Subsonic aero-elastic analysis of a three-dimensional panel supported by an artificial spring boundary

* Xiaodong Liu1) and Dengqing Cao2)

1), 2) The School of Astronautics, Harbin Institute of Technology

PO Box 137, Harbin 150001, China 1)xdliu11b@163.com; 2)

dqcao@hit.edu.cn

ABSTRACT

The subsonic aero-elastic stability and the nonlinear dynamical behavior of a three-dimensional panel supported by an artificial spring boundary are systematically studied in the present study. The problems of the subsonic aero-elasticity of the panel are commonly observed in the aviation and the high speed railway. With the flow velocity increasing, the panel may become instability of the divergence type or flutter type, which may cause damage or fatigue to the structure. The Rayleigh-Ritz method is employed to derive the frequency equations and modal functions of a rectangular panel with elastic constraints by taking the characteristic orthogonal polynomial series as the admissible functions and utilizing artificial springs to simulate the elastic constraints. The incompressible aerodynamic model derived by the linear potential flow theory is adopted. Based on von Karman nonlinear displacement-strain relationship, the governing equations of motion of the panel is established by the Hamilton’s principle. The Galerkin method is employed to truncate the governing partial differential equation into a series of ordinary differential equations. The critical flow velocity of the panel is calculated by analyzing the stability of the equilibrium point of the panel. The generalized eigen-value method is adopted to analyze the variations of the natural frequencies of the panel with respect to the flow velocity. Also the effects of artificial spring stiffness on the critical flow velocity are studied numerically. 1. INTRODUCTION

A series of theoretical and experimental results about two-dimensional panel and

incompressible flow were reported in (Dowell 1963, Gislason 1971, Kornecki 1974, Kornecki 1976, Weaver 1970). A literature review of work on the aeroelasticity of plates and shells is given by Dowell (1974), see also Païdoussis (2003). Recently, Li (2012)

1) Graduate student 2) Professor

presented the dynamics behavior of a two-dimensional panel with nonlinear cubic stiffness and viscous damper in the middle of the panel.

There is a few reports on the three-dimensional subsonic aero-elastic analysis of panel. Ishii (1965) and ELLEN (1973) study the stability of a simply supported flat plate in the three-dimensional generalized pressures with compressible form. Gibbs (2014) study the theory which combines a plate structural model with a three-dimensional vortex lattice aerodynamic mode to create a high-fidelity frequency domain aeroelastic model with various boundary conditions. Song (2014) calculates aero-thermo-elasticity of nonlinear composite laminated panel with aerodynamic heating in the hypersonic flow.

This paper investigates the subsonic aeroelasticity of a three-dimensional nonlinear panel simply supported at three boundaries and supported by an artificial spring at another boundary. The natural frequencies of rectangular plates are derived by using characteristic orthogonal polynomials in Rayleigh-Ritz method proposed in Bhat (1985). The incompressible aerodynamic model is adopted to predict the critical flow velocity of the panel. The generalized eigen-value method is adopted to analyze the variations of the natural frequencies of the panel with respect to the flow velocity and the effects of artificial spring stiffness and structural parameters on the critical flow velocity are studied numerically. 2. FORMULATION FOR SUBSONIC AEROELASTIC MODEL

2.1 The natural characteristic analysis of panel structural

,U

a

z

x

y

b

RKWK

Fig. 1 A three-dimensional panel simply supported at 0 , , 0x x a y and supported by artificial spring at y b in axial flow

In order to analyze the natural properties of the panel supported by artificial springs, the Rayleigh-Ritz method is used, thus characteristic orthogonal polynomial series in y coordinate direction of the rectangular plate is introduced. The material constants and geometrical parameters of the panel are taken as 70GPa ,E 32700kg/m , and

1.0 m , 0.5 ma b , 3.0 mmh respectively. To get the discrete equation of motion, the displacements , ,u v w are expressed in

terms of mode shapes and generalized coordinates as

1 1

( , , ) ( , ) ( ) ( , ) ( )m n

Tij

i j

u x y t x y p t x y p t

(1)

