A Multi-Blade Model for Heliogyro Solar Sail Structural Dynamic Analysis

Abstract

An analytical model is derived to study the structural dynamic stability behavior for the free-flying heliogyro solar sail consisting of a finite set of symmetric or evenly distributed thin blades on a flat surface. Key properties of the flutter instability are described in terms of the relationship among the eigenvalues and eigenvectors associated with a flutter instability frequency. Various simulation cases for an idealized single blade, fixed rotational speed heliogyro model, and a generalized multi-bladed, freely spinning heliogyro model are presented to show how the flutter instability frequencies change as a function of the solar radiation pressure and the size of the dynamic model. Convergence of the flutter instability frequency versus the system order is demonstrated.

Appendices

Appendix A: Order of Variables

The equations of motion for the heliogyro system with multiple blades are expected to be highly nonlinear. It is necessary to neglect higher-order terms for stability analysis. The following is a list of non-dimensional variables of order [3] for use in deriving blade strain energy and system kinetic energy to generate the dynamic equations using Hamilton’s principle. This list is applicable for any number of blades with equal length and so no subscript such as k for any designated blade is shown.

$$ \begin{array}{lll} {\frac{\zeta }{\ell } = {\mathrm{O}}\left( \varepsilon \right);}&{\frac{\eta }{\ell } = {\mathrm{O}}\left( \varepsilon \right);}&{\xi = \frac{x}{\ell } = {\mathrm{O}}\left( 1 \right)}\\ {\bar{v} = \frac{v}{\ell } = {\mathrm{O}}\left( \varepsilon \right);}&{v^{\prime} = \frac{{\partial \left( v\left/ \ell \right. \right)}}{{\partial \left( x\left/\right. \ell \right)}} = \bar{v}^{\prime} = {\mathrm{O}}\left( \varepsilon \right);}&{v^{\prime\prime} = \frac{1}{\ell }\frac{{{\partial^{2}}\left( v\left/ \right. \ell\right)}}{{\partial \left( x^{2}\left/\right. \ell^{2}\right)}} = \frac{1}{\ell }\bar{v}^{\prime\prime} = {\mathrm{O}}\left( {{\varepsilon^{2}}} \right)}\\ {\bar{w} = \frac{w}{\ell } = {\mathrm{O}}\left( \varepsilon \right);}&{w^{\prime} = \frac{{\partial \left( w \left/\right.\ell\right)}}{{\partial \left( x\left/ \right. \ell\right)}} = \bar{w}^{\prime} = {\mathrm{O}}\left( \varepsilon \right);}&{w^{\prime\prime} = \frac{1}{\ell }\frac{{{\partial^{2}}\left( w \left/ \right. L\right)}}{{\partial \left( x^{2}\left/ \right. L^{2}\right)}} \!= \frac{1}{\ell }\bar{w}^{\prime\prime} =\! {\mathrm{O}}\left( {{\varepsilon^{2}}} \right)}\\ {\phi = {\mathrm{O}}\left( \varepsilon \right);}&{\phi^{\prime} = \frac{1}{\ell }\frac{{\partial \phi }}{{\partial \left( x\left/ \right. \ell\right)}} = \frac{1}{\ell }\bar{\phi}^{\prime} = {\mathrm{O}}\left( {{\varepsilon^{2}}} \right);}&{} \end{array} $$

Appendix B: Definitions of Sectional Integrals

Assume that all blades have identical material properties and configuration. The sectional integrals are shown in the following [3].

$$ \begin{array}{@{}rcl@{}} &&\int {{\int}_{A} {\rho d\eta d\zeta } } = m = \rho A\ ({\text{for \ constant }}\ \rho\ {\text{ and }}\ A)\\ &&\int {{\int}_{A} {\rho \eta d\eta d\zeta } } = 0;\int {{\int}_{A} {\rho \zeta d\eta d\zeta } } = 0\\ &&\int {{\int}_{A} {\rho {\eta^{2}}d\eta d\zeta } } = mk_{m2}^{2};\int {{\int}_{A} {\rho {\zeta^{2}}d\eta d\zeta } } = mk_{m1}^{2} \\ &&\int {{\int}_{A} {\rho \left[ {\eta - \zeta } \right]\left[ {\eta + \zeta } \right]d\eta d\zeta } } = m\left( {k_{m2}^{2} - k_{m1}^{2}} \right) = m{\Delta} {k_{m}^{2}}\\ &&\int {{\int}_{A} {\rho \left( {{\eta^{2}} + {\zeta^{2}}} \right)d\eta d\zeta } } = m\left( {k_{m2}^{2} + k_{m1}^{2}} \right) = m{k_{m}^{2}}\\ &&{I_{v}} = \int {{\int}_{A} {{\eta^{2}}d\eta d\zeta } } ;{I_{w}} = = \int {{\int}_{A} {{\zeta^{2}}d\eta d\zeta } } ;A{k_{a}^{2}} = \int {{\int}_{A} {\left( {{\eta^{2}} + {\zeta^{2}}} \right)d\eta d\zeta } } \\ &&A = \int {{\int}_{A} {d\eta d\zeta } } ;\int {{\int}_{A} {\zeta d\eta d\zeta = 0} } ;\int {{\int}_{A} {\eta \zeta d\eta d\zeta } } = 0;J \approx 4{I_{w}} \end{array} $$

Appendix C: Definition of Non-Dimensional Parameters

It is a common practice to use non-dimensional parameters to develop equations of motion [3]. Dimensionless analysis is often used to generalize the problem, because solution of dimensional form is the solution of a particular problem. Non-dimensional equations will reduce the number of variables and provide insight into the controlling parameters. Non-dimensional parameters used in the paper are given as follows.

