the finite--element element method and … finite--element element method and wave analysis part 1:...
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THE FINITETHE FINITE--ELEMENT ELEMENT METHOD AND WAVE METHOD AND WAVE
ANALYSISANALYSIS
Part 1: Introduction to the Finite Part 1: Introduction to the Finite
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Part 1: Introduction to the Finite Part 1: Introduction to the Finite Element Method (FEM)Element Method (FEM)
Part 2: introduction to the Wave Part 2: introduction to the Wave Finite Element Method (WFE)Finite Element Method (WFE)
Elisabetta ManconiUniversity of Parma, Italy
Overview
Aim of this lecture is to give anintroduction to the Wave FiniteElement (WFE) method.
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Background to the WFE Part 1: Introduction to the FEM Part 2: Introduction to the WFE The basic steps of WFE WFE modelling Examples Summary References
Background to the WFE method
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Wave properties
Dispersion relations
Phase velocity
Group velocity
Reflection and transmission
Forced response, acoustic transmission etc
How to calculate wave properties?
Simple structures: analytical solutions
Complex structures?
Numerical methods, spectral elements,spectral finite elements, etc.
Wave/finite element ( WFE) method
The Wave Finite Element Method
The Wave Finite Element method basically is atechnique to investigate wave motion in both periodicstructures or continuous waveguides in 1 dimension(e.g. beams) and 2 dimensions (e.g. plates), [1,2]. Inthis method a “period” of the structure is modelledusing conventional FEs. The equation of motion fortime–harmonic motion is therefore obtained from theFE model in terms of a discrete number of nodalDOFs and forces. Periodicity conditions are then
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DOFs and forces. Periodicity conditions are thenapplied, and an eigenvalue problem is formulatedwhose solutions give the dispersion curves andwavemodes.
Since the WFE method is based on the FiniteElement discretisation of a small segment of awaveguide, a brief introduction to the FE method,together with some examples is given. Then theWFE method is briefly described and applications ofthe method are shown.
Part 1: Introduction to the Part 1: Introduction to the Finite Element Method (FEM) Finite Element Method (FEM)
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Key questionHow do we create mathematical models that allowus to calculate the modal frequencies and modeshapes or the vibration response to applied forcesfor complicated structures?
A bevelled gear rotor.
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The finite element method (FEM) is one of the wellestablished and most employed tool for the dynamicanalysis of structures posed over complicated domains.Basically FEM is a numerical technique for findingapproximate solutions of partial differential equations.“The finite element method is a numerical procedure inwhich a complex structure is considered as anassemblage of a number of smaller elements, whereeach element is a continuous structural member calleda finite element. The elements are assumed to beinterconnected at certain points known as nodes. Sinceit is very difficult, or impossible, to find the exactit is very difficult, or impossible, to find the exactsolution (such as displacements) of the originalstructure under specified loads, a convenientapproximate solution is assumed in each finite element.By requiring that the displacements be compatible andthe forces balance at the joints, the entire structure iscompelled to behave as a single entity” [3].
The basic finite element steps are : the formulation ofthe problem in variational form, the finite elementdiscretisation of this formulation, and the effectivesolution of the resulting finite element equations.
A number of books have been written about the FiniteElement Method. Some good references are [4-6].
Historical background
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The first textbook on FEM was published in 1967 byZienkiewicz and Cheung: Zienkiewicz, O.C. andCheung, Y.K., The finite element method in structuraland continuum mechanics. McGraw-Hill (1967).
Historical background to modern finite element methods, after O. C. Zienkiewicz
The first step in the analysis of any dynamic problemis the formulation of the equations of motion. Thereare a number of methods to obtain the equations ofmotion. One of the basic step in the FEM is theformulation of the problem in variational form. Themost direct and most famous variational principle isthe Hamilton’s principle , which states that “themotion of an arbitrary mechanical system occurs insuch a way that a definite integral, namely the timeintegral of the Lagrangian function, becomesstationary for arbitrary possible variations of the
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stationary for arbitrary possible variations of theconfiguration of the system, provided the initial andfinal configurations of the system are prescribed” [7]
HAMILTON’S PRINCIPLE
T: kinetic energy; V: potential energy; QNC: work doneby non conservative forces
( )( )2
1
0t
NCtT V Q dtδ δ− + =∫
For a conservative system, the principle says that,“along the true time path, the integral of (T – V) isstationary - does not change for small changes ofenergy.” That is, there is a stationary value of
(It can be shown that for mechanical systems thatthe stationary value is always a minimum.)
We choose L = (T – V) as the integrand in ourfunctional. The energy quantity L is called the
( )2
1
0t
tT V dt− =∫
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functional. The energy quantity L is called theLagrangian after the mathematician Joseph-LouisLagrange (1736 – 1813).
Hamilton’s principle leads to a system ofsimultaneous differential equations of the secondorder, the Lagrangian equations of motion.
THE LAGRANGIAN EQUATIONS OF MOTION
1,...,NCkk k
d L LQ k n
dt q q
∂ ∂− = = ∂ ∂ ɺ
THE RAYLEIGH-RITZ METHOD
There are a number of techniques available fordetermining approximate solutions to Hamilton’sprinciple. One of the most widely used procedures isthe Rayleigh-Ritz method [8,9], known also as thefinite displacement method. The method is based onthe premise that a closer approximation to the exactnatural modes can be obtained by superimposing anumber of comparison functions. If the assumedfunctions are suitably chosen, the method provides
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functions are suitably chosen, the method providesnot only the approximate value of the fundamentalfrequencies but also the mode shapes. An arbitrarynumber of functions can be used and the number offrequencies that can be obtained is equal to thenumber of function used. A large number offunctions, although it involves more computationalworks, leads to more accurate results.
Rayleigh-Ritz versus Galerkin’s method
The underlying mathematical basis of FEM lies withR-R method. Although the R-R method works well fora wide class of problems, there are types ofproblems for which the use of classical variationaltheory becomes limited or cannot be applied, e.g.fluid problems. Therefore an extension of themathematical approach was required to cope with awider basis of problems. This was achieved throughthe method of weighted residuals, originallyconceived by Galerkin in the early 20th century.