1 1

( , , ) ( , ) ( ) ( , ) ( )m n

Tij

i j

v x y t x y r t x y r t

(2)

1 1

( , , ) ( , ) ( ) ( , ) ( )m n

Tij

i j

w x y t x y g t x y g t

(3)

A three-dimensional panel simply supported at 0 , , 0x x a y and supported by artificial spring at y b . The mode shapes satisfying the boundary conditions are assumed to be

( , ) ( , ) ( , ) ( , ) sin ( )iij ij ij ij j

i xx y x y x y x y y

a

(4)

( )ij y can be approximated by a linear combination of the basis functions that

satisfy all the geometric boundary conditions, i.e. it can be constructed by Gram-Schmidt orthogonalization.

1

( )NT

i jj f f

f

y d

(5)

Here, NT is the number of terms truncated in practical calculation. It should be noted that, although the approximation goes well as NT approaches infinity, the following convergence study in this paper shows that favorable results can be derived with finite number of terms.

The natural frequencies and corresponding modes in the case of 0, 0W RK K ,

5 5 , 5 5W RK e K e , and 1 10 , 1 10W RK e K e are shown in Fig. 2-4 respectively.

11.87Hz (1,1) 34.38 Hz (2,1) 55.39 Hz (1,2) 70.66 Hz (3,1)

Fig.2 Mode shapes for 0 , 0W RK K .

46.093 Hz (1,1) 62.024 Hz (2,1) 92.337 Hz (3,1) 112.14 Hz (1,2)

Fig.3 Mode shapes for 5 5 , 5 5W RK e K e .

51.03 Hz (1,1) 69.59 Hz (2,1) 103.14 Hz (3,1) 153.28 Hz (1,2)

Fig.4 Mode shapes for 1 10 , 1 10W RK e K e .

2.2 Aeroelastic model

Consider a thin plate which may be used as the skirt panels of high speed vehicles, thus the panel can be modeled using the Kirchhoff plate theory. Under this assumption, the displacement fields ( , , )u v w can be expressed as

0 0, ,w w

u u z v v z w wx y

(6)

where z is the transverse coordinate, 0u and 0v are the in-plane displacements in the

x and y directions. The strain-displacement relation based on the von Karman large deflection plate theory is given as 0 z ε ε κ (7)

where = , ,T

x y xy ε is the strain vector, 0 0 0 0= , ,T

x y xy ε is the membrane strain

vector and = , ,T

x y xy κ is the bending curvature vector, and they are expressed as

0

22

0 0 00

1 1, ,

2 2

T

u vw w wu w

x y x

v

x y y yx

ε (8)

2 2 2

2 2, , 2

Tw w w

x y x y

κ (9)

The equation of motion is formulated using the Hamilton’s principle which can be expressed as

2

1

0t

tK U W dt (10)

where K , U and W are the kinetic energy, potential energy and virtual work done by the external load. They are calculated by

1

21

( )2

V

x x y y xy xyV

A

K u v w dV

U dV

W p wdA

(11)

where , V and A are the mass density, volume and surface area of the panel, respectively; and p is the aerodynamic pressure which is usually calculated by the linear potential theory in the engineering analysis. For low Mach number subsonic flow, the self-induced aerodynamic pressure is approximately given by (Yao 2014)

2 2 2

20 2 2

2w w w

p A U Ut x t x

(12)

where 0 2 2 2 2

abA

i b j a

, is the air density and U is the main flow velocity.

Substituting Eqs.(1), (2) and (3) into the governing equations of motion, the Galerkin method is used to obtain a coupled set of nonlinear ordinary differential equations 0p p L N pM M X C X K K K X (13)

where X is the generalized coordinate vector, pM , pC and pK are the aerodynamic

mass, damping and stiffness matrices. LK is the linear stiffness matrix, and NK is the

nonlinear stiffness matrix. They are listed in APPENDIX A. 3. NUMERICAL SIMULATIONS AND DISCUSSIONS

For the sake of simplicity, we take 3 , 1,m n in the numerical simulations. With the velocity increasing, the vibration frequency of the panel in the subsonic flow decreases down to zero, which corresponds to the critical flow velocities for divergence of the panel (Dowell 1963, Païdoussis 2003). The variations of vibration frequency f versus the airspeed for different artificial spring stiffness are shown in Fig. 5.