$$ \begin{array}{@{}rcl@{}} &&\xi = \frac{x}{\ell };{{\dot {\bar{\vartheta}} }_{x}} = \frac{{{{\dot \vartheta }_{x}}}}{{{{\Omega}_{0}}}};{{\ddot {\bar{\vartheta}} }_{x}} = \frac{{{{\ddot \vartheta }_{x}}}}{{{{\Omega}_{0}^{2}}}};{{\dot {\bar{\vartheta}} }_{y}} = \frac{{{{\dot \vartheta }_{y}}}}{{{{\Omega}_{0}}}};{{\dot {\bar{\vartheta}} }_{y}} = \frac{{{{\ddot \vartheta }_{y}}}}{{{{\Omega}_{0}^{2}}}};{{\dot {\bar{\vartheta}} }_{z}} = \frac{{{{\dot \vartheta }_{z}}}}{{{{\Omega}_{0}}}};{{\ddot {\bar{\vartheta}} }_{z}} = \frac{{{{\ddot \vartheta }_{z}}}}{{{{\Omega}_{0}^{2}}}} \\ &&\bar{T} = \frac{T}{{m{{\Omega}_{0}^{2}}{\ell^{2}}}};\bar{G}\bar{J} = \frac{{GJ}}{{m{{\Omega}_{0}^{2}}{\ell^{4}}}};\bar{E}{{\bar{I}}_{w}} = \frac{{E{I_{w}}}}{{m{{\Omega}_{0}^{2}}{\ell^{4}}}}; \bar{E}{{\bar{I}}_{v}} = \frac{{E{I_{2}}}}{{m{{\Omega}_{0}^{2}}{\ell^{4}}}};{\Delta} \bar{E}\bar{I} = \bar{E}{{\bar{I}}_{v}} - \bar{E}{{\bar{I}}_{w}} \\ &&\bar{u} = \frac{u}{\ell },\bar{v} = \frac{v}{\ell },\bar{w} = \frac{w}{\ell }: \tau = {{\Omega}_{0}}t \Rightarrow \frac{{\partial \left( {} \right)}}{{\partial t}} = \frac{{{{\Omega}_{0}}\partial \left( {} \right)}}{{\partial \tau}};\frac{{\partial \phi }}{{\partial \xi }} = \frac{{\partial \phi }}{{\partial \left( x \left/ \right. \ell\right)}} = \frac{{\ell \partial \phi }}{{\partial x}} \\ &&\bar{v}^{\prime} = \frac{{\partial \bar{v}}}{{\partial \xi }} = \frac{{\partial \left( v\left/ \right.\ell\right)}}{{\partial \left( x\left/ \right. \ell\right)}} = v^{\prime};\bar{v}^{\prime\prime} = \frac{{{\partial^{2}}\left( v\left/ \right. \ell\right)}}{{\partial \left( x^{2} \left/ \right. \ell^{2} \right)}} = \ell v^{\prime\prime}; \ddot{ \bar{v}}^{\prime} = \frac{{{d^{2}}}}{{{{\Omega}_{0}^{2}}d{t^{2}}}}\frac{{\partial \left( v \left/ \right. \ell\right)}}{{\partial \left( x \left/ \right. \ell \right)}} = \frac{1}{{{{\Omega}_{0}^{2}}}}\ddot v^{\prime} \\ &&{{\bar{T}}^{\prime}} = \frac{{\partial {{\bar{T}}}}}{{\partial \xi }} = \frac{{\partial {{\bar{T}}}}}{{\partial \left( x \left/ \right. \ell\right)}} = \frac{1}{{m{{\Omega}_{0}^{2}}\ell }}\frac{{\partial {T}}}{{\partial x}}; \frac{{d\bar{v}}}{{d\tau}} = \frac{{d\left( v\left/ \right. \ell\right)}}{{d{{\Omega}_{0}}t}} = \frac{1}{{\ell {{\Omega}_{0}}}}\frac{{dv}}{{dt}} \Rightarrow \frac{{{{\Omega}_{0}}\dot v}}{{{\ell^{2}}{{\Omega}_{0}^{2}}}} = \frac{{\ell {{\Omega}_{0}^{2}}\dot {\bar{v}}}}{{{\ell^{2}}{{\Omega}_{0}^{2}}}} \Rightarrow \frac{{\dot {\bar{v}}}}{\ell } \\ &&{{\bar{k}}_{m}} = \frac{{{k_{m}}}}{\ell },{{\bar{k}}_{{m_{1}}}} = \frac{{{k_{{m_{1}}}}}}{\ell },{{\bar{k}}_{{m_{2}}}} = \frac{{{k_{{m_{2}}}}}}{\ell } \end{array} $$

Appendix D: Equations of Motion from Kinetic and Strain Energies

The kinetic energy for the k th blade can be calculated by

$$ {{\mathcal{T}}_{k}} = \frac{1}{2}{\int}_{0}^{{\ell_{k}}} {\int {{\int}_{A} {{\rho_{k}}{{\dot{\textbf{r}}}_{k}} \cdot {{\dot{\textbf{r}}}_{k}} d{\zeta_{k}}d{\eta_{k}}d{x_{k}}} } } $$

(D1)

where the position r is defined in Eq. 2. The ratio H_kx/ℓ << 1 is very small, i.e., the distance from the center of hub to the root of the k blade is negligible compared to the blade length. Taking variation of the kinetic energy and integrating from t₀ to t_f yield

$$ {\int}_{{t_{0}}}^{{t_{f}}} {\delta {T_{k}}dt} = {\int}_{{t_{0}}}^{{t_{f}}} {{\int}_{0}^{{\ell_{k}}} {\int {{\int}_{A} {{\rho_{k}}{{\dot{\textbf{r}}}_{k}} \cdot \delta {{\dot{\textbf{r}}}_{k}} d{\zeta_{k}}d{\eta_{k}}} } } d{x_{k}}dt} $$

(D2)

Using the non-dimensional parameters defined in Appendix C, the non-dimensional variation of kinetic energy for the k th blade is

$$ {\int}_{\tau_{0}}^{\tau} \delta\bar{{\mathcal{T}}}_{k} d\tau = - {\int}_{\tau_{0}}^{\tau} \left[ \delta q^{T} {{\int}_{0}^{1}}\left( \hat M_{k} \ddot q + \hat C_{k} \dot q + \hat K_{k} q \right)d\xi_{k} - \sum\limits_{i=1}^{n}{{\int}_{0}^{1}} \varphi_{u_{k}i} \xi_{k} d\xi_{k} \delta q_{u_{i}k} \right] d\tau $$

(D3)

where the generalized coordinate vector is

$$ q = {\left[ {\begin{array}{cccccccccc} {{\bar{R}_{x}}}&{{\bar{R}_{y}}}&{{\bar{R}_{z}}}&{{\bar{\vartheta}_{x}}}&{{\bar{\vartheta}_{y}}}&{{\bar{\vartheta}_{z}}}&{{q_{w_{k} i}}}&{{q_{v_{k} i}}}&{{q_{\phi_{k} i}}}&{{q_{u_{k}i}}} \end{array}} \right]^{T}} $$

and the shape functions are defined in Eq. 4, i.e., $\varphi _{w_{k}i}$, $\varphi _{v_{k}i}$, $\varphi _{u_{\phi } i}$, for i = 1, 2,…,n. The mass matrix $\hat {M}_{k}$, the gyroscopic $\hat {C}_{k}$, and the stiffness matrix $\hat {K}_{k}$ due to the spinning Ω₀ are given as follows.

