Parametric structure optimization of a main rotor blade as an application for FSI analysis in main rotor optimization process

Kocjan, Jakub; Rogólski, Robert

doi:10.1108/AEAT-03-2024-0084

Purpose

Modern warfare and modern battlefield are very demanding. The recent conflicts showed that the usage of the helicopters is very limited and only the best constructions are able to provide support for the operations. The purpose of this research is to show the possibilities of new design tool for main rotor aerostructural optimization. It is a next chapter of the research that is aimed at finding new solutions for rotorcraft constructions.

Design/methodology/approach

This work presents a method of preliminary structure optimization of the main rotor blade using parametric modeling. It is the next step in the main rotor optimization studies. It is the next step after preparing the parametric model for the external shape CFD analysis. As a basis for parametric blade structure calculations, the analytical model is provided in this paper. The equations of rigid blade loads and, as a consequence of the strength elements, stresses are shown. The parametric blade modeling is conducted using the Graphic Integrated Programming language. The parametric design method is shown to be used for various blade planform models and different section airfoils. The structure of a blade is generated automatically after the user enters the parameters. The code-inbuilt analysis systems provide a quick inertia examination of the generated geometry, which is the basis for further optimization. The program calculates the blade loads and verifies them with the given material conditions and proposed safety factors. In the analysis, composite materials for the strength elements were proposed.

Findings

The results of this research showed the application of parametrization into the main rotor blade design loop. It was presented that the main rotor blade structure can be enhanced using fluid-structure interaction (FSI) methods. The time saving with the implementation the process into design loop is shown.

Practical implications

This work can be practically used in the main rotor blade design process. It provides the possibilities to check various blade aerodynamic configuration in a structure strength aspect.

Originality/value

To the best of the authors’ knowledge, there were no published research that combines the main rotor FSI analysis. The method, which is presented in the work, provides a new approach to a rotorcraft design. The application of the parametrization and combining it with the FSI method gives a novel solution for helicopters construction enhancement.

Introduction

The recent military conflicts have prompted a reassessment of the possible operations that helicopters can conduct on the modern battlefield. The need for constructions capable of providing enhanced characteristics for widespread military operations has been acknowledged by manufacturers and national authorities. Consequently, many countries are upgrading current configurations and exploring new solutions for vertical take-off and landing vehicles. Therefore, at the Military University of Technology, studies focused on finding new approaches and design possibilities for rotorcraft are underway. The research began with an evaluation of helicopter construction features and parameters responsible for rotorcraft performance. The analysis considered nearly 70 of the newest helicopter designs from around the world. The results of this comparison were published in Kachel et al. (2021).

The main goal of the entire research is to develop a multicriteria optimization procedure to provide a solution using the newest computer-aided design methods. There are several main approaches to resolving optimization problems nowadays. The results of research focused on main rotor aerodynamic optimization are published in Okumuş et al. (2022), Stalewski (2017), Stalewski and Zalewski (2019), Tamer et al. (2011) and Xie et al. (2017), whereas an example addressing strength problems is shown in Spyropoulos et al. (2021). Some research institutes concentrate on main rotor optimization problems, such as the ONERA French Aerospace Lab and the DLR German Aerospace Centre. Their results are presented in Goerke et al. (2012), Pahlke and Demaret (2017), Tremolet and Basset (2012) and Wilke (2021, 2024). The proposed solution involves using parametrization in CAD geometry design. The benefits of using this approach were presented in Sagimbayev et al. (2021) and MA et al. (2021). Parametrization is also currently used for airfoil design (Allen et al., 2021; Lim, 2018; Vu and Lee, 2015), structural design (Tixadou, 2021; Гребеников et al., 2021) and aerodynamic design (Bailly et al., 2019). Therefore, the second step, following comparative analysis, was the preparation of the main rotor blade shape as a parametric model using the Graphic Integrated Programming (GRIP) language. The evaluation of the model’s application in CFD analysis was published in Kocjan et al. (2022). In addition, in further research, the CFD fluid domains were parametrized and simultaneously generated with a full n-bladed rotor for CFD analysis under different flight conditions. The aim of preparing the CFD environment and parametrizing the rotor model and its enclosures is to obtain a solution for quick preparation of (fluid-structure interaction) analysis.

The second part of preparing the FSI analysis involves setting up a finite element structural loads simulation. Therefore, to integrate it with the optimization procedure, a parametric structural model also needs to be prepared. This paper presents the preparation of a model for further analysis with preliminary strength analytical calculations. The preparation of the CAD parametric model is conducted using the GRIP language, which is implemented in Siemens NX software. Examples of GRIP usage are shown in Grabowik et al. (2015), Ryazanov (2016) and Shabliy and Dmitrieva (2014). This language provides the capability to generate external body or inner structural parts of the airframe that are currently being designed. Within the code, mathematical calculations, logical conditions and geometry creation are possible. The main advantages of the presented solution, which will be demonstrated in this research, include mass and inertia analysis built within the code. The functions offer the possibility to obtain numerical features of any geometry shapes, which can be used to prepare an optimization procedure. The program code presented in this paper, which generates geometry preliminarily prepared and calculated for given flight conditions, is prepared by the authors.