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conceived by Galerkin in the early 20 century.Hence, in many cases the finite element method isintroduced using the Galerkin’s method instead ofthe R-R method to establish approximations to thegoverning equations.
Galerkin’s method
Technically Galerkin’s method is a weighted residualmethod, where the weights are assumed to be thesame as the functions used to define the unknownvariables. In essence the method requires thegoverning differential equation to be multiplied by aset of predetermined weights and the resultingproduct integrated over space. The integral isrequired to vanish.
“Galerkin’s method assumes a solution of theeigenvalue problem in the form of a series of ncomparison functions satisfying all the boundaryconditions. In general the series solution will notsatisfy the differential equation defining theeigenvalue problem. Substituting the series ofcomparison functions in the differential equation anerror will be obtained. At this point one insists thatthe integral of the weighted error over the domain bezero. The weighting functions are exactly the ncomparison functions.” [8]. For the sake of brevitythe Galerkin’s method is not presented in details.
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the Galerkin’s method is not presented in details.There are a number of excellent books where theRayleigh-Ritz method and Galerkin’s method aredescribed and discussed. As an example, a cleardescription and several interesting applications tovibration problems of the methods can be found in[8].
Here the principal features of the R-R method aredescribed . As an example we consider an Eulero-Bernoulli beam . This is a generalised ‘prototype’method for the finite element method.
Illustrative example: Eulero -Bernoulli beam
Energy in a beam bending element
The axial displacement of a point within the beam ata height y is given by
The strain components of this are
( , )w
u x t yx
∂= −∂
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The strain energy is therefore given by
The normal stress is given by
therefore
where
2
2; x xy
u w u wy
x x x xε τ∂ ∂ ∂ ∂= = − = +
∂ ∂ ∂ ∂
1
2 x x
V
U dVσ ε= ∫
; x xE dV dAdxσ ε= =
22
20
1
2
L wU EI dx
x
∂= ∂ ∫
2
A
I y dA= ∫
Lagrange equations
Strain energy
Kinetic energy
kk k
d T VQ
dt q q
∂ ∂+ = ∂ ∂ ɺ
22
20
1
2
L wU EI dx
x
∂= ∂ ∫
21 L w∂
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External force
The Rayleigh-Ritz method approximates the exactsolution with a finite expansion of the form:
where
is a set of prescribed functions of x which are linearlyindependent, and are unknown functions oftime to be determined.
2
0
1
2
L wT A dx
tρ ∂ = ∂ ∫
( , )yQ F x t=
1
( , ) Re ( ) ( )m
k kk
w x t x q tϕ=
= ∑
( ), 1,2,...k x k mϕ =
( )kq t
The prescribed functions ϕk(x) are required to satisfythe following conditions:1) to be linearly independent i.e. one cannot be
described as a linear combination of the others,2) to be p times differentiable, where p is the order
of highest derivative appearing in the expressionof the strain energy,
3) satisfy all the boundary conditions (comparisonfunctions),
4) form a complete series.
The prescribed functions might be:
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The prescribed functions might be:1) Polynomial functions
2) Trigonometric functions
3) Legendre polynomials
4) Chebyshev polynomials
First kind
Second kind
1 2 1( ) ...m mmx x xϕ α α α −= + + +
1 21 2( ) ... mz xz x z xm
mx e e eϕ α α α= + + +
( )1 1 0 11( ) 2 1 ( ) ( ) ; ( ) 1; ( )
1m m mx m xd x md x x x x
mϕ ϕ ϕ+ − = + − = = +
1( ) cos( cos )m x m xϕ −=
1
1
sin(( 1)cos )( )
sin(cos )m m x
xx
ϕ−
−
+=
Strain energy
In matrix form
1
( , ) Re ( ) ( )m
k kk
w x t x q tϕ=
= ∑
[ ]1
21 2( , ) ( ) ( ) ... m
m
q
qw x t x t
q
ϕ ϕ ϕ
= =
φ q⋮
1q
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22
20
2 2
2 20
2 2
2 20
1
2
1 ( ) ( )( ) ( )
2
( ) ( )
L
TLT
TL
wU EI dx
x
x xU t EI dx t
x x
x xEI dx
x x
∂= ∂
∂ ∂ = ∂ ∂
∂ ∂= ∂ ∂
∫
∫
∫
φ φq q
φ φK
1
2 2 2 221 2
2 2 2 2
( , ) ( )( ) ... m
m
q
qw x t xt
x x x x x
q
ϕ ϕ ϕ
∂ ∂ ∂ ∂ ∂ = = ∂ ∂ ∂ ∂ ∂
φq
⋮
Kinetic energy
[ ]1
21 2( , ) ( ) ( ) ... m
m
q
qw x t x t
q
ϕ ϕ ϕ
= =
φ q⋮
[ ]
1
2
1 2
( , ) ( )( ) ... m
q
tq
w x t tx t
t tϕ ϕ ϕ
∂ ∂
∂ ∂ ∂ = = ∂
∂ ∂
qφ
⋮
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[ ]
[ ]
2
0
2
0
2
0
1
2
1( ) ( ) ( ) ( )
2
( ) ( )
L
TLT
TL
wT A dx
t
T t A x x dx t
A x x dx
ρ
ρ
ρ
∂ = ∂
=
=
∫
∫
∫
q φ φ q
M φ φ
ɺ ɺ
[ ]1 2 m
m
tt t
q
t
∂ ∂ ∂ ∂
⋮
With no applied forces, the terms of the Lagrangeequations give
Assuming harmonic motion for the qk(t) parametersin the form q(t) = Rereiωt, the natural frequencies ωj
and corresponding modes can be calculated fromthe eigenvalue problem.
where
Mq + Kq = 0ɺɺ
( )j jλ−K M r = 0
2ω λ=
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and the mode shapes are given by
2j jω λ=
[ ]1
21 2
Re( )
Re( )( ) ( ) Re( ) ...
Re( )
Tj j m
m j
r
rx x
r
φ ϕ ϕ ϕ
= =
φ r⋮
(Believe it or not we are almost at the Finite Elementmethod - we just need to think what our coefficientsmight be and our basis functions/approximationmethod )
The prescribed basis functions in the R-R method• Have to fit the geometry of the problem• Have to fit the boundary conditions
If we want to treat problems in a general way, weneed a general method for constructing basisfunctions. The Finite Element Method provides anautomatic procedure for constructing theseapproximating functions.