Mass Matrix from Variation of Kinetic Energy

The mass matrix for the k th blade from the variation of kinetic energy is

$$ \bar{M}_{k} = {{\int}_{0}^{1}} {\hat M_{k} d\xi_{k} ;} \ \ \ \ \ \hat M_{k} = \left[ {\begin{array}{cc} {{{\hat M}_{k_{RR}}}}&{{{\hat M}_{k_{RB}}}}\\*[5pt] {\hat M_{k_{RB}}^{T}}&{{{\hat M}_{k_{BB}}}} \end{array}} \right] $$

(D4)

where the 6 × 6 submatrix $\hat M_{k_{RR}}$ gives the coupling between the hub translational and rotational coordinates,

$$ {\hat M_{k_{RR}}} = \left( {\begin{array}{cccccc} 1&0&0&0&0&{\xi_{k} s{\theta_{k}}}\\ 0&1&0&0&0&{\xi_{k} c{\theta_{k}}}\\ 0&0&1&{ - \xi_{k} s{\theta_{k}}}&{ - \xi_{k} c{\theta_{k}}}&0\\ 0&0&{ - \xi_{k} s{\theta_{k}}}&{\bar{k}_{m1}^{2} + \bar{k}_{m2}^{2}{c^{2}}{\theta_{k}} + {{\xi_{k}^{2}}}{{\mathrm{s}}^{2}}{\theta_{k}}}&{\left( {{{\xi_{k}^{2}}} - \bar{k}_{m2}^{2}} \right)c{\theta_{k}}s{\theta_{k}}}&0\\ 0&0&{ - \xi_{k} c{\theta_{k}}}&{\left( {{{\xi_{k}^{2}}} - \bar{k}_{m2}^{2}} \right)c{\theta_{k}}s{\theta_{k}}}&\begin{array}{l} \bar{k}_{m1}^{2} + {{\xi_{k}^{2}}}{c^{2}}{\theta_{k}}\\ + \bar{k}_{m2}^{2}{s^{2}}{\theta_{k}} \end{array}&0\\ {\xi_{k} s{\theta_{k}}}&{\xi_{k} c{\theta_{k}}}&0&0&0&{\bar{k}_{m2}^{2} + {{\xi_{k}^{2}}}} \end{array}} \right) $$

(D5)

the 6 × 4n rectangular matrix $\hat M_{k_{RB}}$ shows the coupling of the translational and rotational coordinates with the blade vibrational generalized coordinates,

$$ {\hat M_{k_{RB}}} = \left( {\begin{array}{cccc} 0&{{\varphi_{v_{k}j}}s{\theta_{k}}}&0&{{\varphi_{u_{k}j}}c{\theta_{k}}}\\ 0&{{\varphi_{v_{k}j}}c{\theta_{k}}}&0&{ - {\varphi_{u_{k}j}}s{\theta_{k}}}\\ {{\varphi_{w_{k}j}}}&0&0&0\\ { - s{\theta_{k}}\left( {\varphi {^{\prime}_{w_{k}j}}\bar{k}_{m1}^{2} + \xi {\varphi_{w_{k}j}}} \right)}&0&{{\bar{k}_{m}^{2}}{\varphi_{\phi_{k} j}}c{\theta_{k}}}&0\\ { - c{\theta_{k}}\left( {\varphi {^{\prime}_{w_{k}j}}\bar{k}_{m1}^{2} + \xi {\varphi_{w_{k}j}}} \right)}&0&{ - {\bar{k}_{m}^{2}}{\varphi_{\phi_{k} j}}s{\theta_{k}}}&0\\ 0&{\left( {\varphi {^{\prime}_{v_{k}j}}\bar{k}_{m2}^{2} + \xi {\varphi_{v_{k}j}}} \right)}&0&0 \end{array}} \right) $$

(D6)

and the 4n × 4n matrix $\hat M_{k_{BB}}$ is associated with the blade vibrational generalized coordinates

$$ {\hat M_{k_{BB}}} = \left( \! {\begin{array}{*{20}{c}} {{\varphi_{w_{k}i}}{\varphi_{w_{k}j}} + \bar{k}_{m1}^{2}\varphi {^{\prime}_{w_{k}i}}\varphi {^{\prime}_{w_{k}j}}}&0&0&0\\ 0&{{\varphi_{v_{k}i}}{\varphi_{v_{k}j}} + \bar{k}_{m2}^{2}\varphi {^{\prime}_{v_{k}i}}\varphi {^{\prime}_{v_{k}j}}}&0&0\\ 0&0&{{\bar{k}_{m}^{2}}{\varphi_{\phi_{k} i}}{\varphi_{\phi_{k} j}}}&0\\ 0&0&0&{{\varphi_{u_{k}i}}{\varphi_{u_{k}j}}} \end{array}} \right) $$

(D7)

with i,j = 1,2,…,n and

$$ s{\theta_{k}} = \sin {\theta_{k}};c{\theta_{k}} = \cos {\theta_{k}} $$

Note that the mass matrix is symmetric such that $\bar {M}_{k} =\bar {M_{k}^{T}}$ and positive definite $\bar {M}_{k}>0$.

Gyroscopic Matrix from Variation of Kinetic Energy

The gyroscopic matrix for the k th blade from the variation of kinetic energy is

$$ \bar{C}_{k} = {{\int}_{0}^{1}} {\hat Cd\xi } ;\ \ \ \ \hat C_{k} = \left[ {\begin{array}{cc} {{{\hat C}_{k_{RR}}}}&{{{\hat C}_{k_{RB}}}}\\ { - \hat C_{k_{RB}}^{T}}&{{{\hat C}_{k_{BB}}}} \end{array}} \right] $$

(D8)

where the 6 × 6 matrix $\hat C_{k_{RR}}$ is a skew symmetric matrix,

$$ {\hat C_{k_{RR}}} = \left( {\begin{array}{cccccc} 0&0&0&0&0&{ - \xi_{k} c\theta_{k} }\\ 0&0&0&0&0&{\xi_{k} s\theta_{k} }\\ 0&0&0&0&0&0\\ 0&0&0&0&{ - 2\bar{k}_{m1}^{2}}&0\\ 0&0&0&{2\bar{k}_{m1}^{2}}&0&0\\ {\xi_{k} c\theta_{k} }&{ - \xi_{k} s\theta_{k} }&0&0&0&0 \end{array}} \right) $$

(D9)

and the 6 × 4n rectangular matrix $\hat C_{k_{RB}}$ is

$$ {\hat C_{k_{RB}}} = \left( {\begin{array}{cccc} 0&{ - {\varphi_{v_{k}j}}c\theta_{k} }&0&{{\varphi_{u_{k}j}}s\theta_{k} }\\ 0&{{\varphi_{v_{k}j}}s\theta_{k} }&0&{{\varphi_{u_{k}j}}c\theta_{k} }\\ 0&0&0&0\\ {2\bar{k}_{m1}^{2}{{\varphi }^{\prime}_{w_{k}j}}c\theta_{k} }&0&{2\bar{k}_{m1}^{2}{\varphi_{\phi_{k} j}}s\theta_{k} }&0\\ { - 2\bar{k}_{m1}^{2}{{\varphi }^{\prime}_{w_{k}j}}s\theta_{k} }&0&{2\bar{k}_{m1}^{2}{\varphi_{\phi_{k} j}}c\theta_{k} }&0\\ 0&0&0&{2{\varphi_{u_{k}j}}\xi_{k} } \end{array}} \right) $$

(D10)

and the 4n × 4n matrix $\hat C_{k_{BB}}=-\hat C^{T}_{k_{BB}}$ is

$$ {\hat C_{k_{BB}}} = \left( {\begin{array}{*{20}{c}} 0&0&{ - 2\bar{k}_{m1}^{2}{\varphi_{\phi_{k} j}}{{\varphi }^{\prime}_{w_{k}i}}}&0\\ 0&0&0&{2{\varphi_{u_{k}j}}{\varphi_{v_{k}i}}}\\ {2\bar{k}_{m1}^{2}{\varphi_{\phi_{k} i}}{{\varphi }^{\prime}_{w_{k}j}}}&0&0&0\\ 0&{ - 2{\varphi_{u_{k}i}}{\varphi_{v_{k}j}}}&0&0 \end{array}} \right) $$

(D11)

Note that the gyroscopic matrix is skew symmetric such that $\bar {C}_{k} = - {\bar {C}}_{k}^{T}$.

