MULTI-OBJECTIVE OPTIMIZATION OF VIBRATION CONTROL WITH VISCO-ELASTIC DAMPING
Trang 120 Tran Quang Hung
MULTI-OBJECTIVE OPTIMIZATION OF VIBRATION CONTROL
WITH VISCO-ELASTIC DAMPING
Tran Quang Hung
University of Science and Technology, The University of Danang; tqhung@dut.udn.vn
Abstract - Visco-elastic damping is one of the passive control
methods in structural vibration Multilayer visco-elastic patch is
preferred due to high level of damping and easy implementation In
order to obtain an economical design, many objectives must be
considered and this concept leads to solving multi-objective
optimization problem Genetic algorithm (GA) is an effective tool in
such case This study shows how optimal solutions derived from
NSGA algorithm can be A simple plate coupling with a cavity is
considered Multi-objective optimization is simulated with many
variables regarding geometry and material properties
Key words - vibroacoustic; vibration control; visco-elastic patches;
multilayer plate; multi-objective optimization; genetic algorithm
1 Introduction
In many cases such as vibration of industrial building
floor, machinery vibration, etc., structures may be generally/
partly damaged by periodic load, resulting in high level of
sound This problem can be solved by installing visco-elastic
patches into structures Many authors have utilized such a
method in literature [1, 2, 3, 8]
Visco-elastic patches are bonded on the shell member
of a structure to form a multilayer zone (Figure 1) in which
the structure has the role of a basic layer; restrained layer
is normally made of the same material as the structure
Figure 1 Visco-elastic treatment
As visco-elastic materials possess high capacity of
damping, vibration level can be reduced effectively The
efficiency not only depends on the position of visco-elastic
patches, but also on layer geometry and mechanical
properties of materials
Industrial problems need to be reduced simultaneously
include vibration level and the weight of structure (i.e product
cost) In reality, these goals are always in conflict and require
solving a multi-objective optimization problem
2 Description and formulation of vibroacoustic
problem with visco-elastic damping
Double cavity couple with an aluminum plate is
considered and described in Figure 2 Opening on the plate
assures the interaction between two cavities 3M visco-elastic
patches are chosen and their properties are shown in Table 1
Finite element (FE) model of vibroacoustic problem
lead to unsymmetrical system [5, 6]:
0
s
f
j Z
M
A
Where U and P - structural displacement and air pressure vectors respectively; Ks and Ms-structural stiffness and mass matrices; Kf and Mf – stiffness and mass matrices of fluid; C- fluid-structure coupled matrix; ρf=1,2kg/m3 – mass density of air; Za- acoustic impedance of cavity wall;
- pulsation of periodic load Fs; j-complex number
Table 1 Material properties
Layer Mass density Young modulus Poisson
(kg/m 3 ) (MPa) coefficient
Viscoelastic 1105 Figure 3 0,49
Figure 2 Vibroacoustic problem: FE model (left) and plate
treated by visco-elastic patches (right)
In the presence of a visco-elastic material, stiffness matrix Ks is complex, the function of and temperature T, which can be expressed as [1, 8, 9]:
*
sT = se+ sv= se+G T sv
K K K K K (2)
In which Kse is elastic part, Ksv is visco-elastic part, G*
is the shear modulus of visco-elastic material (Fig 3) and
sv
K is constant matrix It could be noted that the finite element model of multilayer visco-elastic patch was developed by several authors [1, 2, 3]
In order to reduce numbers of DOF, component mode synthesis (CMS) method adapting to coupling system is chosen [4, 10, 11, 12] Let the reduced basis formed by
mselected vectors, the full m DOF can be condensed to m
DOF by projecting equation (1) on basis :
0
s
T
f
j Z
M
A
Where X denotes reduced matrix of X, i.e: X=Φ XΦT
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In this study, full FE model contains 16342 DOF, there are
366 selected vectors introduced in basis The size of
model is reduced to 2%
Figure 3 Shear modulus and lost factor of 112 3M
visco-elastic material
Based on maximum strain energy law, initial position
of visco-elastic patches is detected as shown in Figure 2
(i.e they are installed in zones where shear strain is
maximal) This treatment helps to control the first three
vibration modes of the plate
In the next section, one will find optimal parameters of
theproblem in condition of many cost functions
3 Multiobjective optimization using NSGA algorithm
In this study, cost functions are the following:
- Mean value of square normal velocity of the plate
surface S:
2 1 2
n S n
S
- Mean value of sound level [6]:
c
- Total mass Mp of the whole structure
In the equation (5), V is air volume, p is air pressure
and pis gradient of p
Multi-objective optimization problem can be written as:
{
𝑤𝑖𝑡ℎ:
𝑔(𝑥) ≤ 0 ℎ(𝑥) = 0
(6)
In which x denotes vector of n variables, vector f(x)
collects all cost functions above (i.e 2
n
V , P a and M p), g(x) and h(x) are constrained conditions which will be
described in the next section, Dn is feasible space of the
problem Because of the conflict between cost functions,
this problem has not only one solution, it has in fact an
infinite number of solutions Therefore, designer must find
optimum surface containing these solutions – that is so
called Pareto-optimum
One of the methods to solve effectively multi-objective
optimization problem is the use of genetic algorithms [15],
in which NSGA algorithm adapted to vibroacoustic
problem is described in Figure 4 [13, 14] Variable x is
coded and represented as a string of the biologic gene Initial population must be generated, and then in order to obtain the next generation, genetic operations are applied (selection, mutation, and crossover) Ranking process assures the rate of convergence
4 Optimal Results
Now these variables are considered:
- Basic layer (structure): plate thickness h1; Young modulus E1; mass density 1 Real problem does not allow modify Young modulus and mass density of structure, but this study considers these two parameters to get a general case
- Visco-elastic layer: thickness h2
- Restrained layer: thickness h3
Figure 4 NSGA algorithm
Constrained conditions g(x) of the optimization problem are defined by variation ranges of variables as shown in Table 2 Equal condition h(x)=0 is not considered here
Table 2 Initial value and variation range of variables
The objectives are: sound level Pa at frequency f=35.4Hz (first mode); the mean value Vn at frequency f=168.3Hz (sixth mode); the total mass of treated plate Mp The values of Pa and Vn are presented as dB with the reference of 106
NSGA parameters are: number of initial population P=50; probability of crossover pc=0.5; probability of mutation pm=0.05 A good convergence is obtained after 26 generations If we increase the values of P, pc and pm the rate of convergence is better but the number of population
in each generation is larger, consequently the simulation time is more important
Figure 5 shows 3D Pareto-optimum surface versus the initial design Plots in 2D are also observed in figures 68
Selection
Yes No Evaluation Initial Population
Convergence
Stop
Ranking
Mutation Crossover
Condensation of model
Trang 322 Tran Quang Hung
Figure 5 Pareto-optimum surface; +: all solutions;
o:optimal surface
It is important to note that all points located at
Pareto-optimum surface can be the solution of the problem The
final solution is decided by the designer For example:
- If minimizing of sound level Pa and structural weight
Mp is important, one can choose solution 1, and associated
value of objectives functions are: Pa=77.1dB; Vn=76.55dB
and Mp=2.201kg
- If minimizing of sound level Pa and structural
vibration Vn is important, one can choose solution 2 and
associated value of objectives functions are: Pa=74.29dB;
Vn=67.62dB and Mp=3.676kg
Values of solution 1 and solution 2 are shown in Table 3
Table 3 Optimal solution 1 and 2
Structure
Visco-elastic
patch 1
Visco-elastic
patch 2
Visco-elastic
patch 3
Figure 6 Pareto-optimum, V n versus P a
Figure 7 Pareto-optimum, M p versus P a
Figure 8 Pareto-optimum, M p versus V n
Figure 9 Response corresponding to solution 1 and 2
Trang 4THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(91).2015 23 Finally, Figure 9 shows the responses of system
corresponding to solution 1 and solution 2 versus initial
design over a band of f = [0 300]Hz
5 Conclusion and remarks
This study has solved the multi-objective optimization
of vibroacoustic problem in which visco-elastic damping is
introduced The set of optimal solutions is found and
presented as Pareto-optimum surface by exploring NSGA
algorithm Some solutions are shown to enhance their
optimal effect in detail
It is clear that the result of NSGA algorithm depends on
input parameters: number of the initial population,
probability of mutation or crossover, etc Therefore, if we
would like to obtain more exact solutions, other
simulations with new input data could be run
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(The Board of Editors received the paper on 10/26/2014, its review was completed on 11/10/2014)