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Finite element basis functions(The finite element approximation method)
The basis functions are constructed as follows:
Divide the structure up into a number of elements offinite size. The elements are assumed to be joinedtogether at nodes.
Associate with each node a given number of degreesof freedom.
Construct a set of functions such that each one givesa unit value for one degree of freedom and zerovalues for all the others, viz. shape functions.
Within an element the displacement at each degreeof freedom is approximated as for a single degree offreedom at each node, e.g. for a two-noded elementwith two degrees of freedom at each node then
Substitute the assumed functions for an element
0 0
0 0
i
i j i
i j j
j
u
vu
uv
v
φ φφ φ
≈
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Substitute the assumed functions for an element into the expression for: strain energy,kinetic energy,work done by the applied forcesto give matrix expressions in terms of nodal degreesof freedom for M, K and f.
Add the energies for each element together togive energies for the complete structure.
We now have basis functions that do notautomatically obey any particular boundaryconditions .
Convergence
In the R-R method the accuracy of the solution isincreased by increasing the number of prescribedbasis functions in the assumed series.
In the finite element method the accuracy of thesolution is increased by increasing the number ofelements. There is an alternative option of makingthe shape functions higher order. This will generallyincrease the number of nodes but not of elements.
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increase the number of nodes but not of elements.
Types of analysis
Natural frequencies and normal modes
Response to harmonic excitations
Response to periodic excitations
Response to transient deterministic excitations
Response to transient random excitations
Example: FE model of a road in axial vibration ofroads
Divide the structure into a number of elements offinite size.
Associate with each node point a given number ofdegrees of freedom. We are interested in axial
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degrees of freedom. We are interested in axialvibration of a road, so that we only have onecomponent of displacement, namely axialdisplacement in the x direction.
Construct a set of functions such that each onegives a unit value for one degree of freedom andzero values for all the others.
Element local coordinate
Element strain energy
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
11 2
2
1 2
1 11 1
2 2
e
uu N N
u
N N
ξ ξ ξ ξ
ξ ξ ξ ξ
= =
= − = +
N u
( ) 21
21
1
1 1
2
uU EA ad
a
ξξ
ξ
+
−
+
∂ = ∂
∫
ξ
NB. ξ=x/adx = a dξ
x=-aξ=-1
x=aξ=1
x ξ
u
2a
1 2
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Element stiffness matrix
( ) ( )
( ) ( )
[ ]
1
1
1
1
1
1
1
2
1
2
1/ 21/ 2 1/ 2
1/ 2
1/ 2 1/ 2
1/ 2 1/ 2
TTe e
Te e e
T
e
e
EAU d
a
U
EAd
a
EAd
a
EA
a
ξ ξ ξ
ξ ξ ξ
ξ
+
−
+
−+
−
′ ′=
=
′ ′= =
− = − ⇒
− = −
∫
∫
∫
u N N u
u K u
K N N
K
Element kinetic energy
( )
( ) ( )
( ) ( )
12
1
1
1
1
1
1
2
1
2
1
2
TTe e
Te e e
T
e
T Au ad
T Aa d
T
Aa d
ρ ξ ξ
ρ ξ ξ ξ
ρ ξ ξ ξ
+
−
+
−
+
−
=
=
=
= =
∫
∫
∫
u N N u
u M u
M N N
ɺ
ɺ ɺ
ɺ ɺ
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Element mass matrix
[ ]1
1
1
1/ 2(1 )1/ 2(1 ) 1/ 2(1 )
1/ 2(1 )
2 / 3 1/ 3
1/ 3 2 / 3e
Aa d
Aa
ξρ ξ ξ ξ
ξ
ρ
−+
−
− = − + ⇒ +
=
∫
M
Work done
Assuming the axial load to be constant along theelement
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
1 1
1 1
1
1
1 1 1/ 2(1 ) 1
1/ 2(1 ) 1
e ee x x e
TT e Te x e e
Te e ee x x x
W p u ad p ad
a p d
a p d p a d p a
ξ ξ ξ ξ ξ ξ
ξ ξ ξ
ξξ ξ ξ ξ
ξ
+ +
− −
+
−
+ +
= = =
= =
− = = = +
∫ ∫
∫
∫ ∫
N u
u N u f
f N
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( ) ( )1 1 1/ 2(1 ) 1e x x xa p d p a d p aξ ξ ξ ξ
ξ− −
= = = + ∫ ∫f N
Add the energies for each element together to giveenergies for the complete structure.
The vecotr with all the degrees of freedom of the FEMmodel is required
[ ]T u u u u u=u
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we now relate the displacement vector of the elementto the global displacement vector
[ ]1 2 3 4 5T u u u u u=u
e e=u a u
1 2
3 4
1 0 0 0 0 0 1 0 0 0; ;
0 1 0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0; ;
0 0 0 1 0 0 0 0 0 1
= =
= =
a a
a a
The process of summing the energies then becomes
1 2
4
1
...
1 1
2 2
tot
T T Ttot e e e
e
T T T
T=
= + +
= = ∑u a M a u u Muɺ ɺ ɺ ɺ
[ ]
1 1111 12
1 1 2 2221 22 11 12
2 2 331 2 3 4 5 21 22 11
4
5522
1
2tot
um m
um m m m
uT u u u u u m m m
u
um
+ = +
ɺ
ɺ
ɺɺ ɺ ɺ ɺ ɺ
ɺ⋱
ɺ
1 2
4
...
