Revision as of 10:53, 21 September 2021

Abstract

To investigate the structural dynamic characteristics of a folding wing effectively, a fast structural dynamic modeling approach is proposed. Firstly, the interface compatible relationship of the traditional fixed interface component modal synthesis method is modified, and the internal force of the interface is completely expressed in the structural dynamic equation, so that the influence of the connection stiffness on the wing structure dynamics can be considered. Then, on the basis of the fixed interface component modal synthesis method, the main mode of fixed-loaded interface is introduced to establish the mixed-loaded interface component modal synthesis method, which makes it feasible to accurately reflect the influence of elasticity and inertia of fuselage and outer wing on inner wing. The structural dynamics modeling method based on two different kinds of component modal synthesis method analyzed and deduced in detail. The application of component modal synthesis method in the fast structural dynamics modeling of folding wing is achieved. The whole program is compiled in MATLAB. At the same time, the dynamic characteristics of the folding wing with different folding angles, different connections and different connection positions is investigated. The results of the method proposed in this paper are compared with the results of the repeated finite model established in MSC.NASTRAN to verify the effectiveness from the aspects of natural frequency and vibration mode.

Keywords: Folding wing, component modal synthesis, dynamic analysis

1．Introduction

During the morphing process of morphing aircraft, the wing shape and geometric size change significantly leads to the redistribution of mass and stiffness of aircraft structure, resulting in different dynamic characteristics under different configurations [1]. The analysis of structural dynamic characteristics is the basis of flutter analysis and nonlinear aeroelastic analysis. In addition, the analysis of structural dynamic characteristics is also very important in the dynamic modeling of flexible multi-body system [2]. Therefore, it is particularly important to analyze the dynamic characteristics of the morphing aircraft, which can obtain the natural frequency and modal information of the wing structure and lay the foundation for the subsequent dynamic analysis [3].

Lee et al. [4] used Matlab to generate a high fidelity folding wing model proposed by Lockheed Martin, which is composed of rod, plate and other elements. The dynamic characteristics of the folding wing are analyzed in MSC.NASTRAN. However the effect of the fuselage on the structure is not taken into consideration in Ref.[4]. Snyder et al. [5] used MSC.PATRAN to establish the simplified structure of folding wing with different folding angles. The sensitivity of natural frequency to hinge stiffness and folding angle is analyzed in MSC.NASTRAN. The above research did not verify the correctness of the structural model. Tang et al. [6] established the theoretical model of folding wing structure using linear plate theory. Firstly, the modal analysis of each substructure is carried out by ANSYS, and then the Lagrange equation is used to synthesize to establish the dynamic model of the whole structure, and the correctness of the model is verified. At the same time, the variation rule of natural frequency with folding angle and hinge stiffness is analyzed. Andersen et al. [7] analyzed the natural frequency of the folding wing in MSC.NASTRAN and verified by the ground vibration test. Attar et al. [8] further considered the geometric nonlinearity of the structure and established the dynamic equation of the structure by using Kirchhoff plate theory and virtual work principle. Wang et al. [9] proposed a more general structural dynamic model for predicting the natural frequencies of folding wings, which is based on the beam theory. The natural frequencies of three test models in the whole folding angle range are measured. Zhao et al. [10] established the parametric structural dynamic model of the plate folding wing by the component synthesis method, and compared the results with those by MSC.NASTRAN. The correctness of the structural model is verified, and the influence factors of natural frequency are analyzed. In consideration of the nonlinear factors at the hinge of the folding wing, Hu et al. [11] established the dynamic model of the two wing segments by Lagrange equation, in which the cubic stiffness is considered. Lee et al. [12] used fictitious mass method to establish the structure with hinge free-play for the folding wing. The dynamic response characteristics are studied. Yang et al. [13] proposed a method for modeling multiple degrees of freedom in chord direction with structural nonlinearity by the fixed interface component modal synthesis method. The method introduces the angle difference between the inner and outer wing at the interface. The model synthesis of the two wing segments are achieved. Guo et al. [14-15] assumed that the wing was composed of three carbon fiber composite plates, and established the nonlinear dynamic equation by Hamilton principle and von Karman large deformation theory. The vibration modal function is solved according to the modal analysis, the results of harmonic response by ANSYS and the boundary conditions. The simulation results are compared with those by ANSYS, and the nonlinear dynamic characteristics of 1:2 resonance are further analyzed.