Another approach to main rotor blade optimization is the aeroelastic features optimization. Researches that are conducted nowadays are focused on enhancing features that concerns also the vibration of the blade. One of the most interesting studies are the influence of the BERP-like blade tip on working conditions and aeroelastic response (Johnson et al., 2016; Moffatt and Griffiths, 2009). In this kind of studies also the computer methods are used. Some examples of the CFD analysis including vibrations analysis are shown in Allen et al. (2009) and Cornette et al. (2015). However, the exploration that are aimed at single or multicriteria optimization, that takes into account the main rotor blades aeroelastic attributes, are valuable and shows the procedures to obtain the best main rotor attributes. This is also the aim of research described in this paper. Very good examples of the optimization mentioned above are (Ganguli and Chopra, 1996; Guo and Xiang, 2004; Kizhakke Kodakkattu et al., 2018; Liu and Zhang, 2021; Morris et al., 2008; Straub et al., 1992; Tarzanin et al., 1999; Wang et al., 2013). As it can be spotted there is many research that provides the solution for optimization and vibration reduction; however, there is not many published papers that use the FSI method for that purpose. One example is presented in Sinem and Metin Orhan (2018).

The optimization of the blade structure is described in Barwey and Peters (1994), Chattopadhyay et al. (1995), He and Peters (1992), Tian et al. (2022), and Vu et al. (2016) in which various aspects of the structure are taken into consideration. The most significant for the presented research are the strength considerations in aspects of its aerodynamic loads; however, the proposed procedures have not been developed using modern computational techniques. The example of modern optimization research where the advanced analysis is presented was published in Salh Eddine et al. (2023).

Taking into consideration the conducted analysis of research that were performed in helicopter main rotor design enhanced with optimization techniques, there is lack of the studies that combine the parametrization of the rotor blade for both numerical and analytical analysis, where the analytical model takes into consideration the inertia and mass features of the structure.

The novelty of this study lies in combining analytical research with computer analysis of the blade structure to obtain exact mass and inertia parameters and optimize the shape of the spar across various spar cross-sections.

The analytical analysis for generating the structure is conducted using MATLAB software. Originally developed codes, with implemented mathematical formulas, calculate main rotor loads and bending moments. The generated loads are presented graphically as functions of radius and azimuth. The natural frequencies are also shown in charts. The process of generating the load and frequency shapes is automated and implemented into the code. The program recalculates the inertia parameters at selected cross sections of the blade, providing a comprehensive tool for analyzing structural features and working conditions. Consequently, the designer can quickly evaluate the case and identify possible weak points and faults.

Calculated loads are transferred into a CAD parametric optimization program (prepared using GRIP), where the shape is formed and assessed to meet the given material properties using an optimization procedure. The research also introduces features of modern composite structures.

To the best of the authors’ knowledge, the approach presented in this paper has not been combined in any known research, making it unique in main rotor blade structure design studies.

Analytical model

Main rotor aerodynamic load

To estimate the main rotor aerodynamic loads for structural analysis, it is necessary to calculate the main rotor trim angles. The method of trimming was described in the previous phase of the study (Kocjan et al., 2022) Trimming can be performed for all types of flight conditions. Finding pitch angles and flap angles is the first step in determining the main rotor blade load:

θ (ψ) = θ_{0} - θ_{1} cos (ψ) - θ_{2} sin (ψ)

(1)

β (ψ) = β_{0} - β_{1} cos (ψ) - β_{2} sin (ψ)

(2)

where θ are pith angles, β are flap angles and the ψ is rotor blade azimuth. The rotor motion is shown in Figure 1.

Figure 1

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Rotor velocity and blade motion

The rotor thrust which can be assumed as a rotor lift is given with the equation:

d T = \frac{1}{2} a ρ (θ U_{T}^{2} + U_{p} / U_{U T}) d r

(3)

where a is the lift curve slope and ρ is the air density. However, to calculate the thrust an velocity components at the blade are needed to be estimated:

U_{p} = ΩR (λ' - \frac{x \dot{β}}{Ω} - μ β \cos ψ)

(4)

U_{T} = ΩR (x + μ \sin ψ)

(5)

where Ω is main rotor rotational velocity, R is the rotor radius, and µ is the advance ratio. The inflow ratio is given with:

(λ' = (V \sin α_{D} - ν_{i}) / Ω R

(6)

where V is the rotorcraft velocity α_D is rotor disc angle of attack and ν_i is the rotor induced velocity.

Rotor blade normal modes and frequencies

Estimating the bending shapes of the rotor blade requires calculating the natural frequencies and normal modes. There is no single technique or procedure to determine the values of vibration parameters. For this research, the Rayleigh–Ritz method was chosen, and the algorithm used is based on (Wilde and Price, 1965) In this method, the energy equation of the rotating beam is minimized:

ϵ = \int_{0}^{R} {E I {(\frac{d^{2} y}{d r^{2}})}^{2} + \frac{1}{2} \bar{m} Ω^{2} (R^{2} - r^{2}) {(\frac{d y}{d r})}^{2} - m ω^{2} y^{2}} d r

(7)

where E is the Young Modulus, I is the second moment of area, y is the natural mode shape and ω is the natural frequency.

