Modeling of the Damping Properties of Unidirectional Carbon Fibre Composites
Modeling of the Damping Properties of Unidirectional Carbon Fibre
Composites
Ying Gao, Yibin Li*, Yi Hong, Hongming Zhang, and Xiaodong He
Center for Composite Materials and Structures, Harbin Institute of Technology, Harbin, 150080, China
National Key Laboratory for Advanced Composites in Special Environment, Harbin Institute of Technology, Harbin
150080, P.R. China
Summary
Vibration and noise are harmful to mechanical systems and the environment. Using high damping materials
or structures is a very effective way to control vibration and lower noise. The paper presents modeling of the
damping properties of unidirectional carbon fibre reinforced epoxy resin composite materials, including natural
frequencies, damping loss factors and mode shapes. Damping modeling is implemented using a finite element
method based on mode analysis theory and strain energy method. The influences of fibre orientation on the natural
frequencies and damping loss factors for the first four modes are analyzed in the case of cantilever. The results
showed that natural frequencies decrease with fibre orientation increased. The influences of fibre orientation on
the damping loss factors are different with fibre orientation changed.
1. Introduction
Under outside interference or dynamic
loads, mechanical systems and power
equipments will produce vibration
and noise, which are harmful to
the systems themselves and the
environment. It is necessary to have
a sound methodology to control
vibration and lower noise. Using high
damping materials or structures is one
of the very effective ways to solve
these problems. Polymer composites
have a higher damping property than
metals1. And they can be directly used
as vibration components because of
their excellent performances such as
high specific strength, high specific
stiffness, corrosion resistance, high
damage tolerance, etc.2. Therefore,
the damping of polymer composites
has become a hot spot in the research
field of damping materials, especially
for fibre-reinforced composites.
The initial studies on the damping
m o d e l i n g o f fi b r e - r e i n f o r c e d
composites were reviewed
extensively in the paper by S.H.
Zhang and H.L. Chen3. A threedimensional finite element method
was developed initially by S.J.
Hwang and R.F. Gibson 4 for the
characterization of the effects of
three-dimensional states of stress on
the damping of laminated composites.
The calculation of damping is
performed by a strain energy method.
The study shows that the finite element
method is a powerful technique for
the determination of the damping
of laminated composites. In a later
work5, a finite element method based
on the material complex modulus
was used to determine the damping
of composites. The approach is an
extension of the elastic–viscoelastic
approach, which accounts for the
frequency dependence of the loss
factor. The complex modulus method
was also applied by J.P. Talbot
and J. Woodhouse6 to evaluate the
damping of laminated plates. For the
*corresponding email: [email protected]
damping research of fibre-reinforced
composites, many works1,5,7-10 are
focused on glass and Kevlar fibre
composites. But studies on carbon
fibre composites are relatively less.
This paper presents a study on the
damping properties of unidirectional
carbon fibre composites. The natural
frequencies and mode shapes of the
composites are obtained using the
modal analysis function of ANSYS
software. Damping loss factors are
calculated by the finite element
method based on strain energy. The
influences of fibre orientation on the
damping properties of composite
cantilevers are discussed, which builds
a good foundation for the analysis
and engineering application of the
damping of composites and structures.
2. Material
Carbon fibre reinforced epoxy resin
composite is studied. Orthotropic
elastic and damping properties of
the material are listed in Table 1,
according to11.
Smithers Rapra Technology, 2011
©
Polymers & Polymer Composites, Vol. 19, Nos. 2 & 3, 2011
119
Ying Gao, Yibin Li, Yi Hong, Hongming Zhang, and Xiaodong He
Table 1. Orthotropic elastic and damping properties of the material
E1(GPa)
E2(GPa)
110.0
G12(GPa)
9.0
v12
3.9
3. Damping Modeling Method
Damping modeling is implemented
using a finite element method based
on mode analysis theory and strain
energy method.
The natural frequency and mode shape
for each mode are evaluated by modal
analysis function of ANSYS software.
Modal extraction method is the Block
Lanczos method.
The damping loss factor h is used as
the measure of damping capability in
the study, which is evaluated by the
strain energy method in12.