At present, the structural simulation method of folding wing is mainly based on finite element method, and the analytical method is used in a small number of literatures. This is because the analytical method is only suitable for simple engineering structures, while the finite element method has obvious advantages in dealing with various complex engineering problems. In most literatures, the continuous change process of folding wing of variant aircraft from unfolded state to folded state is usually decomposed into different configurations of wing, and then the repeated modeling of wing structure under different configurations is not conducive to the rapid analysis of structural dynamic characteristics of folding wing.

For the folding wing structure, the shape change exists locally, while the rest parts remain unchanged, it can be divided into different components. Only the changeable components need to be remodeled, and the whole structure model does not need to be reestablished by commercial software, so as to improve the modeling efficiency. The internal degrees of freedom of each component can be condensed, and then integrated into the overall solution equation, which greatly reduces the size of the problem and improves the computational efficiency. Component modal synthesis (CMS) is a natural choice. It enables the dynamic analysis to be performed on highly reduced matrices to save computational time. Also, it can make modifications possible to reduce the number of design iterations [16].

A fast structural dynamic modeling approach is proposed to investigate the structural dynamic characteristics of a folding wing effectively. For the folding wing with the elastic connection, the interface compatible relationship of the traditional fixed interface component modal synthesis method is modified, and the internal force of the interface is completely expressed in the structural dynamic equation, so the influence of the connection stiffness on the wing structure dynamics can be considered. For the folding wing with the rigid connection, a mixed-loaded interface component mode synthesis of a folding wing is proposed based on fixed interface and loaded interface method, which makes it feasible to accurately reflect the influence of elasticity and inertia of fuselage and outer wing on inner wing. The dynamic characteristics of the folding wing with different folding angles, different connections and different connection positions are investigated. The final dynamic equation has a highly reduced size and more concise. To verify the accuracy of this method, the results are compared with those from MSC.NASTRAN.

2．Fast structural dynamics modeling of folding wing

The folding wing is composed of three components, namely fuselage, inner wing and outer wing, denoted A, B and C, respectively. The geometric diagram is shown in Figure 1. The coordinate system with subscript represents the local coordinate system of each component, while the coordinate system without subscript represents the global coordinate system. In order to verify the feasibility of the modeling method, the folding wing model is simplified as aluminum plate with equal thickness. All components are modeled by plates with the Yong’s modulus of ${\displaystyle 7.1\times {10}^{10}Pa}$, Poisson’s coefficient of 0.33 and a density of ${\displaystyle 2.7\times {10}^{3}kg/m^{3}}$, which are discretized using CQUAD4 elements with 2mm thickness in MSC.NASTRAN. Each node of this element has six degrees of freedom: three translations, namely, ${\displaystyle u_{x},u_{y},u_{z}}$ and three rotations, i.e., ${\displaystyle {\theta }_{x},{\theta }_{y},{\theta }_{z}}$. For each component, interface nodes are used at the connection positions of the folding axis to coordinate the boundary displacement. For the folding wing with rigid connection, MPC (Multi - Point Constraint) is used to couple the six degrees of freedom of interface nodes. For the folding wing with elastic connection, a set of torsion spring with a rotational degree of freedom ${\displaystyle {\theta }_{x}}$ are modeled and the other five degrees of freedom are defined via MPC. Among them, the torsional spring stiffness between components A and B is ${\displaystyle K_{A}}$, and that between components B and C is ${\displaystyle K_{B}}$. In this paper the structural dynamics model with two different connection types and different connection positions is established. The simulation method of two different connection types is shown in Figure 2.

(a) Top view	(b) Rear view
Figure 1. Geometry and coordinate system of folding wing

(a) Elastic connection	(b) Rigid connection
Figure 2. Schematic diagram of folding wing connection mode

The dynamic equations of each component with no damp in local coordinates are as follows:

where ${\displaystyle {\boldsymbol {M}}_{l}^{\lambda }}$ and ${\displaystyle {\boldsymbol {K}}_{l}^{\lambda }}$ represent mass and stiffness matrices. ${\displaystyle {\boldsymbol {u}}_{l}^{\lambda }}$ is a displacement vector, and ${\displaystyle {\boldsymbol {f}}_{l}^{\lambda }}$ is the constraints force vector.