The applied method of solution provides possibilities for using the method in numerical software. The equation is transformed into matrix form, with each part being transformed into Fourier series. This allows for the application of nonconstant mass distribution into the procedure and enables the utilization of the orthonormality properties of Fourier series. To further simplify the procedure, the spanwise coordinate is given as x = πr/R. Then the equation’s components are:

\frac{E I (x) π^{2}}{R} = \sum_{1}^{\infty} p_{n} \cos n x

(8)

\frac{\bar{m} Ω^{2} R^{2} (π^{2} - x^{2})}{2 π^{2}} = \sum_{1}^{\infty} q_{n} \cos n x

(9)

y = a_{0} x - \sum_{1}^{\infty} a_{n} \sin n x

(10)

Substituting the parts of energy expression we get:

\begin{matrix} E = \int_{0}^{π} {(\sum_{1}^{\infty} p_{n} \cos n x) {(\sum_{1}^{\infty} n^{2} a_{n} \sin n x)}^{2} + (\sum_{1}^{\infty} q_{n} \cos n x) \\ {(a_{0} - \sum_{1}^{\infty} n a_{n} \sin n x)}^{2} - r_{0} ω^{2} {(a_{0} x - \sum_{1}^{\infty} a_{n} \sin n x)}^{2}} d x \end{matrix}

(11)

These equation can be written in a matrix form and A and B are symmetric matrices with p and q coefficients:

(A^{- 1} B - \frac{I}{Г^{2}}) \underline{a} = 0

(12)

where Γ² = 2ω²r₀ and a is the matrix of coefficients of normal modes shape. Calculating the latent roots of the matrix A⁻¹B the natural frequencies and shapes are obtained.

Rotor flapping blade bending moment

After preparing the earlier results, the bending moment can be estimated for different spanwise coordinates and azimuthal locations. The first step is to establish the external forces on the blade. These external forces consist of:

the elementary lift dT=dF_a;
the blade weight m′gdx;
centrifugal force of rotation m′ω²rβdx;
the force inertia for flapping $- m x \ddot{β}$ ⁠;
the force of inertia of deflection $- m x \ddot{y}$ ⁠;
the centrifugal force of flapping $m' x {\ddot{β}}^{2}$ ⁠;
the Coriolis force due to the simultaneous action of blade deflection and flapping motion $- 2 \dot{β} \dot{y} m'$ ⁠; and
corrective term of the damping of the lift force due to the flexural elastic deformation of the blade $- K \dot{y}$ ⁠.

In this calculation, the Euler–Bernoulli beam theory was chosen. In this theory, the main assumptions are that the cross section of the beam is rigid (i.e. no deformations occur in the plane of the cross section), and shear deformations are neglected. The cross section remains plane, unstretched and normal to the deformed axis of the beam after deformation.

Authors resigned from using the Timoshenko beam theory, which is mostly used in the case of short or thick beams where shear effects can be significant. This theory includes the effects of shear deformation. As a result, the difference in Timoshenko theory is that after deformation, the cross section of the beam is no longer normal to the beam axis. As mentioned above, this effect can be neglected.

The differential equation of the blade is then:

\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} y}{d x^{2}}) - ω^{2} \frac{d}{d x} (\frac{d y}{d x} \int_{x}^{R} m' (a + ξ) d ξ) + m^{' ̈ y} + K \dot{y} = \frac{d F}{d x} - m' g - m x ̈ β - m ’ ω^{2} r β

(13)

The differential equation must be fulfilled by four conditions, which specify the integration constants:

y = 0 for x = 0

(14)

E I \frac{d^{2} y}{d x^{2}} = 0 for x = 0 and for x = R - a

(15)

\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} y}{d x^{2}}) = 0 for x = R - a

(16)

To solve the deflection equation the second member, which is independent of the deflection can be put into function $d F_{d} = dFa - m ’ g - m ’ ω 2 x - m x \ddot{β}$ ⁠. We also put: $\int_{x}^{R} m' (a + ξ) d ξ = S$ so the final form is:

\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} y}{d x^{2}}) - ω^{2} \frac{d}{d x} (s \frac{d y}{d x}) + m' \ddot{y} + K \dot{\dot{y} = F'_{d} (x, t)}

(17)

The applied calculation formula estimates the distribution of bending moment to the third harmonic of natural shape. This method was chosen due to its compatibility with the MATLAB environment.

Initially, we use the natural deflection functions of the rotor blade to solve it. For simplicity, we consider the vibration state to be undamped. The bending moment is assumed to result from the forces acting on the rigid blade. In addition, the time function can be expanded into a series of periodic functions, represented as ψ = ωt, therefore:

\begin{array}{l} M_{r i g} = M_{0} + M_{a} \cos ψ + M_{b} \sin ψ + M_{2 a} \cos 2 ψ + M_{2 b} \sin 2 ψ + M_{3 a} \cos 3 ψ + M_{3 b} \sin 3 ψ \dots \end{array}

(18)

This moment concerns the distribution of external force $F'_{d}$ which constitute the second term of the equation. These forces should also be resolved at each point as a series of deflection distributions correlated with the natural mode shape. The natural functions can always be resolved into series. The proposed series take form:

F'_{d} = g_{0} m' η_{0} + g_{1} m' η_{1} + g_{2} m' η_{2} + \dots

(19)

where g_i is the function a function of time only and η_i is the bending natural function of the order i.

