Evolution of constrained layer damping
using a cellular automaton algorithm
C M Chia, J A Rongong*, and K Worden
Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK
The manuscript was received on 19 February 2007 and was accepted after revision for publication on 27 November 2007.
DOI: 10.1243/09544062JMES638
Abstract: Constrained layer damping (CLD) is a highly effective passive vibration control strat-
egy if optimized adequately. Factors controlling CLD performance are well documented for the
flexural modes of beams but not for more complicated mode shapes or structures. The current
paper introduces an approach that is suitable for locating CLD on any type of structure. It fol-
lows the cellular automaton (CA) principle and relies on the use of finite element models to
describe the vibration properties of the structure. The ability of the algorithm to reach the
best solution is demonstrated by applying it to the bending and torsion modes of a plate. Con-
figurations that give the most weight-efficient coverage for each type of mode are first obtained
by adapting the existing ‘optimum length’ principle used for treated beams. Next, a CA algor-
ithm is developed, which grows CLD patches one at a time on the surface of the plate according
to a simple set of rules. The effectivene ss of the algorithm is then assessed by comparing the
generated configurations with the known optimum ones.
Keywords: passive vibration control, vibration damping, constrained layer damping, cellular
automata
1 INTRODUCTION
A common way to reduce vibrations in plate, shell
and other thin-walled structures is to apply surface
damping treatments. When the damping treatment
is applied as a single-layer coating, sometimes
known as free layer damping, the principal energy
dissipation mechanism involves direct, in-plane
strains induced in the damping material. Viscoelastic
polymers operating in the transition zone are often
used as the damping material as they have a high
material loss factor [1].
Significant increases in energy dissipation can be
achieved by attaching a stiff layer known as the con-
straining layer (CL), on top of the viscoelastic layer
(VL). This occurs because large shear strains are gen-
erated in the damping material. To be effective, how-
ever, such constrained layer damping (CLD) systems
must be optimized in terms of materials used and
configuration of the treatments applied. Key par-
ameters controlling performance were identified sev-
eral decades ago using analytical models developed
for slender beams [2–5]. Since then, many studies
have been carried out to improve the damping of
structures, fully or partially covered with CLD. Exist-
ing theoretical understanding is adequate for opti-
mizing CLD for the flexural vibrations of beams.
However, in structures consisting of plate and shell-
like elements, vibration mode shapes can differ con-
siderably and hence the approach has to be more
general.
The desire to apply CLD to more complicated
structures has encouraged researchers to consider
global optimization methods in conjunction with
finite element (FE) analysis. The genetic algorithm
(GA) for example, has been used in several CLD
design studies [6–8]. This approach [9, 10] is based
on the notion of ‘survival of the fittest’: within a popu-
lation of feasible solutions, those yielding the best
results are retained to the next generation while the
poor solutions are eliminated. The GA is a powerful
*Corresponding author: Department of Mechanical Engineering,
The University of Sheffield, Sir Frederick Mappin Building,
Mappin Street, Sheffield S1 3JD, UK. email: j.a.rongong@sheffield.
ac.uk
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global approach but is computationally expensive for
FE-based optimizat ion because of the large number
of possible treatment locations.
An alternative optimization approach involving FE
analysis is the use of algorithms based on cellular
automata (CA) principles [11]. During the last two
decades, CA algorithms [12–14] have been used to
simulate biological phenomena, which are well
known for having an intrinsic optimization schedule.
Thus, by mimicking the evolution of the biological
phenomena, it is possible for CA to drive a physical
system towards its optimum. For example, Wardle
and Tomlinson [15] used CA, inspired by cell
growth in living organisms to locate a free layer
damping coating on a vibrating plate in a more effi-
cient manner. As the optimum location of a free
layer coating is dictated by the surface strain on the
host structure, its optimization is therefore reason-
ably intuitive. The current paper instead, considers
the use of CA in optimizing the more challenging
case of CLD where several different parameters con-
trol the performance. Note that this study does not
discuss one of the most obvious ways to improve
the performance of a CLD treatment, namely to
increase the loss factor of the damping material
itself. Instead, it focuses on achieving a configuration
that uses a given material as effectively as possible.
