Geometrically Exact Aeroelastic Stability Analysis of Composite Helicopter Rotor Blades in Hover by Updated VABS

The geometrically exact nonlinear beam theory developed by Hodges (2006) is used to establish the equations of motion for composite rotor blades. The schematic of blade deformation from undeformed state to deformed state is shown in Fig. 2. The motion and deformation of the blade are described by three reference frames: the global reference frame A, the local undeformed reference frame b, and the local deformed reference frame T. The frame A is an absolute deformation frame and rotating with the rotor, and its origin is fixed at the center of the rotor disk. The undeformed blade reference line r is composed of the quarter-chord points of each section of the undeformed blade. For the frame b, its orthogonal unit vector ${\mathbf{b}}_1$ is tangent to r; and vectors ${\mathbf{b}}_2$, ${\mathbf{b}}_3$ are along the reference cross section of the undeformed blade. The rotation of the frame b relative to the frame A is caused by the blade collective pitch, initial twist and initial curvatures. The points along r undergo deformation, and their deformed locus is defined as the deformed blade reference line R. The displacement of a point on the reference line is represented as $\mathbf{u}$. For the frame T, its orthogonal unit vector ${\mathbf{T}}_1$ is perpendicular to the reference cross section of the deformed blade; vectors ${\mathbf{T}}_2$ and ${\mathbf{T}}_3$ are along the reference cross section of the deformed blade; and ${\mathbf{T}}_1$ is not necessarily tangent to R due to the transverse shear deformation effect. The rotation of the frame T relative to the frame b is caused by the blade deformation. The deformation of a point in the deformed blade relative to its corresponding frame T is caused by the warping of the blade cross section. The selection of the aforementioned reference frames can exactly represent the displacement of the blade reference line and the rotation of the blade reference cross section, that is, geometrically exact.

Based on the geometrically exact nonlinear beam theory by Hodges (2006), the geometrically exact equations of motion in the mixed variational form for the composite rotor blades can be expressed in the frame A aswhere

In Eqs. (1)–(3), $\mathbf{\delta }{\mathbf{u}}_{\mathrm{A}}$ = virtual displacement vector; $\mathbf{\delta }{\overline{\mathbf{\phi }}}_{\mathrm{A}}$ = virtual rotation vector; ${\mathbf{F}}_{\mathrm{T}}$ and ${\mathbf{M}}_{\mathrm{T}}$ = vectors of the cross-sectional stress and moment resultant measures in the T basis, respectively; ${\mathbf{P}}_{\mathrm{T}}$ and ${\mathbf{H}}_{\mathrm{T}}$ = vectors of the cross-sectional linear and angular momentum measures in the T basis, respectively; $\mathbf{\gamma }$ = vector of the force strain measures corresponding to the extension, edgewise transverse shear and flapwise transverse shear, respectively; $\mathbf{\kappa }$ = vector of the moment strain measures corresponding to the torsion, flapwise bending and edgewise bending, respectively; ${\mathbf{V}}_{\mathrm{T}}$ = vector of the inertial velocity measures of an arbitrary point on the deformed blade reference line in the T basis; ${\mathbf{\Omega }}_{\mathrm{T}}$ = vector of the inertial angular velocity measures of the frame T in the T basis; ${\mathbf{u}}_{\mathrm{A}}$ = vector of the displacement measures of an arbitrary point on the blade reference line in the A basis; ${\mathbf{\theta }}_{\mathrm{A}}$ = vector of the Rodrigues parameters; ${\mathbf{v}}_{\mathrm{A}}$ = vector of the inertial velocity measures of an arbitrary point on the undeformed blade reference line in the A basis; ${\mathbf{\omega }}_{\mathrm{A}}$ = vector of the inertial angular velocity measures of the frame A in the A basis; ${\mathbf{e}}_1={\{1,0,0\}}^{\mathrm{T}}$; $\mathbf{\Delta }$ = identity matrix of order three; ${()}^{\prime }$ denotes the derivative with respect to $x_1$ which is the arclength along the undeformed blade reference line; $(·)$ denotes the derivative with respect to time $t$; $({\tilde{\mathbf{e}}}_1)$ represents an antisymmetric matrix whose components are ${{(\tilde{\mathbf{e}}}_1)}_{ij}=-e_{ijk}{({\mathbf{e}}_1)}_k$ where $e_{ijk}$ represents components of the permutation tensor; and $\tilde{\gamma }$, ${\tilde{\mathbf{\omega }}}_{\mathrm{A}}$, ${\tilde{\mathbf{V}}}_{\mathrm{T}}$, ${\tilde{\mathbf{\theta }}}_{\mathrm{A}}$ have the same form as ${\tilde{\mathbf{e}}}_1$. The single superscript T represents the transpose operator; ${\mathbf{C}}^{\mathrm{Ab}}$ = transformation matrix from frame b to frame A with ${\mathbf{C}}^{\mathrm{bA}}$ as its transpose matrix; $\mathbf{C}$ = finite rotation matrix; ${\mathbf{f}}_{\mathrm{A}}$ = vector of the distributed applied forces per unit length of the blade; and ${\mathbf{m}}_{\mathrm{A}}$ = vector of the distributed applied moments per unit length of the blade. The detailed derivation of Eq. (1), and the physical meanings and detailed expressions of the variables which are not explained here in Eqs. (1)–(3) can be found in Hodges (2006).