The i order deflection of the main rotor blade affects at each point the natural mode function, it is given by function:

y_{i} = h_{i} η_{i}

(20)

where h_i is the function of time. The deflection is obtained by superposing the many natural deflections of the blade vibrating at the corresponding natural frequencies, using the h_i function which defines the amplitude and phase. Therefore, for the i-order frequency:

h_{i} \frac{d^{2}}{d x^{2}} (E I \frac{d^{2} η_{i}}{d x^{2}}) - h_{i} ω^{2} \frac{d}{d x} (s \frac{d η_{i}}{d x}) + h_{i} m^{' \ddot{y}} = g_{i} m' η_{i}

(21)

Next, the natural vibrations of a freely vibrating blade with an angular velocity ω allow an infinite quantity of solutions, using η sinvt a function a limiting conditions and the below function is satisfied:

\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} η_{}}{d x^{2}}) - ω^{2} \frac{d}{d x} (s \frac{d η_{}}{d x}) = ν^{2} m' η_{i}

(22)

As a consequence, for the natural function we obtain a rotating beam function for every natural frequency:

\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} η_{i}}{d x^{2}}) - ω^{2} \frac{d}{d x} (s \frac{d η_{i}}{d x}) = ν_{i, ω}^{2} m' η_{i}

(23)

where ν_i,ω is the corresponding function of natural frequency at the ω speed of rotation.

When the equations are simplified, the result is:

h_{i} ν_{i, ω}^{2} + \ddot{h_{i}} = g_{i}

(24)

The functions given above h_ig_i are periodic with respect to ψ, so there can be transformed into series:

\begin{array}{l} h_{i} = h_{i, 0} + h_{i, a} \cos ψ + h_{i, b} \sin ψ + h_{i, 2 a} \cos 2 ψ + h_{i, 2 b} \sin 2 ψ \\ + h_{i, 3 a} \cos 3 ψ + h_{i, 3 b} \sin 3 ψ \dots \end{array}

(25)

\begin{array}{l} g_{i} = g_{i, 0} + g_{i, a} \cos ψ + g_{i, b} \sin ψ + g_{i, 2 a} \cos 2 ψ + g_{i, 2 b} \sin 2 ψ \\ + g_{i, 3 a} \cos 3 ψ + g_{i, 3 b} \sin 3 ψ \dots \end{array}

(26)

Therefore, the differential equation can be written:

\sum_{i = 0}^{i = \infty} [h_{i} \frac{d^{2}}{d x^{2}} (E I \frac{d^{2} η_{i}}{d x^{2}}) - h_{i} ω^{2} \frac{d}{d x} (s \frac{d η_{i}}{d x}) + h_{i} m' \ddot{y}] = \sum_{i = 0}^{i = \infty} g_{i} m' η_{i}

(27)

Limiting the equation to third harmonic and resolving the identification of the coefficients, it can be reduced into:

h_{i, 0} ν_{i, ω}^{2} = g_{i, 0}

(28)

h_{i, a} (ν_{i, ω}^{2} - ω^{2}) = g_{i, a}

(29)

h_{i, b} (ν_{i, ω}^{2} - ω^{2}) = g_{i, b}

(30)

h_{i, 2 a} (ν_{i, ω}^{2} - 4 ω^{2}) = g_{i, 2 a}

(31)

h_{i, 2 b} (ν_{i, ω}^{2} - 4 ω^{2}) = g_{i 2 b}

(32)

h_{i, 3 a} (ν_{i, ω}^{2} - 9 ω^{2}) = g_{i, 3 a}

(33)

h_{i, 3 b} (ν_{i, ω}^{2} - 9 ω^{2}) = g_{i 3 b}

(34)

To determine the g_i it is needed to multiply the members of the equation (19) by η_i and conduct integration over the blade.

Taking into consideration the orthogonality condition:

\int_{0}^{R} m' η_{i} η_{j} d x = 0 i \neq j

(35)

it provides:

\int_{0}^{R} F'_{d} η_{i} d x = g_{i} \int_{0}^{R} m' η_{i}^{2} d x

(36)

g_{i} = \frac{\int_{0}^{R} F'_{d} η_{i} d x}{\int_{0}^{R} m' η_{i}^{2} d x}

(37)

Next, the bending moment on the deformable main rotor blade is computed. The single harmonic acting the outside forces, produces the elastic deformation:

\sum_{i = 1}^{i = \infty} [g_{i, n a} m' η_{i} \cos n ψ + g_{i, n b} m' η_{i} \sin n ψ]

(38)

which is:

\sum_{i = 1}^{i = \infty} y_{i, n} {\begin{matrix} a \\ b \end{matrix} = \sum_{i = 1}^{i = \infty} h_{i, n} {\begin{matrix} a \\ b \end{matrix} η_{i}

(39)

When the terms h_i are replaced by values which are obtained from (28–34), the corresponding bending moment:

M_{e l a s t n} {_{b}^{a} = \frac{g_{i, n} {_{b}^{a}}{ν_{i, ω}^{2} - n^{2} ω^{2}} - E I \frac{d^{2} η_{i}}{d x^{2}} {\begin{matrix} \cos n ψ \\ \sin n ψ \end{matrix}

(40)

Because the natural functions exhibit minimal variation with changes in angular velocity, the ith order natural function at ω = 0, is comparable to the same natural function for the standard angular velocity. This simplification incurs an error of only 3%, making it suitable for application in this solution.

