Numerical investigation on aerodynamic and inertial couplings of flexible spinning missile with large slenderness ratio

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.1 (1-14)

Aerospace Science and Technology ••• (••••) ••••••

Contents lists available at ScienceDirect

Aerospace Science and Technology www.elsevier.com/locate/aescte

Numerical investigation on aerodynamic and inertial couplings of flexible spinning missile with large slenderness ratio Li Heng ∗ , Ye ZhengYin School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

a r t i c l e

i n f o

Article history: Received 16 July 2019 Received in revised form 25 October 2019 Accepted 21 November 2019 Available online xxxx Keywords: Dynamic response Spinning missile Lagrange equation Six degree of freedom motion Structural vibration Mesh deformation

a b s t r a c t There are aerodynamic and inertial couplings in rigid-body motion and structural vibration of a flexible spinning missile in flight, and when slenderness ratio is larger, the couplings are more obvious. As for the flexible spinning missile with large slenderness ratio, aerodynamic and inertial couplings should be considered. Based on rigid-motion mesh and radial-basis-function (RBF) mesh deformation techniques, unsteady Reynolds-averaged Navier-Stokes (URANS) Equations and flight dynamics equations of flexible spinning missile derived from Lagrange equation are coupled simultaneously to simulate the dynamic response of flexible spinning missile with large slenderness ratio. The URANS equations are solved by Computational Fluid Dynamics (CFD) technique with the in-house code. Not only the aerodynamic coupling between rigid-body six degree of freedom (DOF) motion and structural vibration are included in the flight dynamics equations of flexible spinning missile, but also the variation of inertia tensor and extra moment terms caused by structural vibration are included, too. More importantly, the extra force term caused by angular acceleration of rigid-body motion, Coriolis and centrifugal loading terms caused by angular velocity of rigid-body motion are included. The rigid-motion mesh and RBF mesh deformation techniques are both based on unstructured mesh. The rigid-motion mesh is adopted to treat the large rigid-body motion due to flight dynamics, while the RBF mesh deformation is employed for flexible structural deformation caused by aeroelasticity. Numerical results of free flight case and aeroelastic case are well agreed with the experimental results, which validates the numerical method. A missile model with X-X configuration is constructed to quantitatively investigate the aerodynamic and inertial couplings between rigid-body motion and structural elastic vibration, and aerodynamic force, extra generalized force terms caused by rigid-body motion and extra moment terms caused by structural vibration are all investigated. Final results show that for the critical stable case studied in this paper, the aerodynamic force plays a major role in coupling effects between rigid-body motion and structural vibration, and the inertial coupling terms in rigid-body motion caused by structural vibration are negligible compared with the aerodynamic force. However, the inertial coupling terms in structural vibration due to rigid-body motion are not negligible compared with the aerodynamic force. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Lighter structure and larger slenderness ratio are needed for the design of missile to increase flight range and flight payloads. However, unfortunately, lighter structure and larger slenderness ratio consequently result in lower structural stiffness, smaller natural frequency and larger structural deformation. In general, coupling between rigid-body motion and structural elastic vibration is negligible, and rigid-body motion and structural elastic vibration can

*

Corresponding author. E-mail address: [email protected] (H. Li).

https://doi.org/10.1016/j.ast.2019.105582 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

be separately investigated, and they are called flight dynamics and aeroelasticity, respectively. However, as for the flexible missile with large slenderness ratio, there is aerodynamic coupling between rigid-body motion and structural vibration that is usually need to be considered. That is to say, structural vibration is affected by rigid-body motion, because rigid-body motion changes generalized forces of structural vibration. Similarly, rigid-body motion is affected by structural vibration, because structural vibration changes forces and moments of rigid-body motion. In a word, rigid-body motion and structural vibration are coupled via aerodynamic forces, and the coupling mechanism is called aerodynamic coupling. Furthermore, for the slender flexible missile with large angular velocity, the coupling mechanism is more complicated. Not

JID:AESCTE

2

AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.2 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

only aerodynamic coupling is required to be considered, but also inertial coupling is required to be considered, too. Structural vibration will introduce extra inertial moment terms in rigid-body equations of motion, and rigid-body motion will introduce extra inertial generalized force terms in structural vibration equations. Rigid-body motion and structural vibration are coupled together via aerodynamic coupling and inertial coupling. As a rigid body, a missile flying in air is a flight dynamics problem which can be solved by coupling aerodynamics equations and rigid-body six DOF equations of motion, because the process is associated with aerodynamic force and inertial force [1–6]. And as a flexible body, a missile without rigid-body motion is a pure aeroelastic problem which could be solved by coupling aerodynamics equations and structural dynamics equations [7–14], because the process is only associated with aerodynamics and structural dynamics. However, as for a flexible missile with rigid-body motion, especially a spinning missile with large rotational angular velocity, it is more complicated and more disciplines should be considered in the dynamic process. Since the problem is involved in several interactions among aerodynamics, flight mechanics and aeroelasticity, previous studies mainly handled the problem with simplified models which had various assumptions [15–31]. Waszak et al. [16] developed the nonlinear equations of motion for an elastic airplane from first principles which consider aerodynamic coupling, and presented an example to parametrically address the effects of flexibility for a generic elastic aircraft. Buttrill et al. [17] proposed a mathematical model which integrates nonlinear rigid-body flight mechanics and linear aeroelastic dynamics, and the model considers aerodynamic coupling and inertial coupling, and can be simplified to Waszak’s model when satisfying extra assumptions. Buttrill et al. also presented a modal parameter that characterizes the level of inertial coupling between elastic momentum and rigid-body angular momentum. Zeiler et al. [18] utilized nonlinear strain-displacement relations to improve Buttrill’s model, and examined the effects of flexibility, modal damping and the momentum coupling terms on the tumbling dynamics of the body. Riso et al. [30] developed an integrated formulation of flight dynamics and aeroelasticity allowing large body frame motions coupled with small elastic displacements to analyze free-flying aircraft which are described by detailed finite element models. In addition to this, there were also a large number of other mathematical models that were used to investigate these problems [15,19–29,31]. Over the years, with the development of computer science and numerical algorithm, CFD has been introduced to investigate these problems. Blades et al. [32] coupled CFD and CSD (Computational Structural Dynamics) to investigate the effects of aeroelasticity on the aerodynamic performance of the spinning missile, and pointed out the peak values of side force, yawing moment and pitching moment are 5-10 percent different from that of the rigid model. Schütte et al. [33] coupled aerodynamic, flight mechanics and aeroelastic computations to simulate the unsteady aerodynamics of a free-flight aeroelastic combat aircraft. Abbas et al. [12] coupled CFD and CSD to investigate the effects of structural static deformation of slender rocket on lift coefficient, drag coefficient, pitching moment coefficient and center of pressure location, and pointed out that the flight trajectory may be affected by the change of these aerodynamic coefficient and stability. Hua et al. [34,35] coupled CFD, rigid-body dynamics equations and static aeroelastic equation based on modal superposition method to investigate the effects of elastic deformation on mass location, inertial moment and missile trajectory, furthermore, flight dynamics based on closed-loop feedback control was also studied [36]. Li et al. [37] employed quasi-steady CFD method to construct an aerodynamic model, and coupled the model with flexible dynamics equation derived by Lagrange equation to investigate the effects of elasticity on missile trajectory and vibration. Yin et al. [38,39] employed