Stiffness Matrix from Variation of Kinetic Energy for a Spinning System

The stiffness matrix for the k th blade subject to a nominal spinning rate Ω₀ is

$$ {\bar{K}_{k}} = {{\int}_{0}^{1}} {{\hat K_{k}}d\xi ;} \ \ \ \ {\Delta} \bar{k_{m}^{2}} = \bar{k}_{m2}^{2} - \bar{k}_{m1}^{2};\ \ \ \ \hat K_{k} = \left[ {\begin{array}{cc} {{\hat K_{k_{RR}}}}&{{\hat K_{k_{RB}}}}\\*[5pt] \hat{K}_{k_{RB}}^{T}&{{\hat K_{k_{BB}}}} \end{array}} \right] $$

(D12)

where the 6 × 6 matrix $\hat K_{k_{RR}}$ is symmetric,

$$ {\hat{K}_{k_{RR}}}= \left( {\begin{array}{cccccc} 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&{\bar{k}_{m2}^{2}{c^{2}}\theta_{k} - \bar{k}_{m1}^{2} + {{\xi_{k}^{2}}}{s^{2}}\theta_{k} }&{\left( {{{\xi_{k}^{2}}} - \bar{k}_{m2}^{2}} \right)c\theta_{k} s\theta_{k} }&0\\ 0&0&0&{\left( {{{\xi_{k}^{2}}} - \bar{k}_{m2}^{2}} \right)c\theta_{k} s\theta_{k} }&{{{\xi_{k}^{2}}}{c^{2}}\theta_{k} - \bar{k}_{m1}^{2} + \bar{k}_{m2}^{2}{s^{2}}\theta_{k} }&0\\ 0&0&0&0&0&0 \end{array}} \right) $$

(D13)

and the 6 × 4n rectangular matrix $\hat K_{k_{RB}}$ is

$$ {\hat{ K}_{k_{RB}}} = \left( {\begin{array}{*{20}{c}} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ {\left( {\bar{k}_{m1}^{2}\varphi {^{\prime}_{w_{k}j}} - \xi {\varphi_{w_{k}j}}} \right)s\theta_{k} }&0&{{\varphi_{\phi_{k} j}}{\Delta} {\bar{k}_{m}^{2}}c\theta_{k} }&0\\ {\left( {\bar{k}_{m1}^{2}\varphi {^{\prime}_{wj}} - \xi {\varphi_{wj}}} \right)c\theta_{k} }&0&{ - {\varphi_{\phi j}}{\Delta} {\bar{k}_{m}^{2}}s\theta_{k} }&0\\ 0&0&0&0 \end{array}} \right) $$

(D14)

and the 4n × 4n matrix $\hat K_{BB}=\hat K^{T}_{k_{BB}}$ is

$$ {\hat {K}_{k_{BB}}} = \left( {\begin{array}{*{20}{c}} { - \bar{k}_{m1}^{2}{{\varphi}^{\prime}_{wi}}{{\varphi}^{\prime}_{wj}}}&0&0&0\\ 0&{\left( { - {\varphi_{vi}}{\varphi_{vj}} - \bar{k}_{m2}^{2}\varphi {^{\prime}_{vi}}\varphi {^{\prime}_{vj}}} \right)}&0&0\\ 0&0&{{\varphi_{\phi i}}{\varphi_{\phi j}}{\Delta} {\bar{k}_{m}^{2}}}&0\\ 0&0&0&{ - {\varphi_{ui}}{\varphi_{uj}}} \end{array}} \right) $$

(D15)

Note that the stiffness matrix from the variation of kinetic energy is symmetric, i.e., $\bar {K}_{k} =\bar {K}_{k}^{T}$ but may not be positive definite.

There is a single variation term shown in Eq. D3 associated with the blade elongation displacement, i.e.,

$$ -{{\int}_{0}^{1}}\varphi_{u_{k}i} \xi_{k} \delta q_{u_{i}k} d\xi_{k} $$

(D16)

This term results from the product of ξ_k and $\bar {u}_{k}$. It is an important term for solving the centrifugal force to be discussed later.

Stiffness Matrix from Variation of Strain Energy

The first variation of the strain energy for a blade in terms of engineering stresses and strains is

$$ \delta {\mathcal{V}} = \frac{1}{2}{\int}_{0}^{\ell} {\int {{\int}_{A} {\left( {{\sigma_{xx}}\delta {\varepsilon_{xx}} + {\sigma_{x\eta }}\delta {\varepsilon_{x\eta }} + {\sigma_{x\zeta }}\delta {\varepsilon_{x\zeta }}} \right)d\eta d\zeta dx} } } $$

(D17)

where

$$ \begin{array}{@{}rcl@{}} {\sigma_{xx}} &=& E{\varepsilon_{xx}} = E\left[ \begin{array}{c} \frac{{{{v^{\prime}}^{2}}}}{2} + \frac{{{{w^{\prime}}^{2}}}}{2} + u^{\prime} + \frac{{{{\phi^{\prime}}^{2}}}}{2}\left( {\eta_{}^{2} + \zeta_{}^{2}} \right)\\ - \left( {v^{\prime\prime} - w^{\prime} \phi^{\prime} } \right)(\eta \cos \phi - \zeta \sin \phi ) - \left( {w^{\prime\prime} + v^{\prime}\phi^{\prime} } \right)(\zeta \cos \phi + \eta \sin \phi ) \end{array} \right]\\ {\sigma_{x\eta }} &=& G{\varepsilon_{x\eta }}\\ {\sigma_{x\zeta }} &=& G{\varepsilon_{x\zeta }} \end{array} $$

(D18)

and

$$ \begin{array}{@{}rcl@{}} \delta {\varepsilon_{xx}} & = & \delta u^{\prime} + v^{\prime}\delta v^{\prime} + w^{\prime}\delta w^{\prime} + \left( {\eta_{}^{2} + \zeta_{}^{2}} \right)\phi^{\prime}\delta \phi^{\prime} \!- \left[ {\eta \cos \phi - \zeta \sin \phi } \right]\left( {\delta v^{\prime\prime} + w^{\prime\prime}\delta \phi } \right)\\ && - \left[ {\eta \sin \phi + \zeta \cos \phi } \right]\left( {\delta w^{\prime\prime} + v^{\prime\prime}\delta \phi } \right)\\ \delta {\varepsilon_{x\eta }} & = & - \zeta \delta \phi^{\prime} \\ \delta {\varepsilon_{x\zeta }} & = & \eta \delta \phi^{\prime} \end{array} $$

(D19)

where the terms w^′ϕ^′ and v^′ϕ^′ are one order in magnitude smaller than the other terms in their respective parentheses and thus ignored in the following derivation. Nevertheless, it is still debatable that these ignored terms may have some non-negligible contributions to the stiffness matrix.

