DOI : 10.17577/IJERTV5IS080263
- Open Access
- Total Downloads : 125
- Authors : Zhang Minghao
- Paper ID : IJERTV5IS080263
- Volume & Issue : Volume 05, Issue 08 (August 2016)
- Published (First Online): 23-08-2016
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
on the Non-uniform Torsion of Trapezoidal Thin Wings
Minghao Zhang
College of Aerospace and Civil Engineering Harbin Engineering University Harbin, Heilongjiang , China
AbstractThe trapezoidal thin wing with symmetrical airfoil (e.g. double wedge–airfoil) is of considerable importance in aviation industry. We consider a non-uniform torsion issue that arises in the study of the torsion analysis of a trapezoidal thin wing. Motivated by this new engineering theory, i.e. the plate- beam theory put forward by Prof. W.F. Zhang, this paper will firstly base upon the Kirchhoffs thin plate theory and Vlasovs rigid section assumption to derive two mechanical models, i.e. the energy variational model and differential equation model for the problem of the non-uniform torsion for the trapezoidal thin wing with double symmetrical airfoil. Then the approximate and exact analytical solutions of twist angle for the cantilever case of a trapezoidal thin wing under toque applied at the free end are derived and presented by making use of the energy variational model and differential equation model, respectively. Finally, the correctness of the analytical solutions obtained from the non-uniform torsion theory of the trapezoidal thin wing is verified by the FEM numerical simulations.
Keywords: Trapezoidal Thin Wing; Non-Uniform Torsion; Energy Variational Model; Differential Equation Model; Analytical Solution
-
INTRODUCTION
The trapezoidal thin wing with symmetrical airfoil (e.g. double wedge-airfoil) has been widely used in aviation industry. Therefore, the torsion performance of this kind of wing is of considerable importance in practical aerospace engineering[1,2].
As is known to all, the theory of the non-uniform torsion of a thin wing is the design fundamental of the flutter or aerodynamic instability for high-speed aircraft and hence many researchers have devoted themselves to solving these kinds of problems. For instance, in the aspect of rectangular thin wing, Timoshenko[3,4] obtained the decay index of stress by energy varaional method based upon the Foppl's work on the stress of the non-uniform torsion, and derived the tip rotation angle of narrow rectangular plate under non- uniform torsion; F.V. Chang[5] utilized the double triangle series to derive the tip rotation angle and deflection of a narrow rectangular plate under non-uniform torsion based upon the Kirchhoff's plate theory and the principle of superposition. However, their theoretical derivations are not universal, such as the concept of decay index of stress, and hence cannot be used in the case of a non-rectangular thin wing. This hinders the development of the theory of the non- uniform torsion of thin wings.
Moreover, from the literature review, it is found that few works has been published on the theory of non-uniform torsion of a trapezoidal thin wing.
The current unsatisfied status of research may be due to the complexity of the problem of the non-uniform torsion, coupled with the great difficulty of its mathematics and mechanics. Therefore, the theoretical study on non-uniform torsion (i.e. restrained torsion) member has been developed slowly[6].
In fact, from the point of view on the nature of mechanics, non-uniform torsion exits in any twisted member, namely, the non-uniform torsion and uniform torsion are co-exist in the torsion problem of any shape of cross section, but under extreme conditions, some of the cross section, for example, the torsion of a member with circular section will be dominated by uniform torsion. In other words, in the actual project, "pure torsion" members does not exist. Consequently, developing new theory is of great importance for the theoretical research on the issue of the non-uniform torsion.
Recently, a new non-uniform torsion theory, i.e. the plate- beam theory has been put forward by Prof. W.F. Zhang[7- 13], which is not only simple and easy-to-use, but also powerful and universal. Motivated by this new theory, this paper will firstly present two mechanical models of the non- uniform torsion for the trapezoidal thin wing with double symmetrical airfoil, then the exact and approximate analytical solutions are derived and verified by the FEM simulations.
