Blade element theory (BET) is adopted first to establish the flap equation for the rotor. The blade's cross-section at a radial position y, with azimuth angle ϕ and pitch angle θ, and the radial length of the section dr, are selected for analysis as shown in Fig. 1.

Figure 1

Velocity and aerodynamic environment of blade element.

The velocities of the blade element are shown in Fig. 1(a), where U_R is the radial velocity, and U_T and U_P are tangential and perpendicular to the disk plane, respectively. The velocities are then expressed as follows:

$$\begin{array}{l}\{\begin{array}{l}{U}_T=\Omega R\left(\mu \sin \varphi +r\cos \beta \right)\hfill \\ U_P=\Omega R\left(\mu \sin \beta \cos \varphi +\lambda \cos \beta +r\frac{d\beta }{d\varphi }+\frac{V_C}{\Omega R}\right)\hfill \\ U_R=\Omega R\mu \cos \varphi \hfill \end{array}\hfill \end{array}$$(1)

where Ω is the rotation rate of the blades, R is the blade radius, μ is the advance ratio, and V_C is the climb velocity. λ is the inflow ratio, and it is assumed that the inflow is a uniform (or linear) inflow. The pitch angle is θ = θ₀ + θ_1ccos ϕ + θ_1s sin ϕ. According to the relationship of velocities shown in Fig. 1(a), the inflow angle is $\phi =\arctan \frac{U_P}{U_T}$, and the angle of attack is α = θ - φ.

The majority of the flap model, based on the assumption of small flap angle and inflow angle, is a linearization model that uses the approximate relationships sin β ≈ β and cos β ≈ 1, with arctan φ ≈ φ.

The lift and drag forces are produced by the flow at the blade sections, L and D, which are normal and parallel to U. $L=\frac{1}{2}\rho U^{2}c{C}_l\left(\alpha ,M\right)dr,D=\frac{1}{2}\rho U^{2}c{C}_d\left(\alpha ,M\right)dr,U^2=U_T^2+U_P^2,\rho$ is air density, and c is the blade chord at the position analyzed. The lift coefficient C_l and drag coefficient C_d are intricate functions of the angle of attack and Mach number. The components of the total aerodynamic force are normal to the disk plane F_z = L cos φ - D sin φ. The normal force F_z is typically simplified, as small inflow angle is assumed, in the forms F_z ≈ L cos φ and F_z ≈ L. This method is effective in the case of a small angle, but causes considerable errors in the case of a large angle. To account for this, a nonlinear/general form is adopted in this paper.

This paper uses a rigid blade to build the blade flap equation, focusing primarily on the moment of inertia and the aerodynamic torque related to the center of the disk plane produced by the flapping movement. The forces acting on blade elements are primarily inertial force F_I, centrifugal force F_C, and aerodynamic force F_A, the direction of which is opposite the flapping motion and radially outward, normal to the blade, as shown in Fig. 2.

Figure 2

Force at the flapping blade.

For a blade with an offset flap hinge, the dimensionless flap-hinge offset is e and the radial position is R. The inertial force is $F_I=m{\Omega }^2\left(r-eR\right)\frac{d^2\beta }{d{\varphi }^2}dr$ and the moment arm related to the hinge is r - eR. The centrifugal force is F_C = mΩ²r cos β dr and the moment arm related to the hinge is (r - eR) sin β. The aerodynamic force is F_A = F_z and the moment arm related to the hinge is r - eR. Thus, the moment equilibrium regarding the flap hinge of the full blade can be described by integrating the blade element at different positions along the radius:

$$\begin{array}{l}{\Omega }^2{\displaystyle \underset{eR}{\overset{R}{\int }}\left(m{\left(r-eR\right)}^2\frac{d^2\beta }{d{\varphi }^2}+m\left(r-eR\right)r\cos \beta \sin \beta \right)}dr\\ ={\displaystyle \underset{eR}{\overset{R}{\int }}{F}_A\left(r-eR\right)}dr\end{array}$$(2)

The differential form of the flap equation at radial position r₀ is:

$$\begin{array}{l}\frac{d^2\beta }{d{\varphi }^2}+P\sin \beta \cos \beta =\frac{\left(r_0-eR\right)F_A}{{\Omega }^2{I}_{\Delta \beta }}\Delta R\hfill \end{array}$$(3)

where $P=1+\frac{3e}{2\left(1-e\right)}$ reflects uniform mass distribution, $I_{\Delta \beta }={\displaystyle \underset{r_0-0.5\Delta R}{\overset{r_0+0.5\Delta R}{\int }}m{\left(r-eR\right)}^{2}dr,}\Delta R=\frac{\left(1-e\right)R}{n}$, and the value of n is the determined by precision.

Under increased angle of attack and Mach number, flow separation and stall occur on the airfoil surface, impacting the distribution of the lift. In other words, the relationship between lift coefficient, angle of attack, and Mach number becomes irregular, rendering it impossible to obtain the coefficients of a simple polynomial.

Calculating the load of a helicopter blade under dynamic stall conditions is highly challenging. Vortex theory and Navier-Stokes numerical calculation, which both show reasonable accuracy, are commonly applied to solve this problem. However, the complexity of these calculation processes affects their real-time performance [14] . A simplified stall model that is quick (without sacrificing accuracy) is necessary to remedy this.

