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Proceedings of the 7th International Conference on Mechanics and
Materials in Design
Albufeira/Portugal 11-15 June 2017. Editors J.F. Silva Gomes and
S.A. Meguid.
Publ. INEGI/FEUP (2017)
-313-
PAPER REF: 6801
EXPERIMENTAL EVALUATION OF THE DAMPING PROPERTIES
AND OPTIMAL MODELLING OF COATINGS MADE BY PLASMA
DEPOSITION TECHNIQUES
Stefano Amadori(*), Giuseppe Catania
Department of Industrial Engineering - Ciri MaM, University of
Bologna, Bologna, Italy (*)
Email: [email protected]
ABSTRACT
Coating layers, applied by means of different deposition
technologies, can be used to produce
composites with high damping properties. This is well known in
literature and it is also
verified by some experimental results reported in this work. The
constitutive equivalent
material model of coated specimens is identified and optimized
by means of a generalized
Kelvin model of n-th order, defined by means of the ratio of
polynomials in the frequency
domain. Dynamical measurements data obtained from coated
single-layered and multi-layered
samples obtained with different deposition techniques, are used
to identify the optimal
material model order and parameters. A robust identification
technique that makes use of
Forsythe orthogonal polynomials is employed for the numerical
identification of the model
parameters and a specific technique is introduced to eliminate
the model non-physical
components generated by measurement and model noise.
Keywords: damping, model identification, dynamical measurement,
optimal model fitting.
INTRODUCTION
There is a great demand in industrial, aerospace and automotive
mechanical applications for
non-conventional composite materials suitable to design
components showing high stiffness,
high resistance and high damping behaviour. Coating layer
technology can be employed to
increase the global dissipative properties of an industrial
component with limited influence on
the other mechanical properties (Tassini, 2006; Yu, 2005;
Ustinov, 2001). One or more
coating layers can be deposited or grown in order to obtain a
finished composite component
with specifically designed characteristics. Different techniques
are known (Rongong, 2004;
Blackwell, 2007), for example plasma vapor deposition, plasma
spray or the anodization
process. When designing or analyzing the properties of a single
or multi-layer coating, e.g.
damping efficiency, corrosion resistance, or thermal insulation,
many factors must be taken
into account: layer composition, layer or multi-layer structure,
deposition or growing
technology and adhesion to the substrate.
The dissipative properties of a component can be significantly
tailored by means of the
adoption of layers showing high internal hysteresis or by
maximizing the frictional actions at
the interface between the different layers (Casadei, 2014; Wang,
2013; Kireitseu, 2007). A
considerable number of applications was dedicated to study the
influence of the coating
characteristics, such as the coating material structure
(Casadei, 2014; Guangyu 2014;
Ustinov 2008) or the interface structure (Colorado, 2006) on the
coated component damping
behaviour and its temperature dependence (Khor, 2001).
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Dynamic mechanical measurements can be used as an effective
experimental technique for
estimating the coated component damping properties. Forced and
free vibration experimental
methods were successfully employed to estimate the component
dissipative properties by
means of comparing the coated and uncoated component dynamical
response (Reed, 2008;
Pastias, 2004; Torvik, 2009).
Fig. 1 - Generalized Kelvin model of order n
Single or multi-layered coated materials dynamic behavior can be
considerably complex and
classical material models (Zener, 1948; Nowick, 1972) may not
accurately describe the coated
component constitutive relationship (Amadori, 2016). Different
higher order constitutive
models can be adopted (Finley, 1989; Ramirez, 2007; Vasquez,
2010). Such models may
require the use of not trivial constitutive equations and as a
consequence a robust
identification procedure must be considered for solving the
generally ill-conditioned problem
of parameter identification (Richardson, 1982; Amadori,
2016).
In this work, components made by single-layer and multi-layer
coatings applied on a metal
substrate by means of two different techniques, i.e. by an
anodizing process and by a reactive
plasma vapor deposition (RPVD) process, are considered. Coated
and uncoated specimen
dynamic mechanical measurement test results are compared and
processed by means of a
robust parameter identification and model condensation
technique.