-
NON-UNIFORM TORSION THEORY OF TRAPEZOIDAL THIN WING
-
Problem Descripsions and Asssumptions
-
Problem Descripsions:
Generally, the research object in this paper is a trapezoidal thin wing with double symmetrical airfoil (e.g. double wedge- airfoil). For simplicity, this type of wing may be thought of as a tapered plate with rectangular cross section. One end of the tapered plate is free and subjected to an applied torque Mt. The other end is assumed to be fixed and cannot warp. Moreover, the distributed torque mz is also applied along the centroid of the tapered plate. In this case the thin wing will produce torsion deformation, and can be simplified as a cantilevered, tapered plate subjected to a tip torque and distributed torque as shown in Fig.1a.
Known: The length of trapezoidal thin wing is L, the width of root section is hw, the thickness is tw; The elastic modulus of wing is Es, shear modulus is Gs, Poisson ration is s. When the tapered plate twists, the torsion angle of cross section is assumed to be (z) as shown in Fig.1b.
-
Asssumptions:
-
Ignore the shear strain caused by out-plane bending: This assumption is similar to that of the "plane section" of Euler beam.
-
Ignore the deformation in the mid-plane caused by out-plane bending, namely, the mid-plane cannot be streched: By this assumption, the bending and streching problem of a thin plate can be solved seprately. This is the
deformation decomposition hypothesis, which is the theoretical basis of the plate-beam theory[7-13].
-
normal stress and strain in the normal direction can ommited.
-
Rigid section assumption: This assumption is similar to that of Vlasov[6], which is widely used in the therotical derivation of thin-walled structures.
hw
tw
-
(b)
Fig.1 Calculation diagram and its deformation of narrow tapered thin-plate,
-
-
-
-
Mechanical Models of Trapezoidal Thin Wing
-
Energy variational model:
To facilitate the description of the deformation, the plate –
Accordingly we can see that, when considering only the torsion problem, the displacements of the centroid of cross section are
beam theory introduced two sets of coordinates, i.e. the
uw0 0;
vw0 0;
ww0 0; w0 z
(4)
global coordinate system xyz and the local coordinate system nsz, as shown in Fig.1b. The centroid coordinates of the cross section in the global coordinate system are assumed to be (0,0). In the local coordinate system, the coordinates of arbitrary point in the cross section are denoted as (n,s). While
Based on deformation decomposition hypothesis (b), then the tapered plate only occur out-plane bending. At this point the lateral displacement of arbitrary point caused by out-plane bending deformation are
in the global coordinate system, the coordinates of arbitrary point on the cross section are denoted as (-n,-s).
uw s, z s vw n, z n
(5)
According to the assumption (d), the displacements of arbitrary point are[6]
While the longitudinal displacement of the cross section should be determined based on assumptions (a) [7-13], that
3 x x n y y
s
is
(1)
2 0 0
w n, s, z n
p> uw
ns
(6)
3
3
w z z
r
sin 2
cos
2 n n
s
(2)
Geometric equations (i.e. linear strain) [6-12]
rn cos 3
sin 3
s s
w 2
2 2
L w ns
(7)
cos 3 sin 3 n
z z
z2
vs 2 2 0 n
L vw
v 3 3 0 s
(3)
s 0
(8)
n
sin
cos
s s
2 2
0 0
1
L ww
sz s
-
vw
z
2n
z
z
(9)
Physical equations (i.e. constitutive equations) [7-13]
Based on the assumption (3) and (4), for the classical Kirchhoff 's plate model, there are
Now the total potential energy of the torsion problem of the trapezoidal thin wing can be expressed as[7-13]
2 2
2 ( );
2 ( );
E L L
s
s
EI 2
comp
comp
z z
sz Gs sz
(10)
1 L
z
dz
1 s
Using the above derived relationship, the torsional strain energy of a trapezoidal thin wing can be obtained easily.