unsteady time-accurate simulation to obtain the aerodynamic characteristics of a spinning projectile given continuously elastic deformation. Yang et al. [40,41] investigated the interference aerodynamics caused by the wing elasticity during store separation. Lu et al. [42] studied the aerodynamic coupling effects of spinning and coning motions for a finned vehicle. The aerodynamic coupling effects are mainly produced by the shockwaves and expansion waves around fins, which are significant at low Mach numbers and high coning and spin rates. Chen et al. [13] developed an efficient and accurate computational method by using ANSYS workbench which can be used to analyze the aerodynamic characteristics and static aeroelastic of the slender rocket. Li et al. [43] investigated the effects of rotational motion on dynamic aeroelasticity of a flexible spinning missile based on CFD method, and indicated the mechanism by which the rotational motion leads to the coupling of lateral modes and longitudinal modes and changes the structural natural frequencies. Although couplings of flight dynamics and aeroelasticity of flexible missile and aircraft have been investigated by lots of researches, the aerodynamic loads which dominate the rigid-body six DOF motion and structural vibration were normally predicted with engineering methods such as perturbation method and liftingline method. Furthermore, a large number of researches involved with coupling of rigid-body motion and structural elastic deformation focused on the change of aerodynamic characteristics or flight trajectory due to structural static deformation or given elastic deformation, but the dynamic response of structural vibration were rarely considered. In addition, plenty of investigations involved with rigid-body motion and structural vibration of aircraft and missile focused on aerodynamic coupling, and inertial coupling is usually negligible. That is to say, angular velocity of rigid-body motion is usually small and negligible, only coupling of rigid-body pitching motion and structural longitudinal modes is investigated, while effects of rotational motion are less investigated. Due to the limitations mentioned above, in this paper, based on the frame of unstructured mesh, the rigid-motion mesh and RBF mesh deformation techniques are adopted to treat the large rigid-body motion due to flight dynamics and structural flexible deformation caused by dynamic aeroelasticity, respectively. URANS equations are solved by CFD method to obtain the instant aerodynamic forces at each physical time. Flight dynamics equations of flexible spinning missile that simultaneously consider aerodynamic and inertial couplings of rigid-body motion and structural vibration are solved to simulate the dynamic response of the spinning missile with large slenderness ratio. The investigation is focused on the aerodynamic coupling and inertial coupling between rigidbody motion and structural vibration, and aerodynamic coupling terms, inertial coupling terms in rigid-body motion and structural vibration are all quantitatively investigated. 2. Numerical method 2.1. Coupling strategy and framework of the flight dynamics of flexible spinning missile In this paper, unsteady flow field and the couplings between fluid/rigid-body/structure are investigated using the in-house code [43]. Fig. 1 shows flowchart to solve the flight dynamics problem of the flexible spinning missile. When solving this problem, three modules are mainly employed: grid-treating module, CFD solver module and flight dynamics module of flexible spinning missile. The grid-treating module is employed to obtain new computational domains needed by the CFD solver module, which contains two submodules: rigid-motion mesh submodule and mesh

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.3 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

3

Fig. 1. Flowchart of the solving methodology.

deformation submodule. The rigid-motion mesh submodule is able to adjust the new computational domains according to the large rigid-body displacement due to flight dynamics, while the mesh deformation submodule is capable of adapting boundary deformation of the meshes caused by aeroelasticity. The CFD solver module is used to obtain the numerical solution of the fluid governing equations at every physical time step. In this paper, inviscid Euler equations are solved based on URANS equations. With the solution obtained in CFD solver, the aerodynamic forces and moments on the spinning missile needed by the solution of rigid-body six DOF equations of motion are computed by integrating the pressure on the surface of the missile. At the same time, generalized forces needed by the solution of the structural dynamics equations could be obtained, too. Flight dynamics module of flexible spinning missile is mainly applied to solve aerodynamic and inertial couplings between rigidbody motion and structural elastic vibration, and the module is loosely coupled with the CFD solver module. The module contains two submodules: rigid-body six DOF motion submodule and structural dynamics submodule. These two submodules are solved by iterative method, and finally aerodynamic and inertial couplings between the two submodules can be obtained. The result of this module can be divided into two parts: rigid-body motion result and structural vibration result. The rigid-body motion result contains rigid-body displacement, which can be combined with rigidmotion mesh method to translate or rotate mesh point to new position. While structural vibration result contains boundary deformation, which can be combined with mesh deformation method to adapt volume mesh. The displacement of any mesh point includes two parts: the displacement resulting from rigid-motion mesh and the displacement resulting from mesh deformation. Correspondingly, the flow field is composed of synthetic effects of both rigid-body motion and aeroelasticity. Various components of this framework are described as follows.