As a consequence, the natural function η_i fulfills the equation (23) and:

\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} η_{i}}{d x^{2}}) = ν_{i, 0}^{2} m' η_{i}

(41)

In result:

E I \frac{d^{2} η_{i}}{d x^{2}} = ν_{i, 0}^{2} \int^{} \int_{x}^{R} m' η_{i} d x d ξ

(42)

After replacing in (40) the value of $E I \frac{d^{2} η_{i}}{d x^{2}}$ with (42), the elastic blade bending moment equation is obtained, which is can be integrated along the blade:

M_{e l a s t n} {\begin{matrix} a \\ b \end{matrix} = \sum_{i = 1}^{i = \infty} (\frac{ν_{i, 0}^{2}}{ν_{i, ω}^{2} - n^{2} ω^{2}} g_{i, n} {\begin{matrix} a \\ b \end{matrix} \int^{} \int_{x}^{R} m' η_{i} d x d ξ)

(43)

The value of the g_i is calculated from (37) for the azimuth values for a given acting forces. The expansion of the g_i into Fourier series expresses the coefficient:

g_{i, n} {\begin{matrix} a \\ b \end{matrix}

(44)

Practical bending moment calculation procedure

Considering the equation calculation, a practical calculation method is proposed in de Guillenchmidt (1951). This method offers a step-by-step solution that can be applied to numerical calculations. The first step, as described earlier, involves calculating the natural functions and corresponding natural frequencies of both rotating and nonrotating blades. In the second step, we evaluate the values:

\int_{0}^{R} m' η_{1}^{2} d x \int_{0}^{R} m' η_{2}^{2} d x \int_{0}^{R} m' η_{3}^{2} d x

(45)

Next step is to determine the outside forces on the rigid main rotor blade for different ψ (minimum eight positions with 45° spacing), so the position are evaluated for:

\int_{0}^{R} F^{'}_{d} η_{1} d x \int_{0}^{R} F^{'}_{d} η_{2} d x \int_{0}^{R} F^{'}_{d} η_{3} d x

(46)

Taking into consideration the (37) equation the function are calculated:

g_{1} = \frac{\int_{0}^{R} F'_{d} η_{1} d x}{\int_{0}^{R} m' η_{1}^{2} d x} g_{2} = \frac{\int_{0}^{R} F'_{d} η_{2} d x}{\int_{0}^{R} m' η_{2}^{2} d x} g_{3} = \frac{\int_{0}^{R} F'_{d} η_{3} d x}{\int_{0}^{R} m' η_{3}^{2} d x}

(47)

The function (47) can be developed into Fourier series. The series are produced to third harmonic of ψ:

\begin{matrix} g_{1, 0} & g_{1, a} & g_{1, b} \\ g_{2, 0} & g_{2, a} & g_{2, b} \\ g_{3, 0} & g_{3, a} & g_{3, b} \end{matrix} \begin{matrix} \dots \\ \dots \\ \dots \end{matrix} \begin{matrix} g_{1, 3 b} \\ g_{2, 3 b} \\ g_{3, 3 b} \end{matrix}

(48)

The bending moments are calculated for each harmonic and for different station on the blade span using equation:

M_{x} = \sum_{i = 1}^{i = 3} M_{i, x}

(49)

The inner functions are estimated with:

M_{i, x} = \int^{} \int_{x}^{R} m' η_{i} d x d ξ [A_{i, 0} + A_{i, 1} \cos (ψ - ψ_{i, 1} - φ_{i, 1}) + A_{i, 2} \cos (2 ψ - ψ_{i, 2} - φ_{i, 2}) + A_{i, 3} \cos (3 ψ - ψ_{i, 3} - φ_{i, 3})]

(50)

where

A_{i, 0} = \frac{ν_{i, 0}^{2}}{ν_{i, ω}^{2}} g_{i, 0}

(51)

A_{i, n} = \frac{ν_{i, 0}^{2} \sqrt{g^{2}_{i, n a} + g^{2}_{i, n b}}}{\sqrt{{(ν_{i, ω}^{2} - n^{2} ω^{2})}^{2} + n^{2} Φ^{2} ω^{4}}}

(52)

\tan ψ_{i, n} = \frac{ω^{2} n Φ}{(ν_{i, ω}^{2} - n^{2} ω^{2}})

(53)

\tan φ_{i, n} = \frac{g_{i, n b}}{g_{i, n a}}

(54)

Optimization method

As it was mentioned in introduction this paper presents the part of research that is aimed at finding solutions for main rotor optimization problem. The proposed algorithm for the entire program is presented on Figure 2. The structure optimization is an inner procedure inside the presented loop. It is included inside the “parametric modelling” element, where the outer shape and CFD enclosures are generated, as well as the blade inner structure.

Figure 2

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Main rotor optimization research algorithm

The inner structure optimization is performed using computer techniques; however, some mathematical assumptions must be made. This procedure is a single-criterion mass reduction process while maintaining the required strength. The mathematical description is presented below:

Decision variables: h – spar height, d – honeycomb wall position, t – spar thickness
Parameters: material properties, flight conditions
Constraints: σdop > σmax(h, d, t) – strength condition; h > 0, t > 0, max airfoil thickness position > d >0
Criteria: m(h, d, t) → minimum

The process of resolving main rotor blade stresses and subsequently optimizing the shape is divided into steps, with each step described in consecutive chapters.

