BOUNDARY ELEMENT METHODS FOR
VIBRATION PROBLEMS
by
Ashok D. Belegundu
Professor of Mechanical Engineering
Penn State University
[email protected]
ASEE Fellow, Summer 2003
Colleague at NASA Goddard: Daniel S. Kaufman
Code 542
MY RESEARCH IN LAST FEW YEARS
Noise reduction – passive approaches
V
m
TVA/BBVA
Helmholtz
Resonator
Noisy Structure
Stiffeners
Dynamic analysis
Acoustic analysis
2
MOST RECENT RESEARCH
Speedup
Parallel optimization algorithms
35
30
25
20
15
10
5
0
bracketing
0
16
32
48
number of processors
3
64
interval
reduction
total line
search
linear speedup
(reference)
OBJECTIVES AT NASA: SUMMER 2003
Tutorial ON Boundary Element Analysis (“BEM”)
Computer codes (for tutorial and problem solving)
Study the potential for BEM in Vibration at midfrequency levels
i.e. fill the ‘gap’ between FEA and SEA in view of:
Low freq – FEA is very good
High freq -- SEA seems adequate (although
detailed response is not obtainable)
Assess the state of the art in BEM for vibration
analysis
4
BEM -- SIMPLE 1-D EXAMPLE
d 2u
+ x = 0 , u (0) = 0, u (1) = 0
2
dx
⎛ d 2u
⎞
⎜
w
+
x
∫0 ⎜⎝ dx 2 ⎟⎟⎠ dx = 0
1
Integrate by parts twice:
1
1
d 2w
w u ' 0 − u w' 0 + ∫ u
dx + ∫ w x dx = 0
2
dx
0
0
1
1
d 2w
Choose w such that dx 2 = − δ ( x − xi )
‘Fundamental Solution’ :
w( x) = − step ( x − xi ) ( x − xi )
5
SIMPLE EXAMPLE – cont’d
Then
1
u ( xi ) = w u ' 0 − u w' 0 + ∫ w x dx
1
1
(*)
0
boundary terms
domain term
xi = ‘source ’, and w(x) depends on source point xi
Choose source xi = 0+ = 0 + ε . Thus w(0) = 0 and
remains zero as ε → 0. w(1) = -1, w' (1) = −1, body
force term = -1/3, and we get
u ' (1) = − 1 / 3
We can also recover the solution for all interior
points from Eq. (*) : u(xi) = (xi – xi3)/6
Note: the differential (equilibrium) is exactly
satisfied within the domain in BEM – while boundary
conditions are approximately satisfied, here, the
boundary is only wo points, so BEM gives exact
solution
6
AXIAL VIBRATION
u&& − c 2 u ′′ = p( x, t )
2
2
′
′
u
+
k
u
=
−
p
(
x
,
t
)
/
c
Harmonic loading:
k=
ω
, c = E/ρ
c
∫ (u ′′ + k
)
L
2
u + p / c 2 w dx = 0
0
Integrate by parts twice,
u′ w
L
L
0
L
− u w′ 0 + ∫ u ( w′′ + k w) dx + ∫ pw / c 2 dx
L
2
0
0
Fundamental solution w :
w′′ + k 2 w = − δ ( x, xi )
w=
−1
sin k ( x − x i ) step( x, x i )
k
u ( xi ) = u ′ w
L
0
− u w′
⇒
L
L
0
+ ∫ pw / c 2 dx
0
7
(*)
F
Example
ρA=1
u(0) =0
du/dx (L) = F/c2
F
u
=
From Eq. (*), tip displacement, L k c 2 tan (k L)
This is exact, since boundary conditions are exactly
satisfied.
Resonances occur when cos (kL) = 0, or
kL = π/2, 3π/2, ...
Eq. (*) then gives
u( x i ) =
F
{tan(k L) cos k ( L − x i ) − sin k ( L − x i )}
k c2
Timoshenko’s formula:
uL
=
2F
c2 L
∑
i = 1, 3, 5,...
1
⎛ i2 π 2
2⎞
⎜⎜
⎟⎟
−
k
2
4
L
⎝
⎠
Consider 3 modes in the expansion (FEA needs about
3 elements for each half sine wave) - red color in fig:
8
0.12
0.1
0.08
0.06
0.04
0.02
0
0
500
1000
1500
1
0.5
0
-0.5
-1
-1.5
0
5
10
15
Mode 4 = blue color
9
20
25
BEAM VIBRATION
∂4 v
p
2
−
=
k
v
∂ x4
ρAc 2
∂4 w
− k 2 w = − δ ( x, x i ) ⇒
4
∂x
1
w = − step( x, x i )
sinh
3/ 2
2k
{ [
]
k ( x − x i ) − sin
[
k ( x − xi )
v( xi ) = ...