Define the quantity for the k th blade

$$ {T_{k}} = EA\left( {\frac{{{v^{\prime}}_{k}^{2}}}{2} + \frac{{{w^{\prime}}_{k}^{2}}}{2} + {{u^{\prime}}_{k}} + {k_{A}^{2}}\frac{{{\phi^{\prime}}_{k}^{2}}}{2}} \right) \approx EA\left( {\frac{{{v^{\prime}}_{k}^{2}}}{2} + \frac{{{w^{\prime}}_{k}^{2}}}{2} + {{u^{\prime}}_{k}}} \right) $$

(D20)

which is related to the centrifugal force for a spinning blade. The non-dimensional variation of the strain energy becomes

$$ \begin{array}{@{}rcl@{}} \delta \bar{\mathcal{V}} & = & \sum\limits_{k = 1}^{{n_{b}}} {{{\int}_{0}^{1}} {\left\{ {\left[ {\bar{E}{{\bar{I}}_{w}}{{\bar{w}}^{\prime\prime}_{k}} + {\Delta} \bar{E}\bar{I}{\phi_{k}}{{\bar{v}}^{\prime\prime}_{k}}} \right]\delta {{\bar{w}}^{\prime\prime}_{k}} + {{\bar{T}}_{k}}{{\bar{w}^{\prime}}_{k}}\delta {{\bar{w}^{\prime}}_{k}} + \bar{E}{{\bar{I}}_{w}}{{\bar{\phi}^{\prime}}_{k}}{{\bar{v}^{\prime}}_{k}}\delta {{\bar{w}}^{\prime\prime}_{k}} - \bar{E}{{\bar{I}}_{v}}{{\bar{\phi}^{\prime}}_{k}}{{\bar{v}}^{\prime\prime}_{k}}\delta {{\bar{w}^{\prime}}_{k}}} \right\}} d{\xi_{k}}} \\ && + \sum\limits_{k = 1}^{{n_{b}}} {{{\int}_{0}^{1}} {\left\{ {\left[ {\bar{E}{{\bar{I}}_{v}}{{\bar{v}}^{\prime\prime}_{k}} + {\Delta} \bar{E}\bar{I}\phi {{\bar{w}}^{\prime\prime}_{k}}} \right]\delta {{\bar{v}}^{\prime\prime}_{k}} + {{\bar{T}}_{k}}\bar{v}^{\prime}\delta {{\bar{v}^{\prime}}_{k}} - \bar{E}{{\bar{I}}_{v}}{{\bar{\phi}^{\prime}}_{k}}{{\bar{w}^{\prime}}_{k}}\delta {{\bar{v}}^{\prime\prime}_{k}} + \bar{E}{{\bar{I}}_{w}}{{\bar{\phi}^{\prime}}_{k}}{{\bar{w}}^{\prime\prime}_{k}}\delta {{\bar{v}^{\prime}}_{k}}} \right\}} d{\xi_{k}}} \\ & &+ \sum\limits_{k = 1}^{{n_{b}}} {{{\int}_{0}^{1}} {\left\{ {\left[ {\left( {\bar{G}\bar{J} + {{\bar{T}}_{k}}\bar{k_{A}^{2}}} \right){{\phi}^{\prime}_{k}}} \right]\delta {{\phi}^{\prime}_{k}} + {\Delta} \bar{E}\bar{I}{{\bar{v}}^{\prime\prime}_{k}}{{\bar{w}}^{\prime\prime}_{k}}\delta {\phi_{k}} - \bar{E}{{\bar{I}}_{v}}{{\bar{v}}^{\prime\prime}_{k}}{{\bar{w}^{\prime}}_{k}}\delta {{\bar \phi}^{\prime}_{k}} + \bar{E}{{\bar{I}}_{w}}{{\bar{v}^{\prime}}_{k}}{{\bar{w}}^{\prime\prime}_{k}}\delta {{\bar \phi}^{\prime}_{k}}} \right\}d{\xi_{k}}} } \\ & &+ \sum\limits_{k = 1}^{{n_{b}}} {{{\int}_{0}^{1}} {{{\bar{T}}_{k}}\delta {{u}^{\prime}_{k}}d\xi } } \end{array} $$

(D21)

Assume that all deflection quantities are function separable, substituting Eq. 5 into Eq. D21 produces the non-dimensional variation of strain energy for the k th blade

$$ \delta\bar{\mathcal{V}}_{k} ={\int}_{\tau_{0}}^{\tau} \left[ \delta {q^{T}_{b}} \bar{K}_{k_{S}} q_{b} +\sum\limits_{i=1}^{n}{{\int}_{0}^{1}}{{\bar{T}_{k}}{{\varphi}^{\prime}_{{u_{k}i}}}} d\xi_{k} \delta q_{u_{k}i} \right] d\tau $$

where $q_{b} = {\left [ {\begin {array}{*{20}{c}} {{q_{w_{k} i}}}&{{q_{v_{k} i}}}&{{q_{\phi _{k} i}}}&{{q_{u_{k}i}}} \end {array}} \right ]^{T}}$ are the generalized coordinates associated with the strain energy. The 4n × 4n stiffness matrix $\bar {K}_{k_{S}}$ associated with the generalized coordinates $\left (q_{w_{k}i}, q_{v_{k}i}, q_{\phi _{k}i}, q_{u_{k}i}\right )$ for the k th blade from the strain variation is

