Modelling, Parametric Study, and Optimisation of 3D Model-Scale Helicopter’s Rotor Blade with Piezoelectric Actuators

Abstract

The concept of active blade twisting as a method for reducing helicopter noise and vibration during flight is presented. Active twisting is achieved through piezoelectric actuators embedded in the blade skin, which generate dynamic twist when subjected to an electric field. Such dynamic deformation can lower fuel consumption while also reducing noise and vibration levels. A methodology for determining the optimal geometric dimensions of the cross-section of a helicopter blade, taking into account design constraints, is proposed to achieve the maximum twist angle of the blade under the action of piezoelectric actuators. First, a three-dimensional numerical model of the BO 105 model-scale rotor blade is developed in the finite element software ANSYS 16.0. The effect of the rotor blade’s cross-sectional dimensions on the cross-sectional properties and twist angle is investigated. It is found that skin thickness, spar flange thickness, and spar flange length affect the twist angle, with skin thickness showing a significant effect. Based on these results, an optimisation strategy is formulated to identify the optimal blade cross-section configuration to achieve the maximum twist angle. It was established that with the optimised geometric parameters of the cross-section the maximum active twist reaches 5.2°, while the positions of the elastic axis and the centre of gravity exhibit only minor deviations from those of the reference model. The placement of the piezoelectric actuators has a significant influence on both the flapwise bending stiffness and the torsional stiffness of the blade.

1. Introduction

During helicopter flight, vibration and dynamic load levels are significantly higher than those of aeroplanes. This is due to the advancing and retreating blades operating under different conditions and flowing asymmetrically around them. This unevenness causes periodic changes in aerodynamic forces, leading to increased vibrations and dynamic stresses, a limitation on maximum flight speed, increased operating costs, and a reduction in structural life.

The use of active control systems for helicopter rotor blades is a current scientific trend of research for reducing noise and vibration. One control strategy applied to suppress vibrations is Higher Harmonic Control (HHC). The variable aerodynamic load is realised as an additional higher harmonic, transmitted through the swashplate to all helicopter blades simultaneously. A very similar concept is Individual Blade Control (IBC), in which the actuators in a rotating reference frame act on each individual blade [1].

The development of advanced piezoelectric actuators has enabled the creation of new blade control approaches aimed at improving aerodynamic performance, reducing vibrations, and enhancing rotor efficiency during flight. One of these approaches is active twist, which is based on controlling blade twist using intelligent piezoelectric material embedded within the blade skin. The orientation of the piezoelectric fibres in the piezoelectric actuator on the top and bottom surfaces of the blade is ±45°, which causes dynamic blade twisting when the actuators are activated.

By means of developing the smart technologies of rotor twisting, theoretical and experimental research aimed at improving the aerodynamic performance and reducing the vibration and noise are being conducted rapidly [2,3,4].

Early studies were primarily experimental, used simplified models, and aimed to validate the concept of active torsion control using piezoceramics. The primary objective was to investigate the feasibility of achieving torsion of ±2° to suppress vibrations with minimal power consumption [5].

In the 1990s, Professor Chopra’s group at the University of Maryland (USA) pioneered the study of active blade twisting technology using piezoelectric elements with a d31 effect [6]. Tests showed that placing the piezoelectric elements with an angle of ±45° produced the greatest effect, but the blade twisted only 0.1°, significantly less than the required range of 1–2°.

Later, the team conducted blade twisting tests, achieving a twist angle of approximately 0.5°, which is still insufficient for practical application. These experiments confirmed the potential of active twisting technology but also revealed the limitations of piezoceramics, such as small strains, being insufficient for the desired twist angle, and the material’s brittleness, which hinders its use under high dynamic loads [7].

With the advent of piezoelectric fibre technology, the concept of active twisting was significantly improved. This technology enables the creation of an active piezoelectric layer within laminated composites, such as active fibre composite (AFC) and macro-fibre composite (MFC) actuators. The main difference between these actuators is the cross-section of the piezoelectric fibres. In AFC actuators, the PZT fibres have a circular cross-section, which reduces the contact area with the electrodes and decreases the effectiveness of the electric field inside the fibre. In MFC actuators, the PZT fibres have a rectangular cross-section, which increases the contact area with the electrodes [8,9].

Derham and colleagues used AFC actuators for active blade twisting, which significantly increased the twisting amplitude. In tests on a 1:16 scale model of the CH 47 rotor, the twist angle reached 1.4° [10]. For the 1:6 scale model, the twist angle was 0.77°/m [11].