1 1
tot
T T T
U U U
U
= + +
= =∑u a K a u u Ku
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1
1 1
2 2T T T
tot e e ee
U=
= = ∑u a K a u u Ku
1 2
4
1
...tot
T T Ttot e e
e
W W W
W=
= + +
= = ∑u a f u u f
[ ]
1 1111 12
1 1 2 2221 22 11 12
2 2 331 2 3 4 5 21 22 11
4
5522
1
2tot
uk k
uk k k k
uU u u u u u k k k
u
uk
+ = +
⋱
[ ] [ ]
11
1 22
2 331 2 3 4 5 1 2 3 4 5
3 44
55
x
x x
x xtot
x x
x
pf
p pf
p pfW u u u u u u u u u u
f p p
f p
+ + = = +
Use the energy expression to derive the equations ofmotion of the complete structure in terms of the nodaldegrees of freedom
Equation of motion
Impose the boundary conditions
ii i
d T Uf
dt u u
∂ ∂+ = ∂ ∂ ɺ
( ) ( ) ( )t t t+ =Mu Ku fɺɺ
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Impose the boundary conditionse.g. 1 0u =
The FE software
Commercial programsTypically available for a whole range of ‘operatingsystems and processors.Support from established companies.Evolved over several decades with a changing userinterface.Big industry users committed to use.Large ‘user base’.Nowadays tend to be integrated with otherengineering software and Company-wide systems.
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engineering software and Company-wide systems.
Examples
In general a finite element solution maybedivided into these three steps:
1) Preprocessing: defining the problemdefine geometrydefine keypoints/lines/areas/volumesdefine element type and material/geometric
propertiesmesh lines/areas/volumes as requiredThe amount of detail required will depend on
the dimensionality of the analysis (i.e. 1D, 2D, axi-
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the dimensionality of the analysis (i.e. 1D, 2D, axi-symmetric, 3D).2) Solution: assigning loads, constraints andsolving
specify the loads (point or pressure),constraints (translational and rotational)choose the type of solutionsolve the resulting set of equations
3) Post-processing: further processing andviewing of the results
Lists of nodal displacementsElement forces and momentsDeflection plotsStress contour diagrams
Exercise
Write a MATLAB code to find the first 4 naturalfrequencies of a simply supported Eulero-Bernoullibeam using the FE procedure described. Comparethe results with analytical results, e.g. [3,8]. FE massand stiffness matrices of a beam bending elementare given in the next pages.
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A = 1.006×10-2 m2 ;I = 1.774×10-4 m4
Height of beam, h = 0.3063 mLength of the beam L=9.144 mE = 207×109 Nm-2
ρ = 7860 kgm-3
yx
Beam bending element
It is now necessary to take
as degrees of freedom at each node of the beamelement. The displacement function can thus berepresented by a polynomial having four constants
2 31 2 3 4
1
22 3( ) 1 ( ) ( )
w
w
α α ξ α ξ α ξαα
ξ ξ ξ ξ ξ α
= + + +
= = p a
2a
x=-aξ=-1
x=aξ=1
x ξ
w
1 2
and w w x∂ ∂
NB. dx = a dξ
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Express the nodal values for the specific value ofin order to get a matrix equation which can be solvedfor the coefficient alpha
2 3
3
4
21 3 4
( ) 1 ( ) ( )
2 3
w
w wa a
x
ξ ξ ξ ξ ξ ααα
θ α α ξ α ξξ
= =
∂ ∂= = = + +∂ ∂
p a
x aξ =
1 1
1 2
2 3
2 4
1 1 1 1
0 1 2 3 or
1 1 1 1
0 1 2 3
e e
w
a
w
a
αθ α
αθ α
− − − = =
w A α
1 e e e e−= =α A w C w
Hence
Which can be expressed in the form
The shape function are given by
( ) ( ) e ew ξ ξ= p C w
[ ]1 2 3 4
( ) ( )
( ) ( ) ( ) ( ) ( )ew
a a
ξ ξξ ξ ξ ξ ξ
==
N w
N N N N N
( ) ( )( ) ( )
3 2 31 2
3 2 3
1 1( ) 2 3 ( ) 14 41 1( ) 2 3 ( ) 1
ξ ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξ ξ
= − + = − − +
= + − = − − + +
N N
N N
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Element kinetic energy
element mass matrix
( ) ( )3 2 33 4
1 1( ) 2 3 ( ) 14 4ξ ξ ξ ξ ξ ξ ξ= + − = − − + +N N
( ) ( )
( ) ( )
1 12
1 1
1
1
1 1
2 2
TTe e
T
e
T Aw ad Aa d
Aa
ρ ξ ρ ξ ξ ξ
ρ ξ ξ
+ +
− −
+
−
= =
=
∫ ∫
∫
w N N w
M N N
ɺ ɺ ɺ
2 2
2 2
78 22 27 13
22 8 13 6
27 13 78 22105
13 6 22 8
e
a a
a a a aAa
a a
a a a a
ρ−
− = − − − −
M
Element strain energy
element stiffness matrix
( ) ( )
( ) ( )
21 12
4 2 31 1
1
31
2 2
3
1 1 1
2 2
3 3 3 3
3 4 3 2
3 3 3 32
TTe e
T
e
e
w EIU EI ad d
a a
EId
a
a a
a a a aEI
a aa
ξ ξ ξ ξξ
ξ ξ ξ
+ +
− −
+
−
∂ ′′ ′′= = ∂
′′ ′′=
− − = − − −
∫ ∫
∫
w N N w
K N N
K
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2 2
3 3 3 32
3 2 3 4
a aa
a a a a
− − − −
Work done
Assuming the transverse load to be constant alongthe element
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
1 1
1 1
1
1
1
1
3
33
e ee y y e
TT e Te x e e
Te ee y y
W p w ad p ad
a p d
aaa p d p
a
ξ ξ ξ ξ ξ ξ
ξ ξ ξ
ξ ξ ξ
+ +
− −
+
−
+
−
= = =
= =
= = −
∫ ∫
∫
∫
N w
w N w f
f N
There are a number of methods which can beaddressed to study wave propagation using finitediscretisation. Amongst them, one of the wellestablished is the Wave Finite Element method. Themethod was originally developed to study wavecharacteristics and vibro-acoustics of periodicstructure, e.g. [10]. Periodic structures can beconsidered as an assemblage of identical elements,
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Part 2: Introduction to the Wave Part 2: Introduction to the Wave Finite Element (WFE) MethodFinite Element (WFE) Method
considered as an assemblage of identical elements,called cells or periods, which are coupled to eachother on all sides and corners by identical junctions.This characteristic is indeed observable in manyengineering real systems. Examples include flat orcurved panels regularly supported, such as stringerstiffened panels, fluid filled pipes with regular flangesand so on.