Firstly, with the following expression of strain energy
2 0
GJk
2
z
z
comp
mz z
-
Mt z L
-
(16)
U 1 L L dndsdz (11)
This is the total potential energy for the non-uniform
2 Vwc
z z sz sz
torsion problem of the trapezoidal thin wing. It is consistent with the result of traditional torsion theory in the form.
However, the derivation of this paper is more natural, and
and substituting Eq. (7) – Eq. (10) into Eq.(11), we can get the torsional strain energy of the trapezoidal thin wing as follows[7-13]
U 1 L L dndsdz
only the Kirchhoffs thin plate theory and Vlasov's rigid section assumption are used.
Using expression of the total potential energy (16), the non-uniform torsion problem of the trapezoidal thin wing
2
2
2 Vwc
z z sz sz
subjected to torsion loads can be translated into such an energy variational model: Within the range, 0zL, looking
1 E
2
2
for a function (z) to make it satisfy the specified geometric
s
s
2 ns
2
Gs 2n
dndsdz
boundary conditions, i.e. the endpoint constraints, and the
2 V 1 s
wc
wc
If setting
z
z
energy functional defined by the following formula is minimum.
EI
Es
s
s
1 2
I
1 2
n2 s2dnds
Where
L
' ' '
' ' '
F , dz
0
(17)
Es
Es
s s Awc
E t3p z
(12)
3
' '' 1
' '2 '2
(18)
s c w 1 2( )tan
F , EI GJk
s
s
w
w
1 2 144 h
as the non-uniform torsional rigidity or warping torsion rigidity of the trapezoidal thin wing, and
2
2
-
-
Differential equation model:
-
The differential equation, along with boundary conditions, can be readily obtained by the principle of energy variational, which states that the true state of deformation is distinguished
GJ G J
4G n2dnds
from all other statically correct state of deformation by the
k s s s
Awc
condition that the energy functional be a minimum, i.e.
0
(19)
t3h z
(13)
s
s
G c w 1 2( )tan
Then we get
3 hw
L EI ' ' ' '
dz M
0 (20)
as the uniform torsional rigidity of the trapezoidal thin
0 GJ ' ' m
z
t zL
k z
wing, now the torsional strain energy of can be simplified as[7-13]
Use integration by parts, we get
EI
EI
2 ''
2 2 2
L z2
1 L
(14)
dz
2 z
2 z
U 0 EI 2
-
GJk z dz
0 GJ ' m z
z k z
In the case of a concentrated torque is applied at the free end and a distributed torque is applied along the length of a cantilevered, trapezoidal thin wing , the corresponding load
EI ' ' '
z L
z L
z 0
z 0
z L
(21)
potential energy is
EI '' GJ '
z k
L z 0
zL 0
zL 0
0
0
V Mt mz z dz
(15)
Mt
z L
Due to the arbitrariness of in the above formula, we could get the following differential equation
In order to obtain an approximate analytic solution, we can choose the trail function for the rotation of cross section
2
EI ''
GJ ' m
z 0
(22)
as follows
2
z2 z k z
A0 x x
and the corresponding boundary conditions
z
2 L 3 L
(26)
-
Fixed end (cross section cannot freely rotate, nor
freely warp)
Obviously, the above trial function satisfies the following boundary conditions of the cantilever plate
0 ' 0
-
Free end (with tip toque, cross section can freely warp)
(23)
0 ' 0 0 '' L 0
If Eq.(26) is substituted into Eq.(25), then we get
(27)
E
t3p z
3
EI ' ' GJ ' M
0
c c w 1 2( )tan
c
c
w
w
z
k t
(24)
1 2 144 h
EI '' 0
2 2
2
A0 x 3 x
Thus, the non-uniform torsion issue of the trapezoidal thin
z2
2 L
L
wing can also be expressed as follows: Within the range, 0z
1
L
dz
L, looking for a function (z), it satisfies the differential
2 0
t3h z
equation (i.e. the equilibrium equation) (22), and at the same time meets the boundary conditions (25) and (26).