2.2. Grid-treating module There are two submodules in grid-treating module: rigidmotion mesh submodule and mesh deformation submodule. The techniques used in the two submodules are introduced, respectively. 2.2.1. Rigid-motion mesh method The basic idea of the rigid-motion mesh method is to update the location of all nodes of the computational mesh according to the translational displacement of the center of mass and the rotational displacement around the center of mass. Define r0 = [x0 , y 0 , z0 ]T as initial position of a mesh node, and suppose the initial attitude angle φ , θ , ψ are all 0. New position of the mesh node at any time r = [x, y , z]T can be defined as

r = r0 + rcg + rrot

(1)

Here, rcg is the translation of the center of mass, which is same for all nodes in mesh. rrot is the displacement owing to rotation around the center of mass, which is different at different position, and can be written as

rrot = TB−I (r0 − rcg,0 ) − (r0 − rcg,0 )

(2)

Here, rcg,0 is the position of the center of mass, the transition matrix from body coordination to inertial coordination TB−I is

⎡

cos ψ TB−I = ⎣ sin ψ 0

⎡

1

⎤⎡ − sin ψ 0 cos θ cos ψ 0 ⎦⎣ 0 0 1 − sin θ ⎤ 0

⎤

0 sin θ 1 0 ⎦ 0 cos θ

0

× ⎣ 0 cos φ − sin φ ⎦ 0 sin φ cos φ

(3)

Here, φ , θ and ψ are rolling angle, pitching angle and yawing angle, respectively. The distinct advantage of the rigid-motion mesh

JID:AESCTE

AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.4 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

4

method is that the computational mesh can be obtained directly without being reconstructed during the numerical simulation process and what we need to do is just some transformation on coordinate position. The method is very efficient, and is able to keep the topological relationship and the quality of the computational meshes. 2.2.2. Mesh deformation method The main purpose of the mesh deformation method is to diffuse the displacements of wall surfaces to the interior mesh nodes of the computational zones. In this paper, mesh deformation method is based on RBF mesh deformation method [44,45]. In general, the number of nodes in boundary is large, and there will be a large amount of computational cost in the process of mesh deformation. In order to improve the mesh deformation efficiency, a “double-edge” greedy supporting point selection algorithm [46] using a multi-level subspace method is adopted to reduce data. In the condition of meeting the requirement of the grid precision, this method can effectively reduce computational cost. The RBF mesh deformation method with greedy point selection algorithm is robust and efficient, and the mesh quality of volume element in computational domains can be well kept.

3. The total rotational displacement due to elastic deformation of any lumped mass with respect to its undeformed orientation is small. 4. Deformation is described by a linear sum of mode shapes multiplied by their time-dependent participation coefficients. 5. Gravity is constant over the aircraft. 6. The sea-level local Earth frame is assumed to be an inertial frame. The six assumptions are easy to be satisfied and are appropriate for most of cases. The equations in practical mean-axes reference frame [47] can be written as:

˙ = F + mg − m(ω × V) mV

(5)

˙ + h jk η j η¨ k = L − ω × Jω − ˙Jω − h jk η˙ j η˙ k − ω × h jk η j η˙ k Jω

(6)

˙ · h jk ηk = Q j − K jk ηk + 2ω · h jk η˙ k M jk η¨ k − ω

 1 + ωT J j + 2 J jk ηk ω

(7)

2

Here, the terms such as x jk yk follows Einstein summation convention, which means indicial sum over k, that is to say

2.3. CFD solver module

x jk yk = x j1 y 1 + x j2 y 2 + ... + x jN y N

The flow governing equations used in this paper are threedimensional URANS equations. In order to consider the rigid-body motion and structural elastic vibration, the flow control equations described by arbitrary Lagrange-Euler method (ALE) are employed. For a grid cell with arbitrary motion and deformation, the integral form of URANS equations in Cartesian coordinate system is given as

x˙ = is time rate of change of variable x and x¨ = is time rate of change of variable x˙ ; m = i dmi is total mass of the aircraft; i indicates sum over the lumped mass; the subscript i indicates lumped mass id i; dmi is mass of lumped mass id i; V is velocity ˙ is acceleration in body frame; F is aerodynamic in body frame; V force; g is acceleration of gravity; ω is angular velocity in body frame; J is real-time two-order inertia tensor, and can be expressed as:

∂ ∂t

˚

¨









¨

FV (Q) · ndS

F Q, Vgrid · ndS =

QdV + ∂

T

(4)

∂

Here, Q = ρ ρ u ρ v ρ w e is conservation variable; ρ is density; u, v, w are velocity components in three different directions; e is total energy; F(Q, Vgrid ) is inviscid flux and FV (Q) is viscous flux; Vgrid is mesh velocity of a control volume due to rigid-body motion and structural deformation;  represents a control volume and ∂  represents the boundary of the control volume ; n is outer normal direction of the boundary face ∂  of the control volume . In this paper, the finite volume method based on cell-center method are employed on hybrid unstructured mesh. Physical time is marched by dual time stepping method and pseudo time is marched by Gauss-Seidel iteration. Since flow field of computational domains has been obtained, the aerodynamic forces can be obtained by integrating the pressure and skin friction over the body, and meanwhile, the generalized forces used by aeroelasticity can be obtained, too. 2.4. Flight dynamics module of flexible spinning missile Flight dynamics equations of flexible spinning missile are based on Buttrill’s equations [17] derived from Lagrange equation. The Buttrill’s equations are full governing equations of flight dynamics of flexible aircraft, which are based on the following six assumptions: 1. The aircraft is idealized as a collection of lumped-mass elements, each being a finite rigid body, and each having an associated mass and moments of inertia. 2. The elastic restoring force resulting from displacement of any mass element is linear and proportional to that displacement.

(8) d x˙ dt

dx dt

J = J0 + J j η j +

1 2

2 J jk η j ηk

(9)

Here, the subscripts j and k indicate elastic modes j and k, respectively; J0 is two-order inertia tensor in undeformed condition, and is defined as

J0 =

{(ri · ri )I − ri rTi }dmi

(10)

i

where, ri is position vector locating the center of the lumped mass i respect to the body frame when the body is in the undeformed reference condition; I is two-order unit tensor. J j is first partial derivative of the inertia tensor with respect to elastic mode j, and it is a two-order tensor defined as:

J j =

{(2ri · φ i j )I − φ i j rTi − ri φ Ti j }dmi

(11)

i

2 J jk indicates the effect of the second-order deformation on the total time-varying inertia matrix of the aircraft, and it is defined as:

2 J jk = A jk + Akj

(12)

where  J jk =  Jkj . A jk is a two-order tensor defined as: 2

A jk =

2

{(φ i j · φ ik )I − φ i j φ Tik }dmi

(13)

i T where A jk = Akj . η j is generalized displacement of mode j; η˙ j is generalized velocity of mode j; η¨ j is generalized acceleration of ˙ is angular acmode j; φ i j is mode j shape at lumped mass i; ω celeration in body frame; h jk indicates energy coupling between

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.5 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

5

rigid-body angular velocity and modal velocity in mode k due to deformation in mode j, and it is a vector defined as:

h jk =

φ i j × φ ik dmi

(14)

i

where h jk = −hkj . L is total applied moment on the aircraft; ˙J = J j η˙ j + 2 J jk η j η˙ k is time derivative of inertia tensor J; M jk = i φ i j · φ ik dmi is generalized mass coupling between elastic mode j and mode k; K jk is generalized stiffness coupling between elastic mode j and mode k; Q j is generalized aerodynamic force of mode j. What should be mentioned here is mode shapes are all orthogonal to each other, and the matrices M and K are diagonal, and the relationship between the elements of principle diagonal in the two matrices is