But now, also need to differentiate v wrt xi to get additional equation(s) for solution
10
]}
STATIC PLATES BY DIRECT BEM
∇4 w =
∫∫ (w ∇
4
p
D
w*
Kirchoff theory for thin plates
−
)
(
)
w* ∇ 4 w dA = ∫ w* V − V * w − θ * M + M * θ dS
A
S
+ corner terms
Choose the ‘concentrated force’ fundamental solution
w* such that
∇ 4 w* ( xi ) = δ ( x, xi )
1
w =
*
which gives
4π D
r 2 ln r
Thus
ci w( xi ) =
1
D
∫∫ p w
A
*
dA +
∫ (w
*
)
V − V * w − θ * M + M * θ dS
S
K*
(
+ ∑ w* F − F * w
)
k =1
where ci equals 0.5 if xi is on smooth S, and equals 1
if interior.
11
A second equation is obtainable as:
cξ
∂w
∂w
+ cη
∂ξ i
∂η i
=
1
D
∫∫ p w'
*
∫ (w'
dA +
*
A
)
V − V '* (w − wi ) − θ '* M + M '* θ dS
S
K
*
(
)
+ ∑ w'* F − F '* (w − wi )
k =1
where the ‘concentrated moment’ fundamental
*
solution w' is given by
w' =
1
*
2π D
r ln r cos(α )
Expressions for V*, V’* etc are given in the literature.
Note:
Corners (eg. rectangular plates) require
special attention
Distributed load requires domain integration
Here, a function s is defined such that
∂
1 ∂
∇ s=w , ∇ =
+
and then converting to a
r ∂r
∂r
2
2
*
2
2
contour integral as
p ∫∫ w* dA = p ∫
A
S
∂s
cos( β ) dS
∂r
where β = angle between r and n. Similarly
*
for w' .
12
BEM WITH CONSTANT ELEMENTS :
CONTOUR INTEGRATION
W(S) = Wj = constant for e
element k
n
j
r
Gauss point
source i
*
*
V
w
dS
=
w
V
j
∫ dS which is evaluated using
Thus, ∫
e
e
gauss quadrature
Linear elements will provide more accuracy
Finally: A x = b
A is square, unsymmetric, dense, dim 2N+3K
where N = no. of regular nodes, K = no of corner
nodes
Clamped Plate: x = [shear, moment] per unit length
Simply-Supported Plate: x = [slope, shear]
13
DYNAMIC ANALYSIS OF PLATES
∇4 w +
ρh
D
&& =
w
p
D
For general transient loading, apply Laplace
transform and then use BEM. Here, we assume
harmonic loading. Thus,
w(x, t ) = w(x) e i ω t ,
w(x) = amplitude
p
∇ w −k w=
D
ω
4
2
, k=
c
Choose the ‘concentrated force’ fundamental solution
w* such that
∇ 4 w* ( xi ) − k 2 w = − δ ( x, xi )
[
i
( 2)
(1)
w =
H 0 ( k r ) + H 0 (i k r )
8k
*
]
*
The ‘concentrated moment’ fundamental solution w'
is given
by
⎡i c1 J1 ( k r ) + c1 Y1 ( k r ) +
w' = cos(α ) ⎢
⎢⎣c2 K1 ( k r )
*
14
⎤
⎥
⎥⎦
CORNER EFFECTS
When source point is a corner, two concentrated
moment fundamental solutions, and corresponding
equations, need to be written. There are a total of
three unknowns at each corner
When a plate is loaded, corners tend to curl up –
corner reactions keep these clamped
Details are omitted here for brevity
15
HANDLING DOMAIN (PRESSURE) LOADS IN
DYNAMICS
1
D
∫∫ p w
A
*
dA
,
1
D
∫∫ p w'
*
dA
A
Approach 1: Create a domain mesh, and integrate
(careful mesh design, and integration are necessary
for accuracy)
m
Approach 2: Express
p w' = ∑ ai ψ i ,
*
1
where ai are
determined through regression, and each of the
known functions ψi can be written as ∇ s . Then, the
Divergence theorem allows us to convert the domain
integral to a contour integral.
2
i
16
DIFFICULTIES IN BEM
1. Fundamental solutions are hard to derive for
complicated differential equations
2. Domain integration requires special care
3. Numerical integration involving bessel functions
need special care
17
BEM WITH MULTIPLE RECIPROCITY
(BEM-MRM)
∇4 w − k 2 w =
p
D
Choose the ‘concentrated force’ fundamental solution
w* such that
Instead of ∇ 4 w * ( x i ) − k 2 w = − δ ( x , x i )
Choose w* as the static fundamental solution :
∇ 4 w* ( xi ) = δ ( x, xi )
w =
1
*
which gives
4π D
r 2 ln r
We then have (ignoring corner terms):
ci w( xi ) = k 2
∫∫
A
1
D
∫∫ p ws
A
*
dA +
w w s dA +
*
∫ (w
s
*
)
V − Vs w − θ s M + M s θ dS
*
*
*
S
*
*
*
*
*
*
4
4
4
Let ∇ s1 = ws , ∇ s 2 = s1 , ∇ s3 = s 2 etc
and repeatedly use, with s = si ,
∫∫ (w ∇
A
4
s*
−
(
)
)
s * ∇ 4 w dA = ∫ s * V − Vs w − θ s M + M s θ dS
*
S
18
*
*
BEM-MRM (Cont’d)
to obtain an expression for the displacement
ci w( xi ) =
1
D
*
ˆ
p
w
dA +
∫∫
A
∫ (wˆ V − Vˆ
*
*
)
w − θˆ* M + Mˆ * θ dS
S
K*
(
+ ∑ wˆ * F − Fˆ * w
k =1
)
where
p
wˆ * = ws + ∑ k 2 i si
*
i =1
p
*
*
, Vˆ * = Vs + ∑ k 2 i Vsi
*
i =1
etc, along with a similar equation for for the slope.