Examples of periodic structures
For these structures the dynamic behaviour of thecomplete structure can be predicted through theanalysis of a single period. One of the classical bookwhere the mathematic of wave propagation inperiodic structures has been discussed is that ofBrillouin [11].
The Wave Finite Element method basically is atechnique to investigate wave motion in periodicstructures. In this method a period of the structure ismodelled using conventional FEs. The equation ofmotion for time–harmonic motion is therefore
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motion for time–harmonic motion is thereforeobtained from the FE model in terms of a discretenumber of nodal DOFs and forces in the same formas the dynamic stiffness method [12]. Periodicityconditions are then applied, and an eigenvalueproblem is formulated whose solutions give thedispersion curves and wavemodes.
In what follows the WFE method is briefly describedfor 1 dimensional waveguides and 2 dimensionalwaveguides. In the cases considered the structuresare homogeneous, hence the periodicity of arbitrarylength. Therefore what it is shown is a generalapproach to study wave propagation in trulyperiodic or continuous structures.
1-Dimensional Structural Waveguides
The waveguide is assumed to have uniform crosssection, which means that the cross-section has thesame physical and geometrical properties at allpoints along the axis wave propagation. Waves canpropagate in both directions along the waveguides.
Uniform in one direction (x-axis)
Cross-section may be 0, 1 or 2-dimensional
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Examples of 1-dimensional waveguides
x
Beam of arbitrary section
Plate strip
Isotropic laminate
Tyre
A wave propagates as ( ) ( ) ( ), , , , i t kxw x y z t W y z eω −=
The WFE for 1 -Dimensional Structural Waveguides
Take a small segment (or a period) of the waveguideand mesh it using conventional finite element (if thestructure is continuous, the length of the segment isarbitrary). To obtain good accuracy in the waveanalysis, the “periodic” length should be less thenone sixth of the expected wavelength. Thiscorrespond to the more familiar requirement of using6-10 elements per wavelength. However if the finite
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6-10 elements per wavelength. However if the finiteelements are too shorts compared to the wavelength,computational problems can arise. A condition for thewave analysis is to have an equal number of identicaldistribution of degrees of freedom on each side of thesegment.
The number of elements over the cross section hasto be high enough to characterise the wavemodes.The shape of the elements, viz. width-to-height ratioshould comply with standard FE modellingtechniques.
In what follows continuous structures are considered,but truly periodic structures can be studied in thesame way..
FEA of short section of waveguide
In order to apply the method a 2-noded superelement is defined, whose nodes and nodalforces are obtained by concatenating all DOFsand nodal forces through the thickness
∆
,R Rq f,L Lq f
x
L
R
L
R
=
=
q
ff
f
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and nodal forces through the thickness
Find mass and stiffness matrices M, K Include damping (C, K complex)
Equations of motion:
Periodicity condition:
2ω − = K M q f
;R Lλ=q q ikeλ − ∆=
Apply periodicity condition:
Equilibrium at node L
R Lλ=q q
R R = ; = [ ]TL λΛ Λq q I I
L 0; = [ 1/ ]TL λΛ = Λf I I
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Consider the equation of motion
Substitute
Premultiply both sides of the equation by
2L R L Lω Λ − Λ = Λ K M q f
2ω − = K M q f
LΛ
R R = ; = [ ]TL λΛ Λq q I I
Hence we obtain an eigenvalue problem in terms of degrees of freedom of node left only
where and L R L R= Λ Λ = Λ ΛK K M Mare the reduced stiffness and mass matrices, i.e. theelement matrices projected onto the degrees offreedom of node only.
2L R L R Lω Λ Λ − Λ Λ = K M q 0
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The eigenvalue problem can also be written as
where is the reduced dynamic stiffness matrix(DSM). If there are n DOFs per node (viz. nodes Land R), the nodal displacement and force vectors aren×1, the element mass and stiffness matrices are2n×2nwhile the reduced matrices are n × n.
It can be seen that the mathematical formulation ofthe method is fairly simple. Standard FE packagescan be used to obtain the mass and stiffnessmatrices of the segment of the structure. No newelements or new “spectral” stiffness matrices mustbe derived on a case–by–case basis.
( , ) Lλ ω =D q 0D
The fact that standard FE packages can be used toobtain the mass and stiffness matrices of one periodof the structure is a great advantage sincecomplicated constructions such as sandwich andlaminated constructions can be analysed in asystematic and straightforward manner. As a furtheradvantage, if the elements used in the discretisationare brick solid elements, the present method isformulated within the framework of a 3–dimensionalapproach, that is the stress and displacementassumptions are the one used in the 3–dimensionalFE analysis.
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FE analysis.
The form of the eigenproblem obtained depends onthe nature of the problem at hand. If thewavenumber is real and assigned, the frequencies ofthe waves that propagate in the structure can beobtained from a standard eigenvalue problem while,if the frequency is prescribed, the waves areobtained from a polynomial eigenvalue problem. Forthese kinds of problems efficient algorithms, whichinclude error and condition estimates, are wellestablished.
Free wave propagation (no damping) λ real and known → find ω (real) Standard linear eigenvalue problem
Real dispersion curves
Frequency prescribed (damping can be included)
2L R L R Lω Λ Λ − Λ Λ = K M q 0
Forms of the eigenvalue problem
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Frequency prescribed (damping can be included) ω real and known → find λ (complex
wavenumber) Polynomial eigenvalue problem
Partition the original dynamic matrix
[ ]
2
2 ( )
LL LR
RL RR
L R L
LR LL RR RR L
ω
λ λ
= − =
Λ Λ =
+ + + =
D DD K M
D D
D q 0
D D D D q 0
Properties of eigenvalues and associated waves
In general the eigenvalue are complex and can bewritten
where µ represents the change in amplitude and krepresents the change in phase, which has beenpreviously introduced as the wavenumber.