0.34
ψ = 2πη (1)
The specific damping coefficient y is
defined as the ratio of the total energy
dissipated in the structure to the total
strain energy of the structure per cycle
of vibration. It can be expressed as:
ψ=
ΔU
U (2)
The total strain energy stored in the
element k is given by:
N
6
Uk = ∑ ∑ U kp,m
P=1 m=1
(3)
With:
U kp,1 =
1
1
∫ σ11ε11dVpk , U kp,2 = 2 ∫k σ 22ε22 dVpk , 2 Vpk
Vp
U kp,3 =
1
1
∫ σ 33ε33dVpk , Ukp,4 = 2 ∫k σ12ε12 dVpk , 2 Vpk
Vp
U kp,5 =
1
1
∫ σ 23ε23dVpk , Ukp,6 = 2 ∫k σ13ε13dVpk , 2 Vpk
Vp
120
(4)
0.75
Ψ22(%)
5.95
Ψ12(%)
ρ(kg/m3)
6.79
1513
where N is the layer number of the element k. Ukp,m, sij and eij are the strain
energy, stress and strain related to the material direction, respectively. Vkp is the
volume of layer p in the element k.
The total strain energy stored in the structure is obtained by summation of the
elements:
n
U = ∑ Uk
k=1
(5)
where n is the number of the elements in the structure.
The energy dissipated by damping in the element k can be written in the form:
ΔUk =
N
∑ (ψ
U kp,1 + ψ22 U kp,2 + ψ33U kp,3 + ψ12 U kp,4 + ψ23U kp,5 + ψ13U kp,6 )
11
p=1
The damping loss factor h is related
to the specific damping coefficient y
by the relation:
Ψ11(%)
(6)
introducing the specific damping coefficients yij related to the material directions.
For unidirectional fibre reinforced composite, Ψ22＝Ψ33，Ψ12＝Ψ23＝Ψ13.
Then, the total energy dissipated in the structure is given by:
n
ΔU = ∑ ΔU k
k=1
(7)
Finally, the specific damping coefficient y and the damping loss factor h of the
structure can be obtained by Expressions (2) and (1), respectively.
4. Influences of Fibre Orientation on the Damping Properties of the Composites
The damping properties of unidirectional carbon fibre composites were evaluated
in the case of cantilevers with fibre orientation θ varying from 0° to 90°. The
cantilevers have a nominal length of 190 mm, a nominal width of 10 mm and a
nominal thickness of 2 mm.
The variation of the natural frequencies for the first four modes with fibre
orientation is given in Figure 1. It shows that the influences of fibre orientation on
the natural frequencies for the first four modes are similar. The natural frequencies
for the first four modes decrease when the fibre orientation is increased, due to
the decrease of the stiffness of the cantilevers with fibre orientation increased.
Figure 2 reports the results of the variation of the damping loss factors as function
of fibre orientation. The results for the first three modes are similar. The value
Polymers & Polymer Composites, Vol. 19, Nos. 2 & 3, 2011
Modeling of the Damping Properties of Unidirectional Carbon Fibre Composites
of damping loss factor is the highest
at fibre orientation of about 45°. For
the fourth mode, a high damping is
observes at fibre orientation of 0°,
which can be explained by the mode
shapes of the cantilevers. The mode
shapes for the first four modes of the
cantilevers with fibre orientation of
0° and 15° are showed in Figures 3
and 4, respectively. The results show
that twisting mode induces an increase
of damping, because of the notable
increase of in-plane shear deformation
due to twisting. While bending mode has
a relatively low damping. As a result,
shear deformation is the main factor of
damping for each mode.
5. Conclusions
Figure 1. Variation of the natural frequencies as function of fibre orientation
Figure 2. Variation of the damping loss factors as function of fibre orientation
The damping properties of
unidirectional carbon fibre reinforced
composite materials were evaluated
based on mode analysis theory and
strain energy method. The influences
of fibre orientation on the natural
frequencies and damping loss factors
are analyzed. The results showed that
natural frequencies decrease when
fibre orientation is increased. The
influences of fibre orientation on the
damping loss factors are different
with fibre orientation changed. The
damping properties of the cantilever
depend on the mode shapes. The mode
with more in-plane shear deformation
has a higher damping.
Figure 3. The mode shapes of the cantilever with fibre orientation of 0°
Polymers & Polymer Composites, Vol. 19, Nos. 2 & 3, 2011
121
Ying Gao, Yibin Li, Yi Hong, Hongming Zhang, and Xiaodong He
Figure 4. The mode shapes of the cantilever with fibre orientation of 15°
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