The first phase of the proposed method involves resolving the mathematical model presented in Chapter 2. This model was implemented in the MATLAB environment to solve equations and compute values for rotor parameters. The results serve as the basis for a customized GRIP program to assess the best structural shape.

Analytical solving

The problem was divided into two separate codes. The first one was prepared for the main rotor trim, involving calculations of blade loads. The second one focused on blade mode shapes, natural frequencies, and finally, the determination of bending moment values and distribution.

To initiate the calculations, the first code begins with declaring the geometric and aerodynamic parameters of the main rotor. Basic helicopter parameters, especially weight, are also included. In the input section, initial blade mass parameters are declared. These parameters can be based on an existing blade or estimated using statistical data from similar constructions. Such statistics were compiled in the initial phase of the research program (Kachel et al., 2021).

In addition, the user declares the flight conditions under which the main rotor blade stresses are evaluated. These conditions describe the flight with the forward speed and climb, and include the statistical change of the center of gravity as a function of velocity. The forward speed is also calculated in the form of an advance ratio.

The next step involves parameterizing the rotor chord and twist so that their values can vary along the span. The calculated coefficients will also be used in the program that generates the geometry for structural evaluation.

The subsequent stage of the procedure involves trimming the main rotor pitch and flap angles. The code has been partially prepared in previous phases of the research. Trim angles are determined by resolving acting forces horizontally and vertically with reference to the rotor-disc plane. After acquiring the trim angles, the blade aerodynamic loading can be determined. By solving the equations of velocity components, the lift distribution is obtained. Examples of aerodynamic loading distribution generated using the MATLAB environment are shown in Figure 3, whereas the total aerodynamic loading is depicted in Figure 4. These figures illustrate that the calculated loading offers a means to evaluate the rotor’s working conditions. It can be seen that the behavior of aerodynamic forces slightly changes with inflow ratio increase. Especially on advancing blade position (azimuth 135° represents the transition position), where the influence of maximum forward speed, due to the inflow angle change, is reducing the aerodynamic force. The blade geometric twist is provided to reduce the diminish the negative force value.

Figure 3

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Aerodynamic loading distribution for different inflow ratio (flight speed)

Figure 4

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Aerodynamic loading in function of radius for: μ = 0.1

The second calculation program, which receives the aerodynamic loading from the first procedure, is used to find first the natural modes and frequencies and next solve the bending moment equation. It also includes the declaration of inputs part of the code, where the geometry and mass dimensions of the rotor are given. In this phase, the stiffness declaration is provided as a function of material Young modulus and moment of inertia. The stiffness alters along the span due to the change of cross section of the blade spar. The moments of inertia are established using the GRIP program, which will be described later.

Furthermore, the program calculates the natural mode shapes and frequencies. The calculation procedure implemented in the code is described in section Rotor blade normal modes and frequencies. Matrices are prepared to compute the vibration quantities up to the 8th harmonic. Using the built-in functions of MATLAB, the results are easily obtained. Functions are transformed into Fourier series using the Fourier Series integral definition, and then matrices of coefficients are constructed. The eigenvalues of the matrices equation are determined using MATLAB’s built-in functions. The characteristics can be evaluated for every shape of a blade and materials that are used. Exemplary results for the rotor blade (W-3 helicopter rotor blade shape) are depicted in Figure 5. The natural function value is normalized and the radius is parameterized as a function of π.

Figure 5

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Natural modes and frequencies

In the next step of the code execution, the bending moment for the first three harmonics is obtained. The outside forces, established in the first program and described in section Main rotor aerodynamic load, are input for the aerodynamic loading. The damping coefficient is estimated, and then the outside force and mass distribution are correlated with the natural vibration shape. The procedure is applied as follows: the time functions g_i are calculated for each harmonic, and the resultant equations are substituted into the bending moment function, respectively, to the harmonic number. The calculations end with summing the total bending moment, which is a function of radius and azimuth. An example of the values of bending moment depending on the radius and azimuth is shown in Figure 6. The maximum bending moment values are written into a file to serve as input for geometry strength analysis.

Figure 6

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Main rotor blade bending moment function

Graphics integrated programming

The primary objective of this study is to develop a new method for preparing the main rotor blade geometry for FSI analysis and optimization. The aim is to reduce the time consumption of complex simulations and find the best strength structure by combining analytical calculations with the capabilities of open GRIP inertia analysis code, which is an integral part of Siemens NX. The program has been partially used because the initial stages of the study, as the external geometry needs to be exact for further FSI simulations, ensuring connections of corresponding faces.

The blade model is parametrized in the program using main rotor geometric features. The designer can input values into a popup input window, corresponding with the input values from MATLAB programs. The first parameter is the blade chord and rotor radius. The airfoil is shaped from a text file with the coordinates, allowing the operator to prepare the geometry for any chosen airfoil. The next phase of parametrization involves polynomial coefficients for twist and chord of the blade, allowing them to vary along the span. The outer model is filled with the structure. The inner assembly of the blade is based on existing solutions, with the main spar and filling in the leading edge part, and a honeycomb structure in the trailing edge part. An illustration of the CAD model is shown in Figure 7.

Figure 7

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Main rotor blade CAD model for calculation

With the prepared solution, the blade shape can be freely changed in accordance with the analytically calculated values. The shape building process relies on built-in GRIP commands for lines and splines, which are used to create the requested shape.