K jj = M jjω jω j

j = 1, 2, ..., N

(15)

Here, ω j is undamped structural natural frequency of mode j without rotational motion, and N is the number of modes. Equations (5), (6) and (7) are flight dynamics equations of flexible aircraft, which not only consider the aerodynamic coupling between rigid-body six DOF motion and structural vibration, but also consider the variation of inertia tensor and extra moment terms caused by structural vibration, and even more importantly, the extra generalized force term caused by angular acceleration of rigid-body motion, Coriolis and centrifugal loading terms caused by angular velocity of rigid-body motion are considered, too. It’s easy to find that there is no inertial coupling between translational equation (Eq. (5)) and elastic equation (Eq. (7)) because of the linearized mean-axes constraints [17], but there is inertial coupling between angular equation (Eq. (6)) and elastic equation (Eq. (7)). Every term in angular equation means:

˙ : Angular acceleration term in rigid-body six DOF equations 1. Jω of motion. 2. h jk η j η¨ k : Extra moment term caused by generalized acceleration of structural vibration, which is an inertial coupling term. 3. L − ω × Jω : Aerodynamic moment and angular motion term in rigid-body six DOF equations of motion. 4. −˙Jω : Extra term caused by variation of inertia tensor with time. In general, it is zero, however, it’s not zero when the effect of elastic deformation on mass distribution is considered, because inertia tensor will be different with different mass distribution. And it can be regarded as an inertial coupling term. 5. −h jk η˙ j η˙ k − ω × h jk η j η˙ k : Extra moment term caused by generalized velocity of structural vibration, which is an inertial coupling term. Elastic equation (7) could be also written as

˙ · h jk − M jk η¨ k − 2ω · h jk η˙ k + ( K jk − ω 1

= Q j + ω T J j ω 2

1 2

ωT 2 J jk ω)ηk

Fig. 2. Geometric parameter and inertia properties of spinning projectile.

5. 2ω · h jk η˙ k : Extra Coriolis term caused by angular velocity of rigid-body motion, acting as a damping term, and it is an inertial coupling term. 6. 12 ωT J j + 2 J jk ηk ω = 12 ωT J j ω + 12 ωT 2 J jk ωηk : Extra centrifugal loading term caused by angular velocity of rigid-body motion. The term 12 ωT J j ω acts as a generalized force term, while the term

1 2

ωT 2 J jk ωηk acts as a stiffness term.

When elastic deformation and elastic vibration are negligible, Eq. (5)–(7) can be simplified to standard rigid-body six DOF equations of motion:

˙ = F + mg − m(ω × V) mV ˙ = L − ω × J0 ω J0 ω

(17)

Similarly, when rigid-body motion is negligible, Eq. (5)–(7) can be simplified to structural dynamics equation:

M jk η¨ k = Q j − K jk ηk

(18)

Structural damping term is not included in Eq. (18). And Eq. (17) and Eq. (18) are used by verification cases described in next section. The Eq. (5) and Eq. (6) both are first-order ordinary differential equations, while the Eq. (7) is a second-order ordinary differential equation, and all the equations can be solved by hybrid linear multi-step (HLM) scheme [48] in time domain. In addition, there is coupling between Eq. (6) and Eq. (7), therefore, in this paper, Eq. (6) and Eq. (7) are recursively solved in every physical time ˙ is converstep until the residual error of angular acceleration ω gent. 3. Verification cases of numerical method Because of the limitation of experimental facility and the space of the test section in wind tunnel, it is extremely difficult to conduct flight experiments of flexible vehicles for a long time. In the open literatures, there are few cases involved in the aerodynamics/rigid-body/structure coupled simulation. Accordingly, the coupling issue related to aerodynamics, flight dynamics and structural dynamics is decomposed into the two issues related to aerodynamics/rigid-body coupling and aerodynamics/structure coupling, respectively. 3.1. Supersonic spinning projectile

(16)

According to Eq. (16), the meaning of each term in elastic equation can be understood more clearly. Every term in Eq. (7) means: 1. M jk η¨ k : Inertial term in structural dynamics equations. ˙ · h jk ηk : Extra force term caused by angular acceleration 2. −ω of rigid-body motion, acting as a stiffness term, and it is an inertial coupling term. 3. Q j : Aerodynamic force term in structural dynamics equations. 4. − K jk ηk : Stiffness term in structural dynamics equations.

ARL supersonic spinning projectile [3] is chosen as a verification case to verify aerodynamics/rigid-body coupling. The geometric parameter (unit is millimeter) and inertia properties are depicted in Fig. 2, computational mesh is depicted in Fig. 3, and initial flight conditions are depicted in Table 1. The equivalent free flight initial Mach number is 3.04, and the Reynolds number per unit length is 7.08e7/m. One equation SA turbulent model is used to enclose the turbulence equations. Fig. 4 shows pitching angle with flight range x. The pitching angle is of cyclical fluctuation with amplitude decreasing. Fig. 5

JID:AESCTE

AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.6 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

6

Fig. 3. Computational mesh of spinning projectile. Table 1 Initial conditions of finned projectile.

X Y Z

Distance of C.G. (m)

Speed of C.G.(m/s)

Euler angle (rad)

Angular velocity (rad/s)

4.593 0.2 0.159

1030.81

−22.06 −86.28

0.00 −0.088 0.023

2518.39 52.802 −22.233

Fig. 5. Yawing angle with flight range.

Fig. 4. Pitching angle with flight range.

shows yawing angle with flight range x. Yawing angle is smaller than pitching angle, and it also changes periodically. Fig. 6 and Fig. 7 show the distances of y and z with flight range x, respectively. The distances of y and z both increase with the flight range. The oscillation of the distance in Z direction is larger than that in Y direction, this is mainly because the amplitude of the pitching angle is larger than that of the yawing angle in the flight process. Fig. 8 and Fig. 9 indicate the angle of attack and sideslip angle with flight range, respectively. The calculated results are in good agreement with experimental results. Therefore, the code can give a reliable result for the rigid-body motion simulation, and can be used in simulations related to aerodynamics/rigid-body coupling.

Fig. 6. The distance of y with flight range.