19
Advantages of BEM_MRM
Simple fundamental solutions regardless of
complexity of differential equation
All domain integrals can be easily converted to
contour integrals
Claimed that number of terms, p, are small for
convergence – is this true for all frequencies ?
Integrals independent of k can be done once and
stored within the frequency loop
20
COMPUTER CODE
Circular plate (done), Rectangular plate (in progress)
Concentrated load, Constant boundary element
Damping: E = E0 (1 + i η)
Input: pressure as SPL (converted to point load)
Outputs:
Center displacement Vs frequency
PSD b
=
1
Nb
∑ [acc ( n ) ]
2
,
n
N b = ( f u − f l ),
where
f u and f l = upper & lower freq in band, at 1Hz inc.
acc =
accelerati on in g' s
(- ω
acc(n) =
2
n
)
,
w n / g , ω n = f n / 2π
1 ∞
2
GRMS = ∑ [acc(n)]
2 n =1
2
Validation: with Timoshenko’s formulas and with
Ansys
21
VALIDATION EXAMPLE
Clamped Circular Steel Plate, R=1m, η=.03
Concentrated load = 1 N
2.5
x 10
-5
Circular Plate
Center Displacement
2
1.5
1
0.5
0
0
20
40
60
80
frequency, Hz
22
100
120
140
ANSYS RESULT
Circular plate concentrated
23
SAMPLE PROBLEM
Clamped Circular Steel Plate
1m radius, 1 cm thick, η = .01
Pressure corresponds to about 50 Pa
22 1/3-octave bands, ranging from 16Hz to 2000 Hz
SPL in 1/3 Octave Bands
150
SPL, dB
100
50
0
0
5
10
15
24
20
25
CENTER DISPLACEMENT
Vs HZ
6
x 10
-3
Circular Plate
Center Displacement
5
4
3
2
1
0
0
500
1000
1500
frequency, Hz
25
2000
2500
MEAN ACCELERATION IN G’S IN EACH
1/3RD OCTAVE BAND
Circular Plate
80
70
Center Acceleration in g"s
60
50
40
30
20
10
0
0
5
10
15
26
20
25
PSD IN EACH 1/3RD OCTAVE BAND
2
x 10
4
Circular Plate
1.8
2
2
g /Hz = int(g(f) )/deltaf/deltaf
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
27
20
25
DRAWBACK IN MASS MATRIX
CALCULATIONS IN FINITE ELEMENTS
u=Nq
u = displacement field in element e
q = nodal displacements of e
N = shape functions – polynomials
Mass matrix:
1
m = ∫ N T N dV
2 e
M is assembled from each m
Choice of N critical in dynamics, since inertial force
equals ω2 M , and solution is obtained from
[K - ω2 M]-1 F
Comment: for large ω , polynomial shape functions
are inadequate
Also, shear effects must be properly captured
28
A FEW KEY REFERENCES
Book: “Boundary elements – an introductory course”,
2nd ed, C.A. Brebbia and J. Dominguez
Paper: “Boundary integral equations for bending of
thin plates”, Morris Stern
Paper: “free and forced vibrations of plates by
boundary elements”, C.P. providakis and D.E.
Beskos, Computer methods in applied mechanics and
engineering, 74, (1989), 23-250
Papers on BEM-MRM: see J. Sladek.
Example: “Multiple reciprocity method for harmonic
vibration of thin elastic plates”, V. Sladek, J. Sladek,
M. Tanaka, Appl. Math. Modelling, 17, (1993), 468475.
29
SUMMARY OF WORK COMPLETED
(MAY 19TH – JULY 25TH, 2003)
Tutorial has been developed for BEM in vibration
-- rods, beams, Laplace eq., plate bending
Computer code for circular plate vibrations
clamped or simply-supported
validated with Closed-form solutions and Ansys
SPL input, PSD and and other metrics output
State of the art for vibration analysis has been
assessed
Rectangular (and other geometries) on-going
Research issues identified (see next slide)
30
FUTURE WORK IN APPLYING BEM FOR
PLATE VIBRATIONS
at MID-FREQUENCIES
Accurate and efficient numerical integration of body
force terms
Develop BEM-MRM and compare with ‘standard’
BEM; more validation
Include shear effects (BEM-MRM is attractive)
Generalize geometries (any planar shape, thick
plates, shells)
Compare BEM, FEM, Experiment on a specific
problem;
Spectral element methods
Other research: (shock loading, optimization )
31