IDENTIFICATION PROCEDURE
The procedure used to identify the composite constitutive
relationship, defined by the authors
of this work in a previous paper (Amadori, 2016), is briefly
outlined here. A generalized
Kelvin model (also cited in literature as the standard linear
solid model) is chosen to model
the composite material constitutive equation. Since low order
material models such as the
Zener and Kelvin-Voight models can show limitations in
predicting the dynamic behaviour of
complex materials, high order models are taken into account in
this work. A generalized
Kelvin model of order n, obtained by connecting n single Kelvin
model units in series is
shown in Fig.1, and starting from the relationship between the
Fourier transform ( )$ of σstress and ε strain of the single Kelvin
unit in the frequency domain:
ˆˆ ( ) , 1, ... ,i i iE j i nσ ω β ε= + ⋅ ⋅ = . (1)
The constitutive relationship for the generalize Kelvin model
is:
( ) ( )ˆ ˆˆ E E Cσ ω ε ω ε= ⋅ = ⋅ ⋅ (2)
where:
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( )( )
( )
1
1
11
1
11
, =
1
ns
s ns
nr i i
r
r
C EE
α ωω
β ω
−
=−
=
=
+ ⋅
= + ⋅
∑∑
∑ (3)
Fig. 2 - Slender beam in flexural excitation experimental
set-up
with clamped-sliding boundary conditions
The constant �� is the modulus of the material when the
frequency value is zero and it can be
obtained from static experiments or extrapolated from dynamical
measurements estimates as
the frequency goes to zero. C(ω) is a rational function with a
n-degree numerator polynomial
and a n-1 degree denominator polynomial. The estimate value of
C(ω) in a wide frequency
range can be obtained from dynamic mechanical measurements in a
forced flexural excitation
experimental set-up.
The contribution of the inertial actions is also taken into
account and for a slender beam in a
flexural experimental set-up with clamped-sliding boundary
condition (Fig.2) and with a
periodic force ( ) ( )expq t Q j tω= ⋅ applied at the sliding
end, the relationship in the frequency domain between the
transverse displacement ( )ν̂ ω and applied force ( )q̂ ω at the
beam sliding end is approximated by:
( )
( )
( )( )
2
4 3 21
2 1 cosh cos
sinh sinˆ ˆ
m
s s
ns s
s s
k k
k kv q
E C k I L Mω ω
ω ω=
− ⋅ + ⋅⋅ ⋅ ⋅ − ⋅∑� (4)
M is the beam mass, I is the beam section moment, L is the beam
length, ks satisfy the
following equation:
sinh cos cosh sin 0k k k k+ = (5)
Using Eq.4 and the corresponding displacement and applied force
data, the material
constitutive relationship Ci=C(ωi) at ωi excitation frequency
can be estimated by finding the
solution of the non-linear equation Eq.6:
( )
( )( )( )
2
34 21
2 1 cosh cos
ˆsinh sin0
ˆ
m
s s
ns s i
i
s ii s i
k k
k k vC
qE C k I L M
ωωω=
− ⋅ + Θ = − =⋅ ⋅ ⋅ − ⋅
∑ (6)
L 0
q(t)
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Topic-B: Experimental Mechanics
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The solution is found by means of the Newton-Raphson method, the
procedure showed to be
stable and only a few iterations are generally needed for
convergence.
The Ci experimentally estimated values are used to identify the
parameters of the constitutive
relationship by means of an approach based on the least square
error technique. The rational
function C(ω) is defined in Eq.7 by means of the ratio of two
polynomials expressed in a
basis of orthogonal polynomials ( )s jη ω and ( )r jµ ω :
( )( )
( )1
1
1
1
1
n
s s i
si i n
r r i
r
j
C C
j
γ η ωω
δ µ ω
=−
=
+ ⋅= =
+ ⋅
∑
∑ (7)
Starting from Eq.6 and applying the least square error method
the unknown γs and δr
coefficients can be obtained as the solution of two uncoupled
systems of linear equations
(Amadori, 2016). With the adoption of an orthogonal polynomial
base, and in the specific
case of a Forsythe polynomial base (Forsythe, 1957), high
accuracy for the identification of
both low and high order model can be obtained. The polynomial
base coefficients γs and δr
can be unsuitable for further processing. It is more convenient
to convert them to an
equivalent monomial base, this can be done by means of known
algorithms (Kelly, 1967).
This procedure has proved to be robust and the material
constitutive relationship C(ω) can be
obtained as the ratio of two polynomials defined by coefficients
in a monomial base, as in
Eq.3.
MODEL CONDENSATION
Experimental measurements are affected by noise and the basic
least-square error technique is
not able to distinguish between system poles and virtual,
non-physical poles resulting from
such noise. High order model fitting estimates then generally
result. This model can be further
condensed, by finding an optimized model order and by
eliminating the non-physical model
components. The rational function C(ωi) is manipulated in order
to outline the system poles
and residues. By inverting C(ωi) and defining it in standard
partial fraction form:
( ) ( )11 n s
si i s
R
C jω ω ψ==
−∑ (8)
where residues Rs are computed from the rζ zeros and sψ poles of
1/C(ω). Since from Eq.2,8:
( )( )
( )1
1ˆ ˆ
ns
s s
R
E jε ω σ ω
ω ψ== ⋅ ⋅
−∑ (9)
computational and physical sψ pole and Rs residues subsets can
be found by observing that poles and residues are expected to
appear as complex conjugated pairs or real values. Poles
with positive real part can be considered as computational poles
and can be discarded.