3.2. AGARD 445.6 dynamic aeroelastic case The dynamic aeroelastic response of AGARD 445.6 weakened modal wing case [49] is chosen to verify aerodynamics/structure coupling. The incoming free flow Mach number is 0.901, the angle of attack and the sideslip angle are both 0 degree. Euler equations are used as flow governing equations, and spatial discretization method is Roe scheme. The first four modes are considered during the simulation. Physical time step is 1/40 of the natural period of the highest order mode, and the sub-iteration number is 200 in

Fig. 7. The distance of z with flight range.

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.7 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

order to make sure the residual convergent, therefore, the simulation precision can be ensured. As shown in Fig. 10, when non-dimensional velocity V cf is about 0.342, the generalized displacement response is in equal amplitude oscillation, therefore, the non-dimensional flutter critical speed is 0.342, which agrees well with experimental result 0.37. Therefore, the code can give a reliable result for the simulations related to aerodynamics/structure coupling.

7

Table 2 Inertia properties of the spinning missile. Centroid position relative to head (m)

2.07, 0.00, 0.00

Length (m) Diameter (m) Mass (kg) Moment of inertia (kg·m2 )

4.74 0.20 264.90 1.79 500.85 500.85

Jxx Jyy Jzz

4. Computational results and analysis 4.1. Physical properties of the model

Fig. 8. Angle of attack with flight range.

Fig. 9. Sideslip angle with flight range.

Fig. 10. Generalized displacements history at V cf = 0.342.

The geometric shape of the spinning missile is presented in Fig. 11, while inertia properties are shown in Table 2. The diameter of the missile body is 0.2 m, the total length of the missile is 4.74 m, and the slenderness ratio is 23.7. The sweep angle of head fins are 45 degree, while the sweep angle of tail fins are 35 degree. The far boundary mesh, wall boundary mesh and space mesh slice are shown in Fig. 12. The number of the volume cells of the mesh is about 425 thousand, while the number of the volume nodes of the mesh is about 79 thousand. The incoming free flow Mach number is 2.5, and flight altitude is 8 km. Spatial scheme is AUSM+, while pseudo time is marched by Gauss-Seidel iteration, and Euler equations are used as flow governing equations. Both two submodule equations of flexible flight dynamics are solved by second-order precision of HLM scheme [48], and they are recursively solved until the residual error of angular acceleration is convergent to a specific value such as 1.0e-13 used in this paper. The structural dynamics analysis software ANSYS is used for modal analysis, and the first two modes are chosen for further calculation and analysis. The reason for only the first two modes are chosen is that lower modes usually play a dominant role in structural deformation and the previous work [43] about the effects of rotational motion on dynamic aeroelasticity verifies this assumption. Fig. 13 shows natural frequencies and mode shapes, and here the generalized displacements are both 1.0. Structural mode shapes are calculated only for the missile body, and the head and tail fins are supposed as rigid bodies, and the mode shapes of the fins are interpolated from the missile body. Viewed in body frame, the first mode mainly vibrates in lateral direction, while the second mode mainly vibrates in longitudinal direction, and they are both first bending modes. When solving the flight dynamics equations of flexible spinning missile depicted by Eq. (5), Eq. (6) and Eq. (7), there are three terms h, J and 2 J that need to be evaluated beforehand. The terms h and J are both three order tensors, while the term 2 J is a four order tensor. The three terms are very complicated and are related to mass distribution and mode shapes. In this paper, the three terms are evaluated by numerical method. The mesh used by the software ANSYS in modal analysis is exported and used to discretize the missile, then a collection of lumped-mass elements are obtained, and the mode shapes at lumped-mass center can be obtained by interpolation method. In this paper, the RBF interpolation method (Radial basis function for interpolation/smoothing scattered data) is chosen to interpolate mode shapes at each

Fig. 11. Geometric parameter (unit is millimeter).

JID:AESCTE

AID:105582 /FLA

8

[m5G; v1.261; Prn:28/11/2019; 8:32] P.8 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

Fig. 12. Computational mesh.

Fig. 13. The first two modes of structure.

lumped-mass center. Now, since the location and mode shapes of each lumped mass have already been obtained, the three terms can be evaluated according to the definitions. 4.2. Results and analysis Initial conditions are important when simulating dynamic response of a flexible spinning missile. Initial flight angle of attack is 2.07 degree with small tail fin deflection angle. In this state, the missile is in rigid-body longitudinal force equilibrium. In other words, as a rigid body, lift of the missile is equal to gravity, and pitching moment (in body frame) of the missile relative to the center of mass is zero. Initial pitching angular velocity as a disturbance is 0.2 rad/s, and initial yawing angular velocity is 0 rad/s.

Initial structural generalized displacements and velocities of the two modes are all zero. Throughout the simulation, the spinning missile keeps flight velocity stable. That is to say, the missile’s flight velocity and flight altitude is invariable in the whole simulation process. The rotational angular velocity is 31.3 rad/s, which is 1/4 of first structural vibration frequency, and it’s invariable in the whole simulation process, too. Therefore, the aerodynamic and inertial couplings between rigid-body motion and structural vibration at specific flight velocity and rotational angular velocity can be investigated. Physical time step should be chosen according to flow field, rotational angular velocity and structural natural frequencies. Fig. 14 indicates the effect of real time step on dynamic response of pitching angle (Y direction) and yawing angle (Z direction), while Fig. 15

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.9 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

Fig. 14. History of pitching angle and yawing angle with different real time step.

Fig. 15. History of the first generalized displacement with different real time step.

Fig. 16. History of the second generalized displacement with different real time step.

and Fig. 16 indicate the effect of real time step on dynamic response of first and second generalized displacements, respectively. As shown in Fig. 14, Fig. 15 and Fig. 16, when the time step is 0.14e-3 second, time step refinement has negligible effects on dynamic response of pitching angle, yawing angle and structural generalized displacements. Therefore, in this paper, physical time step is chosen as 0.14e-3 second, which is about 1/360 the structural natural period of the first mode, and the spinning missile will rotate 0.25 degree in each real time step. The simulation of the dynamic response starts from a converged flow field calculated by steady CFD method, and the pseudo iteration number of CFD solver module in the simulation of the dynamic response is 100 per real time step in order to ensure pseudo iterations convergent.

9

Fig. 17. History of pitching angle and yawing angle.

Fig. 18. History of generalized displacements.