When the local coordinate system in Figure 1 is adopted, the displacement vectors of components A and C are the same in the global coordinate system as in the local coordinate system, and no transformation is needed. For component B, the relationship between the displacement vector in the global coordinate system and that in the local coordinate system is as follows:

where ${\displaystyle {\boldsymbol {\overline {T}}}=blockdiag({\boldsymbol {T}},\cdots ,{\boldsymbol {T}})}$, ${\displaystyle {\boldsymbol {T}}=\left[{\begin{array}{cccccc}1&0&0&0&0&0\\0&cos\theta &-sin\theta &0&0&0\\0&sin\theta &cos\theta &0&0&0\\0&0&0&1&0&0\\0&0&0&0&cos\theta &-sin\theta \\0&0&0&0&sin\theta &cos\theta \end{array}}\right]}$. The structural dynamic equation of component B is transformed from the local coordinate system to the global coordinate system by pre-multiplying by ${\displaystyle {\boldsymbol {\overline {T}}}}$ on both sides of the structural dynamic equation of component B.

In order to facilitate the application of component modal synthesis method, firstly, the displacement of each component in the global coordinate system is divided into internal node displacement and interface node displacement, and the mass matrix and stiffness matrix can be partitioned, which is similar to Ref. [10]. In this case, the structural dynamic equation of each component in the global coordinate system can be expressed as follows:

where subscript i is the internal node displacement, the node displacement with MPC is denoted by subscript m, and the node displacement with torsion spring is denoted by subscript s. the subscript a represents the interface between components A and B, and b donates the interface between components B and C.

2.1 Fixed interface component mode synthesis for folding wing with elastic connection

At present, when the fixed interface substructure modal synthesis method is applied to the modeling of complex structures, the elastic connection is rarely considered. Therefore, based on the fixed interface substructure modal synthesis method, a fast structural dynamic model of the folding wing with elastic connection is established.

For component A, the displacements is transformed into the modal displacement by introducing the modal displacement and modal matrix:

where ${\displaystyle {\boldsymbol {\varphi }}_{N}^{A}}$ is the main mode, which can be obtained by solving the dynamic equation when the interface is fixed, i.e. ${\displaystyle {\boldsymbol {K}}_{ii}^{A}{\boldsymbol {\varphi }}_{n}^{A}={\omega }_{A}^{2}{\boldsymbol {M}}_{ii}^{A}{\boldsymbol {\varphi }}_{n}^{A}}$, ${\displaystyle {\boldsymbol {\varphi }}_{C}^{A}}$ is the constrained mode, i.e. ${\displaystyle {\boldsymbol {\varphi }}_{cm}^{A}=-{\left({\boldsymbol {K}}_{ii}^{A}\right)}^{-1}{\boldsymbol {K}}_{im}^{A}}$, ${\displaystyle {\boldsymbol {\varphi }}_{cs}^{A}=-{\left({\boldsymbol {K}}_{ii}^{A}\right)}^{-1}{\boldsymbol {K}}_{is}^{A}}$. Through coordinate transformation, the displacement at the interface does not change, and it is still expressed in the physical space, which is conducive to the introduction of torsional spring stiffness. If the higher order main mode is not ignored, the coordinate transformation is the complete transformation, and the accurate result can be obtained. But the order of matrix after transformation has not changed, and the purpose of reducing the degree of freedom is not achieved. Therefore, the higher-order of the main mode is usually ignored, and only the low-order mode is retained. The first coordinate transformation is introduced into the mass matrix and stiffness matrix of component A, and the following results can be obtained:

where ${\displaystyle {\boldsymbol {M}}_{CB}^{A}={\boldsymbol {\varphi }}_{CB}^{AT}{\boldsymbol {M}}^{A}{\boldsymbol {\varphi }}_{CB}^{A}}$, ${\displaystyle {\boldsymbol {K}}_{CB}^{A}={\boldsymbol {\varphi }}_{CB}^{AT}{\boldsymbol {K}}^{A}{\boldsymbol {\varphi }}_{CB}^{A}}$, ${\displaystyle {\boldsymbol {f}}_{CB}^{A}={\boldsymbol {\varphi }}_{CB}^{AT}{\boldsymbol {f}}^{A}=}$${\displaystyle {\left[{\begin{array}{ccc}{\boldsymbol {0}}^{T}&{\boldsymbol {f}}_{m}^{AT}&{\boldsymbol {f}}_{s}^{AT}\end{array}}\right]}^{T}}$.

