Frequency response of cylindrical variable stiffness composite

Frequency response of cylindrical variable stiffness composite

Frequency response of cylindrical
variable stiffness composite
laminated shells
S. Stoykov, P. Ribeiro
Institute of Information and Communication
Technologies, Bulgarian Academy of Sciences
Faculty of Engineering,
University of Porto
Outline
1. Introduction
2. Mathematical model of cylindrical shells with variable
stiffness materials
–
–
p - FEM
Stress-strain relations
3. Method of solution
– Harmonic balance method
– Continuation method
4. Results
– Convergence study
– Steady-state periodic responses
5. Future work
2
Introduction
y
• Composite laminates reinforced by
curvilinear fibres are analysed here.
x
• Curvilinear fibres paths allow one to
tailor fibre-reinforced laminated
composites so that the stiffness is a
function of the location, leading to a
Variable Stiffness Composite Laminate.
• The practical interest of VSCL is large,
particularly in aeronautics, because it
offers the possibility to reduce weight
without compromising safety.
3
Mathematical model
z
• Let R1 and R2 are the principal curvature
radius, then, before deformation, the
x
middle surface is defined as:
2
2
1
𝑥
𝑦
𝑤 𝑖 𝑥, 𝑦 = −
+
2 𝑅1 𝑅2
• In shallow shells:
2
𝜕𝑤 𝑖 𝑥, 𝑦
≪1
𝜕𝑥
𝜕𝑤 𝑖 𝑥, 𝑦
𝜕𝑦
y
b
R2
a
R1
2
≪1
• a/R  0.5 or a/R  0.4 are accepted domains for shallow shell
theory (Leissa, Kadi; J. of Sound and Vib. 1971; Leissa, Narita; J. of
Sound and Vib. 1984)
4
Mathematical model
• In shallow shells, Cartesian
coordinates x and y can
replace the curvilinear
coordinates α and β.
Coordinate
lines a
z
b=bj
x
y
W
a=ai
• Lamé parameters, A and B, which relate changes
in curvilinear coordinates with changes in arclengths over the surface (ds1=Adα, ds2=Bdβ)
become: A=1 and B=1
Coordinate lines b
• One arrives at simpler kinematic relations than the ones
for general shells.
5
Mathematical model
Thin Shell Theory
𝑢 𝑥, 𝑦, 𝑧, 𝑡 = 𝑢0 𝑥, y, 𝑡 − 𝑧
𝜕𝑤0 𝑥,𝑦,𝑡
𝜕𝑥
𝑣 𝑥, 𝑦, 𝑧, 𝑡 = 𝑣0 𝑥, y, 𝑡 − 𝑧
𝜕𝑤0 𝑥,𝑦,𝑡
𝜕𝑦
𝑤 𝑥, 𝑦, 𝑧, 𝑡 = 𝑤0 𝑥, 𝑦, 𝑡
𝜕𝑤0 𝑥, 𝑦, 𝑡
𝜃𝑥 𝑥, 𝑦, 𝑡 = −
𝜕𝑦
𝜃𝑦 𝑥, 𝑦, 𝑡 = −
𝜕𝑤0 𝑥, 𝑦, 𝑡
𝜕𝑥
6
Mathematical model
• Membrane strains (other strains are neglected):
𝜕𝑢0 𝑤0 1 𝜕𝑤0
𝜀𝑥 =
+
+
𝜕𝑥
𝑅1 2 𝜕𝑥
2
𝜕 2 𝑤0
−𝑧
𝜕𝑥 2
𝜕𝑣0 𝑤0 1 𝜕𝑤0
𝜀𝑦 =
+
+
𝜕𝑦 𝑅2 2 𝜕𝑦
2
𝜕 2 𝑤0
−𝑧
𝜕𝑦 2
𝛾𝑥𝑦
𝜕𝑢0 𝜕𝑣0 𝜕𝑤0 𝜕𝑤0
𝜕 2 𝑤0
=
+
+
− 2𝑧
𝜕𝑦
𝜕𝑥
𝜕𝑥 𝜕𝑦
𝜕𝑥𝜕𝑦
• In the numerical tests cylindrical shells are considered, where R2 = .