It is notable that the unknown variables in Eq. (1) include not only the blade displacement and rotation but also the blade cross-sectional stress resultant, moment resultant, linear momentum, and angular momentum. One advantage of this mixed variational form is that simple shape functions can be used for finite element spatial discretization, which simplifies the solving process of the equations of motion for the blade. Moreover, the blade root loads can be directly obtained as unknown variables by passing the need of the conventional force summation or modal summation calculation, which makes the hub loads calculation not only simpler but also more accurate compared with those conventional models. In addition, all the variational terms in Eq. (1) are represented in the global frame A. With this representation, when the blade is not straight, the finite element spatial discretization of Eq. (1) can be completed without complex coordinate transformations between adjacent elements. The matrix $\mathbf{S}$ in Eq. (2) and the matrix $\mathbf{M}$ in Eq. (3) are the $6\times 6$ cross-sectional stiffness and mass matrices of the blade, respectively. For composite blades with complex geometry and material distribution, $\mathbf{S}$ is a $6\times 6$ fully coupled matrix. In this paper, $\mathbf{S}$ is obtained from the updated VABS (Hodges 2015), which can accurately deal with the arbitrary cross-sectional geometry and material distribution of the blade, the effect of the blade initial twist and curvatures, and the nonclassical effects caused by anisotropic material properties, such as cross-sectional warping, transverse shear deformation, and elastic couplings. The matrix $\mathbf{M}$ is also obtained from the updated VABS (Hodges 2015) by numerical integration over the reference cross section of the blade.

By the Peters finite state airloads theory (Peters et al. 2007), the aerodynamic forces ${\mathbf{f}}_{\mathrm{T}}={\{0,f_{\mathrm{T}2},f_{\mathrm{T}3}\}}^{\mathrm{T}}$ and aerodynamic moments ${\mathbf{m}}_{\mathrm{T}}={\{m_{\mathrm{T}1},0,0\}}^{\mathrm{T}}$ in the T basis can be expressed in terms of the aforementioned structural variables aswhere $\rho$ = air density; $c$ = blade chord; $a$ = lift curve slope of airfoil; $c_{\mathrm{d}}$ = profile drag coefficient; $W_{\mathrm{T}2}$ and $W_{\mathrm{T}3}$ = components of the resultant air velocity along the vectors ${\mathbf{T}}_2$ and ${\mathbf{T}}_3$, respectively, which depend on the blade motion and the induced velocity at the rotor disk; and ${\mathrm{\Omega }}_{\mathrm{T}1}$ = component of ${\mathbf{\Omega }}_{\mathrm{T}}$ along the vector ${\mathbf{T}}_1$. In hover, $U_{\mathrm{T}3}$ = part of $W_{\mathrm{T}3}$ due to the blade motion. Hencewhere $\mathrm{\Omega }$ = rotor angular speed; $R$ = rotor radius; ${\mathbf{e}}_2={\{0,1,0\}}^{\mathrm{T}}$; ${\mathbf{e}}_3={\{0,0,1\}}^{\mathrm{T}}$; and $\overline{\lambda }$ = dimensionless induced velocity at the rotor disk, and is obtained from the Peters-He finite state dynamic wake model (Peters et al. 1989) by solving a set of inflow equations. For the presented model, the pressure integrals in the inflow equations are expressed in terms of the aforementioned structural variables aswhere $m$ and $n$ = harmonic number and polynomial number, respectively. The superscripts $c$ and $s$ denote the cosine part and sine part of the pressure integrals, respectively; $Q$ = number of blades.; $\overline{r}=r/R$ = dimensionless radial coordinate of blade; ${\mathrm{\Phi }}_n^m(\overline{r})$ = radial expansion function; ${\widehat{\psi }}_q$ = azimuth of the $q$th blade; and ${\overline{L}}_{\mathrm{c}}$ = sectional circulatory lift of the $q$th blade and for the presented model

When $k_3=0.0$, $\pm 0.1$, $\pm 0.2$, $\pm 0.3/\mathrm{m}$, the variations of lag mode damping versus collective pitch angle for the five hingeless composite rotors are calculated. The results for the hingeless composite rotors with FBT-0 and FBT-N/0/P blades are shown in Figs. 6 and 7, respectively. To observe these variations more clearly, the variations at collective pitch angles between $-4°$ and 4° are also shown in a separate figure. Similar results are obtained for the FBT-P, FBT-N, and FBT-N/P blades. When $k_3=0.0$, the blade is straight; when $k_3\mathrm{>}0.0$, the blade is swept forward; when $k_3\mathrm{<}0.0$, the blade is swept back. For each hingeless composite rotor, compared with the straight blade, the negative curvature increases the blade lag mode damping, and the smaller the curvature, the larger the lag mode damping; the positive curvature decreases the blade lag mode damping, and the larger the curvature, the smaller the lag mode damping. When $k_3\ne 0$, the blade is swept forward or back, which changes the inertia of the blade bending modes and torsional modes on the one hand and causes additional coupling between the blade flap bending and torsional modes on the other hand. These changes will certainly affect the blade lag mode damping. From Fig. 5, the positive coupling of flap bending and torsion increases the lag mode damping, and the negative coupling of flap bending and torsion decreases the lag mode damping. Therefore, the forward sweep of the blade provides additional negative coupling of flap bending and torsion, and the backward sweep of the blade provides additional positive coupling of flap bending and torsion.