4.2.1. Dynamic responses of flexible spinning missile Fig. 17 indicates dynamic responses of pitching angle and yawing angle. Under the action of initial pitching angular velocity, pitching angle increases firstly, and then is gradually into cyclical oscillation with invariable amplitude. However, as for yawing angle, there is no initial yawing angular velocity, and the yawing angle gradually converges to oscillation with invariable yawing amplitude. In final stage, amplitude and period of oscillation of pitching angle and yawing angle are almost the same. Fig. 18 indicates structural generalized displacement responses of the two modes. The initial generalized displacements are both zero, and there is no disturbance. They increase firstly, and then are both in cyclical oscillation with invariable amplitude, and more details are shown in Fig. 19 and Fig. 20. In final stage, the amplitude and period of oscillation of generalized displacement of the two modes are almost the same. However, the peak value of oscillation of the generalized displacement of the two modes are not critically equal, the reason is mainly due to the geometric asymmetry caused by the existence of fin deflection angle. What should be mentioned here is that the stability of the dynamic response is affected by the rotational angular velocity. For this case, the dynamic responses are critical stable, this is to say, in final stage, the amplitude of the pitching angle, yawing angle and generalized displacements neither increase nor decrease. However, when the rotational angular velocity varies, the amplitudes of the pitching angle, yawing angle and structural generalized displacements may all gradually increase or decrease, in other word, they are divergent or convergent. For this paper, the critical stable state is selected to quantitatively investigate the aerodynamic and inertial couplings in the dynamic response of a flexible spinning missile. Fig. 21 indicates distribution of amplitude of pitching angle and yawing angle, and there is no obvious difference in amplitude and

JID:AESCTE

10

AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.10 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

Fig. 19. Detailed figure of history of generalized displacements in initial stage.

Fig. 20. Detailed figure of history of generalized displacements in final stage.

Fig. 21. Distribution of amplitude of pitching angle and yawing angle.

frequency between pitching angle and yawing angle. There is only a main peak frequency 0.54 Hz, which reveals short-period mode of rigid-body motion. Fig. 22 indicates distribution of amplitude of generalized displacement of the two modes, and the characteristics of amplitude and frequency of the first and second modes are almost the same, because structural longitudinal and lateral modes are coupled by rotational motion, and a more detailed reason could be found in our previous work [43]. There are three main peak frequencies in Fig. 22, and they are 4.41 Hz, 15.15 Hz, and 25.06 Hz, respectively. The peak frequency 4.41 Hz reflects the frequency of rotational motion whose value is 4.98 Hz. The peak frequencies 15.15 Hz and 25.06 Hz reflect structural natural frequencies with rotational motion. What should be mentioned here is that structural natural frequencies are affected by rotational motion,

Fig. 22. Distribution of amplitude of generalized displacement of the two modes.

Fig. 23. Variation of relative change rate of moments of inertia.

including angular velocity and angular acceleration. As Eq. (16) shown, angular velocity and angular acceleration of rigid-body motion introduce extra inertial terms in structural dynamics equations that have effects on structural damping and structural stiffness matrices, and of course will have effects on structural natural frequencies. A more comprehensive mechanism by which structural natural frequencies are affected by rotation motion could be found in our previous work [43]. Furthermore, the peak frequencies of structural vibration are not pure, the reason can be explained as follows. In this case, rotational angular velocity around X axis of rigid-body motion is invariable and large. When there is no angular velocities around Y and Z axes of rigid-body motion, structural natural frequencies are only affected by rotational angular velocity around X axis, and can be obtained, and they are different from the case without rotational angular velocity. However, when there are small angular velocities and angular acceleration around Y and Z axes, the variation of angular velocity and angular acceleration around Y and Z axes will have small and direct effects on structural damping and structural stiffness matrices, and of course will have small and indirect effects on structural natural frequencies. In another word, the structural natural frequencies are in oscillation with small amplitude due to variation of angular velocity and angular acceleration around Y and Z axes. Therefore, there are small peak frequencies around main peak frequencies in structural vibration. Fig. 23 indicates variation of relative change rate of moments of inertia of the flexible spinning missile, and Fig. 24 and Fig. 25 give more details in initial and final stages, respectively. The change of moments of inertia is caused by structural deformation, and just as shown in Fig. 18, structural generalized displacements are small, therefore, the relative change rate of moments of inertia is small,

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.11 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

Fig. 24. Detailed figure of variation of relative change rate of moments of inertia in initial stage.

11

Fig. 27. Every force term of 2nd mode in elastic equation.

Fig. 28. Every force term of 1st mode in elastic equation in initial stage. Fig. 25. Detailed figure of variation of relative change rate of moments of inertia in final stage.

Fig. 29. Every force term of 2nd mode in elastic equation in initial stage. Fig. 26. Every force term of 1st mode in elastic equation.

too. The relative change rate Jxx /Jxx is always larger than 0, while J y y /J y y and Jzz /Jzz are in cyclical oscillation around zero. 4.2.2. Aerodynamic and inertial coupling terms in structural vibration Fig. 26 and Fig. 27 indicate dynamic responses of aerodynamic and inertial coupling terms in elastic equation (Eq. (7)), and more details in initial and final stages are shown in Figs. 28–31. In the calculated state, aerodynamic force plays a major role among all generalized force terms. Comparing to aerodynamic force, Coriolis term and centrifugal loading term are relatively smaller, and the generalized force term caused by angular acceleration of rigidbody motion is even smaller. The attention must be paid is that Coriolis term and centrifugal loading term will increase with the

increase of angular velocity of rigid body. When angular velocity is small, Coriolis term and centrifugal loading term are small, too, and they may be less than aerodynamic force. However, when angular velocity increases, Coriolis term and centrifugal loading term may be larger than aerodynamic force. The generalized force term caused by angular acceleration of rigid-body motion will increase with the increase of angular acceleration. These three inertial coupling terms have effects on structural damping, structural stiffness, generalized forces and structural natural frequencies, and should be considered carefully in the simulation related to structural vibration of spinning missile. 4.2.3. Aerodynamic and inertial coupling terms in rigid-body motion Fig. 32 and Fig. 33 indicate every moment term in angular equation (Eq. (6)) in Y (pitching) and Z (yawing) directions in

JID:AESCTE

12

AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.12 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

Fig. 30. Every force term of 1st mode in elastic equation in final stage.

Fig. 33. Every moment term in Z direction in angular equation.

Fig. 31. Every force term of 2nd mode in elastic equation in final stage.

Fig. 34. Inertial coupling terms in Y direction in angular equation.

Fig. 32. Every moment term in Y direction in angular equation.