G
G
c
c w 1 2(
3 hw
)tan
2
2
It should be noted that in the equilibrium equation (22)
A
x 2
x
given herein, the first term is the internal twisting moment
0
3
caused by the non-uniform torsion, and the second term is the
z
2 L
L
internal twisting moment caused by the uniform torsion, i.e. the free torsion (St. Venant torsion). It can be seen that, according to the proposed plate-beam theory, two types of
L
L
t
t
A x x
A x x
2
M 0 3
torsion, i.e. non-niform torsion and uniform torsion, could be integrated in one mixed torsion equation naturally, namely,
L
L
2
z L
the "separated" traditional torsion theory is included in one theoretical framework, Therefore, the new mixed torsion theory has important theoretical and practical value.
and its integration result is
1 6 A2 33tan A2 1
0 0 GJ
-
-
-
ANALYTICAL SOLUTION OF TRAPEZOIDAL THIN WING SUBJECTED TO TIP TORQUE
2 5L 20h
3A2 9tan A2
k E 2
0 0
(28)
-
Approximate Analytical Solution Based on Energy
L3
2hL2
EI
-
A M
Variation Model
In this section, we will base on the energy variation model
18tan2 A2 6tan3 A2 E 0 t
0 0
to examine the non-uniform torsion issue of the trapezoidal thin wing. For the purposes of simplifying this discussion, we consider here the case where only concentrated torque is
applied in the free end torque. In this case the total potential
5p L
Where
E
5p
t3h 3
t3h
energy simplifies to
EI
c c w ; GJ
G c w
c
c
E 1 2 144
k E 3
c
c
1 L 2 2
2
(29)
2 z
2 z
0 EI 2
-
GJk z dz
(25)
Based upon energy variational method, there must be
-
M
0
(30)
t z L
A0
After arrangement, we get
This is a more complex, third-order differential equations with variable coefficients. In this paper, based on its unique
18
L 2
mathematical structure constituted, along with the remaining
3
9
tan2
5 h
L 2
three boundary conditions, its exact analytical solution is derived.
Firstly, Eq.(37) is rewritten in dimensionless form.
tan
2 h
KE
Setting
A
Mt L 0
(31)
0 6
L 3
GJ z
5
tan3 k E
z 1 2( )tan
hw
(38)
6 33 tan L
h
h
After arrangement, we get
h
h
5 20
2
2
z
z
z 3 z
Mt hw
E
E
Then the solution can be obtained
z
z 2
z 2
2tan GJk
A Mt L 1
(32)
(39)
0 GJ K 2
where
where
k E 1 E 2
K
2L tan K
EI
(40)
E
E
2 3 E h
E GJ L2
3 18 tan2 L 9 tan L 6 tan3 L
w k E
1 5 h 2
h 5
h
Then, using the theory of differential equations, the exact
(33)
analytical solution can be derived as follows
6 33 tan L
(34)
z A
1 z A
1 z
2 5 20
h p p
M h 1
(41)
EI
E
E
t w
ln z A
KE 2
(35)
2tan GJ
2 1 0
E
E
GJk L
k E
-
-
Exact Analytical Solution Based on differential equation model
where,
1 2
2
; A0 ,
Ap
and
Ah 2
are three
In this section, we will base on the differential equation model to examine the non-uniform torsion issue of the trapezoidal thin wing. For the purposes of simplifying this discussion, we consider here the case where only
constants of integration, whose expression are as follows, which can be obtained according to the ends of the boundary conditions
concentrated torque is applied in the free end torque. In this case the simplified differential equation is
A
A
p
Mt hwcot ; 2(1 2 ) GJ
2 '' '
E
E
k
k
A 0;
(42)
EI GJk 0
2 comp
comp
2 comp
comp
(36)
h 2
2
2
z z
M h cot
Using the first condition of the boundary condition (24) , we can obtain
A0 t w
E
E
k
k
2(1 ) GJ
EI
z 3
1 2( )tan
''
Finally, the maximum rotation (tip rotation in the free end) of the cantilevered, tapered plate can be obtained
E
E
w
w
z h
z
z
(37)
GJ
1 2( )tan ' M
k E
hw
t
max
L
cot h
control the length of element, which is 50mm. Then, the freedoms of all nodes at the fixed end are restrained to simulate the fixed constraint conditions, a unit of torque is
w
applied at the centroid of the cross section at the free end. In
M L
2(1 2 ) L
addition, the CERIG command is used in the finite element
t
GJk E 1 L
(43)
simulation in order to define rigid region for each cross section. Last, the torsion angle at the free end is extracted
h
h
1 2( )tan
w
after the analysis. The corresponding modeling, meshing, load and constraint and torsion deformation are shown in
-
w
-
w
1 h
1 L
1 L
2 ln 1 2( )tan
Fig.2.