In 1997, NASA Langley Research Centre, the Army Research Laboratory, and the MIT Active Materials and Structures Laboratory formed a joint team for the active twist rotor (ATR) project. Its goal was to test the feasibility of active blade twist technology. The project aimed to demonstrate the operation of an AFC-driven blade in a wind tunnel and evaluate how this technology could enhance aerodynamics, reduce vibrations, and minimise noise. ATR tests showed that the maximum twist angle was achieved when the AFC actuators were located at the angle of ±45°. The measured twist angle was approximately 1° [12,13].

After the completion of the ATR project in 2003, the research began on a new generation of smart rotor—the advanced active twist rotor (AATR)—using MFC actuators [14]. Rig tests showed that with MFC actuators, the blade twist angle in static mode reached approximately 2°.

The German Aerospace Centre (DLR) also conducted research into active twist. Blade testing was conducted using MFC, and demonstration models of the ATB (Advanced Technology Blade) blades were developed and tested on the DLR rotor tower [15,16]. In total, DLR developed five blade models, AT1–AT5. It is particularly noteworthy that for the AT2 blade under quasi-steady excitation (0.15 Hz) and with a voltage range of –500 to 1300 V and various rotation speeds the twist angle was approximately 4° [17,18].

Since 2005, the international project STAR has been running. STAR (smart twisting active rotor) is an international scientific research project aimed at developing and experimentally testing actively twisted helicopter rotor blades. The project uses active blade twisting technology with piezoelectric actuators (MFC—macro-fibre composite). It is one of the largest international projects dedicated to active blade control and reducing helicopter vibration and noise. The team includes representatives from universities and companies involved in the design, manufacture, and calculation. Main participants include ONERA, DLR, the US Army, NASA, the Japan Aerospace Exploration Agency (JAXA), Konkuk University, the Korea Aerospace Research Institute (KARI), DNW (German–Dutch Wind Tunnels), and the University of Glasgow. A research model of a helicopter blade used in this project is similar to the Bo 105 rotor blade. A total of 30 MFC actuators are attached to the upper and lower surfaces of the blade, allowing it to be twisted by 2 degrees. Research on the blade is conducted in static and aerodynamic modes. In studies of the new-generation STAR, a comparative analysis of predictive data from various project participants revealed that smart twisting has strong potential to reduce noise and vibration during low-speed descents, enhance aerodynamic performance at high speeds, and improve aerodynamic performance and vibration control under heavy loads. The experimental results confirmed the simulations in most respects [19].

In 2022, the Korea Aerospace Research Institute (KARI), as one of the participants in the STAR project, collaborated with Konkuk University to investigate the application of MFC actuators in active twist technology for controlling vibration loads [20]. Researchers from the University of Glasgow reported on a study of the effect of blade twist on rotor performance using the Hover Validation Acoustic Baseline (HVAB) and smart twisting active rotor (STAR). This work was a numerical study based on computational aerodynamics and aeroelastic simulations [21,22].

Reducing energy consumption, vibration, and noise remains a key challenge in helicopter development. Smart rotor technology with active twisting has great development potential and could help address these issues [23]. The last publication about the study of a rotor blade with active twist was presented by the authors of ref. [24]. The segment control of the active twist rotor was introduced to reduce rotor power consumption and lower vibration levels. A numerical model was employed to predict rotor power characteristics and vibratory loads. A multi-objective genetic algorithm was used to solve the optimisation problem. The aerodynamic load distributions were analysed and compared.

Despite the complexity of researching and implementing this technology in a real helicopter blade, research in this area continues. The use of piezoelectric actuators for geometry control is being explored for aircraft wing control [25].

Currently, there are relatively few publications devoted to the study of three-dimensional (3D) models of active twist rotor blades under quasi-static conditions. In work [26], the authors present the design of a new integral blade with controlled twist, incorporating single-crystal piezoelectric–fibre composites. The cross-section is based on the NACA0012 airfoil and includes the nose section, active spar, web, and fairing. To maximise the twist-actuation effect, the layers of single-crystal MFCs are oriented at ±45° relative to the blade’s longitudinal axis.

The authors of paper [27] present preliminary results on the design and optimisation of a model blade with active torsion, achieved using macro-fibre composite actuators. A numerical study was carried out for the well-known BO105 rotor blade model, whose geometric parameters and material characteristics are described in detail in the literature and are widely used as reference data. The paper presents a study of three scientific groups. The POLIMI (Politecnico di Milano) research group demonstrated the possibility of numerically modelling and determining the characteristics of the composite beam sections subject to induced deformations. The approach implemented at DLR (German Aerospace Center) is based on three-dimensional finite element modelling (3D FEM). The authors present the optimisation of the blade cross-section. RTU presented another 3D model used for a parametric study of the influence of cross-section parameters on the twist angle under the action of piezoelectric actuators.