When $k_2=0.0$, $\pm 0.1$, $\pm 0.2$, $\pm 0.3/\mathrm{m}$, the variations of lag mode damping versus collective pitch angle for the five hingeless composite rotors are calculated. When $k_2=0.0$, the blade is straight; when $k_2\mathrm{>}0.0$, the blade is dihedral; and when $k_2\mathrm{<}0.0$, the blade is anhedral. The investigations show that the effect of the initial out-of-plane curvature on the lag mode damping of the FBT-0 and FBT-P blades are similar. As shown in Fig. 8 for the hingeless composite rotors with FBT-P blades, at the positive collective pitch angles, the negative curvature increases the blade lag mode damping, and the smaller the curvature, the larger the lag mode damping; the positive curvature decreases the blade lag mode damping, and the larger the curvature, the smaller the lag mode damping. At the negative collective pitch angles, the negative curvature decreases the blade lag mode damping, and the smaller the curvature, the smaller the lag mode damping; the positive curvature increases the blade lag mode damping, and the larger the curvature, the larger the lag mode damping. In addition, the effect of the initial out-of-plane curvature on the lag mode damping of the FBT-P blade is greater than that of the FBT-0 blade. The predicted results also show that the effect of the initial out-of-plane curvature on the lag mode damping of the FBT-N, FBT-N/P, and FBT-N/0/P blades is similar. As shown in Fig. 9 for the hingeless composite rotors with FBT-N/P blades, at the positive collective pitch angles, the positive curvature increases the blade lag mode damping, and the larger the curvature, the larger the lag mode damping; the negative curvature decreases the blade lag mode damping, and the smaller the curvature, the smaller the lag mode damping. At the negative collective pitch angles, the positive curvature decreases the blade lag mode damping, and the larger the curvature, the smaller the lag mode damping; the negative curvature increases the blade lag mode damping, and the smaller the curvature, the larger the lag mode damping. When $k_2\ne 0$, the blade is anhedral or dihedral, which changes the inertia of blade bending modes and torsional modes on the one hand and causes additional coupling between blade lag bending and torsional modes on the other hand. These changes will certainly affect the blade lag mode damping. Overall, the effect of the initial out-of-plane curvature on the lag mode damping of the FBT-N, FBT-P/N, and FBT-P/0/N blades is similar, and the effect on the lag mode damping of the FBT-P and FBT-N blades is totally different. In Fig. 5, the effect of the FBT-N, FBT-P/N, and FBT-P/0/N elastic couplings on the lag mode damping is similar, and the effect of the FBT-P and FBT-N elastic couplings on the lag mode damping is totally different. Therefore, it is inferred that the effect of the initial out-of-plane curvature on the lag mode damping is related to blade elastic couplings.

The effect of combined initial in-plane and out-of-plane curvatures on the lag mode damping of the five hingeless composite rotors are investigated. The collective pitch angle = ${\theta }_0=8°$; the range of the initial out-of-plane curvature = $-0.3\le k_2\le 0.3$; and the range of the initial in-plane curvature = $-0.3\le k_3\le 0.3$. The investigations show that the effect of the combined initial in-plane and out-of-plane curvatures on the lag mode damping of the FBT-0 and FBT-P blades are similar. As shown in Fig. 10 for the hingeless composite rotors with FBT-0 blades, the highest lag mode damping is obtained when $k_2=-0.3$ and $k_3=-0.3$ (i.e., the blade is simultaneously anhedral and swept back), and the lowest lag mode damping is obtained when $k_2=0.3$ and $k_3=0.3$ (i.e., the blade is simultaneously dihedral and swept forward). The predicted results also show that the effect of the combined initial in-plane and out-of-plane curvatures on the lag mode damping of the FBT-N, FBT-N/P, and FBT-N/0/P blades are similar. As shown in Fig. 11 for the hingeless composite rotors with FBT-N blades, the highest lag mode damping is obtained when $k_2=0.3$ and $k_3=-0.3$ (i.e., the blade is simultaneously dihedral and swept back), and the lowest lag mode damping is obtained when $k_2=-0.3$ and $k_3=0.3$ (i.e., the blade is simultaneously anhedral and swept forward). Therefore, a combination of different initial in-plane and out-of-plane curvatures can change the lag mode damping of the investigated hingeless composite rotors, and with appropriate values of these curvatures, the lag mode damping of the blade can be increased.