Fig. 35. Inertial coupling terms in Z direction in angular equation.

body frame. In this case, inertial coupling moment terms are much smaller than aerodynamic moment. In final stage, pitching moment and yawing moment are both in cyclical oscillation with invariable amplitude, and the amplitude and period are similar. The average pitching moment is larger than zero, while average yawing moment is about zero, the reason can be explained as follows. At initial time, the attack of angle and pitching angle are both 2.07 degree, while yawing angle is 0 degree, and the missile is in longitudinal force equilibrium due to the deflection angle of fins, that is to say, pitching moment is 0, and yawing moment is 0, too. However, in final stage, average pitching angle and yawing angle are both 0 (Fig. 17), and average pitching angle is different with its initial value, therefore extra moment in longitudinal direction are introduced due to the different pitching angle.

Fig. 34 and Fig. 35 indicate dynamic response of inertial coupling terms in angular equation (Eq. (6)). Extra moment caused by variation of inertia tensor is larger than other two terms, and the extra moment terms caused by structural vibration velocity and structural vibration acceleration are relatively smaller. The amplitude and period of extra moment caused by variation of inertia tensor in Y and Z direction are almost same in the whole history. In this case, structural deformation is small, extra moment caused by variation of inertia tensor is relatively smaller than aerodynamic moment, however, when structural deformation is larger, the extra inertial coupling moment terms will be larger, too. Fig. 36 and Fig. 37 indicate moment terms response caused by generalized velocity and acceleration of structural vibration in angular equation (Eq. (6)). For the case used in this paper, extra moment caused by

JID:AESCTE AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.13 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

13

forces. In addition, Structural elastic deformation changes moments of inertia of rigid body, and structural deformation is small, therefore, the change of moments of inertia is small, too. Acknowledgement This work could not be accomplished successfully without the financial support from National Natural Science Foundation of China under the grant No. 11732013. Declaration of competing interest There is no conflict of interest. Fig. 36. Moment terms in Y direction in angular equation caused by generalized velocity and acceleration of structural vibration.

Fig. 37. Moment terms in Z direction in angular equation caused by generalized velocity and acceleration of structural vibration.

structural vibration velocity is larger than extra moment caused by structural vibration acceleration. 5. Concluding remarks Based on rigid-motion mesh and RBF mesh deformation method, URANS equations and flight dynamics equations of flexible spinning missile derived from Lagrange equation are coupled simultaneously to simulate the dynamic response of the flexible spinning missile with large slenderness ratio. Both the rigid-body six DOF equations of motion and structural dynamics equations are considered in the flight dynamics equations of flexible spinning missile. The investigation is focused on the aerodynamic coupling and inertial coupling between rigid-body motion and structural vibration, and aerodynamic coupling terms and inertial coupling terms in rigid-body motion and structural vibration are quantitatively investigated. The following conclusions can be drawn: 1. Angular motion introduces extra inertial coupling terms in structural vibration equations. For the case studied in this paper, these inertial coupling terms are not negligible compared with aerodynamic force. These inertial terms have effects on structural damping matrix, structural stiffness matrix, and structural generalized forces. As a result, angular motion will change structural natural frequencies. Furthermore, structural natural frequencies are not so pure due to pitching motion and yawing motion of rigid-body motion. 2. Structural vibration introduces extra inertial coupling terms in rigid-body angular equation. For the case studied in this paper, these inertial coupling terms are much less than aerodynamic

References [1] L.E. Lijewski, N.E. Suhs, Time-accurate computational fluid dynamics approach to transonic store separation trajectory prediction, J. Aircr. 31 (1994) 886–891. [2] J. Sahu, Time-accurate computations of free-flight aerodynamics of a spinning projectile with flow control, in: AIAA Atmospheric Flight Mechanics Conference and Exhibit, Colorado, USA, 2006. [3] J. Sahu, Time-accurate numerical prediction of free-flight aerodynamics of a finned projectile, J. Spacecr. Rockets 45 (2008) 946–954. [4] J. Sahu, Unsteady flow computations of a finned body in supersonic flight, in: 25th AIAA Applied Aerodynamics Conference, Florida, USA, 2007. [5] V.T. Luu, C. Grignon, F. Plourde, Toward a CFD/6 DOF coupled model enhancing projectile trajectory prediction, in: 21st AIAA Computational Fluid Dynamics Conference, San Diego, CA, 2013. [6] R. McCoy, Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric Projectiles, Schiffer Pub., 1999. [7] P. Reisenthel, D. Lesieutre, D. Nixon, Prediction of aeroelastic effects for seaskimming missiles with flow separation, in: 32nd Structures, Structural Dynamics, and Materials Conference, 1991. [8] M. Karpel, S. Yaniv, D.S. Livshits, Integrated solution for computational static aeroelasticity of rockets, J. Spacecr. Rockets 35 (1998) 612–618. [9] D. Raveh, M. Karpel, S. Yaniv, Nonlinear design loads for maneuvering elastic aircraft, J. Aircr. 37 (2000) 313–318. [10] S. Chae, D. Hodges, Dynamics and aeroelastic analysis of missiles, in: 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2003. ¸ A. Akgül, Development of a coupling procedure for static aeroelastic [11] E. Baskut, analyses, Sci. Tech. Rev. 61 (2011) 39–48. [12] L.K. Abbas, D. Chen, X. Rui, Numerical calculation of effect of elastic deformation on aerodynamic characteristics of a rocket, Int. J. Aerosp. Eng. 2014 (2014). [13] D. Chen, L.K. Abbas, X. Rui, G. Wang, Aerodynamic and static aeroelastic computations of a slender rocket with all-movable canard surface, Proc. Inst. Mech. Eng., Part G, J. Aerosp. Eng. 232 (2018) 1103–1119. [14] D. Tihomirov, D.E. Raveh, Nonlinear aerodynamic effects on static aeroelasticity of flexible missiles, in: AIAA Scitech 2019 Forum, 2019. [15] G. Reis, W. Sundberg, Calculated aeroelastic bending of a sounding rocket based on flight data, J. Spacecr. Rockets 4 (1967) 1489–1494. [16] M.R. Waszak, D.K. Schmidt, Flight dynamics of aeroelastic vehicles, J. Aircr. 25 (1988) 563–571. [17] C. Buttrill, P. Arbuckle, T. Zeiler, Nonlinear simulation of a flexible aircraft in maneuvering flight, in: Flight Simulation Technologies Conference, Monterey, CA, USA, 1987. [18] T. Zeiler, C. Buttrill, Dynamic analysis of an unrestrained, rotating structure through nonlinear simulation, in: 29th Structures, Structural Dynamics and Materials Conference, 1988. [19] L. Meirovitch, Hybrid state equations of motion for flexible bodies in terms of quasi-coordinates, J. Guid. Control Dyn. 14 (1991) 1008–1013. [20] M.R. Waszak, C.S. Buttrill, D.K. Schmidt, Modeling and Model Simplification of Aeroelastic Vehicles: An Overview, 1992. [21] D. Platus, Aeroelastic stability of slender, spinning missiles, J. Guid. Control Dyn. 15 (1992) 144–151. [22] D.K. Schmidt, D.L. Raney, Modeling and simulation of flexible flight vehicles, J. Guid. Control Dyn. 24 (2001) 539–546. [23] N. Nguyen, I. Tuzcu, Flight dynamics of flexible aircraft with aeroelastic and inertial force interactions, in: AIAA Atmospheric Flight Mechanics Conference, 2009. [24] E.J. Oliveira, P. Gasbarri, I. Milagre da Fonseca, Flight dynamics numerical computation of a sounding rocket including elastic deformation model, in: AIAA Atmospheric Flight Mechanics Conference, Dallas, TX, USA, 2015. [25] Z. Shi, L. Zhao, Effects of aeroelasticity on coning motion of a spinning missile, Proc. Inst. Mech. Eng., Part G, J. Aerosp. Eng. 230 (2016) 2581–2595.