ike eµλ − ∆ − ∆=
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Different types of waves
Eigenvalues, [13]
The energy related quantities here always refer toone wave.Using complex notation, the time average kineticenergy can be written as
where * denotes the complex conjugate.The time overage potential energy can be found as
( ) ( ) 2** 21 1 1 1Re Re
2 2 4 4kinE m q q m i q i q m qω ω ω= = =ɺ ɺ
*1 1ReE k q q=
Energy related quantitiesKinetic, potential and total energy
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In order to find the time-average kinetic and potentialenergies of the section of the waveguide under thepassage of a wave, the previous equations can bewritten in matrix form using the mass and stiffnessmatrices and displacement vector of the section.Thus
The time average total energy associated with asection is then given by
*1 1Re
2 2potE k q q=
( ) ( ) ( ) ( )
*
*
1Re
41
Re4
kin
pot
E i i
E
ωφ ωφ
φ φ
=
=
M
M
L
L
φλ
=
q
q
tot kin potE E E= +
The transmitted power at a point is given by
Using complex notation, the time average power flowis given by
The power-flow through the section of the waveguidecan be calculated either at the left or at the rightborder. The power-flow is constant for freelypropagating waves and positive in the direction of
P fq= ɺ
*1Re ( )
2P f i qω=
Power flow and group velocity
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propagating waves and positive in the direction ofwave propagation. It is given by
The speed at which energy travels along thewaveguide is called group velocity and is defined by
The preceding equations show a simple way tocalculate the group velocity.
These two sections are taken from [13]. For furtherdiscussion about WFE energy related quantities, groupvelocity etc. see also [14].
tot g gtot
PP E c c
E= ⇒ =
* *1 1Re Re
2 2L L R RP i iω ω= = −f q f q
Numerical issues
Issues arise because the original structure iscontinuous while the WFE model is a lumped,discrete, spring–mass structure which is spatiallyperiodic. In particular at high frequencies, or forshort wavelengths, there are substantial differencesbetween the behaviour of the continuous structureand the periodic structure. Hence the issue is one ofdetermining which solutions to the eigenvalueproblem are artifacts of the spatial discretisation andwhich are valid estimates of wavenumbers in the
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which are valid estimates of wavenumbers in thecontinuous structure.As with conventional FEA, FE discretisation errorsbecome significant if the size of the element is toolarge. As a rule–of–thumb, there should be at least 6or so elements per wavelength. if the size of theelement is too small then care must be taken innumerical computations because round–off errorscan occur if the dynamic stiffness matrix is to becalculated.Aliasing effect is a consequence of considering adiscretised structure instead of the continuous oneand represents a substantial difference to theanalytical approach. Detailed discussion about theperiodicity conditions can be found in [1].
Periodic structures are known to exhibit a pass- andstop-band structure, in that disturbances canpropagate freely only in certain frequency ranges,otherwise they decay with distance. For a one-dimensional element with n DOFs per node there willbe n propagation surfaces. It is worth noting that forelements with rigid body modes (i.e. those for whichthe stiffness matrix is singular), ω=0 is a cut-offfrequency so that at least one wave must propagatefrom 0, and this wave must represent a wave in thecontinuous structure. A more detailed discussionabout numerical issues can be found in [1,15].
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about numerical issues can be found in [1,15].
Example: real dispersion curves of a thin isotropic plate
Analytical solution
WFE, band 1
WFE, band 2
WFE, band 3
-3 -2 -1 0 1 2 30
20
30
40
50
60
kxL
x/π
Ω
Ωa
Ωb
Ωc
Take a small segment of a beam of length L (e.g. L=5mm). Model it using a standard beam element – use
as degrees of freedom at each nodeof the beam element:
and w w x∂ ∂
LL
L
w
θ
=
q RR
R
w
θ
=
q
L
L
R
=
q
Illustrative example:Wave propagation in a thin beam
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Obtain the mass and stiffness matrices
2 2
2 2
2 2
3
2 2
/ 2
78 22 27 13
22 8 13 6
27 13 78 22105
13 6 22 8
3 3 3 3
3 4 3 2
3 3 3 32
3 2 3 4
a L
a a
a a a aAa
a a
a a a a
a a
a a a aEI
a aa
a a a a
ρ
=−
− = − − − −
− − = − − − −
M
K
Apply periodicity condition:
obtain an eigenvalue problem in terms of degrees offreedom of node left only
Where
R Lλ=q q
2 0L R Lω Λ − Λ = K M q
ikLeλ −=
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1 0
0 1 = [ 1/ ]
1/ 0
0 1/
TL λ
λλ
Λ =
I I
R
1 0
0 1= [ ]
0
0
Tλλ
λ
Λ =
I I
Real dispersion curves: wavenumber k prescribed (viz. λ prescribed )
solve the standard eigenvalue problem
Complex dispersion curves: frequency
2L R L R Lω Λ Λ − Λ Λ = K M q 0
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Complex dispersion curves: frequency prescribed
solve the standard polynomial eigenvalue problem
2
2
( )LR LL RR RR L
LL LR
RL RR
λ λ
ω
+ + + =
= − =
D D D D q 0
D DD K M
D D
Flexural waves in a thin beam: comparison between analytical
and WFE results
1
5/4
3/2
7/4
kL π
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0 2 4 6 8 10 12 14 16 18 20
0
1/4
1/2
3/4
Analytical solution
WFE, real part
WFE, imag part
2 hL
D
ρωΩ =
WFE accurate, kL/π < 1/3 or so
Example: Waves in a laminated plate [16]
FE mesh 17 rectangular plane-strain elements
∆ = 1mm 0.6mm
15mm
x
y
aluminium skins
viscoelastic core
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17 rectangular plane-strain elements 2 DOFs per node (x,y displacements)
36 DOFs per cross-section, 36 wave pairs
Many wave modes
Dispersive
Complex cut-off effects0 0.5 1 1.5 2 2.5 3 3.5 4
10-1
100
101
102
103
Frequency (Hz) x 104
Wavenum
ber
(m-1)
-
2-Dimensional Structural Waveguides
The waveguide is assumed to be uniform andhomogeneous in two directions, but properties canvary in an arbitrary manner through the crosssections. This analysis considers time–harmonicwaves propagating in a 2–dimensional structure asplane waves at frequency ω.