The structure of this paper is as follows: important
and relevant findings reported in the literature
regarding CLD and the CA approach are briefly
described in sections 2 and 3 respectively; section 4
contains a description of the way in which the per-
formance of the model used in this study is evaluated
and includes optimum CLD for a representative but
easily defined case (namely first bending and torsion
modes of a plate); an appropriate algorithm utilizing
CA is introduced in section 5 and its performa nce is
tested by applying to the plate; finally, conclusions
regarding the suitability of the CA approach for this
problem and possible future studies are summarized
in section 6.
2 OPTIMIZATION OF CLD
Over the years, many have studied the optimization
of CLD on beam-like structures. Energy dissipation
in CLD occurs through deformations induced in the
viscoelastic material (VEM). In most cases, shear is
the dominant mechanism. Though high levels of
damping can be achieved where out-of-plane defor-
mations occur, this regime occurs in a relatively
narrow (and unusual) design space, which is easy to
define [16]. It is therefore not considered further
here. Instead, this section briefly considers relevant
findings in the literature regarding the optimization
of shear-dominated damping.
Analytical work on beams treated with CLD [2 –5]
has identified two parameters that govern damping
effectiveness. The geometric parameter is related to
the stiffness increase caused by the addition of the
CL. It controls the maximum damping achievable.
For most commonly used configurations of conven-
tional CLD, the geometric parameter can be simpli-
fied to show that
h
H
h
v
E
c
t
c
E
h
t
h
þ E
c
t
c
ð1Þ
where
h
is the modal loss factor, H a constant,
h
v
the
material loss factor at the specific frequency and
temperature, E the Young’s modulus, t the thickness,
and the subscripts h and c refer to the host structure
and CL, respectively.
The shear parameter is the relative shear stiffness
of the damping layer normalized by the extensional
stiffness of the CLs. The effectiveness of CLD is extre-
mely sensitive to this parameter and its optimization
has been the focus of several papers [5, 17, 18]. For
example, in 1987, Lifshitz and Leibowitz [18] devel-
oped a numerical program to optimize the uniform
thickness of each layer of a beam for a large variety
of boundary conditions. Note that the shear par-
ameter does not alter the maximum damping
achievable–its value controls whether or not the
maximum (for a given geometric parameter) is
achieved.
In the literature, the shear parameter is expressed
in a number of different ways as researchers adapted
its definition to suit the needs of their particular
study. In order to avoid confusion, in this work refer-
ence will instead be made to the stiffness ratio, C,
which is defined as
C ¼
shear stiffness of VL
in-plane stiffness of CL
¼
G
v
L
2
c
E
c
t
c
t
v
ð2Þ
where G is the shear modulus, L the length, t the
thickness, and subscripts v and c refers to the visco-
elastic and CLs, respectively. Note that C is identical
to the shear parameter used in a number of other
works [19, 20], and is generally used when the CL is
relatively flexible in comparison to the host structure.
By considering a CLD treatment attached to a host
structure under uniform strain, Plunkett and Lee [19]
showed that an optimum configuration occurs when
C 10. They demonstrated this on a beam with CLD,
which had the treatment cut into appropriate length
segments [19]. An extension of this work by Demoret
and Torvik [20] showed that as the non-uniformity of
the strain on the host increases, the optimum stiff-
ness ratio C can be as high as 40. An examination of
results showing damping levels achieved for plates
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treated with different sized patches of CLD has shown
that optimum performance on plate bending modes
is achieved for the same range of C values [21].
To help with the design of CLD for box-section
beams, Marsh and Hale [22] introduced the three-
spring model shown in Fig. 1. They set k
v
to be the
shear stiffness of the VL and k
c
to be the in-plane stiff-
ness of the CL. It can be seen that the stiffness ratio C
used in the current paper is equal to k
v
/k
c
.
