decision analysis on the limit scheme of system engineering [15]. It ranks the alternative scheme $X$ through the ideal solution and negative ideal solution of the multi-attribute problem. The alternative scheme set of a multi-objective decision problem is assumed to be $X=\{x_1,x_2,\dots ,x_m\}$. The attribute vector is assumed to be $Y=\{y_1,y_2,\dots ,y_n\}$. The vector $Y=\{y_{i1},y_{i2},\dots ,y_{in}\}$ is constituted by the $n$ attribute values in alternative scheme $x_i$. The ideal solution is the assumed optimal solution. Each of its attribute values is the single objective optimal solution. The negative ideal solution is the assumed worst solution. Each of its attribute values is the single objective worst solution. The principle for ranking the scheme is to evaluate the alternative scheme by means of the Euclidean distance between the ideal solution and the negative ideal solution in the alternative scheme. TOPSIS method with the visual principle is calculated simply. Therefore, it is widely applied to the environmental assessment, site selection of logistic distribution and medical service evaluation [16-18] etc. See reference [15] for the detailed solving flow of TOPSIS method.

For the scheme with the maximum comprehensive satisfaction provided by TOPSIS method, the change of its design variables which were compared with the initial design scheme was shown in Table 1.

Part III is the minimum section of the flexible beam. It is also the key part of aero-elastic cutting. The scale factors were changed to meet the structural and dynamic design requirements. See Fig.9 for the section change before and after optimization.

Fig. 9Diagram of the typical part III for the flexible beam before and after optimization

The solid line in Fig. 9 was the initial design section of part III, and the dashed line was the optimized section. According to the sectional view, the section width $a$ after optimization was reduced, while the section height $b$ was increased. The area of the section changes not so much. Sizes of the section for the flexible beam before and after optimization were shown in Table 2.

A rotor blade model was built based on the optimized scheme to analyze the changes of the modal frequency and modal shape as well as the vibration loads.

4.3.1. Changes of modal frequency and modal shape before and after optimization

The changes of the blade vibration loads would be caused by the changes of the modal frequency and modal shape of the rotor blades. The reduction vibration effect of the multi-objective optimization design was shown in the changes of modal frequency and modal shape. See Table 3 for the changes of each order frequencies before and after optimization.

Table 3Changes of the frequencies for the blades before and after optimization

Modal frequency	Original data	Optimized data
First order lead-lag	0.6418	0.6171
First order flapping	1.0870	1.0882
Second order lead-lag	3.5956	3.4867
Second order flapping	2.5685	2.5724

According to Table 3, the changes of the frequencies after optimization were deviated from frequency ratio 3, 4 and 5. Therefore, resonance of the blades was avoided. See Fig. 10 to Fig. 13 for the changes of each order modal shape before and after optimization.

Change curve of the vibration modal shapes before and after optimization were shown in Fig. 10 to Fig. 13. The figures showed that the magnitudes of the curves for the initial vibration modal shapes were reduced after optimization. The optimization scheme was feasible.

Fig. 10The changes of the first order flapping modal shape

Fig. 11The changes of the first order lead-lag modal shape

Fig. 12The changes of the second order flapping modal shape

Fig. 13The changes of the second order lead-lag modal shape

4.3.2. Changes of the vibration loads before and after optimization

The multi-objective optimization was carried out on the rotor hub system with the vibration loads as the emphasis. The changes of the loads were shown in Fig. 14.

Fig. 14The changes of the vibration loads before and after optimization

It could be seen from Fig. 14 that the lateral loads and vertical load have been reduced obviously after optimization. Moreover, the reduction degree of the vertical load was larger than the lateral one, which was more beneficial for reducing the vibration displacement during the work process of the rotor hub through the mentioned experimental comments.