Later, Barkanov et al. [28] presented numerical modelling of a 3D rotor blade. Finite element solutions and optimisation results were presented for four design solutions with different piezoelectric actuator locations. A disadvantage of this work is that the authors did not account for the rotor blade’s stiffness parameters in the optimisation. The active twist can be integrated into existing rotor blades without significant structural modifications. The dynamic twist improves aerodynamic load distribution, reduces vibratory loads, and enhances overall rotor performance. However, the placement of piezoelectric actuators on the blade skin can alter its rigidity. Thus, in the present research, the author decided to modernise the numerical model and calculate the twist angle of a composite blade with piezoelectric actuators, accounting for the rotor blade’s stiffness. The acquired skills and results are planned to be used to construct a full-scale BO105 helicopter blade and optimise the blade cross-section using piezoelectric actuators.

2. Materials and Methods

This chapter proposes a multi-stage approach to achieving the maximum twist angle of a composite helicopter blade using piezoelectric actuators. First, a realistic three-dimensional finite element model of the blade is created at scale, and its characteristics are validated by comparing them with those of the BO105 rotor blade. Then, a parametric study is performed to assess the influence of cross-sectional geometry on the twist angle. Based on the obtained data, an optimisation problem is formulated to determine the optimal locations of piezo actuators and the dimensions of the cross-sections, subject to constraints set by design requirements. The results of numerical solutions are used to construct approximation functions, after which a nonlinear global optimisation problem is solved.

2.1. Configuration of Model-Scale Rotor Blade

The object of this research is a model-scale composite rotor blade of a BO105 helicopter (Figure 1). The composite helicopter blade is a composite structure that includes the main load-bearing structure of the blade—the spar; composite skin with multilayered layup in different directions to withstand torsional and bending loads; a foam core ROHACELL 51 FX by EVONIK (Essen, Germany) that provides rigidity with minimal weight; and the nose which contains a lead balance weight to ensure proper mass balance and dynamic stability of the rotor system. The helicopter’s nose is protected with abrasion-resistant material to prevent erosion. The blade is equipped with NACA 23012 airfoils and is not pre-twisted.

Figure 2 presents the cross-sectional dimensions selected for the numerical modelling of a model-scale rotor blade with controlled geometry. The length of the investigated blade, excluding the root, is 1560 mm. The chordwise length, including the tab, is 121 mm. The spar cross-section is simplified to facilitate the subsequent optimisation analysis, which investigates how variations in blade cross-sectional geometry influence the resulting twist.

In the scaled rotor blade, the spar consists of equal-strength layers of unidirectional (UD) fibreglass-reinforced material (GFRP) with a [0°] reinforcement pattern. The skin consists of four layers of UD GFRP material with the reinforcement scheme of [+45/−45]. The layer thickness is 0.125 mm.

To maximise twist actuation, the piezoceramic fibres in the MFC actuators are aligned with the outer GFRP skin layers on the top and bottom surfaces of the rotor blade. In the MFC actuators, the piezoceramic fibres are embedded in an epoxy matrix and placed between polyamide films containing integrated electrodes (Figure 3). The interdigitated electrodes apply the electric field needed to activate the piezoelectric effect along the fibre direction, producing a strong longitudinal response. As a result, shear deformation within the laminated skin is amplified, generating a smoothly distributed twist torque along the rotor blade. The thickness of the MFC actuators is 0.3 mm. The distance between electrodes is 0.5 mm. The piezo actuators are applied as a rectangular active layer of 1560 mm length [29].

2.2. Finite Element Model

A 3D finite element (FE) model of the investigated composite rotor blade in model-scale was developed using ANSYS MECHANICAL 16.0 software (Figure 4). The multilayered composite skin incorporating the MFC actuators was modelled with 8-node SHELL281 elements, while the balance weight, UD spar, and foam core were represented using 20-node SOLID186 elements. The erosion protection was not modelled in the finite element model. One end of the FE model was fixed using a clamped boundary condition. Several simplifications were introduced to the model to facilitate numerical analysis. In the rear part of the rotor blade, the foam material was removed. This allows for a reduction in the finite element model’s dimensions while maintaining a proportional mesh for the blade skin, ensuring no loss of accuracy in the results obtained. Before beginning the analysis, convergence tests were performed on the finite element results using different mesh densities. The material properties of the blade materials used for numerical calculation are presented in

Table 1. Material properties of the rotor blade used in numerical modelling [30].

Title	UD GFRP	Foam	Lead	MFC
Elastic modulus, E1 [GPa]	45.166	0.035	13.790	30.000
Elastic moduli, E2 = E3 [GPa]	11.981	0.035	13.790	15.500
Shear moduli, G12 = G13 [GPa]	4.583	0.014	2.000	5.700
Shear modulus, G23 [GPa]	1.289	0.014	2.000	10.700
Poisson’s ratios, υ12 = υ13	0.238	0.25	0.44	0.31
Poisson’s ratio, υ23	0.325	0.25	0.44	0.16
Density, ρ [kg/m3]	2008	52	11,300	4700
Piezoelectric constant, d33 [m/V]	-			4.18×10−10
Piezoelectric constants, d31 = d32 [m/V]	-			−1.98×10−10

.