7
Mathematical model
• The stress-strain relation has the following form:
(𝑘)
𝜎𝑥 𝑥, 𝑦, 𝑧, 𝑡
𝜎𝑦 𝑥, 𝑦, 𝑧, 𝑡
𝜏𝑥𝑦 𝑥, 𝑦, 𝑧, 𝑡
𝑄11 (𝑥) 𝑄12 (𝑥) 𝑄16 (𝑥)
= 𝑄12 (𝑥) 𝑄22 (𝑥) 𝑄26 (𝑥)
𝑄16 (𝑥) 𝑄26 (𝑥) 𝑄66 (𝑥)
(𝑘)
𝜀𝑥 𝑥, 𝑦, 𝑧, 𝑡
𝜀𝑦 𝑥, 𝑦, 𝑧, 𝑡
𝛾𝑥𝑦 𝑥, 𝑦, 𝑧, 𝑡
(𝑘)
• Unlike in straight-fibre reinforced laminates, the transformed
reduced stiffness matrix is not constant. This occurs, because the
fibre path is a function of x:
y
𝜃𝑘
𝑥 =
2
𝑇1𝑘
− 𝑇0𝑘
𝑎
𝑥 + 𝑇0𝑘
q (x)
T1
T0
a/2
x
8
Mathematical model
• The displacements on the middle surface are expressed in the form:
𝐟 𝑢 (𝑥, 𝑦)T
𝑢0 (𝑥, 𝑦, 𝑡)
𝑣0 (𝑥, 𝑦, 𝑡) =
𝟎
𝑤0 𝑥, 𝑦, 𝑡
𝟎
Displacements
on the middle surface
𝟎
𝐟 𝑣 (𝑥, 𝑦)T
𝟎
Displacement shape
functions
𝟎
𝟎
𝐟 𝑤 (𝑥, 𝑦)T
𝐪𝑢 (𝑡)
𝐪𝑣 (𝑡)
𝐪𝑤 (𝑡)
Generalised
coordinates
• Accuracy is improved by increasing the number of shape functions
and generalised coordinates in each element.
9
Mathematical model
• The displacements on the middle surface are expressed in the form:
𝐟 𝑢 (𝑥, 𝑦)T
𝑢0 (𝑥, 𝑦, 𝑡)
𝑣0 (𝑥, 𝑦, 𝑡) =
𝟎
𝑤0 𝑥, 𝑦, 𝑡
𝟎
Displacements
on the middle surface
𝟎
𝐟 𝑣 (𝑥, 𝑦)T
𝟎
Displacement shape
functions
𝟎
𝟎
𝐟 𝑤 (𝑥, 𝑦)T
𝐪𝑢 (𝑡)
𝐪𝑣 (𝑡)
𝐪𝑤 (𝑡)
Generalised
coordinates
9
Mathematical model
• The displacements on the middle surface are expressed in the form:
𝐟 𝑢 (𝑥, 𝑦)T
𝑢0 (𝑥, 𝑦, 𝑡)
𝑣0 (𝑥, 𝑦, 𝑡) =
𝟎
𝑤0 𝑥, 𝑦, 𝑡
𝟎
Displacements
on the middle surface
𝟎
𝐟 𝑣 (𝑥, 𝑦)T
𝟎
Displacement shape
functions
𝟎
𝟎
𝐟 𝑤 (𝑥, 𝑦)T
𝐪𝑢 (𝑡)
𝐪𝑣 (𝑡)
𝐪𝑤 (𝑡)
Generalised
coordinates
9
Mathematical model
•
The equation of motion is derived by the principle of virtual work
𝛿𝑊𝑉 + 𝛿𝑊𝑖𝑛 + 𝛿𝑊𝐸 = 0
• The equation of motion has the following form:
𝐌𝐮11
𝟎
𝟎
𝟎
𝐌𝐯22
𝟎
+
𝟎
𝟎
33
𝐌𝐰
𝐊11
𝐪𝐮 (𝑡)
𝐋𝐮𝐮
𝐪𝐯 (𝑡) + 𝐊 21
𝐋𝐯𝐮
𝐪𝐰 (𝑡)
𝐊 31
𝐋𝐬𝐮
𝟎
𝟎
𝟎
𝟎
𝐊 31
𝐍𝐋𝟑
𝐊 32
𝐍𝐋𝟑
𝐊12
𝐋𝐮𝐯
𝐊 22
𝐋𝐯𝐯
𝐊 32
𝐋𝐬𝐯
𝐊13
𝐋𝐮𝐬
𝐊 23
𝐋𝐯𝐬
33
𝐊 33
𝐋𝐬𝐬 + 𝐊 𝐋𝐛
𝐊13
𝐍𝐋𝟐
𝐊 23
𝐍𝐋𝟐
33
33
𝐊 33
𝐍𝐋𝟒 + 𝐊 𝐍𝐋𝟐𝐒 + 2𝐊 𝐍𝐋𝟐𝐒
T
𝐪𝐮 (𝑡)
𝐪𝐯 (𝑡) +
𝐪𝐰 (𝑡)
𝐪𝐮 (𝑡)
𝟎
𝟎
𝐪𝐯 (𝑡) =
𝐏𝐰 (𝑡)
𝐪𝐰 (𝑡)
10
Mathematical model
•
The equation of motion can be written in the following form:
𝐌𝐪 𝑡 +
𝛽
𝐌𝐪 𝑡 +𝐊 𝐪 𝑡
𝜔
𝐪 𝑡 = 𝐅(𝑡)
• Periodic solutions are considered, thus 𝐪 𝑡 is expressed
in Fourier series:
⋮
𝑘=∞
𝐪 𝑡 =
𝐜𝑘
𝑘=−∞
𝑒 𝑖𝑘ω𝑡
𝐜−1
𝐐 = 𝐜0
𝐜1
⋮
• An algebraic nonlinear system is obtained, it is solved by
continuation method.