The lift coefficient under steady state is easy to obtain through experimentation, so a semi-empirical model is often adopted based on data obtained from experiments. Many semi-empirical dynamic stall models have been applied to aircraft research -- the two most commonly utilized are the Beddoes-Leishman model and the ONERA model. The Beddoes-Leishman model mainly focuses on simplifying calculations, deriving a series of ordinary differential equations to calculate the lift coefficient. The ONERA semi-empirical model uses both a first and second-order differential equation to describe the unstable behavior of a rotor. The inviscid contribution is described by the first-order differential equation, and the second-order differential equation calculates stall delay plus lift increments. These two models often contain some of the necessary parameters, but determination of these parameters is typically based on dynamic or static measurement data for specific airfoils, which makes the models all together rather inconvenient. The Snel stall model is adopted in this paper to remedy this [15].

The Snel stall model, first proposed by H. Snel [16, 17], is typically applied to the elastic stall effect of wind turbines. The model, which is based on lift coefficient under steady state, consists of two parts: one describing the forcing frequency response, and one describing the higher frequency of a self-excited nature. The lift coefficient is expressed as follows:

$$\begin{array}{l}{C}_l=C_l\left(steady\right)+\Delta C_{l,1}+\Delta C_{l,2}\hfill \end{array}$$(4)

where ∆C_l,1 and ∆C_l,2 are obtained by first and second-order differential equations, respectively [18] . The differential equations are:

$$\begin{array}{l}\{\begin{array}{l}\Omega \tau \frac{d\Delta C_{l,1}}{d\varphi }+C_{10}\Delta C_{l,1}=F_1\left(\varphi \right)\hfill \\ {\left(\Omega \tau \right)}^2\frac{d^2\Delta C_{l,2}}{d{\varphi }^2}+C_{21}\Omega \frac{d\Delta C_{l,2}}{d\varphi }+C_{20}\Delta C_{l,2}=F_2\left(\varphi \right)\hfill \end{array}\hfill \end{array}$$(5)

where time is a constant parameter $\tau =\frac{c}{2V}$, c denotes the chord of the airfoil, and V is the relative velocity, $V=\Omega r\left(1+\frac{R}{r}\mu \sin \varphi \right)$.

According to the Snel model, C₁₀ can be denoted:

$$\begin{array}{l}{C}_{10}=\{\begin{array}{ll}\frac{1+0.5\Delta }{8\left(1+60\Omega \tau \frac{d\alpha }{d\varphi }\right)},\hfill & 2\pi \sin \left(\alpha -{\alpha }_Z\right)\frac{d\alpha }{d\varphi }\le 0\hfill \\ \frac{1+0.5\Delta }{8\left(1+80\Omega \tau \frac{d\alpha }{d\varphi }\right)},\hfill & 2\pi \sin \left(\alpha -{\alpha }_Z\right)\frac{d\alpha }{d\varphi }>0\hfill \end{array}\hfill \end{array}$$(6)

where ∆ = 2π sin(α - α_Z) - C_l(steady), α_Z is the zero lift angle of attack, C_l(steady) is the lift coefficient in the steady state, and F₁(ϕ) is the frequency of the forcing term of the first-order differential equation, obtained as follows:

$$\begin{array}{l}{F}_1\left(\varphi \right)=\Omega \tau \frac{d\Delta }{d\varphi }\hfill \end{array}$$(7)

According to the description by V. K. Truong and H. Snel, C₂₁ can be written as follows:

$$\begin{array}{l}{C}_{21}=\{\begin{array}{ll}12\tau \left[-0.01\left(\Delta -0.5\right)+2{\left(C_{l,2}\right)}^2\right],\hfill & \frac{d\alpha }{d\varphi }>0\hfill \\ 0.4\tau ,\hfill & \frac{d\alpha }{d\varphi }\le 0\hfill \end{array}\hfill \end{array}$$(8)

C₂₀ is expressed:

$$\begin{array}{l}{C}_{20}=0.04\left[1+3{\left(C_{l,2}\right)}^2\right]\left[1+3{\Omega }^2{\left(\frac{d\alpha }{d\varphi }\right)}^2\right]\hfill \end{array}$$(9)

The forcing term of the first-order differential equation concerns the difference between lift coefficient and potential flow and steady state:

$$\begin{array}{l}{F}_2\left(\varphi \right)=-0.018\left(\Delta \right)+0.006\Omega \frac{d\Delta }{d\varphi }\hfill \end{array}$$(10)

According to the model, it is possible to determine the lift coefficient even in a stall condition. The magnitude of drag is small compared to lift, according to J. G. Leishman [19] . As for the definition of the drag coefficient, an approximate expression is effective:

$$\begin{array}{l}{C}_D=\{\begin{array}{ll}{a}_0+a_1{C}_l+a_2{C}_l^2,\hfill & \left|\alpha \right|<{\alpha }_S\hfill \\ 1135-105\cos 2\left(\alpha -{\alpha }_z\right),\hfill & \left|\alpha \right|\ge {\alpha }_S\hfill \end{array}\hfill \end{array}$$(11)

where α_S is the threshold value for the stall, and the values of a₀, a₁, and a₂ can be obtained dependent on the specific airfoil.