2.3. Parametric Study

Controlling the twist angle of the blade reduces the impact of turbulent flow from the other blade and reduces noise and vibration in the helicopter’s main rotor. Therefore, to reduce blade vibrations, the twist angle function f(a) for a given electrical voltage applied to the piezo actuators was chosen as the objective function in the design optimisation problem.

When solving an optimisation problem, it is necessary to conduct a preliminary parametric study of the blade’s cross-section geometry and location of MFC actuators. Parametric analysis allows us to determine the geometric parameters influencing the twist angle and those which can be eliminated. This reduces the dimensionality of the optimisation problem and narrows the search space.

For the parametric study of the cross-section of the helicopter blade, the following parameters were selected (Figure 5): spar circular fitting (l_CF), spar flange length (l_spar), spar flange thickness (t_spar), thickness of skin (t_skin), and MFC actuators’ chordwise length (l_MFC).

Thus, the main condition of the parametric study problem can be written as follows:

Objective function:	f(a) ⟹ max
Design variables:	16.0 ≤ lCF ≤ 22.0 mm
	24.0 ≤ lspar ≤ 46.0 mm
	0.5 ≤ tspar ≤ 2.0 mm
	0.25 ≤ tskin ≤ 1.0 mm
	0 ≤ lMFC ≤ 80 mm

Additionally, one is required to study the influence of design constraints on structural limitations. The location of the centre of gravity (y_cg), elastic axis location (y_ea), mass per unit span length (m), first torsional frequency (f_T1), flap bending stiffness (EI_F), lag bending stiffness (EI_L), and torsion stiffness (GJ) of the blade are considered as the design constraints for the optimisation of a composite blade with controllable geometry. The behaviour of the design constraints enables the evaluation of the approximation equations. The location of the centre of gravity (y_cg) and elastic axis location (y_ea) are described as a percentage of the chord length (c).

2.4. Static Behaviour of the Rotor Blade

To model the inverse piezoelectric effect in numerical analysis, where mechanical deformation of the blade occurs due to the operation of piezo actuators placed on its surface shell, a temperature analogy is used. This analogy is based on the equivalence between piezoelectrically induced strains and thermal expansion, where the piezoelectric coefficients of the actuator are interpreted as linear thermal expansion coefficients and the applied electrical voltage is treated as an equivalent temperature change, ΔV = ΔT. This approximation allows the control voltage to be directly introduced into the finite element model as a temperature effect.

The converse effect of a piezoelectric material in the constitutive equations has the following stress–charge form [31]:

$$\left\{T\right\}=\left[c^E\right]\left\{S\right\}-{\left[e\right]}^T\left\{E\right\}$$

(1)

where {T}, {S}, and {E} are the stress, strain, and electric field vectors, and [c^E] and [e] are the elastic stiffness coefficient and the piezoelectric stress coefficient matrix. The superscript T denotes a transposed matrix.

In the strain–charge form, this equation can be written as follows:

$$\left\{S\right\}=\left[s^E\right]\left\{T\right\}-{\left[d\right]}^T\left\{E\right\}$$

(2)

where [s^E] and [d] are the elastic stiffness and piezoelectric strain coefficient matrix.

The relations between [e] and [d] can be written as follows [31]:

$${\left[e\right]}^T=\left[c^E\right]{\left[d\right]}^T$$

(3)

Using Equation (3), Equation (1) can be rewritten as follows:

$$\left\{T\right\}=\left[c^E\right]\left\{S\right\}-\left[c^E\right]{\left[d\right]}^T\mathsf{\Delta }T$$

(4)

Considering the thermal effect, the generalised Hooke’s Law equation can be written as follows:

$$\left\{T\right\}=\left[c^E\right]\left\{S\right\}-\left[c^E\right]\left\{\alpha \right\}\mathsf{\Delta }T$$

(5)

where $\left\{\alpha \right\}$ is the thermal expansion coefficient vector and $\mathsf{\Delta }T$ is the change in temperature relative to the reference temperature.

If we accept that S_voltage = S_thermal, the relation between piezoelectric and thermal strains is written as follows:

$${\left[d\right]}^T\left\{E\right\}=\left\{\alpha \right\}\mathsf{\Delta }T$$

(6)

Further, the electric field E in the piezoelectric actuator is defined as follows:

$$E={\displaystyle \frac{\mathsf{\Delta }V}{{\mathsf{\Delta }}_{es}}}$$

(7)

where $\mathsf{\Delta }V$ is the voltage difference between electrodes and ${\mathsf{\Delta }}_{es}$ is the distance between electrodes.