Uniform in one direction (x,y-axis)
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Example of 2-dimensional waveguides
Uniform in one direction (x,y-axis)
Cross-section may be complicated, e.g. laminate
A wave propagates as
( ) ( ) ( ), , , x yi t k x k yw x y z t W z e
ω − −=
x
zy
The WFE for 2-Dimensional Structural Waveguides [1,2]
Take a small rectangular segment (or a period) of thewaveguide and mesh it using conventional finiteelement (If the structure is continuous, the length ofthe segment is arbitrary in the x and y directions). Toobtain good accuracy in the wave analysis, the“periodic” lengths along the x and y directions shouldbe less then one sixth of the expected wavelength.This correspond to the more familiar requirement of
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This correspond to the more familiar requirement ofusing 6-10 elements per wavelength. However if theelement is too short compared to the wavelength,computational problems can arise. A condition for thewave analysis is to have an equal number of identicaldistribution of degrees of freedom on each side of thesegment.
The number of elements over the cross section has tobe high enough to characterise the wavemodes. Theshape of the elements, viz. width-to-height ratio shouldcomply with standard FE modelling techniques.
In what follows continuous structures are considered,but truly periodic structures can be studied in the sameway.
FEA of small rectangular segment
Waves propagate as Bloch waves
x
y
Lx
Ly
x x x
y y y
k L
k L
µµ
==( ) ( ) ( ), , , x yi t
x yw z t W z eω µ µµ µ − −=
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In order to apply the method a 4-noded super element is defined, whose nodes and nodal forces are obtained by concatenating all DOFs and nodal forces through the thickness
FE equation of motion
yLx
Ly
q1
q3
q2
q4
1 2 3 4
1 2 3 4
TT T T T
TT T T T
=
=
q q q q q
f f f f f
2ω − = K M q f
Apply periodicity condition:
Equilibrium at node 1
Consider the equation of motion
R 1 R = ; = [ ]Tx y x yλ λ λ λΛ Λq q I I I I
L 0; = [ 1/ 1/ 1/ ]TL x y x yλ λ λ λΛ = Λf I I I I
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Substitute
Premultiply both sides of the equation by
This equation represents an eigenvalue problem that can be solved to obtain the phase constant surface, dispersion curves, contour curves and wavemodes
21L Rω Λ − Λ = K M q 0
2ω − = K M q f
LΛ
R 1 = Λq q
Forms of the eigenvalue problem
Free wave propagation (no damping) λx, λy real and known → find ω (real) Standard linear eigenvalue problem
Reflection at a boundary; waves in closed cylinders λx and ω known → find λy (real or complex) Polynomial eigenvalue problem
Wave propagation in given direction θ
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Wave propagation in given direction θ θ = tan-1(λy /λx) and ω known → find λx, λy (real
or complex) Polynomial or transcendental eigenvalue
problem
NB. The eigenvalue problem In terms of the partitionsof original DSM
( ) ( ) ( )( ) ( )
111 22 33 44 12 34 21 43
1 1 113 24 31 42 14 41
1 132 23 1
[
] 0
x x
y y x y x y
x y x y
λ λ
λ λ λ λ λ λ
λ λ λ λ
−
− − −
− −
+ + + + + + + +
+ + + + + +
+ + =
D D D D D D D D
D D D D D D
D D q
Take a small segment of a plate of lengths(e.g. ). Model it using a standardshell element (the element has six degrees offreedom at each node: translations in the nodal x, y,and z directions and rotations about the nodal x, y,and z axes.
1u
v
1 q
and x yL L= 5x yL L L mm= =
q3 q4
Illustrative example: wave propagation in a thin plate plate
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Obtain the mass and stiffness matrices M and K.
Apply periodicity conditions
1
11
1
1
1
x
y
z
v
w
θθθ
=
q
1
2
3
4
=
q
q
qy
L
L
q1
q3
q2
q4
x
2 2 4 1 3 1; ; ;
; yx
x y x y
ik yik xx ye e
λ λ λ λ
λ λ −−
= = =
= =
q q q q q q
Obtain an eigenvalue problem in terms of degrees offreedom of node 1 only
where
and I is the identity matrix of size 6x6.
21 0L Rω Λ − Λ = K M q
R
1= =
1
1
x xL
y y
x y x x
λ λλ λ
λ λ λ λ
Λ Λ
I II II I
I I
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Since the plate is isotropic, wave propagationcharacteristics are the same for any heading θ, e.g. θ=0.
Real dispersion curves for θ=0 : prescribed and (viz. )
solve the standard eigenvalue problem
21L R L Rω Λ Λ − Λ Λ = K M q 0
xk0yk = 1yλ =
Complex dispersion curves: frequency prescribed, and
Solve the standard second order polynomialeigenvalue problem
1yλ =
( ) ( ) ( )( ) ( )
111 22 33 44 12 34 21 43
113 24 31 42 14 41
32 23 1
[
] 0;
x x
x x
x y
λ λ
λ λλ λ
−
−
+ + + + + + + +
+ + + + + ++ + =
D D D D D D D D
D D D D D D
D D q
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where
11 12 13 14
21 22 23 242
31 32 33 34
41 42 43 44
;
here are of size 6x6ij
ω
= − =
D D D D
D D D DD K M
D D D D
D D D D
D
Isotropic plateComparison with Love-Kirchoff theory
1 SHELL63 element in ANSYS, 6DOFs after WFEreduction
0.1
0.12
WFE....Analytic
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0 0.5 1 1.5 2
x 104
0
0.02
0.04
0.06
0.08
kL
Frequency[Hz]
Illustrative example: isotropic sandwich plate. Comparison with analytical solutions
60 SOLID45 elements in ANSYS, 5 for each skin and 50 for the core
183DOFs after reduction
Skin: E=16.7 GPa, ν=0.3, ρ=1730 kg/m3 5 mm
Core : E=0.13 GPa, ν=0.3, ρ=130 kg/m3
Skin: E=16.7 GPa, ν=0.3, ρ=1730 kg/m3 5 mm
50 mm
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k [1
/m]
0 1000 2000 3000 4000 50000
10
20
30
40
50
60
70
Mead-Markus....WFE
Isotropic sandwich plateComparison with Mead-Markus theory [17]
Frequency [Hz]
Anti-symmetric laminated sandwich panel
Two glass/epoxy skins with a foam core.
FE model: 4 solid elements for each skin and 10 solid elements for the core
4 sheets of glass/epoxy, 0/45/45/0 2 mm
ROHACELL foam core 10 mm
4 sheets of glass/epoxy, 45/0/0/45 2 mm
Complex dispersion curves
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Imag
[µ]
Frequency[Hz]
Rea
l[µ]
Frequency[Hz]
Anti-symmetric laminated sandwich panelWAVE MODES at 2kHz
(a) branch 1;
(b) branch 2;
(c) branch 3.