Similarly, introducing the modal coordinates ${\displaystyle {\boldsymbol {p}}^{B}}$ and modal matrices ${\displaystyle {\boldsymbol {\varphi }}_{CB}^{B}}$, the modal analysis is carried out for component B, then:

where ${\displaystyle {\boldsymbol {K}}_{ii}^{B}{\boldsymbol {\varphi }}_{n}^{B}={\omega }_{B}^{2}{\boldsymbol {M}}_{ii}^{B}{\boldsymbol {\varphi }}_{n}^{B}}$, ${\displaystyle {\boldsymbol {\varphi }}_{cm}^{Ba}=-{\left({\boldsymbol {K}}_{ii}^{B}\right)}^{-1}{\boldsymbol {K}}_{im}^{Ba}}$, ${\displaystyle {\boldsymbol {\varphi }}_{cs}^{Ba}=-{\left({\boldsymbol {K}}_{ii}^{B}\right)}^{-1}{\boldsymbol {K}}_{is}^{Ba}}$, ${\displaystyle {\boldsymbol {\varphi }}_{cm}^{Bb}=-{\left({\boldsymbol {K}}_{ii}^{B}\right)}^{-1}{\boldsymbol {K}}_{im}^{Bb}}$, ${\displaystyle {\boldsymbol {\varphi }}_{cs}^{Bb}=-{\left({\boldsymbol {K}}_{ii}^{B}\right)}^{-1}{\boldsymbol {K}}_{is}^{Bb}}$. One can obtain the first coordinate transformation of component B:

where ${\displaystyle {\boldsymbol {M}}_{CB}^{B}={\boldsymbol {\varphi }}_{CB}^{BT}{\boldsymbol {M}}^{B}{\boldsymbol {\varphi }}_{CB}^{B}}$, ${\displaystyle {\boldsymbol {K}}_{CB}^{B}={\boldsymbol {\varphi }}_{CB}^{BT}{\boldsymbol {K}}^{B}{\boldsymbol {\varphi }}_{CB}^{B}}$, ${\displaystyle {\boldsymbol {f}}_{CB}^{B}={\boldsymbol {\varphi }}_{CB}^{BT}{\boldsymbol {f}}^{B}=}$${\displaystyle {\left[{\begin{array}{ccccc}{\boldsymbol {0}}^{T}&{\boldsymbol {f}}_{m}^{BaT}&{\boldsymbol {f}}_{s}^{BaT}&{\boldsymbol {f}}_{m}^{BT}&{\boldsymbol {f}}_{s}^{BbT}\end{array}}\right]}^{T}}$.

Similar expressions can be written for component C:

where ${\displaystyle {\boldsymbol {K}}_{ii}^{C}{\boldsymbol {\varphi }}_{n}^{C}={\omega }_{C}^{2}{\boldsymbol {M}}_{ii}^{C}{\boldsymbol {\varphi }}_{n}^{C}}$ , ${\displaystyle {\boldsymbol {\varphi }}_{cm}^{C}=-{\left({\boldsymbol {K}}_{ii}^{C}\right)}^{-1}{\boldsymbol {K}}_{im}^{C}}$ , ${\displaystyle {\boldsymbol {\varphi }}_{cs}^{C}=-{\left({\boldsymbol {K}}_{ii}^{C}\right)}^{-1}{\boldsymbol {K}}_{is}^{C}}$ . The first coordinate transformation is introduced into the mass matrix and stiffness matrix of component C, and the following results can be obtained:

where ${\displaystyle {\boldsymbol {M}}_{CB}^{C}={\boldsymbol {\varphi }}_{CB}^{CT}{\boldsymbol {M}}^{C}{\boldsymbol {\varphi }}_{CB}^{C}}$ , ${\displaystyle {\boldsymbol {K}}_{CB}^{C}={\boldsymbol {\varphi }}_{CB}^{CT}{\boldsymbol {K}}^{C}{\boldsymbol {\varphi }}_{CB}^{C}}$, ${\displaystyle {\boldsymbol {f}}_{CB}^{C}={\boldsymbol {\varphi }}_{CB}^{CT}{\boldsymbol {f}}^{C}=}$${\displaystyle {\left[{\begin{array}{ccc}{\boldsymbol {0}}^{T}&{\boldsymbol {f}}_{m}^{C}{}^{T}&{\boldsymbol {f}}_{s}^{C}{}^{T}\end{array}}\right]}^{T}}$.