−𝜔2 𝐌 𝐇𝐁𝐌 + 𝐂 𝐇𝐁𝐌 + 𝐊 𝐇𝐁𝐌 𝐐
𝐐 = 𝐅 𝐇𝐁𝐌
11
Mathematical model
Complex form of the Fourier series is preferred, because the structure
of the nonlinear matrices can be expressed independent on the
number of harmonics.
⋱
⋮
𝑘
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰−𝑖
⋯
𝐊 33
𝐍𝐋𝟒
HBM
𝑖=−𝑘
𝑘
𝑖=−𝑘
𝑘
𝑖=−𝑘+1
𝑘
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰−𝑖
𝑖=−𝑘
𝑘
𝑖=−𝑘+2
𝐜𝑘
𝑘=−∞
⋯
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰−1−𝑖
⋯
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰−𝑖
⋯
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰1−𝑖
⋮
𝑖=−𝑘
⋮
⋱
⋮
𝑘=∞
𝐪 𝑡 =
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰−2−𝑖
𝑖=−𝑘
𝑘
𝑖=−𝑘+1
⋮
⋰
𝑖=−𝑘
𝑘−1
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰2−𝑖
⋯
⋮
𝑘−2
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰−1−𝑖
𝐊 33
𝐍𝐋𝟒 𝐜𝐰𝑖 , 𝐜𝐰1−𝑖
𝐐, 𝐐 = ⋯
⋰
⋮
𝑘−1
𝑒 𝑖𝑘ω𝑡
𝐜−1
𝐐 = 𝐜0
𝐜1
⋮
12
Properties of VSCL shells
Properties (of a carbon fibre reinforced epoxy):
-
Longitudinal modulus E1 = 126.3109 GPa;
Transverse modulus E2 = 8.765109 GPa;
In-plane shear modulus G12 = 4.92109 GPa;
Mass density r= 1600 kg/m3,
Poisson’s ratio n12=0.3.
a=b=1 m. The thickness is h=a/100.
Curvature radius: R1 = 2.5 m (therefore, a/R1 =0.4)
Uniformly distributed harmonic forces is applied
Clamped BC are assumed
y
<𝑇0𝑘 |𝑇1𝑘  = [< 90⁰|45 ⁰ , < -90 ⁰ |-45 ⁰ ]sym
q (x)
T1
T0
a/2
x
13
Linear natural frequencies
Variable Stiffness Composite Laminated Shell
𝜔1 = 851.93 rad/s
𝜔2 = 950.86 rad/s
𝜔3 = 1429.8 rad/s
𝜔4 = 1691.2 rad/s
Constant Stiffness Composite Laminated Shell
𝜔1 = 1143.7 rad/s
𝜔2 = 1253.1 rad/s
𝜔3 = 1501.4 rad/s
𝜔4 = 1775.9 rad/s
14
Nonlinear frequency response
Convergence with the number of shape functions
𝑊1
ℎ
𝜔 𝜔𝑙
7 membrane and 5 transverse sh. f.
5 membrane and 3 transverse sh. f.
15
Nonlinear frequency response
Convergence with the number of shape functions
1
2
3
4
5
(5 5 3)
shape f.
857.57
1036.30
1700.08
1736.54
1935.32
(7 7 5)
shape f.
851.93
950.86
1429.84
1691.18
1823.39
Linear natural frequencies (rad/s)
3rd mode shape
16
Nonlinear frequency response
Convergence with the number of harmonic functions
𝑊1
ℎ
3 harmonics
𝜔 𝜔𝑙
17
Nonlinear frequency response
Convergence with the number of harmonic functions
𝑊1
ℎ
3 harmonics
●●●●
5 harmonics
𝜔 𝜔𝑙
17
Nonlinear frequency response
Convergence with the number of harmonic functions
𝑊1
ℎ
3 harmonics
𝜔 𝜔𝑙
●●●●
5 harmonics
●●●●
7 harmonics
17
Nonlinear frequency response
𝑊1
ℎ
Shape of vibration of model
considering 3 harmonics
𝜔 𝜔𝑙
3 harmonics
●●●●
5 harmonics
Shape of vibration of model
considering 5 harmonics
18
Nonlinear frequency response
First harmonic
Second harmonic
Fourth harmonic
Fifth harmonic
Third harmonic
Total shape of vibration
19
Nonlinear frequency response
𝑊1
ℎ
𝜔 𝜔𝑙
●●●●
●●●●
Variable Stiffness Shell
Constant Stiffness Shell
20
Future Work
• Run the continuation method in parallel processors.
• Compute nonlinear frequency response curve with more shape
functions.
• Determine the bifurcation points and follow secondary
branches.
21
Thank you for your attention!
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