The reliance of damping performance on treatment
location and coverage area has been demonstrated
numerically for beams [23, 24], frames [25] and
plates [21, 26]. The optimum location occurs where
the surface strain energy of the host structure is high-
est (i.e. at point of high modal curvature). By altering
the thickness of an initially uniform VL, Lunden [23]
showed that a non-uniform distribution could
improve performance–amplitude reductions of 40
per cent are reported. An optimization approach
that allowed both treatment layers (CL and VL) to
be redistributed is presented by Lumsdaine and Pai
[27]. They applied the sequential quadratic program-
ming (SQP) algorithm to a treated beam and showed
that for a fixed amount of VEM coverage, there exists
an optimum thickness for the base layer of the CLD
structure.
For the context of the work presented here, import-
ant findings from the literature are given below.
1. The stiffness of the CL and the loss factor of the VL
are the most significant parameters that affect the
maximum damping achievable.
2. It is usually desirable to locate CLD at points of
high modal curvature.
3. The stiffness ratio (C) controls the efficiency of a
given CLD treatment. It is a function of the thick-
ness and moduli of both layers in the CLD treat-
ment and thus controls factors such as the
optimum patch size.
In practice, a designer often has a limited choice of
CLD materials available due to environmental factors
(thermal and chemical), fabrication issues (for
example, doubly curved surfaces) and the fact that
commercial CLD is supplied in a few preset thickness
combinations. As a result, optimized performance is
not always the main factor affecting the decision
regarding the nature of the treatment chosen. Thus
to provide good performance, it is important to
select carefull y, the correct shape and location of
the treatment to be applied.
3 CA ALGORITHMS
Biological phenomena are well known for their intrin-
sic optimization schedules. In fact, over the last 30
years or so, biological metaphors have proved invalu-
able in the design of powerful computational pro-
cedures for optimization. The most well known
example of this is of course the genetic algorithm
and its various evolution-based variants [9]. More
recent developments include the Ant Colony Meta-
phor [28] and various approaches motivated by the
human immune system [29]. In fact, despite the
fact that these algorithms are formally optimization
routines in the sense that they aim to maximize or
minimize a given objective function, there is now a
body of evidence that nature does not always seek
to optimize. Ben-Haim [30], among others, argues
that in many cases, nature will not actually seek to
optimize, but will rather seek a solution which
simply satisfies appropriate performance criteria.
This strategy of ‘satisficing’ rather than optimizi ng
has the significant advantage that it can provide sol-
utions, which are robust against uncertainty in the
specification of the problem; it can be proved for-
mally that ‘optimal’ solutions are fragile against
uncertainty [30 ]. The cellular automata approach dis-
cussed in this paper falls into the class of algorithms
which satisfice rather than optimizing; the algorithm
in this case is designed to drive the solution towards a
satisfactory performance rather than optimizing a
formal objective.
The cellular automaton (CA) achieves its perform-
ance objectives through the interactions between
entities (individual cells of the ‘organism’), respon-
sible for ensuring the appropriate performance of
the system as a whole. A simple example from
nature is the balancing of cells in bone through the
twin processes of birth and death. The birth process
produces new cells (and strengthens existing ones)
around the parts that are highly stressed. The death
process weakens and eventually kills cells that are
not being utilized fully. In this way, bone adapts to
perform its mechanical function while maintaining
an appropriate weight [31]. The process of bone
remodelling does not then formally maximize the
‘strength-to-weight’ ratio of the bone, but iterates
towards a system, which satisfies appropriate per-
formance criteria. The power and generality of the
CA approach arise from the fact that a set of simple
interacting processes with limited individual capa-
bility are able to construct arbitrarily complicated
Fig. 1 Three-spring model of a CLD system
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systems. In fact, it is known that particular CA
schemes are Turing machines and thus universal
computers [14].
Often, this complicated system is homogeneous in
terms of complexity of its constituent processes, i.e.
the rules for evolution of the individual cells are the
same throughout the organism. This is the case for
bone.