Furthermore, the program conducts the analysis and strength calculation. Using integral commands, the mass and inertia analysis of the cross sections and the entire 3D geometry are performed. As a result, the moment of inertia of cross sections is obtained. The density of the parts is programmed by the user, allowing the 3D analysis to provide the total mass of the blade and the mass of the components. The moment acting on a blade in respective sections is loaded from the file generated previously. The strength equations are programmed within the code. Stresses are calculated for each cross section using bending moment values and the moment of inertia from the GRIP analysis. Finally, it is checked whether the required stiffness is sufficient for the given loads. The program adjusts the spar dimensions to meet the stiffness requirement and iterates the calculation until the requirements are met. The iterative procedure modifies the dimensions in each step and evaluates the new shape. A safety factor of 1.2 is proposed, but it can be changed by the user. As a result, a CAD geometry is provided, which can be applied in further complex analysis. An example of the code with analysis options is shown in Figure 8.

Figure 8

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Code example

Example of results

The steps that were described in the previous chapters were combined into an optimization procedure. The entire procedure is shown in Figure 9.

Figure 9

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Initial structure optimization procedure

To demonstrate the capabilities of the procedure, the W-3 helicopter main rotor blade was recalculated using the original blade dimensions and modern materials applied to the proposed structure. The properties of the composite materials used are presented in Table 1.

Table 1

Material properties

Material property	Gr/E woven cloth	S-glass/E woven cloth	Nomex honeycomb
Young’s modulus (E11) [Mpa]	1.69E + 03	6.45E + 02	4.06
Young’s modulus (E22) [Mpa]	1.69E + 03	6.26E + 02	4.06
Young’s modulus (E33) [Mpa]	2.36E + 02	1.60E + 02	4.06
Shear modulus (G12) [Mpa]	1.32E + 02	1.00E + 02	6.82
Shear modulus (G13) [Mpa]	1.32E + 02	1.00E + 02	6.82
Shear modulus (G23) [Mpa]	1.32E + 02	1.00E + 02	6.82
Poisson ratio (v12)	6.76E − 06	1.65E − 05	4.35E − 05
Poisson ratio (v13)	4.64E − 05	3.92E − 05	4.35E − 05
Poisson ratio (v23)	4.64E − 05	3.92E − 05	4.35E − 05
Density [kg/m³]	1.60E + 03	1.80E + 03	6.41E + 01

Material property	Gr/E woven cloth	S-glass/E woven cloth	Nomex honeycomb
Young’s modulus (E11) [Mpa]	1.69E + 03	6.45E + 02	4.06
Young’s modulus (E22) [Mpa]	1.69E + 03	6.26E + 02	4.06
Young’s modulus (E33) [Mpa]	2.36E + 02	1.60E + 02	4.06
Shear modulus (G12) [Mpa]	1.32E + 02	1.00E + 02	6.82
Shear modulus (G13) [Mpa]	1.32E + 02	1.00E + 02	6.82
Shear modulus (G23) [Mpa]	1.32E + 02	1.00E + 02	6.82
Poisson ratio (v12)	6.76E − 06	1.65E − 05	4.35E − 05
Poisson ratio (v13)	4.64E − 05	3.92E − 05	4.35E − 05
Poisson ratio (v23)	4.64E − 05	3.92E − 05	4.35E − 05
Density [kg/m³]	1.60E + 03	1.80E + 03	6.41E + 01

The procedure was conducted within six iteration loops. The mass reduction and change of the center of mass are shown in Table 2.

Table 2

Iteration results

Step	Mass [kg]	Center of mass [m from rotor axis]
1	60	4.73
2	40.01	4.91
3	35	4.76
4	34.43	4.74
5	34.70	4.79
6	34.70	4.79

In each step, the change of the cross section and consequently the change of the moment of inertia is considered. The geometry is generated automatically, and the output is provided as the blade CAD model. The spar cross section varies along the span in accordance with the calculated bending moments.

Evaluation of the model

To validate the generated model, a one-way FSI simulation is conducted. One-way FSI was chosen to check the optimized structure in simulated blade working conditions. The CFD simulation is proposed as stationary, so the forces produced are transferred one-way to the structure. This approach allows for the assessment of stresses and deformations. In the next research step, a two-way FSI for transient n-blade simulation will be established, taking into consideration the evaluated structure as described in this paper. The CFD part of the simulation is configured for a single-blade fluid enclosure, as demonstrated in the previous part of the research published in Kocjan et al. (2022). The pressure distribution from the mentioned CFD simulation is applied to the FEM model. According to MATLAB calculations, the flight conditions for a blade at an azimuthal position of 360° are simulated. The inflow is parameterized using calculated functions. An example of pressure distribution is shown in Figure 10.

Figure 10

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Pressure distribution on the blade

The pressure is transferred to the model generated from the CAD geometry prepared using the algorithm described in this study. The mesh was prepared using the tetrahedral mesher and thin mesher options. The base element size was set at 500 mm, with a target surface size of 0.025 m and a minimum surface size of 0.005 m. The meshing process was fully automated for the imported geometry. The results of the simulation confirm the parameters of the construction and provide information that the model is fully applicable in the FEM analysis environment. Examples of deformation and stresses are shown in Figures 11 and 12.