B. Comparison of theoretical and FEM solutions
2tan L 1 hw
cot h
There are 3 different sizes of the trapezoidal thin wings,
usi
w
ng the above theoretical method and finite element method
2(1 2 ) L
-
-
FEM SIMULATION AND VERIFICATION
-
FEM Model
In order to verify the correctness of the above analytical solutions, the finite element software ANSYS is used to establish the model for the analysis of the non-uniform torsion of the cantilevered, tapered plate subjected to a tip torque. The elastic shell element SHELL63 with 4 nodes is chosen to simulate the trapezoidal thin wings. The elastic modulus Es=2.06×105MPa, Poisson ratio s=0.3.
When the model is built, the key points of the left and right ends are established, and the tapered plate is formed through these key points, using the ESIZE command to
(FEM) to calculate the torsion angles for all wings. The comparison results are shown in Table I and Table II.
From the analysis of data of Table I and Table II, it can be seen that: (1) The results given by the exact analytical solutions are almost the same as those obtained from FEM simulations. This proves the correctness of the theory of non- uniform torsion presented herein. (2) The results given by the approximate analytical solution and those given by the finite element analysis are basically consistent, and the error is within the range of -6.45% and -2.76%. However, the expression of the approximate analytical solution is more simple than the eact one. Therefore, the approximate analytical solution is suitable for the engineering design personnel in the approximate calculation or estimation.
Fig.2 FEM model and its deformation
TABLE I. COMPARISON OF TORSION ANGLES BETWEEN EXACT SOLUTIONS AND FEM SOLUTIONS
Number
L(m)
hw (m)
FEM solutions (10-3rad)
Exact solutions (10-3rad)
Error1 (%)
1
8.6
1.2
1.7003
1.6983
-0.12
2
4.3
1.2
0.82828
0.827094
-0.14
3
2.2
1.2
0.40093
0.401075
-0.04
TABLE II. COMPARISON OF TORSION ANGLES BETWEEN APPROXIMATE SOLUTIONS AND FEM SOLUTIONS
Number
L(m)
hw (m)
FEM solutions (10-3rad)
Approximate solutions (10-3rad)
Error2 a
(%)
1
8.6
1.2
1.7003
1.59042
-6.45
2
4.3
1.2
0.82828
0.787992
-4.86
3
2.2
1.2
0.40093
0.389851
-2.76
-
Erro2=(Approximate solution- FEM solution)/ FEM solution×100%
-
-
-
CONCLUTION
Theoretical studies and numerical simulation practice prove that:
-
The plate-beam theory proposed by Prof. W.F. Zhang [7-13] is universal, and can be easily used to solve the non- uniform torsion problem of the trapezoidal thin wing with double symmetrical airfoil;
-
Deriving the differential equation model from the energy variation model has a clear concept of mechanics.
-
Approximate analytical solution presented here is simple and practical, and is basically agree with the FEM analysis results. While the exact analytical solution is almost the same as the results obtained from the FEM analysis. This proves the correctness of the solution of differential equation model.
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-
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