Formally, a CA is a mathematical idealization of a
physical system in which space and time are discrete
[14]. The design domain is divided into a lattice of
cells, each one capable of performing only a set
of simple operations. Also, each cell may be in one
of the finite number of states, S. These states are
updated synchronously in discrete time steps, t,
according to identical local rules, R; and these rules
depend on the present states of the cell and its neigh-
bours within a certain proximity (neighbourhood).
Equation (3) shows the evolution of the state of
each cell at discrete position r, where r þ D designates
the cells belonging to a given neighbourhood of the
CA
Sðr; t þ 1Þ¼RðSðr; tÞ; Sðr þ D
1
; tÞ ; ...; Sðr þ D
N
; tÞÞ
ð3Þ
Figure 2 shows commonly used CA neighbour-
hoods. The CA neighbourhood does not have any
restrictions on size or location, except that it has to
be the same throughout the entire lattice. The lattice
structure, however, is not limited to regular shape; an
irregular shape of lattice is also possible (in this case,
the neighbourhood is defined in terms of connectivity
rather than ‘shape’). The class of algorithms specified
by the rule of equation (3) is very large and encom-
passes, among others, the class of finite difference
algorithms used to evolve the solutions of partial
differential equations.
The principle of a CA-based algorithm is that over-
all global behaviour of a system can be computed by
cells that only interact with their neighbours based on
local conditions. In general, because of their univer-
sal nature [14], it is possible to simulate any system
using CA by modifying the structural and local
rules; where the structural rules are the shape of the
cell, number of dimensions, and the type of neigh-
bourhood. One can approach the construction of a
CA model for a given problem in two ways. In the
first approach, one has a non-formalized task,
which requires a thorough understanding on the
nature of the corresponding problem and some
experience in dealing with CA. In the case of bone
remodelling for example, the approach would be to
generate rules, which encapsulate the engineering
objective of preserving strength while controlling
weight. This is the approach taken here. A more
formal approach is based on a type of system identi-
fication where rules can be learnt from data [12].
In the current paper, an algorithm based on the CA
principle is used to locate CLD treatments on a plate
with free boundary conditions. The effectiveness
(damping per unit added mass) is evolved in accord-
ance with appropriate performance criteria for the
lowest vibration modes of the plate. Starting with a
single cell, the approach used causes the CLD patch
to grow until a satisfactory size and location are
achieved. While the process is not formally one of
optimization, one can obtain solutions better than
those previously observed in the literature, simply
by making the performance criteria appropriately
stringent.
4 DEVELOPMENT OF AN APPROACH TO TEST
THE EFFECTIVENESS OF CA FOR CLD
As this piece of work aims to show how the CA
approach can be used to assist in the design of CLD
treatments for general structures, care has been
taken to ensure that methods used are readily avail-
able and resource-efficient. To achieve this, the
approximate modal strain energy method [32], in
conjunction with a commercial FE analysis software
package, is used to estimate the performance of
CLD treatments considered. It is important to note,
however, that the CA method described in this
paper is equally compatible with more accurate cal-
culation methods: the CA rules do not depend on
the method chosen to evaluate the cost function.
This section provides an explanation of the modelling
approach used and the basis for testing the perform-
ance of the developed algorithm.
The host structure chosen for this study is a
rectangular plate with free boundaries. The reason
for this choice is that a plate provides a good compro-
mise between a system with independently verifiable
optimum coverage and the need to apply the CA
approach to a structure with a two-dimensional
dynamic strain field. The lowest frequency vibrat ion
modes of plate-like structures involve out-of-plane,
flexural deformations. Two types of mode shape are
Fig. 2 Commonly used neighbourhoods; hatched areas
show (a) the Von Neumann neighbourhood, and
(b) the Moore neighbourhood of the black
elements
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most common: those for which all node lines are
parallel and those for which at least one node line is
perpendicular to the others. For ease of identifi-
cation, these are referred to as bending and torsion
modes respectively in this paper.