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Anti-symmetric laminated sandwich panelWAVE MODES
(a) branch 4 at 10 kHz;
(b) branch 5 at 10kHz;(c) branch with 2nd
largest wavenumberat 17.5 kHz.
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SOME APPLICATION OF THE WFE METHOD
WAVE PROPAGATION IN AXISYMMETRIC STRUCTURES FROM FINITE ELEMENT
ANALYSIS
[2] E. Manconi, B.R. Mace, Wave characterisation ofcylindrical and curved panels using a finite element
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cylindrical and curved panels using a finite elementmethod, Journal of the Acoustical Society ofAmerica, 125,154–163, 2009.
[18] E. Manconi, B. R. Mace and R. Garziera, Wavefinite element analysis of fluid–filled pipes. NOVEM2009,3th Conferece on Noise and Vibration:Emerging Methods, Oxford, UK, 2009.
WAVE PROPAGATION IN DAMPED WAVEGUIDES FROM
FINITE ELEMENT ANALYSIS
[19] E. Manconi, B. R. Mace, Estimation of the lossfactor of viscoelastic laminated panels from finiteelement analysis, Journal of Sound and Vibration,329, 3928–3939, 2010.
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329, 3928–3939, 2010.
[20] E. Manconi, B. R. Mace, R. Garziera, The loss-factor of pre-stressed laminated curved panels andcylinders using a wave and finite element method,Journal of Sound and Vibration, 332, Pages 1704-1711, 2013.
FORCED RESPONSE OF WAVEGUIDES USING THE WAVE AND
FINITE ELEMENTE METHOD
[21] J.M. Renno, and B.R. Mace, On the forcedresponse of waveguides using the wave and finiteelement method. Journal of Sound and Vibration,329, 5474-5488. 2010.
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[22] J.M. Renno, Jand B.R. Mace, Calculating theforced response of two-dimensional homogeneousmedia using the wave and finite element method.Journal of Sound and Vibration, 330, 5913-5927,2011.
SUMMARYIn this lecture a short introduction to the WFEmethod was given togheter with an introduction tothe FE method. The background and the basic stepsof the WFE approach were discussed and fewexamples were given.
In essence:
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WFE method: post-process small, conventional FEmodel periodicity condition, eigenvalue problem;
WFE uses full power of commercial FE packages,element libraries etc;
results found for negligible computational cost.
References[1] B. R. Mace and E. Manconi, Modelling wavepropagation in two-dimensional structures, usingfinite element analysis. Journal of Sound andVibration, 318, 884–902, 2008.[2] E. Manconi and B.R. Mace, Wavecharacterisation of cylindrical and curved panelsusing a finite element method, Journal of theAcoustical Society of America, 125,154–163, 2009.[3] S Rao, Mechanical Vibrations, Addison-WesleyPublishing Company, 1995.
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Publishing Company, 1995.[4] O. C. Zienkiewicz and R. L. Taylor. The finiteelement method. Vol I-II, Butterworth–Heinemann,2000.[5] M. Petyt. Introduction to finite element vibration analysis. Cambridge, University Press, 1990.[6] K. J. Bathe, Finite Element Procedures inEngineering Analysis, Prentice-Hall, 1982.[7] C. Lanczos, The Variational Principles ofMechanics, Dover Publications, 1986.[8] L. Meirovitch, Analytical Methods in Vibrations,Macmillan Publishing Co., 1967.[9] R. Courant and D. Hilbert, Methods ofMathematical Physics, Vol. I-II, Wiley-Interscience,1953.
[10] R. M. Orris and M. Petyt. A finite element studyof harmonic wave propagation in periodic structures.Journal of Sound and Vibration, 33, 223–236, 1974.[11] L. Brillouin. Wave propagation in periodicstructures. Dover Publications, 1953.[12] J. R. Banerjee. Dynamic stiffness formulation forstructural elements: a general approach. Computersand Structures, 63, 101–103, 1997.[13] L. Hinke, B. R. Mace and M. J. Brennan, FiniteElement Analysis of Waveguides, ISVR TechnicalMemorandum n. 932, April 2004.
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Memorandum n. 932, April 2004.[14] E. Manconi, B. R. Mace, Estimation of the lossfactor of viscoelastic laminated panels from finiteelement analysis, Journal of Sound and Vibration,329, 3928–3939, 2010.[15] Y. Waki, B. R. Mace and M.J. Brennan,Numerical issues concerning the wave and finiteelement method for free and forced vibrations ofwaveguides, Journal of Sound and Vibration, 327,92–108, 2009.[16] B. R. Mace, D. Duhamel, M. J. Brennan, and L.Hinke. Finite element prediction of wave motion instructural waveguides. Journal of the AcousticalSociety of America, 117, 2835–2843, 2005.
[17] D. J Mead and S. Markus, The forced vibrationof a three-layer, damped sandwich beam witharbitrary boundary conditions, Journal of Sound andVibration, 10, 163-175, 1969.[18] E. Manconi, B. R. Mace and R. Garziera, Wavefinite element analysis of fluid–filled pipes. NOVEM2009,3th Conference on Noise and Vibration:Emerging Methods, Oxford, UK, 2009.[19] E. Manconi, B. R. Mace, Estimation of the lossfactor of viscoelastic laminated panels from finiteelement analysis, Journal of Sound and Vibration,329, 3928–3939, 2010.
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329, 3928–3939, 2010.[20] E. Manconi, B. R. Mace, R. Garziera, The loss-factor of pre-stressed laminated curved panels andcylinders using a wave and finite element method,Journal of Sound and Vibration, 332, Pages 1704-1711, 2013.[21] Renno, J.M. and B.R. Mace, On the forcedresponse of waveguides using the wave and finiteelement method, Journal of Sound and Vibration,329, 5474-5488. 2010.[22] Renno, J.M. and B.R. Mace, Calculating theforced response of two-dimensional homogeneousmedia using the wave and finite element method,Journal of Sound and Vibration, 330, 5913-5927,2011.