In this case, the uncoupled dynamic equation of the whole structure can be written as follows:

where ${\displaystyle {\boldsymbol {M}}=diag({\boldsymbol {M}}_{CB}^{A},{\boldsymbol {M}}_{CB}^{B},{\boldsymbol {M}}_{CB}^{C})}$, ${\displaystyle {\boldsymbol {K}}=diag({\boldsymbol {K}}_{CB}^{A},{\boldsymbol {K}}_{CB}^{B},{\boldsymbol {K}}_{CB}^{C})}$

${\displaystyle {\boldsymbol {p}}={\left[{\begin{array}{ccc}{\boldsymbol {p}}^{AT}&{\boldsymbol {p}}^{BT}&{\boldsymbol {p}}^{CT}\end{array}}\right]}^{T}={\left[{\begin{array}{ccccccccccc}{\boldsymbol {p}}_{n}^{A}{}^{T}&{\boldsymbol {u}}_{m}^{A}{}^{T}&{\boldsymbol {u}}_{s}^{A}{}^{T}&{\boldsymbol {p}}_{n}^{B}{}^{T}&{\boldsymbol {u}}_{m}^{Ba}{}^{T}&{\boldsymbol {u}}_{s}^{Ba}{}^{T}&{\boldsymbol {u}}_{m}^{Bb}{}^{T}&{\boldsymbol {u}}_{s}^{Bb}{}^{T}&{\boldsymbol {p}}_{n}^{C}{}^{T}&{\boldsymbol {u}}_{m}^{C}{}^{T}&{\boldsymbol {u}}_{s}^{C}{}^{T}\end{array}}\right]}^{T}}$,

${\displaystyle {\boldsymbol {f}}={\left[{\begin{array}{ccc}{\boldsymbol {f}}_{CB}^{AT}&{\boldsymbol {f}}_{CB}^{BT}&{\boldsymbol {f}}_{CB}^{CT}\end{array}}\right]}^{T}={\left[{\begin{array}{ccccccccccc}{\boldsymbol {0}}^{T}&{\boldsymbol {f}}_{m}^{A}{}^{T}&{\boldsymbol {f}}_{s}^{A}{}^{T}&{\boldsymbol {0}}^{T}&{\boldsymbol {f}}_{m}^{Ba}{}^{T}&{\boldsymbol {f}}_{s}^{Ba}{}^{T}&{\boldsymbol {f}}_{m}^{Bb}{}^{T}&{\boldsymbol {f}}_{s}^{Bb}{}^{T}&{\boldsymbol {0}}^{T}&{\boldsymbol {f}}_{m}^{C}{}^{T}&{\boldsymbol {f}}_{s}^{C}{}^{T}\end{array}}\right]}^{T}}$.

The connection relationship between the components does not expressed in Eq. (12). The components of modal coordinates ${\displaystyle {\boldsymbol {p}}}$ are not independent and cannot be directly used for the dynamic characteristics analysis of folding wing. It is necessary to eliminate the dependent modal coordinates according to the compatible conditions of the interface to obtain the whole structural dynamic model. The displacement compatible condition of the interface is as follows:

Considering the equilibrium condition of the force at the interface, the compatible condition of the force can be expressed as:

The torque generated by the torsion spring can be expressed as:

The independent mode coordinate transformation matrix is introduced, which can be expressed as follows:

One obtains the following expression:

where ${\displaystyle {\boldsymbol {\tilde {M}}}={\boldsymbol {\Pi }}^{T}{\boldsymbol {M\Pi }}}$ , ${\displaystyle {\boldsymbol {\tilde {K}}}={\boldsymbol {\Pi }}^{T}{\boldsymbol {K\Pi }}}$ , ${\displaystyle {\boldsymbol {q}}={\left[{\begin{array}{ccccccccc}{\boldsymbol {p}}_{n}^{A}{}^{T}&{\boldsymbol {u}}_{m}^{A}{}^{T}&{\boldsymbol {u}}_{s}^{A}{}^{T}&{\boldsymbol {p}}_{n}^{B}{}^{T}&{\boldsymbol {u}}_{s}^{Ba}{}^{T}&{\boldsymbol {u}}_{s}^{Bb}{}^{T}&{\boldsymbol {p}}_{n}^{C}&{\boldsymbol {u}}_{m}^{C}&{\boldsymbol {u}}_{s}^{C}\end{array}}\right]}^{T}}$ ,