$$ \begin{array}{@{}rcl@{}} {{\bar{K}}_{{k_{S}}}} &=& {{\int}_{0}^{1}} {{{\hat K}_{k_{S}}}d{\xi_{k}}} ;{\Delta} EI = \bar{E}{{\bar{I}}_{v}} - \bar{E}{{\bar{I}}_{w}} \\ {{\hat K}_{k_{S}}} &=& \left[ {\begin{array}{cccc} {\left( \begin{array}{l} \bar{E}{{\bar{I}}_{w}}{{\varphi }^{\prime\prime}_{{w_{k}}i}}{{\varphi }^{\prime\prime}_{{w_{k}}j}}\\ + {{\bar{T}}_{k}}{{\varphi^{\prime}}_{{w_{k}}i}}{{\varphi^{\prime}}_{{w_{k}}j}} \end{array} \right)}&{{q_{{\phi_{k}}\gamma }}{\Delta} EI{\varphi_{\phi \gamma}}{{\varphi }^{\prime\prime}_{{w_{k}}i}}{{\varphi }^{\prime\prime}_{{v_{k}}j}}}&{{q_{v_{k} \gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{{v_{k}}\gamma}}{{\varphi }^{\prime\prime}_{{w_{k}}i}}{\varphi_{{\phi_{k}}j}}}&0\\ {{q_{{\phi_{k}}\gamma }}{\Delta} EI{\varphi_{{\phi_{k}}\gamma}}{{\varphi }^{\prime\prime}_{{v_{k}}i}}{{\varphi }^{\prime\prime}_{{w_{k}}j}}}&{\left( \begin{array}{l} \bar{E}{{\bar{I}}_{v}}{{\varphi }^{\prime\prime}_{{v_{k}}i}}{{\varphi }^{\prime\prime}_{{v_{k}}j}}\\ + {{\bar{T}}_{k}}{{\varphi^{\prime}}_{{v_{k}}i}}{{\varphi^{\prime}}_{{v_{k}}j}} \end{array} \right)}&{{q_{w_{k}\gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{{w_{k}}\gamma}}{{\varphi }^{\prime\prime}_{{v_{k}}i}}{\varphi_{{\phi_{k}}j}}}&0\\ {{q_{{v_{k}}\gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{{v_{k}}\gamma}}{\varphi_{{\phi_{k}}i}}{{\varphi }^{\prime\prime}_{{w_{k}}j}}}&{{q_{{w_{k}}\gamma}}{\Delta} EI\varphi_{{w_{k}}\gamma}^{\prime \prime }{\varphi_{{\phi_{k}}i}}\varphi_{{v_{k}}j}^{\prime \prime }}&{\left( \begin{array}{l} \bar{G}\bar{J}{{\varphi^{\prime}}_{{\phi_{k}}i}}{{\varphi^{\prime}}_{{\phi_{k}}j}}\\ + \bar{k}_{a}^{2}{{\bar{T}}_{k}}{{\varphi^{\prime}}_{{\phi_{k}}i}}{{\varphi^{\prime}}_{{\phi_{k}}j}} \end{array} \right)}&0\\ 0&0&0&0 \end{array}} \right]\\ \end{array} $$

(D22)

where double subscript integer index γ implies summation from γ = 1,2,…,n. Note that all the off-diagonal submatrices are nonlinear terms. The subscript integers i and j mean the i th row and j th column of the corresponding submatrices.

Similar to the case for the kinetic energy variation, the single term in the strain energy variation,

$$ {{\int}_{0}^{1}}{{\bar{T}_{k}}{{\varphi}^{\prime}_{{u_{k}i}}}}\xi_{k} \delta q_{u_{k}i} d\xi $$

(D23)

is used to solve for the quantity $\bar {T}_{k}$. Assume that the blade elongation is negligible, i.e., $\ddot q_{u_{k}i} = \dot q_{u_{k}i} = q_{u_{k}i} =0$. The last rows of $\hat M_{k_{BR}} = \hat M^{T}_{k_{RB}}$ in Eq. D6, $\hat C_{k_{BR}}= - \hat C^{T}_{k_{RB}}$ in Eq. D10, and ${\hat {K}_{k_{BB}}}$ in Eq. D13, and the quantities shown in Eqs. D16 and D23 produce the following equation

$$ {{\int}_{0}^{1}} {\left[ {{\varphi_{{u_{k}}i}}\left( {{{\ddot R}_{x}}{\mathrm{c}} {\theta_{k}} - {{\ddot R}_{y}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{x}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{y}}{\mathrm{c}} {\theta_{k}} - 2{\xi_{k}}{{\dot \vartheta }_{z}} - 2{\varphi_{{v_{k}}j}}{{\dot q}_{{v_{k}}j}} \!- {\xi_{k}}} \right) + {\bar{T}_{k}}{{\varphi^{\prime}}_{{u_{k}}i}}} \right]} d{\xi_{k}}\delta {q_{{u_{k}}i}} = 0 $$

Setting the terms in the bracket to zero yields

$${\bar{T}^{\prime}_{k}} = \left( {{{\ddot R}_{x}}{\mathrm{c}} {\theta_{k}} - {{\ddot R}_{y}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{x}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{y}}{\mathrm{c}} {\theta_{k}} - 2{\xi_{k}}{{\dot \vartheta }_{z}} - 2{\varphi_{{v_{k}}j}}{{\dot q}_{{v_{k}}j}} - {\xi_{k}}} \right)$$

after the following integration by parts is used

$$\begin{array}{l} {{\int}_{0}^{1}} {{{\bar{T}}_{k}}{{\varphi^{\prime}}_{{u_{k}}i}}\delta {q_{{u_{k}}i}}} d{\xi_{k}} = \left. {{{\bar{T}}_{k}}{\varphi_{{u_{k}}i}}\delta {q_{{u_{k}}i}}} \right|_{0}^{1} - {{\int}_{0}^{1}} {\bar{T}^{\prime}{\varphi_{{u_{k}}i}}\delta {q_{{u_{k}}i}}} d\xi \\*[5pt] \Rightarrow {{\int}_{0}^{1}} {{T_{k}}{{\varphi^{\prime}}_{{u_{k}}i}}\delta {q_{{u_{k}}i}}} d\xi = - {{\int}_{0}^{1}} {{{\bar{T}^{\prime}}_{k}}{\varphi_{{u_{k}}i}}\delta {q_{{u_{k}}i}}} d{\xi_{k}} \end{array}$$

with the boundary conditions, $\bar {T}_{k} = 0$ at ξ = 1 and $\delta {q_{{u_{k}}i}} = 0$ at ξ = 0. The centrifugal force can then be solved by

$$ \begin{array}{lcl} {{\bar{T}}_{k}} &=& - {\int}_{{\xi_{k}}}^{1} {\left( {{{\ddot R}_{x}}{\mathrm{c}} {\theta_{k}} - {{\ddot R}_{y}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{x}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{y}}{\mathrm{c}} {\theta_{k}} - 2{\xi_{k}}{{\dot \vartheta }_{z}} - 2{\varphi_{{v_{k}}j}}{{\dot q}_{{v_{k}}j}} - {\xi_{k}}} \right)} d{\xi_{k}}\\ &=& - \left( {1 - {\xi_{k}}} \right)\left[ {{{\ddot R}_{x}}{\mathrm{c}} {\theta_{k}} - {{\ddot R}_{y}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{x}}{\mathrm{s}} {\theta_{k}} - {{\dot R}_{y}}{\mathrm{c}} {\theta_{k}}} \right] + \left( {1 - {\xi_{k}^{2}}} \right){{\dot \vartheta }_{z}}\\ &&+ 2\left( {{\int}_{{\xi_{k}}}^{1} {{\varphi_{{v_{k}}j}}d{\xi_{k}}} } \right){{\dot q}_{{v_{k}}j}} + \frac{1}{2}\left( {1 - {\xi_{k}^{2}}} \right) \end{array} $$

(D24)