Finally, the piezoelectric coefficients of the actuator can be formulated as follows:

$${\alpha }_{ij}={\displaystyle \frac{d_{ij}}{{\mathsf{\Delta }}_{es}}}$$

(8)

where subscripts 1, 2, and 3 correspond to the directions along the X-, Y-, and Z-axis, respectively. The verification of the thermal analogy for MFC actuators is presented in ref. [32,33]. The operating voltage for the actuator ranges from −500 to +1500 volts [34]. A preliminary study of the effect of voltage on the twist angle showed a linear relationship. Therefore, a voltage of 1000 volts is used in the numerical model. Figure 6 shows an example of helicopter rotor blade twist using thermal analogy.

Numerical modelling determines the displacements of the leading and trailing edges of the blade. The maximum axial displacements $\mathsf{\Delta }$z are used to calculate the twist angles of the blade with controlled geometry, which is as follows:

$$\alpha ={\displaystyle \frac{\mathsf{\Delta }z}{c}}·{\displaystyle \frac{180}{\pi }},$$

(9)

where $\mathsf{\Delta }z$ is the sum of displacements of the nose and tab parts of the blade and c is the chord length.

The structural analysis parameters are the mass per unit span length m and the centre of gravity location y_cg, which are determined during finite element analysis. Modal analysis is used to determine the first natural frequency of torsion.

Determining the location of the elastic axis y_ea involves several static calculations with the force applied at the beginning (see Figure 7a) and end of the structure (see Figure 7b). Using the resulting displacements from each applied force, the location of the elastic axis y_ea is determined [35].

Based on the similarity of the triangles (see Figure 7c), the location of the elastic axis is determined as follows:

$$y_{ea}=\left(l_0-l_1\right)=l_0+{\displaystyle \frac{l}{1+\frac{{\mathsf{\Delta }}_2}{{\mathsf{\Delta }}_1}}}.$$

(10)

Lag and flap bending stiffnesses are determined as follows [18]:

$$EI={\displaystyle \frac{F{L}^3}{3\mathsf{\Delta }}},$$

(11)

where F is the unit load, L is the length of the composite blade, and $\mathsf{\Delta }$ is the displacement of the composite blade under unit load (Figure 8a,b).

Torsion stiffness is calculated as follows [18]:

$$GJ=2{\displaystyle \frac{FlL}{\alpha }},$$

(12)

where F is the unit force, l is the distance between the location of the elastic axis and the applied force F. L is the length of the composite blade, and $\alpha$ is the twist angle (see Figure 8c).

2.5. Optimisation

Based on the formulation and numerical algorithm for solving the optimisation problem, a methodology for the optimal design of a helicopter main rotor blade with controlled geometry made of composite materials is developed [36].

Based on the results of the parametric study, the influence of the geometric dimensions of the blade cross-section on the twist angle is determined. Additionally, the location of the centre of gravity, elastic axis location, mass per unit span length, first torsional frequency, and stiffness of the blade with controllable geometry are evaluated. After selecting the most significant parameters that affect the twist angle by more than 5%, an experiment plan is constructed (Figure 9).

After determining the maximum and minimum geometric dimensions of the cross-section of the blade that has influence on the twist angle, a Latin hypercube (LH) experimental design was developed. The construction methodology was described by the authors of refs. [37,38].

Developing the experimental plan begins by specifying the number of factors, n, and the total number of experiments, k. The experimental points are distributed as uniformly as possible within the factor space to ensure balanced domain coverage. Accordingly, the following criterion is applied:

$$\mathsf{\Phi }={\sum }_{i=1}^k{\sum }_{j=i+1,\dots ,n}^k{\displaystyle \frac{1}{l_{ij}^2}}\Rightarrow \mathrm{m}\mathrm{i}\mathrm{n},$$

(13)

where $l_{ij}$ is the distance between the points having numbers $i$ and $j$ ($i\ne j$). Physically, this corresponds to a configuration that minimises the potential energy of repulsive forces between unit mass points, assuming that these forces decrease inversely with the distance between the points.

The plan of the experiment is characterised by the matrix of plan $B_{ij}$. The domain of the experiments is determined as $x_j\left[x_j^{min};x_j^{max}\right]$ and the points of the experiments are calculated by the following expression:

$$x_j^{\left(i\right)}=x_j^{min}+{\displaystyle \frac{1}{k-1}}\left(x_j^{max}-x_j^{min}\right)\left(B_{ij}-1\right).$$

(14)

where $i=\mathrm{1,2},\dots ,k$ and $j=\mathrm{1,2},\dots ,n$.