${\displaystyle {\boldsymbol {\tilde {f}}}={\left[{\begin{array}{ccccccccc}{\boldsymbol {0}}^{T}&{\boldsymbol {0}}^{T}&K_{A}{\left({\boldsymbol {u}}_{s}^{Ba}-{\boldsymbol {u}}_{s}^{A}\right)}^{T}&{\boldsymbol {0}}^{T}&-K_{A}{\left({\boldsymbol {u}}_{s}^{Ba}-{\boldsymbol {u}}_{s}^{A}\right)}^{T}&K_{B}{\left({\boldsymbol {u}}_{s}^{Bb}-{\boldsymbol {u}}_{s}^{C}\right)}^{T}&{\boldsymbol {0}}^{T}&{\boldsymbol {0}}^{T}&-K_{B}{\left({\boldsymbol {u}}_{s}^{Bb}-{\boldsymbol {u}}_{s}^{C}\right)}^{T}\end{array}}\right]}^{T}}$ .

Obviously, the right-hand term ${\displaystyle {\boldsymbol {\tilde {f}}}}$ in Eq. (17) is related to the torsional spring stiffness ${\displaystyle K_{A}}$ , ${\displaystyle K_{B}}$ and the modal coordinates ${\displaystyle {\boldsymbol {q}}}$ . The dynamic analysis of wing structure is usually based on free vibration. Therefore, it is necessary to take ${\displaystyle {\boldsymbol {\tilde {f}}}}$ into account in the stiffness matrix of the structure. From the previous coordinate transformation, it can be seen that the displacement at the interface is still expressed in the physical space, which is conducive to the introduction of torsional spring stiffness. Here ${\displaystyle {\boldsymbol {\tilde {f}}}}$ is written in the form of a matrix:

where ${\displaystyle {\boldsymbol {K}}^{'}=\left[{\begin{array}{ccccccccc}{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&-{\boldsymbol {K}}_{A}&{\boldsymbol {0}}&{\boldsymbol {K}}_{A}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {K}}_{A}&{\boldsymbol {0}}&-{\boldsymbol {K}}_{A}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {K}}_{B}&{\boldsymbol {0}}&{\boldsymbol {0}}&-{\boldsymbol {K}}_{B}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}&-{\boldsymbol {K}}_{B}&{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {K}}_{B}\end{array}}\right]}$ , ${\displaystyle {\boldsymbol {K}}_{A}=diag(K_{A},\cdots ,K_{A})}$ ,

${\displaystyle {\boldsymbol {K}}_{B}=diag(K_{B},\cdots ,K_{B})}$ .In the independent modal coordinates, the complete dynamic equation is as follows:

The global mass matrix is tridiagonal and the stiffness matrix is quasi diagonal. The natural frequency and modal information of the original structure can be obtained by relevant matrix characteristic analysis methods, such as Lanczos method. After two coordinate transformations, the system can return to the physical coordinates, so as to obtain the vibration mode of the whole system in the physical coordinates, namely, the reconstruction of the component.

2.2 Mixed-loaded interface component mode synthesis for folding wing with rigid connection

The advantage of the fixed interface component modal synthesis method is that the method is relatively simple, which is conducive to compile, and the calculation accuracy is generally high, but the disadvantage is that the degree of freedom of the interface is retained after the synthesis. For a complex structure, the interface degree of freedom of substructure is often very high, and the reduced degree of freedom is still high. Benfield [17] proposed fixed-loaded interface component mode synthesis. It is necessary to divide the folding wing into master-slave substructures. Although some suggestions are given for the selection of the main substructure, the specific model should be treated concretely.

Therefore, this paper assumes that the inner wing is the master component, and the fuselage and outer wing are the slave components. Here, main mode of fixed-loaded interface is adopted for the master component, and the main mode and constraint mode of fixed interface are adopted for the slave component. Finally, the mixed-loaded interface component modal synthesis for folding wing with rigid connection is established.