Couples of physical poles and residues are also assumed to
present stability if the model order
is increased. Unstable poles with respect to model order n can
be considered not physical and
discarded. The model, condensed to a subset of nr physical
pole-residues sR% and sψ% , is
expressed in the form:
( )( )
( ) ( ) ( )1 0
1 + ;
nr mrs
m m r
s rs
Rj
C jω ω ω λ ω
ω ψ= =Λ Λ = ⋅
−∑ ∑%
� (8)
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EXPERIMENTAL SET-UP AND TESTS
Specimens in the form of slender beams of rectangular cross
section are tested with a dynamic
mechanical analyzer (Fig.3) in a forced flexural excitation
experimental set-up with
clamped-sliding boundary conditions.
(a) (b)
Fig. 3 - Experimental apparatus. Dynamic mechanical analyzer (a)
and specimen
mounted in the flexural excitation experimental set-up (b).
Λm(ω) is a m order (m=1,2 generally works well) polynomial
function used to estimate the
contribution of the nc=n-nr eliminated computational
poles-residue pairs. The calculation
procedure to obtain coefficients λr can be found in the authors
previous work
(Amadori, 2016).
Two different coating solutions are considered in this work. In
both cases coating layers are
applied on a Al alloy (Al 1000) beam, length 10-2
m, width 3×10-3
m, thickness 5×10-4
m and
density 2700 kg/m3. One set of specimens is obtained by means of
the reactive plasma vapor
deposition (RPVD) technique. A dual-layer coating is applied on
two opposite faces of the
beam. First a Ti metallic bond coat is deposited and then coated
with an external hard coating
layer formed by TiO2 (Fig.4a)..
(a) (b)
Fig. 4 - Schematic representation of the dual-layered structure
of Ti+TiO2 coating (a)
and of the single layer Al oxide coating (b).
Both the metallic bond coat and the hard coating layer thickness
is 1µm. Other coated
specimens are obtained by means of an anodizing process, the Al
1000 beam is coated on the
two opposite faces of the beam by a single, 4µm thick, Al oxide
layer (Fig.4b). Dynamic
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Topic-B: Experimental Mechanics
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mechanical test on the coated specimen sets and homogeneous Al
beams are made at 0.01%
maximum strain over a [0.01-200] Hz frequency range and 35°C
constant temperature.
RESULTS AND DISCUSSION
The ratio of the imaginary and real part of E(ω) (Eq.2) is
considered as a measure of the
material dissipative properties (Nowick, 1972), and the results
obtained for the uncoated and
the coated specimens are compared. Fig.5 shows the estimated
Im(E(ω))/Re(E(ω)) ratio of the
dual-layered (Fig.4a) and of a homogeneous Al 1000 specimen, and
the relative increase of
the same ratio.
Fig. 5 - Estimates of the ratio Im(E(ω))/Re(E(ω)) for dual-layer
(Ti+TiO2) coated specimen (red)
and homogeneous Al 1000 specimen (green), and their
Im(E(ω))/Re(E(ω)) relative difference % (blue).
Fig.6 shows the same results obtained for the single-layered
(Fig.4b) specimen. It is shown
that the coated specimen Im(E(ω)/Re(E(ω)) ratio increases with
respect to the uncoated case.
The dual-layer coated specimen only exhibits a meaningful effect
at frequencies above 70 Hz,
while a significant increase at all measured frequencies can be
observed for the single-layer
coated specimen. Since for both coating technologies a very
small coating layer thickness,
compared to the beam substrate thickness, is adopted, it can be
assumed that this results are
mainly due to the interface frictional actions acting during the
imposed flexural test cycle.
The identification and the model condensation procedure are used
to process experimental
measurements data, and the results are shown in Fig.7 for the
dual-layer specimen and in
Fig.8 for a single-layer specimen. Condensed model fitting
estimates of order n=11 are shown
for both specimen solutions and compared with the estimated
measurement values. In both
cases a n=43 model order was taken into account as the starting
point of the condensation
procedure and a n=11 condensed model is obtained by eliminating
all of the poles with
positive real part and the ones not exhibiting stability with
respect to the model order increase.