The plate is modelled using quadratic FEs: eight
noded offset shells are used to represent the host
structure and CL while the viscoelastic material is
modelled using 20 noded solid elements. While
many researchers have developed special shell
elements for more efficient modelling CLD on struc-
tures, these are not used here because they are cur-
rently not available in general purpose commercial
FE code. The chosen FE mesh gave a grid of 10
10 mm
2
elements on the surface of the plate. This is
the result of initial studies aim ed at achieving a com-
promise between the need for computational
efficiency and the need to get good calculation accu-
racy and spatial resolution for the CA. Dimensions
and materials properties for the host structure and
CLD treatment are given in Table 1.
A variety of different approaches can be used to
represent the viscoelastic properties of the damping
material. While it is well known that the complex
modulus of a viscoelastic material varies dramatically
with temperature and frequency, most analytical
work is carried out under isothermal conditions. In
the w ork presented here, the complex modulus is
also assumed constant over all frequencies. For the
purposes of demonstrating the CA approach and
when vibration modes are considered individually,
this is a reasonable simplification as the exact
design of CLD rarely changes natural frequencies by
more than 10 per cent. The consequences of ignoring
this change are minimal as the level of uncertainty
arising from material characterization (including
batch variability and the use of the temperature–
frequency superposition method) is usually at least
20 per cent. However, shoul d several vibration
modes spanning a significant frequency range be
considered simultaneously, a calculation strategy
that takes frequency dependence in to account
would be necessary.
The final approximation is associated with the use
of the modal strain energy approach itself. It is well
known that this approach, in its simplest form, can
provide overestimates of damping and underesti-
mates of natural frequency for systems with high
material loss factors (
h
v
) as it ignores the complex
part of the stiffness matrix during the eigenvalue
extraction routine. When applied to a CLD patch,
the approximation leads to an underestimation of
the magnitude of the stiffness ratio C. For example,
a typical polymer used in CLD is ISD112 (from 3M)
whose material loss factor
h
v
, at room temperature,
is in the range 0.8 to 1 depending on the frequency.
Equation (2) implies that the optimum length of a
given patch might be overestimated by no more
than 20 per cent if the effect of material loss factor
is ignored. This is consistent with reports in the litera-
ture that the optimum length is only weakly depen-
dent on the loss factor of the VL [19, 20]. Methods
for improving the accuracy of the modal strain
energy approach are described elsewhere [33, 34]
and can easily be used if required. However, in the
work presented here, this is not considered necessary
as the aim is to demonstrate the CA approach.
To allow direct comparison between patches of
different size, the effectiveness of a particular CLD
treatment is quantified as the loss factor ratio per
unit added mass. The Modal Strain Energy approach
gives
Loss factor ratio ¼
h
h
v
¼
U
visc
U
total
ð4Þ
where
h
is the modal loss factor, U
visc
is the modal
strain energy in the VL and U
total
is the total strain
energy for that mode.
In order to assess the ability of the CA approach
developed to generate satisfactory treatments, identi-
fication of the characteristics of a weight-efficient
coverage is necessary. The general 3-spring model
of Marsh and Hale [22] (see Fig. 1) suggests that an
optimum stiffness ratio exists for modes of any struc-
ture treated with CLD. While there is abundant infor-
mation relating to the optimization of flexural modes
for beams (and hence by analogy, a reasonable start-
ing point for the plate bending modes), guidelines for
the optimization of CLD for plate torsion modes are
not available in the literature. This section therefore
contains a brief study in which the concept of the
optimum stiffness ratio is applied to bending and tor-
sion modes in plates.
CLD treatments are located near the centre of the
plate and calculations are carried out to understand
the effect that patch size and VL modulus had on
the damping performance achieved. Practically this
is achieved in the FE analysis by altering the
Young’s modulus of the VL (and hence the shear
modulus G
v
).
Table 1 Geometric and material properties used
Host
structure
Viscoelastic
layer
Constraining
layer
Young’s
modulus (GPa)
70 Various 70
Poisson’s ratio 0.3 0.45 0.3
Density (kg/m
3
) 2700 1100 2700
Thickness (mm) 3 0.25 0.3
Width (mm) 300 Various Various
Length (mm) 450 Various Various
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