From Eqs. D15, D22 and D24, the overall stiffness matrix becomes

$$ {{\mathcal{K}}_{k}} = \left[ {\begin{array}{ccc} { \begin{array}{l} \bar{E}{{\bar{I}}_{w}}{{\varphi }^{\prime\prime}_{w_{k}i}}{{\varphi }^{\prime\prime}_{w_{k}j}} \\*[3pt] + \frac{1}{2}\left( {1 - {\xi_{k}^{2}}} \right){{\varphi }^{\prime}_{w_{k}i}}{{\varphi }^{\prime}_{w_{k}j}}\\ - \bar{k}_{m1}^{2}{{\varphi }^{\prime}_{w_{k}i}}{{\varphi }^{\prime}_{w_{k}j}} \end{array} }&{{q_{\phi_{k} \gamma}}{\Delta} EI{\varphi_{\phi_{k} \gamma}}{{\varphi }^{\prime\prime}_{wi}}{{\varphi_{k} }^{\prime\prime}_{vj}}}&{{q_{v_{k}\gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{v_{k}\gamma}}{{\varphi }^{\prime\prime}_{w_{k}i}}{\varphi_{\phi_{k}j}}}\\*[3pt] {{q_{\phi_{k} \gamma}}{\Delta} EI{\varphi_{\phi_{k} \gamma}}{{\varphi }^{\prime\prime}_{v_{k}i}}{{\varphi }^{\prime\prime}_{w_{k}j}}}&{ \begin{array}{l} \bar{E}{{\bar{I}}_{v}}{{\varphi }^{\prime\prime}_{v_{k}i}}{{\varphi }^{\prime\prime}_{v_{k}j}} - {\varphi_{v_{k}i}}{\varphi_{v_{k}j}}\\ + \frac{1}{2}\left( {1 - {\xi_{k}^{2}}} \right){{\varphi }^{\prime}_{v_{k}i}}{{\varphi }^{\prime}_{v_{k}j}}\\*[3pt] - \bar{k}_{m2}^{2}{{\varphi }^{\prime}_{v_{k}i}}{{\varphi }^{\prime}_{v_{k}j}} \end{array} }&{{q_{w_{k}\gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{w_{k}\gamma}}{{\varphi }^{\prime\prime}_{v_{k}i}}{\varphi_{\phi_{k} j}}}\\ {{q_{v_{k}\gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{v_{k}\gamma}}{\varphi_{\phi_{k} i}}{{\varphi }^{\prime\prime}_{w_{k}j}}}&{{q_{w_{k}\gamma}}{\Delta} EI{{\varphi }^{\prime\prime}_{w_{k}\gamma}}{\varphi_{\phi i}}{{\varphi }^{\prime\prime}_{v_{k}j}}}& \begin{array}{l} \bar{G}\bar{J}{{\varphi }^{\prime}_{\phi_{k} i}}{{\varphi }^{\prime}_{\phi_{k} j}} + {\varphi_{\phi_{k} i}}{\varphi_{\phi_{k} j}}{\Delta} {\bar{k}_{m}^{2}}\\ + \frac{1}{2}\left( {1 - {\xi_{k}^{2}}} \right){\bar{k}_{a}^{2}}{{\varphi }^{\prime}_{\phi_{k} i}}{{\varphi }^{\prime}_{\phi_{k} j}} \end{array} \end{array}} \right] $$

(D25)

where double subscript integer γ implies summation from γ = 1,2,…,n. The elements in the off-diagonal submatrices are nonlinear and time varying in the sense that each term has involved with a time-dependent generalized coordinate. These nonlinear terms are one order in magnitude smaller than the other constant terms. However, the nonlinear terms are not negligible when the blade is subject to a constant force producing a static deflection.

System Matrices from Variation of Kinetic and Strain Energies

From Eqs. D4 to D7, the system mass matrix for the multiple blades is

$$ \bar{\mathcal{M}} = {{\int}_{0}^{1}} {\tilde Md\xi ;} \ \ \ \tilde M = \left[ {\begin{array}{cccc} \bar{M}_{h}+{\sum\limits_{k = 1}^{{n_{b}}} {{{\hat M}_{{k_{RR}}}}} }&{{{\hat M}_{{1_{RB}}}}}& {\cdots} &{{{\hat M}_{n{b_{RB}}}}}\\ {\hat M_{{1_{RB}}}^{T} + {{\hat M}_{1{N_{BR}}}}}&{{{\hat M}_{{1_{BB}}}}}& {\cdots} &0\\ {\vdots} & {\vdots} & {\ddots} &0\\ {\hat M_{n{b_{RB}}}^{T} + {{\hat M}_{nb{N_{BR}}}}}&0& {\cdots} &{{{\hat M}_{n{b_{BB}}}}} \end{array}} \right] $$

(D26)

where $\bar {M}_{h}$ is the non-dimensional mass matrix of the hub and it is a diagonal matrix if the the center of hub is chosen to be the center of mass, and the hub axes are the principal axes of inertia, i.e., $\bar {M}_{h}=\text {diag}[\bar {m}_{h},\bar {m}_{h},\bar {m}_{h}, \bar {I}_{xx},\bar {I}_{yy},\bar {I}_{zz}]$. Note that all the submatrices are smaller in size than the ones shown earlier because the columns and rows associated with the neglected elongation displacement of the blades u_k for k = 1,2,…n_b are deleted. The size of the stiffness matrix $\mathcal {M}$ is (6 + 3nn_b) × (6 + 3nn_b).

The submatrices introduced by the centrifugal force from the strain energy are nonlinear, i.e,

$$ {\hat M_{k{N_{BR}}}} = \left[ {\begin{array}{cccccc} { - \left( {1 - {\xi_{k}}} \right){q_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}j}}{\mathrm{c}} {\theta_{k}}}&{\left( {1 - {\xi_{k}}} \right){q_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}j}}{\mathrm{s}} {\theta_{k}}}&0&0&0&0\\ { - \left( {1 - {\xi_{k}}} \right){q_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}j}}{\mathrm{c}} {\theta_{k}}}&{\left( {1 - {\xi_{k}}} \right){q_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}j}}{\mathrm{s}} {\theta_{k}}}&0&0&0&0\\ { - \left( {1 - {\xi_{k}}} \right){\bar{k}_{a}^{2}}{q_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}j}}{\mathrm{c}} {\theta_{k}}}&{\left( {1 - {\xi_{k}}} \right){\bar{k}_{a}^{2}}{q_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}j}}s{\theta_{k}}}&0&0&0&0 \end{array}} \right] $$

where the double subscript integer i implies summation from i = 1,2,…,n. These additional nonlinear and time-varying terms may be neglected without static deflections of the blades subject to external forces.