Stability diagrams used to evaluate poles stability are shown in
Fig.9.
0 20 40 60 80 100 120 140 160 180 2000
0.05
0.1
0.15
0.2
Im(E
)/R
e(E
)
Frequency [Hz]
0 50 100 150 200
0
100
200
300
400
Frequency [Hz]
Im(E
)/R
e(E
)
Rel
ativ
e D
iffe
ren
ce %
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Materials in Design
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Fig. 6 - Estimates of the ratio Im(E(ω))/Re(E(ω)) for
single-layer (Al oxide) coated specimen (red)
and homogeneous Al 1000 specimen (green), and their
Im(E(ω))/Re(E(ω)) relative difference % (blue).
Fig. 7 - Dual-layer (Ti+TiO2) coated specimen E(ω) estimates
(green) and n=11 condensed
model fitting estimate (red).
0 20 40 60 80 100 120 140 160 180 2000
0.05
0.1
0.15
0.2Im
(E)/
Re(
E)
Frequency [Hz]
0 50 100 150 200
0
200
400
600
800
Frequency [Hz]
Im(E
)/R
e(E
)
Rel
ativ
e D
iffe
ren
ce %
0 20 40 60 80 100 120 140 160 180 2005.5
6
6.5
7
7.5
8x 10
10
Re(
E)
[Pa]
Frequency [Hz]
0 20 40 60 80 100 120 140 160 180 200
0
5
10
15
20x 10
9
Im(E
) [P
a]
Frequency [Hz]
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Topic-B: Experimental Mechanics
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Fig. 8 - Single-layer (Al oxide) coated specimen E(ω) estimates
(green) and n=11
condensed model fitting estimate (red).
(a) (b)
Fig. 9 - Stability diagrams with selected poles (blue and red)
of n=11 condensed models
of the dual-layered (Ti+TiO2) (a) and single-layered (Al oxide)
(b) specimens.
0 20 40 60 80 100 120 140 160 180 200
6
7
8
9
x 1010
Re(
E)
[Pa]
Frequency [Hz]
0 20 40 60 80 100 120 140 160 180 2000
1
2
3
4x 10
10
Im(E
) [P
a]
Frequency [Hz]
0 500 10000
10
20
30
40
50
60
Poles absolute value [rad/s]
Mo
del
ord
er
0 500 10000
10
20
30
40
50
60
Poles absolute value [rad/s]
Mo
del
ord
er
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To assess the effectiveness of the model condensation step,
Figs.10-11 report the
computational model fitting estimate results of the same order
(n=11). It can be observed that
in both the dual-layer and the single-layer solutions the
condensed model is a significantly
more accurate approximation of the experimental estimates and
that the un-condensed model
fail to accurately model the experimental estimates.
These results show that both coating solutions may affect the
specimen dissipative properties.
The integrity of the specimen layer interface showed to be
nevertheless affected by cyclic
loading, and preliminary results showed that damages and cracks
may appear at the interface
between the different layers, making the solution partially
non-effective. Different coating
process parameters and technologies will be investigated in
future work to solve this specific
problem.
CONCLUSIONS
A composite solution adopting many coating layers of different
materials, may strongly
increase the component structural damping, according to some
engineering design
specification constraints.
Two simple solutions are investigated, a dual-layer obtained by
means of a thin coating of Ti
and of Ti oxide and a single-layer obtained by means of a thin
coating of Al oxide. Coated
specimens show improved damping behaviour with respect to
uncoated ones for both the
solutions under study. Preliminary cyclic loading tests show
that the component dissipative
properties may decrease due to the formation of damage and
cracks at the layer interfaces.
The obtained results are positive enough to justify further
investigations. Different
technological and processing solutions will be examined in order
to obtain increased damping
behavior by means of frictional actions at the layer
interfaces.
Fig. 10 - Dual-layer (Ti+TiO2) coated specimen coated specimen
E(ω) estimates (green)
and computational n=11 model fitting (blue)
0 20 40 60 80 100 120 140 160 180 2005.5
6
6.5
7
7.5
8x 10
10
Re(
E)
[Pa]
Frequency [Hz]
0 20 40 60 80 100 120 140 160 180 200
0
5
10
15
20x 10
9
Im(E
) [P
a]
Frequency [Hz]
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Fig. 11 - Single-layer (Al oxide) coated specimen E(ω) estimates
(green)
and computational n=11 model fitting (blue)
ACKNOWLEDGMENTS
This study was developed within the MAM-CIRI with the
contribution of the Regione
Emilia Romagna, progetto POR-Fesr-Tecnopoli.
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