JID:AESCTE

14

AID:105582 /FLA

[m5G; v1.261; Prn:28/11/2019; 8:32] P.14 (1-14)

H. Li, Z. Ye / Aerospace Science and Technology ••• (••••) ••••••

[26] T. Xu, J. Rong, D. Xiang, C. Pan, X. Yin, Dynamic modeling and stability analysis of a flexible spinning missile under thrust, Int. J. Mech. Sci. 119 (2016) 144–154. [27] D. Zhang, S. Tang, Q.-j. Zhu, R.-g. Wang, Analysis of dynamic characteristics of the rigid body/elastic body coupling of air-breathing hypersonic vehicles, Aerosp. Sci. Technol. 48 (2016) 328–341. [28] F. Saltari, C. Riso, G.D. Matteis, F. Mastroddi, Finite-element-based modeling for flight dynamics and aeroelasticity of flexible aircraft, J. Aircr. 54 (2017) 2350–2366. [29] C. Xie, L. Yang, Y. Liu, C. Yang, Stability of very flexible aircraft with coupled nonlinear aeroelasticity and flight dynamics, J. Aircr. 55 (2017) 862–874. [30] C. Riso, F. Mastroddi, C.E. Cesnik, Coupled flight dynamics and aeroelasticity of very flexible aircraft based on commercial finite element solvers, in: 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2018. [31] S.A. Keyes, P. Seiler, D.K. Schmidt, Newtonian development of the mean-axis reference frame for flexible aircraft, J. Aircr. (2018) 1–6. [32] E. Blades, J. Newman, Aeroelastic effects of spinning missiles, in: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2007. [33] A. Schütte, G. Einarsson, A. Raichle, B. Schöning, W. Mönnich, M. Orlt, J. Neumann, J. Arnold, T. Forkert, Numerical simulation of maneuvering aircraft by aerodynamic, flight mechanics and structural mechanics coupling, J. Aircr. 46 (2009) 53–64. [34] R.H. Hua, C.X. Zhao, Z.Y. Ye, Y.W. Jiang, Effect of elastic deformation on the trajectory of aerial separation, Aerosp. Sci. Technol. 45 (2015) 128–139. [35] R.H. Hua, Z.Y. Ye, J. Wu, Effect of elastic deformation on flight dynamics of projectiles with large slenderness ratio, Aerosp. Sci. Technol. 71 (2017) 347–359. [36] R. Hua, X. Yuan, Z. Tang, Z. Ye, Study on flight dynamics of flexible projectiles based on closed-loop feedback control, Aerosp. Sci. Technol. (2019). [37] M.J. Li, X.T. Rui, L.K. Abbas, Elastic dynamic effects on the trajectory of a flexible launch vehicle, J. Spacecr. Rockets 52 (2015) 1–17.

[38] J. Yin, J. Lei, X. Wu, T. Lu, Effect of elastic deformation on the aerodynamic characteristics of a high-speed spinning projectile, Aerosp. Sci. Technol. 45 (2015) 254–264. [39] J. Yin, J. Lei, X. Wu, T. Lu, Aerodynamic characteristics of a spinning projectile with elastic deformation, Aerosp. Sci. Technol. 51 (2016) 181–191. [40] L. Yang, Z.Y. Ye, The interference aerodynamics caused by the wing elasticity during store separation, Acta Astronaut. 121 (2016) 116–129. [41] L. Yang, Z.Y. Ye, J. Wu, The influence of the elastic vibration of the carrier to the aerodynamics of the external store in air-launch-to-orbit process, Acta Astronaut. 128 (2016) 440–454. [42] T. Lu, X. Wu, J. Lei, J. Yin, Numerical study on the aerodynamic coupling effects of spinning and coning motions for a finned vehicle, Aerosp. Sci. Technol. 77 (2018) 399–408. [43] H. Li, Z. Ye, Effects of rotational motion on dynamic aeroelasticity of flexible spinning missile with large slenderness ratio, Aerosp. Sci. Technol. 94 (2019). [44] T.C. Rendall, C.B. Allen, Efficient mesh motion using radial basis functions with data reduction algorithms, J. Comput. Phys. 228 (2009) 6231–6249. [45] T. Rendall, C. Allen, Parallel efficient mesh motion using radial basis functions with application to multi-bladed rotors, Int. J. Numer. Methods Eng. 81 (2010) 89–105. [46] G. Wang, H.H. Mian, Z.-Y. Ye, J.-D. Lee, Improved point selection method for hybrid-unstructured mesh deformation using radial basis functions, AIAA J. 53 (2014) 1016–1025. [47] M.R. Waszak, D.K. Schmidt, On the flight dynamics of aeroelastic vehicles, in: Astrodynamics Conference, 1986. [48] W.W. Zhang, Y.W. Jiang, Z.Y. Ye, Two better loosely coupled solution algorithms of CFD based aeroelastic simulation, Eng. Appl. Comput. Fluid Mech. 1 (2007) 253–262. [49] E.C. Yates Jr., AGARD Standard Aeroelastic Configurations for Dynamic Response. Candidate Configuration I.-Wing 445.6, 1987.