From Eqs. D8 to D11, the system gyroscopic matrix for the multiple blades is

$$ \bar{\mathcal{C}} = {{\int}_{0}^{1}} {\tilde Cd\xi ;} \ \ \ \tilde C = \left[ {\begin{array}{cccc} \bar{C}_{h}+{\sum\limits_{k = 1}^{{n_{b}}} {{{\hat C}_{{k_{RR}}}}} }&{{{\hat C}_{{1_{RB}}}}}& {\cdots} &{{{\hat C}_{n{b_{RB}}}}}\\ { - \hat C_{{1_{RB}}}^{T} + {{\hat C}_{1{N_{BR}}}}}&{{{\hat C}_{{1_{BB}}}} + {{\hat C}_{1{N_{BB}}}}}& {\cdots} &0\\ {\vdots} & {\vdots} & {\ddots} &0\\ { - \hat C_{n{b_{RB}}}^{T} + {{\hat C}_{nb{N_{BR}}}}}&0& {\cdots} &{{{\hat C}_{n{b_{BB}}}} + {{\hat C}_{nb{N_{BB}}}}} \end{array}} \right] $$

(D27)

where $\bar {C}_{h}$ is the non-dimensional gyroscopic matrix of the hub and it it is a zero matrix except, i.e., $\bar {C}_{h}(4,5)=\bar {I}_{zz}-\bar {I}_{xx}-\bar {I}_{yy}$ and $\bar {C}_{h}(5,4)= - \bar {C}_{h}(4,5)$. The off-diagonal nonlinear and time varying submatrices are defined by

$$ {\hat C_{k{N_{RB}}}} = \left[\!\!\! {\begin{array}{*{20}{c}} {\left( {1 - {\xi_{k}}} \right){q_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}j}}s{\theta_{k}}}&{\left( {1 - {\xi_{k}}} \right){q_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}j}}c{\theta_{k}}} & 0&0&0&{\left( {1 - {\xi_{k}^{2}}} \right){q_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}j}}}\\ {\left( {1 - {\xi_{k}}} \right){q_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}j}}s{\theta_{k}}}&{\left( {1 - {\xi_{k}}} \right){q_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}j}}c{\theta_{k}}}&0&0&0&{\left( {1\! - {\xi_{k}^{2}}} \right){q_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}j}}}\\ {\left( {1 - {\xi_{k}}} \right){\bar{k}_{a}^{2}}{q_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}j}}s{\theta_{k}}} & {\left( {1 - {\xi_{k}}} \right){\bar{k}_{a}^{2}}{q_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}j}}c{\theta_{k}}}&0&0&0&{\left( {1 - {\xi_{k}^{2}}} \right){\bar{k}_{a}^{2}}{q_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}j}}} \end{array}} \!\right] $$

and the diagonal nonlinear submatrices are

$$ {\hat C_{k{N_{BB}}}} = \left[ {\begin{array}{ccc} 0& - 2{q_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}i}}{{\varphi }^{\prime}_{{w_{k}}j}}\left( \varphi^{\prime\prime\prime}_{{v_{k}}j} \left/\beta_{j}^{4}\right)\right.&0\\ 0&- 2{q_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}i}}{{\varphi }^{\prime}_{{v_{k}}j}}\left( \varphi^{\prime\prime\prime}_{{v_{k}}j}\left/\beta_{j}^{4}\right)\right.&0\\ 0&- 2{q_{{\phi_{k}}i}}{\bar{k}_{a}^{2}}{{\varphi }^{\prime}_{{\phi_{k}}i}}{{\varphi }^{\prime}_{{\phi_{k}}j}}\left( \varphi^{\prime\prime\prime}_{{v_{k}}j}\left/ \beta_{j}^{4}\right)\right.&0 \end{array}} \right] $$

for k = 1,2,…,n_b and j = 1,2,…,n. Note that double subscript integer i implies summation from i = 1,2,…,n. These nonlinear matrices destroy the anti-symmetry of the gyroscopic matrix. They represent only one-way coupling between the hub coordinates and the blade generalized coordinates.

From Eqs. D12 and D25, the system stiffness matrix for multiple blades is

$$ \bar{\mathcal{K}} = {{\int}_{0}^{1}} {\hat {\mathcal{K}}} d\xi ; \ \ \ {\hat {\mathcal{K}}} = \left[ {\begin{array}{*{20}{c}} \bar{K_{h}}+{\sum\limits_{k = 1}^{{n_{b}}} {{\hat K_{{k_{RR}}}}} }&{{\hat K_{{1_{RB}}}}}& {\cdots} &{{\hat K_{n{b_{RB}}}}}\\ {\hat {K}_{{1_{RB}}}^{T}}&{{{\mathcal{K}}_{1}}}& {\cdots} &0\\ {\vdots} & {\vdots} & {\ddots} &0\\ {{\hat K}_{n{b_{RB}}}^{T}}&0& {\cdots} &{{{\mathcal{K}}_{{n_{b}}}}} \end{array}} \right] $$

(D28)

where $\bar {K}_{h}$ is the non-dimensional stiffness matrix of the hub and it it is diagonal for the principal axes, i.e., $\bar {K}_{h}=\text {diag}[0,0,0, \bar {I}_{zz}-\bar {I}_{yy} ,\bar {I}_{zz}-\bar {I}_{xx},0]$.

Stiffness Matrix with Solar Radiation Pressure

$$ \bar{\mathcal{K}} = {{\int}_{0}^{1}} {\tilde {K}} d\xi ; \ {\tilde{K}} = \left[ {\begin{array}{cccc} {\sum\limits_{k = 1}^{{n_{b}}} {{\hat K_{{k_{RR}}}}}+\hat{K}_{kP_{RR}} }&{{\hat K_{{1_{RB}}}}} +\hat{K}_{1P_{RS}} & {\cdots} &{{\hat K_{n{b_{RB}}}}}+\hat{K}_{nbP_{RS}}\\ {\hat{ K}_{{1_{RB}}}^{T}}&{{{\mathcal{K}}_{1}}}+\hat{K}_{1P_{SS}}& {\cdots} &0\\ {\vdots} & {\vdots} & {\ddots} &0\\ {\hat{ K}_{n{b_{RB}}}^{T}}&0& {\cdots} &{{{\mathcal{K}}_{{n_{b}}}}}+\hat{K}_{nbP_{SS}} \end{array}} \right] $$

(D29)

where

$$ {\hat K_{k{P_{RR}}}} = \left( {\begin{array}{cccccc} 0&0&0&0&{ - \bar{p}}&0\\ 0&0&0&{\bar{p}}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&{\bar{p}{\xi_{k}}{\mathrm{c}}{\theta_{k}}}&{ - \bar{p}{\xi_{k}}{\mathrm{s}} {\theta_{k}}}&0 \end{array}} \right) $$

and

$$ {\hat K_{k{P_{RB}} = }}\left( {\begin{array}{ccc} {\bar{p}{{\varphi^{\prime}}_{{w_{k}}j}}c\theta }&0&{\bar{p}{\varphi_{{\phi_{k}}j}}s\theta }\\ { - \bar{p}{{\varphi^{\prime}}_{{w_{k}}j}}s\theta }&0&{\bar{p}{\varphi_{{\phi_{k}}j}}c\theta }\\ 0&0&0\\ 0&{ - \bar{p}{\varphi_{{v_{k}}j}}c\theta }&0\\ 0&{\bar{p}{\varphi_{{v_{k}}j}}s\theta }&0\\ 0&0&{\bar{p}{\xi_{k}}{\varphi_{{\phi_{k}}j}}} \end{array}} \right) $$

and

$$ {\hat K_{k{P_{SS}}}} = \left( {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&{\bar{p}{\varphi_{{v_{k}}i}}{\varphi_{{\phi_{k}}j}}}\\ 0&0&0 \end{array}} \right) $$

for i,j = 1,2,…,n.