Following the experimental plan, a numerical calculation is performed at each point and the data obtained is restored. After performing the calculations for the entire set of points, approximating dependencies are formed for the twist angle and each constraint condition. The approximation is performed using the following second-degree polynomial:

$$y=b_0+{\sum }_{i=1}^m{b}_i{x}_i+{\sum }_{i=1}^m{b}_{ii}{x}_i^2+{\sum }_{i=1}^m{\sum }_{j=1}^m{b}_{ij}{x}_i{x}_j,$$

(15)

where $y$ is the response, $x_i$ and $x_j$ are the variables, $b_0$ is the constant coefficient, $b_i$, $b_{ii},$ and $b_{ij}$ are the linear, quadratic, and interactive coefficients, respectively, and $m$ is the number of the factors.

The adequacy of the approximation models is quantified using the coefficient of determination $R^2$ and the adjusted coefficient of determination $R_{adj}^2$. These coefficients characterise the goodness of fit of the model. The coefficient of determination $R^2$ is calculated as follows:

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{\sum _{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}},$$

(16)

where $\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2$ is the residual sum of squares, $\sum _{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2$ is the total sum of squares, $y_i$ is the actual observed values, ${\widehat{y}}_i$ is the predicted value from the model, and ${\overline{y}}_i$ is the mean of the observed values.

In situations where the number of fitting parameters p is smaller than the total number of experimental runs n, the adjusted coefficient of determination provides an additional measure for evaluating the approximation quality, which is as follows:

$$R_{adj}^2=1-{\displaystyle \frac{n-1}{n-p-1}}\times {\displaystyle \frac{\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{\sum _{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}}.$$

(17)

The coefficients of determination used to validate a regression model should reflect the model’s significant reliability and demonstrate values approaching 1.

The nonlinear optimisation problem was solved via a random search method based on response surfaces. Experimental design, data approximation, and optimisation were performed using the EdaOpt programme (RTU, Riga, Latvia) [39].

3. Results

3.1. Model Validation

To validate the numerical model, several characteristics of a BO105 helicopter rotor GFRP blade were compared by means of the model-scale and FEM models. Creating an FE model that accurately reproduces the parameters of the reference design is one of the most complex and labour-intensive processes.

Table 2. Comparative characteristics of the BO105 model-scale rotor blade and FE blade.

Characteristics	Symbol	Units	Reference Blade [40]	FEM Blade	Δ, %
Centre of gravity location	ycg	[%c]	25.1	25.1	0.0
Elastic axis location	yea	[%c]	20.6	20.0	3.1
Mass per unit span length	m	[kg/m]	0.95	0.95	0.0
First torsional frequency	fT1	[Hz]	67.9	72.03	5.7
Flap bending stiffness	EIF	[Nm2]	250	243	3.0
Lag bending stiffness	EIL	[Nm2]	5200	6905	33.0
Torsion stiffness	GJ	[Nm2]	160	194	21.4

shows that most numerical model characteristics agree well with the reference blade. The difference does not exceed 6%. Unfortunately, not all characteristics of the studied blade were close to the reference blade. The lag bending stiffness of the numerical model was 33% higher than that of the reference blade.

3.2. Parametric Study and Analysis

To select the significant parameters of the optimisation problem, preliminary numerical studies were conducted to examine the influence of the composite blade’s cross-sectional geometric dimensions and the location of the piezo actuators on the skin surface. When studying the influence of geometric parameters on the objective function value, the remaining parameters were held constant. The initial geometric parameters before the calculation were set as follows: l_MFC = 80 mm, l_CF = 22.0 mm, l_spar = 46.0 mm, t_spar = 1.0 mm, and t_skin = 0.5 mm. Additionally, the voltage applied was 1000 V.

The influence of the location of MFC actuators on the twist angle, stiffness characteristics, as well as the locations of the elastic axis and the centre of gravity are illustrated in Figure 10. The maximum twist angle of the numerical composite rotor blade with MFC actuators is achieved by ~3.7° (Figure 10a). The mass per unit length span of the structure linearly increases with increase in MFC actuators installed sequentially along the chordwise length of the blade (Figure 10b). The increase amounted to ~24%.

Changing the mass of the composite helicopter blade depending on the MFC actuators installed changes the locations of the centre of gravity (Figure 10c) and the elastic axis (Figure 10d). Figure 10c illustrates that the centre of gravity shifts initially to the blade nose but then to the blade tail. The percentage change between the minimum (24.6%) and maximum (28.2%) values of the centre of gravity locations is ~15%. It is seen that the location of the centre of gravity, when the MFC actuators’ length along the chord is 40 mm, coincides with that of the reference blade without MFC actuators (see Table 2). Figure 10d shows that the installation of MFC actuators does not show significant changes in the elastic axis location. At the beginning, the elastic axis location decreases by ~5% and then increases by ~7% compared to the parameters of the reference rotor blade.

The first torsional frequency showed the smallest percentage influence (~4%) of the MFC actuators’ location on the rotor blade’s skin surface (Figure 10e). The effect of the MFC actuators’ chordwise length on the flap bending stiffness of the blade is illustrated in Figure 10f. Placing MFC actuators on the surface of the composite blade increases its flap bending stiffness by ~23%. The lag bending stiffness of the composite blade begins to increase when the MFC actuator chordwise length exceeds 30 mm and reaches a difference of up to ~23% (Figure 10g). The torsional stiffness shows a ~17% increase compared to the reference blade without MFC actuators (Figure 10h).

Then, the dependence of cross-section parameters on the twist angle was studied in detail when l_MFC = 80 mm (Figure 11). It is assumed that the cross-sectional dimensions can be reduced or increased when designing the optimal blade cross-section to achieve the maximum twist angle.

Figure 11a shows that decreasing the spar flange length from 46 mm to 22 mm increases the twist angle by ~14%. Decreasing the spar thickness by 0.5 mm increases the twist angle by ~6%. Increasing the thickness of the spar flange reduces the twist angle by ~6% (Figure 11b). A decrease in the spar circular fitting gives an increase in twist angle of only ~3.0% (Figure 11c). Skin thickness significantly changes the twist angle of the helicopter blade. Decreasing the skin thickness by 0.25 mm increases the twist angle by ~30%. Increasing the thickness of the spar flange to 2 mm reduces the twist angle by ~27% (Figure 11d). The influences of the geometric parameters of the cross-section on design constraints are presented in Appendix A.1.

Analysis of the parametric study results showed that the spar circular fitting l_CF has the least effect on the twist angle (3%). The thickness of the spar flange (t_spar) and the spar flange length (l_spar) have a moderate effect (12% and 14%, respectively). The thickness of the skin t_skin has the greatest effect on the twist angle (48%). Based on the results of parametric studies, it was proposed to exclude the spar circular fitting l_CF from the optimisation design variables (Figure 12).

3.3. Optimisation

As a result of the parametric study of the composite rotor blades, the following design variables and constraints were adopted:

Objective function:	Design variables:		Constraints:
f(a) ⟹ max	24.0 ≤ lspar ≤ 46.0	[mm]	22 ≤ ycg ≤ 30	[%]
	0.5 ≤ tspar ≤ 2.0	[mm]	10 ≤ yea ≤ 25	[%]
	0.25 ≤ tskin ≤ 1.0	[mm]	m ≤ 1.35	[kg/m]
	14 ≤ lMFC ≤ 90	[mm]	fT1 ≥ 67.9	[Hz]
			100 ≤ EIFlap ≤ 500	[Nm2]
			2600 ≤ EILag ≤ 15,400	[Nm2]
			64 ≤ GJ ≤ 320	[Nm2]

Thus, the optimisation problem for the composite design of a helicopter main rotor blade with controlled geometry was formulated, including the selection of the objective function and the determination of the blade design optimisation parameters. The constraints used to solve the design optimisation problem for a blade with controlled geometry are formulated based on the design requirements of the blade under investigation [27].

Four design variables were generated using a Latin hypercube sampling method in the EdaOpt programme. The design consisted of 30 runs, and an example of the experimental setup is illustrated in Figure 13.

In the next stage, static and modal finite element analyses were performed for all 30 points in the experimental design. To obtain the required data at each design point, eight separate calculations were performed. The data obtained from numerical calculations were approximated in EdaOpt v2.96 software.

The data of

Table 3. Approximation estimates.

	a [°]	ycg [%c]	yea [%c]	m [kg/m]	fT1 [Hz]	EIFlap [Nm2]	EILag [Nm2]	GJ [Nm2]
$R^2$.	99.53%	99.92%	98.68%	100.00%	94.09%	99.73%	99.97%	99.94%
$R_{adj}^2$.	99.09%	99.85%	97.45%	100.00%	88.58%	99.48%	99.94%	99.88%

demonstrate that the second-order polynomial equation of research values of the twist angle and constraints has high accuracy, with a high value of the coefficient of determination ($R^2$ > 98.68%) and the adjusted coefficient of determination ($R_{adj}^2$ > 97.45), respectively. Low values of coefficients of determination are observed only for the first torsional frequency ($R^2$ > 94.09% and $R_{adj}^2$ > 88.58).

Additionally, the approximation equations were validated using finite element solutions at points not included in the experimental design. Examples of the finite element verification for the helicopter rotor blade approximation equations are shown in Figure 14. Graphical representation demonstrates very good agreement between the approximations and the finite element results.

The optimal result obtained for the scaled rotor blade using the response surface method (RSM) is presented in

Table 4. Results of optimisation.

Title	Design Parameters				Constraints							Twist Angle
	lspar [mm]	tspar [mm]	tskin [mm]	lMFC [mm]	ycg [%c]	yea [%c]	m [kg/m]	fT1 [Hz]	EIFlap [Nm2]	EILag [Nm2]	GJ [Nm2]	a [°]
RSM	40.0	1.8	0.25	88.0	26.3	18.6	1.14	67.9	297.0	6798	175.0	5.2
FEM	40.0	1.8	0.25	88.0	26.3	18.2	1.14	67.1	304.0	6860	174.0	5.0
Δ, %	-	-	-	-	0	2.2	0	1.2	2.3	0.9	0.6	3.7

. Using linearity, the maximum twist angle of the blade can be interpolated within the range from −500 V to 1500 V (total 2000 V), which corresponds to the operational voltage of MFC actuators. Thus, the obtained twist angle values can be multiplied twice. In this case, the twist angle is 10.4°.

In the work of [41], the authors conducted a parametric study and optimisation of the cross-section of a blade with piezoelectric actuators for a similar numerical model. The thickness of the spar flanges was excluded from the experimental plan. As a result, the twist angle was only 3.69° at a voltage of 1000 V. This shows that the correct choice of geometric parameters has a decisive influence on the maximal twist angle.

Optimum results were verified using finite element solutions. The geometric dimensions of the blade cross-section obtained from the optimum solution are inserted into the finite element model of the helicopter blade. The obtained twist angle and constraint values are compared with the optimal values from the optimisation process. In most cases, the percentage change did not exceed 2%, indicating a strong correlation between the approximating functions.

Table 5. Compare the reference blade and the optimal blade.

Notation	Symbol	Units	Optimum Blade	Reference FEM Blade	Δ, %
Centre of gravity location	ycg	[%c]	26.3	25.1	4.6
Elastic axis location	yea	[%c]	18.6	20.0	7.5
Mass per unit span length	m	[kg/m]	1.14	0.95	16.7
First torsional frequency	fT1	[Hz]	67.9	72.03	6.1
Flap bending stiffness	EIF	[Nm2]	297	243	18.2
Lag bending stiffness	EIL	[Nm2]	6798	6905	1.6
Torsion stiffness	GJ	[Nm2]	175	194	10.9

presents the parameters compared between the reference numerical model and the optimal rotor blade model. It was observed that the locations of the centre of gravity and elastic axis did not change significantly (4.6% and 7.5%, respectively). Mass per unit span length increases by 16.7%. The first torsional frequency was closer to the lower bounds of constraints and decreased by 6.1% compared with the reference numerical model of the rotor blade. Lag bending stiffness has hardly changed (1.6%). Flap bending stiffness provided a significant increase of 18.2%, but torsion stiffness decreased by 10.9%.

4. Conclusions

The present investigation was conducted to develop an optimisation methodology for the design of the cross-section of helicopter rotor blades with piezoelectric actuators. A 3D finite element model of a helicopter rotor blade was created based on the geometric specifications and material properties of a reference model-scale BO 105 blade. The main results of this work can be summarised as follows.

• The 3D numerical model was developed by means of the finite element software ANSYS and compared with the reference rotor model-scale blade. Most characteristics of the experimental rotor blade were in good agreement with those predicted by the numerical model. The differences from the experimental ones were no more than 6.5%, except for lag bending and torsion stiffness.
• The influences of the blade’s cross-sectional dimensions were investigated without changing the aerodynamic profile. The spar flange thickness and spar flange length showed a negligible effect on the twist angle. Skin thickness had the greatest influence, accounting for ~57% of the twist angle. To activate the piezoelectric effect, the thermal analogy was used.
• An optimal design of the model-scale rotor blade was carried out to demonstrate the capabilities of the developed optimisation methodology. Verification of the approximation equations was confirmed using the coefficient of determination, and validation was performed using finite element solutions at points outside the experimental design. The optimum results were also verified by finite element solutions.
• It was established that, with the optimised geometric parameters of the cross-section, the maximum active twist reaches 5.2°, while the positions of the elastic axis and the centre of gravity exhibit only minor deviations from those of the reference model. The placement of the piezoelectric actuators significantly influences both the flapwise bending and torsional stiffness of the blade.
• Parametric studies using response surfaces can be used by designers to investigate the influence of different design parameters on the behaviour functions. The use of approximation equations enables the engineer to find an optimal compromise between the required blade twist angle, the dimensions of its cross-section, and the specified structural constraints.