130 135 140 145 1500

2

4

6

U

f (H

z)

230 235 240 245 2500

2

4

6

8

U

f (H

z)

(a) 0 , 0W RK K (b) 5 5 , 5 5W RK e K e

270 275 280 2850

5

10

U

f (H

z)

(c) 1 10 , 1 10W RK e K e

Fig.5 Variation of frequency f versus flow velocity

As is shown in Fig. 6, with the increase of the artificial spring stiffness, the critical flow velocity for divergence increases, which indicates that the aeroelastic stability is strengthen by the supporting stiffness at the boundary.

0 2 4 6 8 10100

150

200

250

300

lg K

Ucr

Fig.6 The critical flow velocities for divergence with respect to artificial spring stiffness 4. CONCLUSIONS

The subsonic aero-elastic stability for a three-dimensional panel supported by an artificial spring boundary has been systematically investigated. The Rayleigh-Ritz method was employed to derive the frequency equations and modal functions of a rectangular panel with elastic constraints by taking the characteristic orthogonal polynomial series as the admissible functions and utilizing artificial springs to simulate the elastic constraints. A multi-degree-of-freedom nonlinear dynamical model has been established in terms of the incompressible aerodynamic model for the subsonic flow, the von Karman theory for the structure, and the Galerkin method for truncating the system into a series of ordinary differential equations. The critical flow velocity of the panel was calculated by analyzing the stability of the equilibrium point of the panel and the effects of artificial spring stiffness on the critical flow velocity are studied numerically. ACKNOWLEDGEMENTS

This research is supported by the national basic research program of China (No.2011CB71110). APPENDIX A The matrices in Eq.(13) are written as

0 0

0 0

0 0

u

v

w

M

M M

M

(14)

where

0 0

1

2

b a TuM h dxdy ,

0 0

1

2

b a TvM h dxdy (15)

3

0 0 0 0

1 1

2 8

b a b aT T Tw x x y yM h dxdy h dxdy (16)

and

uu uv uw

L vu vv vw

wu wv ww

K K K

K K K K

K K K

(17)

where

2 0 0 0 0

1 1

2 1 2

b a b aT Tuu x x y y

EhK dxdy Gh dxdy

(18)

2 0 0 0 0

1 1

2 1 2

b a b aT Tvv y y x x

EhK dxdy Gh dxdy

(19)

3

2 0 0

33

2 0 0 0 0

1

8 1

1 1

4 1 2

b a T Tww xx xx yy yy

b a b aT Txx yy xy xy

EhK dxdy

E hdxdy Gh dxdy

(20)

2 0 0 0 0

1

1 2

b a b aT Tuv x y y x

E hK dxdy Gh dxdy

(21)

, 0 , 0T T Tvu uv uw wu vw wvK K K K K K (22)

0 0

0 0uw

N vw

wu wv ww

K

K K

K K K

(23)

in which

2 0 0

2 0 0 0 0

1

2 1

1 1

2 1 2

b a T Tuw x x x

b a b aT T Tx y y y x y

EhK g dxdy

E hg dxdy Gh g dxdy

(24)

2 0 0

2 0 0 0 0

1

2 1

1 1

2 1 2

b a T Tvw y y y

b a b aT T T Ty x x x x y

EhK g dxdy

E hg dxdy Gh g dxdy

(25)

2 0 0

2 0 0 0 0

1

8 1

1 1

4 1 2

b a T T T T T Tww x x x x y y y y

b a b aT T T T T Tx x y y x y x y

EhK gg gg dxdy

E hgg dxdy Gh gg dxdy

(26)

,T Twu uw wv vwK K K K (27)

0 0 0 0 0 0 0 0 0

0 0 0 , 0 0 0 , 0 0 0

0 0 0 00 0p p p

p pp

M C K

M KC

(28)

where

20 0 00 0 0 0 0 0

, 2 ,b a b a b aT T T

p p x p xxM A dxdy C U A dxdy K A U dxdy (29)

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