Figure 11

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Blade deformation

Figure 12

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Stress distribution in the blade structure

Discussion

This research introduces a new methodology for preparing geometry for subsequent FEM and FSI analyses. The first section outlines the analytical assumptions for the proposed solution, including calculations for loads and vibrations. The second part provides details on solving analytical problems using modern computer environments, accompanied by a description of the structural optimization method. Furthermore, an example of the resultant geometry with mass effects is presented, along with an evaluation using FSI calculations.

It has been demonstrated that the generated CAD model is fully compatible with FEM software and that meshes can be generated for even extremely complex blade shapes.

The proposed solution offers a novel approach to helicopter design. First, it addresses procedures for resolving natural mode shapes and vibrations of rotating blades, followed by blade trimming under various flight conditions. Furthermore, it provides aerodynamic loading of the blade as a function of radius and azimuth. Finally, the research includes the generation and preparation of a preliminary optimized main rotor blade structural model. These solutions can be used independently or combined for any blade shape and flight conditions.

The results of this research are valuable for reducing the time required to create and evaluate final models for various blade configurations. It is crucial to note that the model has been validated and is ready for implementation in further two-way FSI simulations for full rotor geometry.

To evaluate the solution, the proposed geometry is based on the existing rotor blade shape and dimensions. The forces were calculated using real mission objective parameters (e.g. velocity, height and payload). The flight parameters were also based on data from the Flight Data Recorder. An example of a 1-h flight is shown in Figure 13. The indicated velocity presented in the figure is effective from 50 km/h, due to the stream velocity from the rotor; however, it was not cut out to include the moments when the rotorcraft is on the ground.

Figure 13

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Indicated air speed, barometrical height and g-force

The proposed method is part of wider research; however, the studies described in this paper can be used as a stand-alone procedure during rotorcraft design. The tool can be used to assess the inner structure without integrating it with the proposed rotorcraft optimization procedure. Nevertheless, in this phase, it is limited to the Sikorsky helicopter layout. Nevertheless, the challenge of covering of the various blades structure was achieved for that layout.

Compared to existing design procedures, the main advantage of the presented solution is the versatility of the tool for various blade shapes, materials and environmental conditions, which can be examined in a short time. Practically, it can be implemented in the preliminary design phase of the rotorcraft, allowing many options to be evaluated and some to be chosen for the next design steps. It can also be implemented into the rotor blade lifecycle to assess the existing construction using data acquired during the rotorcraft’s life – flight data, maintenance data, airworthiness data, etc.

Conclusion

The initial optimization of the structure is presented in this paper, showcasing a combination of analytical calculations and computer numerical analysis. The optimization of the spar shape was conducted in six steps, with the process lasting approximately 40 min. By parametrizing the spar dimensions in subsequent selected sections, shape optimization was achieved, resulting in the minimum mass of the spar, which has a significantly greater density compared to the remaining elements of the blade.

Considering the potentially time-consuming nature of computer calculations, particularly when simulating an articulated main rotor using FSI, the method outlined in the paper offers an original method to reducing the time required for model preparation for analysis. By inputting desired performance parameters and previously obtained aerodynamic features, the structure can be adapted to fit any proposed blade.

The evaluation of the structure, provided with one-way FSI analysis, demonstrates that the geometry is suitable for CAD/CAE software. Introducing a tool, such as the one presented in this paper, which automates mesh preparation, ensures repeatability of the process for future designs.

In conclusion, the research objectives have been achieved. The preparation method of the model using optimization techniques is ready to be used independently or can be integrated into a comprehensive optimization loop, as intended by the authors for further research endeavors. What is more, even though it is part of a research, the tool is ready to use and can be implemented into the design process as described and used to calculate and assess the blade structure. It can be used in industry to optimize not only the structure but also provide more efficient design and lifecycle analysis.

This work was financed by the Military University of Technology (Warsaw, PL) under the two following research projects: No. UGB 22-732/2024/WAT entitled: Aerodynamic design of aircraft in terms of optimizing selected performance parameters; No. UGB 22-734/2024/WAT entitled: Design of light aircraft lifting structures in terms of optimizing strength and aeroelastic properties of the airframe. Both grants were carried out in the Institute of Aviation Technology (Faculty of Mechatronics, Armament and Aerospace of the Military University of Technology) in the year 2024 as university statutory activities funded by the Ministry of National Defence of the Republic of Poland.

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Parametric structure optimization of a main rotor blade as an application for FSI analysis in main rotor optimization process

Introduction

Analytical model

Main rotor aerodynamic load

Rotor blade normal modes and frequencies

Rotor flapping blade bending moment

Practical bending moment calculation procedure

Optimization method

Analytical solving

Graphics integrated programming

Example of results

Evaluation of the model

Discussion

Conclusion

References

Further reading

New and popular articles

Email Alerts

Cited By

Parametric structure optimization of a main rotor blade as an application for FSI analysis in main rotor optimization process

Introduction

Analytical model

Main rotor aerodynamic load

Rotor blade normal modes and frequencies

Rotor flapping blade bending moment

Practical bending moment calculation procedure

Optimization method

Analytical solving

Graphics integrated programming

Example of results

Evaluation of the model

Discussion

Conclusion

References

Further reading

New and popular articles

Email Alerts

Suggested Reading

Related Chapters

Recommended for you

Cited By

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