For the sake of simplicity, it is assumed that the fuselage and the inner wing, the inner wing and the outer wing are rigidly connected, that is, all the degrees of freedom of the interface nodes are defined by MPC. Different components have been partitioned according to internal and interface degrees of freedom. The subscripts m and s in Eqs. (3), (4) and (5) are all represented by subscript j, so the simpler expression is as follows:

For components A and C, the main mode and constraint mode of fixed interface are adopted, and the expression is similar to Eqs. (6) and (10). A more concise form is given here:

For the main mode of component B, the influence of components A and C should also be considered. The internal degrees of freedom of components A and C can be condensed to the interface by static condensation. When the influence of inertia is not considered, the first row of Eqs.(20) and (22) can be used to calculate the static equilibrium:

Eqs.(25) and (26) also show that the process of static condensation is to establish the relationship between internal degrees of freedom and interface degrees of freedom by the constraint modes of the fixed interface. The displacement compatibility condition of the interface is as follows

Consequently, one obtains:

Eqs.(28) and (29) show that all the degrees of freedom of components A and C can be expressed by the degrees of freedom of component B. So component B is the master component, and A and C are the slave components, which well reflects the connection between components. By substituting the two equations above into the dynamic equations of components A and C respectively, namely Eqs. (20) and (22), and pre-multiplying ${\displaystyle {\boldsymbol {T}}_{1}^{T}}$ and ${\displaystyle {\boldsymbol {T}}_{2}^{T}}$ on both sides , one can get the following results:

where ${\displaystyle {\boldsymbol {\overline {M}}}^{A}=\left[{\begin{array}{ccc}{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {\overset {\mbox{ˆ}}{M}}}_{jj}^{A}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\end{array}}\right]}$ , ${\displaystyle {\boldsymbol {\overline {K}}}^{A}=\left[{\begin{array}{ccc}{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {\overset {\mbox{ˆ}}{K}}}_{jj}^{A}{}_{jj}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\end{array}}\right]}$ , ${\displaystyle {\boldsymbol {\overline {M}}}^{C}=\left[{\begin{array}{ccc}{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {\overset {\mbox{ˆ}}{M}}}_{jj}^{C}\end{array}}\right]}$ , ${\displaystyle {\boldsymbol {\overline {K}}}^{C}=\left[{\begin{array}{ccc}{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {0}}&{\boldsymbol {\overset {\mbox{ˆ}}{K}}}_{jj}^{C}\end{array}}\right]}$ , ${\displaystyle {\boldsymbol {\overset {\mbox{ˆ}}{M}}}_{jj}^{A}={\boldsymbol {\varphi }}_{c}^{AT}{\boldsymbol {M}}_{ii}^{A}{\boldsymbol {\varphi }}_{c}^{A}+}$${\displaystyle {\boldsymbol {\varphi }}_{c}^{AT}{\boldsymbol {M}}_{ij}^{A}+}$${\displaystyle {\boldsymbol {M}}_{ji}^{A}{\boldsymbol {\varphi }}_{c}^{A}+{\boldsymbol {M}}_{jj}^{A}}$ , ${\displaystyle {\boldsymbol {\overset {\mbox{ˆ}}{K}}}_{jj}^{A}={\boldsymbol {\varphi }}_{c}^{AT}{\boldsymbol {K}}_{ii}^{A}{\boldsymbol {\varphi }}_{c}^{A}+}$${\displaystyle {\boldsymbol {\varphi }}_{c}^{AT}{\boldsymbol {K}}_{ij}^{A}+}$${\displaystyle {\boldsymbol {K}}_{ji}^{A}{\boldsymbol {\varphi }}_{c}^{A}+{\boldsymbol {K}}_{jj}^{A}}$ ,

${\displaystyle {\boldsymbol {\overset {\mbox{ˆ}}{M}}}_{jj}^{C}={\boldsymbol {\varphi }}_{c}^{CT}{\boldsymbol {M}}_{ii}^{C}{\boldsymbol {\varphi }}_{c}^{C}+}$${\displaystyle {\boldsymbol {\varphi }}_{c}^{CT}{\boldsymbol {M}}_{ij}^{C}+}$${\displaystyle {\boldsymbol {M}}_{ji}^{C}{\boldsymbol {\varphi }}_{c}^{C}+{\boldsymbol {M}}_{jj}^{C}}$ , ${\displaystyle {\boldsymbol {\overset {\mbox{ˆ}}{K}}}_{jj}^{C}={\boldsymbol {\varphi }}_{c}^{CT}{\boldsymbol {K}}_{ii}^{C}{\boldsymbol {\varphi }}_{c}^{C}+}$${\displaystyle {\boldsymbol {\varphi }}_{c}^{CT}{\boldsymbol {K}}_{ij}^{C}+}$${\displaystyle {\boldsymbol {K}}_{ji}^{C}{\boldsymbol {\varphi }}_{c}^{C}+{\boldsymbol {K}}_{jj}^{C}}$ .

Derivation of the right end term of Eq. (30) is as follows: