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IntroductiontoIwanModelsandModalTestingforStructureswithJoints
Matthew S. AllenAssociate Professor
University of Wisconsin-Madison
Thanks to Brandon Deaner (now at Mercury Marine) for his work in this area and for creating the initial version of these slides.
Much of the damping in built up structures comes from the interfaces (joints).
Joints are nonlinear and influenced by physics at multiple length scales.
However, the response of structures with joints is often quasi-linear.
Motivation
size
Quantummechanics,chemistry
Moleculardynamics
Meso,micro,asperitymodels
FEA,detailedjointmechanics
Assembly/StructuralscaleAssembly/StructuralscaleFEA+jointmodels
0.1nm
1nm
1m
0.1mm
10mm
1m
500 1000 1500 2000 2500 3000 3500 4000
104
105
106
107
Frequency (Hz)
Mag
nit
ud
e
0
13.0927.04
42.18
58.35
82.28 107
129.6
1340 1360 1380 1400 1420 144Frequency (Hz)
0
13.0927.04
42.18
58.35
82.28
107
129.6
2
Outline
Discrete Damping Model
Modal Damping Model
Apply the Modal Damping Model Academic System
Finite Element Model
Beam Structure
Exhaust Structure
Conclusions
M0
u3
K∞
M0M0
K∞ K∞
u2u1
F
FS, KT, χ, β
Wewanttoaccuratelymodelthenonlineardampingseeninjoints.
Structures with joints exhibit increased damping.
Jointed Structure vs. Monolithic Structure
Damping depends on the amplitude of the force
0 200 400 600 800 1000-1
0
1
Res
pons
e
Time0 200 400 600 800 1000
-1
0
1
Res
pons
e
Time
3
Thenonlineardampingofjointsisclassifiedintotworegions.
“macro-slip” – occurs when the stick region vanishes and there is large relative motion between the two parts.
“micro-slip” – nonlinearity associated with the slip region at the outskirts of the contact patch.
D. J. Segalman, "An Initial Overview of Iwan Modeling for Mechanical Joints," Sandia National Laboratories, Albuquerque, New Mexico SAND2001-0811, 2001.
F
F
Slip Region
Stick Region
AJenkinselementisoftenusedtoprovidestick‐slipbehavior.
Consists of: Coulomb Friction Damper Linear Elastic Spring
Provides Stick-Slip behavior Stuck state has stiffness and no energy dissipation Slipped state has no stiffness and energy dissipation
Fx(t, ϕ)
k
4
OneapproachistomodelajointwithseveralJenkinselementsoveranarea.
Advantage: Predictive!
Disadvantage: Many degrees-of-freedom, no guarantee that power law behavior is captured.
* C. W. Schwingshackl, D. D. Maio, I. Sever, and J. S. Green, "Modeling and validation of the nonlinear dynamic behavior of bolted flange joints," Journal of Engineering for Gas Turbines and Power, vol. 135, 2013.
(see also) Bograd, S., Reuss, P., Schmidt, A., Gaul, L., and Mayer, M., “Modeling the Dynamics of Mechanical Joints,” Mechanical Systems and Signal Processing, 25, pp. 2801-2826, 2011.
AparallelarrangementofJenkinselementsisknownasanIwanmodelandisusedtomodelbothregionsofslip.
D. J. Segalman, "A Four-Parameter Iwan Model for Lap-Type Joints," Journal of Applied Mechanics, vol. 72, pp. 752-760, September 2005.
x(t, ϕ1)
x(t, ϕ2)
x(t, ϕ3)
F
x(t, ϕ4)
Micro-Slip
Macro-SlipIwan Model
5
A4‐ParameterIwanmodelisusedtocapturebothmicro‐ andmacro‐slip.
The Iwan Model consists of spring and frictional damper elements in parallel
D. J. Segalman, "A Four-Parameter Iwan Model for Lap-Type Joints," Journal of Applied Mechanics, vol. 72, pp. 752-760, September 2005.
Population Density
ρ(ϕ) = R ϕχ[H(ϕ) – H(ϕ – ϕmax)]+Sδ(ϕ – ϕmax)
x(t, ϕ1)
x(t, ϕ2)
x(t, ϕ3)
F
x(t, ϕ4)
ϕρ(ϕ
)
ϕmax
micro-slip macro-slip
KT
FS
micro-slip macro-slip
Sti
ffn
ess
Joint Force
Wecanrepresentthe4‐ParameterIwanmodelgraphicallytodescribeshowstiffnessanddissipationchangewithforcelevel.
Slope = 3 + χ
β
micro-slip macro-slip
log(Joint Force)
log(
Dis
sip
atio
n/C
ycle
)
FS
{R, χ, ϕmax, S} {FS, KT, χ, β}
Converted to physical parameters
Slope = 1
ThisisthekeytotheIwanmodel:itcapturesaspecificevolutionofdampingandstiffnesswithamplitudethathasbeenobservedinmanyexperiments!!
6
Stiffnesschangesaretypicallysmall– dampingeffectisofprimaryinterest.
Slope = 1 + χ
micro-slip macro-slip
Log Joint Force
Log
Dam
pin
g R
atio
FS
Slope = –1
no-slip (material damping)
2
/
2
diss cyc
n
E c X
m X
2
/ /diss cyc d nE m X
2 2212
2
( ) ( )d
d n
V t V t
X
0 cos( )mx cx kx F t
Ediss/cyc goes as |X|2 !!
Whywouldwewanttoapplythedampinginamodalframework?
Simplify computations for structure with many joints
1 1 1 1
2 2 2 2
3 3 3 3
S T
S T
S T
F ,K , χ ,β
F ,K , χ ,β
F ,K , χ ,β
Many parameters needed to characterize each joint
separately.
7
Whywouldwewanttoapplythedampinginamodalframework?
Simplify computations for structure with many joints
M = 1
q
K∞
FS, KT, χ, β
2 T,
ˆΦ FX Jr r r r rq q + F
Whywouldwewanttoapplythedampinginamodalframework?
Because tests in the micro-slip regime frequently reveal that the structure as a whole often behaves modally!
8
Linear Modes of the system are preserved No geometric nonlinearities due to large deflection or
material nonlinearities
TheproposedmodalIwanmodelcomeswithseveralassumptions.
F
D
No coupling among modes
rth Modal Force applied = rth Modal Response only
Doesthemodalapproachworkforasimplemassspringsystem?
M0
u3
K∞
M0M0
K∞ K∞
u2u1
F
FS, KT, χ, β
Discrete Model:
D. J. Segalman, "A Modal Approach to Modeling Spatially Distributed Vibration Energy Dissipation," Sandia National Laboratories, Albuquerque, New Mexico and Livermore, California SAND2010-4763, 2010.
Modal Models:
M = 1
q1
K∞1
FS1, KT
1, χ1, β1
M = 1
q2
K∞2
FS2, KT
2, χ2, β2
M = 1
q3
K∞3
FS3, KT
3, χ3, β3
???
9
Comparisonprocedurefordiscreteandmodalsimulations.
Apply Impact Force
0 20 400
0.5
1
Time (s)
Fo
rce
Convert physicalresponse to modal
Discrete Simulation:
q = -1x
Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
104
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Optimize Analytical Modelto frequency and energydissipation to each mode.
Min f
Apply Impact Force
0 20 400
0.5
1
Time (s)
Fo
rce
Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
10
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Compare Modal andDiscrete simulations
Modal Simulation:
101
102
103
0.188
0.189
0.19
0.191
0.192
0.193
0.194
Modal Force
Nat
ura
l F
req
uen
cy (
Hz)
101
102
103
10-2
100
102
104
Modal Force
Mo
dal
En
erg
y D
iss
ipat
ion
/Cyc
le
FS, KT, χ, β
FS1, KT
1, χ1, β1
HilbertTransformwithpolynomialfittingisusedtoextractfrequencyandenergydissipationdata.
0 50 100 150 200 250 300 350 400 450 500-500
0
500
1000
An
gle
(ra
d)
Time (s)
Measurement
Polynomial Fit
0 50 100 150 200 250 300 350 400 450 500
102
|Vel
oci
ty|
Time (s)
Academic Model 3rd Mode
|Measurement|
|Hilbert Transform|Polynomial Fit
Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
10
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Instantaneous damping and
frequency are related to the derivative of the amplitude and phase, respectively.
10
Optimizationcanbeusedtofittheanalyticalmodeltomeasuredchangesindissipationandfrequency.
ˆ 3
0
0
ˆ4 ˆ ˆif ˆ ˆ ˆ3 2
ˆ ˆ ˆ4 if
JS
Model
JS S
RqF F
D
q F F F
ˆ 1ˆˆ ˆ1ˆ ˆ( 1)( 2) ˆ ˆˆ
2
ˆˆ ˆ
2
if
if
T
JS
Model
JS
rK K
F Ff
KF F
micro-slip macro-slip
Nat
ura
l Fre
qu
ency
Modal Joint Force
micro-slip macro-slip
log(Modal Joint Force)
log(
Mod
alD
issi
pat
ion
/Cyc
le)
Optimize the ‘Analytical Modal Model’ in a least squares sense
Approximate Frequency:
Approximate Dissipation:
Optimize analytical modelto match frequency and
energy dissipation to eachmode.
22ˆ ˆˆ ˆ
Min ˆ ˆ ˆ ˆmax max
Exp Model Exp ModelD f
Exp Model Exp Model
D D f ff f f
D D f f
HowwelldoesamodalIwanmodelpredicttheresponseofaspringmasssystem.
vs.
101
102
103
10-1
100
101
102
103
104
105
Academic Model 1st Mode
Modal Force
Mo
da
l En
erg
y D
iss
ipa
tio
n/C
yc
le
Discrete SimAnalytical ModelModal Sim
101
102
103
0.0672
0.0672
0.0673
0.0673
0.0674
0.0674
0.0675
0.0675
0.0676Academic Model 1st Mode
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Discrete SimAnalytical ModelModal Sim
macro-slipmicro-slip
macro-slipmicro-slip
ˆ 3
0
0
ˆ4 ˆ ˆif ˆ ˆ ˆ3 2
ˆ ˆ ˆ4 if
JS
Model
JS S
RqF F
D
q F F F
ˆ 1ˆˆ ˆ1ˆ ˆ( 1)( 2) ˆ ˆˆ 2
ˆˆ ˆ
2
if
if
T
JS
Model
JS
rK K
F Ff
KF F
vs.
11
HowwelldoesamodalIwanmodelpredicttheresponseofaspringmasssystem.
101
102
103
100
Academic Model 2nd Mode
Modal Force
Mo
dal
En
erg
y D
issi
pat
ion
/Cyc
le
Discrete Sim
Analytical ModelModal Sim
101
102
103
0.188
0.189
0.19
0.191
0.192
0.193
0.194Academic Model 2nd Mode
Modal Force
Nat
ura
l F
req
uen
cy (
Hz)
Discrete Sim
Analytical ModelModal Sim
101
102
103
100
Academic Model 3rd Mode
Modal Force
Mo
dal
En
erg
y D
issi
pat
ion
/Cyc
le
Discrete Sim
Analytical ModelModal Sim
101
102
103
0.272
0.274
0.276
0.278Academic Model 3rd Mode
Modal Force
Nat
ura
l F
req
uen
cy (
Hz)
Discrete Sim
Analytical ModelModal Sim
Limitations The previous results all excited the system with a force that
was spatially distributed to excite a single linear mode.
Forces with other spatial distribution can cause macro-slip at a lower force level.
100
102
104
106
0.422
0.4225
0.423
0.4235
0.424
Frequency vs Log Modal Amplitude
Log Modal Disp. Amplitude
Nat
ura
l Fre
qu
en
cy (
rad
)
100
102
104
106
10-5
100
105
1010Log Dissipation vs Log Modal Amplitude
Modal Disp. Amplitude
Mo
dal
En
erg
y D
iss
ipat
ion
/Cyc
le
100
102
104
106
10-2.9
10-2.7
10-2.5
Log Damping Ratio vs. Modal Amplitude
Modal Disp. Amplitude
Dam
pin
g R
ati
o
Linear region
governed by
Material Damping
12
Limitations– Mode2
10-1
100
101
102
103
10-2
10-1
Damping Ratio vs Log Modal Amp.
Modal Disp. Amplitude
Dam
pin
g R
ati
o
10-1
100
101
102
103
1.18
1.19
1.2
1.21
Frequency vs Log Modal Amplitude
Log Modal Disp. Amplitude
Nat
ura
l Fre
qu
ency
(ra
d)
Limitations– Mode3
10-1
100
101
102
103
1.71
1.72
1.73
1.74
Frequency vs Log Modal Amplitude
Log Modal Disp. Amplitude
Nat
ura
l Fre
qu
ency
(ra
d)
10-1
100
101
102
103
10-1
Damping Ratio vs Modal Amplitude
Modal Disp. Amplitude
Dam
pin
g R
atio
13
CananIwanmodelbeusedtopredicttheresponseofaFEmodel?
Academic simulations: micro-slip region = Excellent Representation transition region = may be inaccurate macro-slip region = Excellent Representation
Finite Element models?
Comparisonprocedurefordiscreteandmodalsimulations.
Apply Impact Force Convert physicalresponse to modal
Discrete Simulation:
q = *-1x
Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
10
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Optimize Analytical Modelto frequency and energydissipation to each mode.
Min f
Apply Impact Force Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
10
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Compare Modal andDiscrete simulations
Modal Simulation:10
110
210
3
0.188
0.189
0.19
0.191
0.192
0.193
0.194
Modal Force
Nat
ura
l F
req
uen
cy (
Hz)
101
102
103
10-2
100
102
104
Modal Force
Mo
dal
En
erg
y D
iss
ipat
ion
/Cyc
le
14
HowisadiscreteIwanjointappliedinafiniteelementmodel?
Discrete Iwan (Fs, KT, χ, β)
x
y
z
Contact Region
FE model
Determine the contact surfaces
Use an Iwan model for each joint
AsimpleFEmodelwascreatedforsimulations.
Mesh: • Hexahedron elements• Relatively course mesh (1,554 nodes)
Constraints:• Free-Free Boundary conditions• Two washers are merged with the link• Washers are constrained to move and rotate along the surface of the beam Link
Beam
Washers
Discrete Iwan Models
3.5"
20"
2"
MSA3
Slide 28
MSA3 "For Simulations" isn't too specific. How to improve? "To evaluate the modal approach" ??
Did you sufficiently explain the motivation - Why did we create an FEA model? What are we hoping to learn from these simulations?Matt Allen, 1/31/2013
15
102
103
104
105
124
124.5
125
125.5
126
126.5
127
127.5
128
128.5
129
1 lbf
5 lbf10 lbf
50 lbf
100 lbf
500 lbf1000 lbf 5000 lbf
10000 lbf
Finite Element Model 1st Mode
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Discrete SimAnalytical ModelModal Sim
102
103
104
105
10-6
10-4
10-2
100
102
104
1 lbf
5 lbf
10 lbf
50 lbf
100 lbf
500 lbf1000 lbf
5000 lbf
10000 lbf
Finite Element Model 1st Mode
Modal Force
Mo
dal
En
erg
y D
issi
pat
ion
/Cyc
le
Discrete Sim
Analytical ModelModal Sim
HowwelldoesthemodalIwanmodelpredicttheresponseofthestructure?
macro-slipmicro-slip
= -.28
macro-slipmicro-slip
102
103
104
105
106
344.94
344.945
344.95
344.955
344.96
344.965
344.97
344.975
344.98
1 lbf
5 lbf
10 lbf
50 lbf
100 lbf 500 lbf
1000 lbf
5000 lbf
10000 lbf
Finite Element Model 2nd Mode
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Discrete SimAnalytical ModelModal Sim
102
103
104
105
106
10-10
10-8
10-6
10-4
10-2
100
102
1 lbf
5 lbf
10 lbf
50 lbf
100 lbf
500 lbf
1000 lbf
5000 lbf
10000 lbf
Finite Element Model 2nd Mode
Modal Force
Mo
da
l En
erg
y D
iss
ipa
tio
n/C
yc
le
Discrete SimAnalytical ModelModal Sim
HowwelldoesthemodalIwanmodelpredicttheresponseofthestructure?
micro-slip
micro-slip
= -.99
Mode 2
16
102
103
104
105
106
344.94
344.945
344.95
344.955
344.96
344.965
344.97
344.975
344.98
1 lbf
5 lbf
10 lbf
50 lbf
100 lbf 500 lbf
1000 lbf
5000 lbf
10000 lbf
Finite Element Model 2nd Mode
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Discrete SimAnalytical ModelModal Sim
102
103
104
105
106
10-10
10-8
10-6
10-4
10-2
100
102
1 lbf
5 lbf
10 lbf
50 lbf
100 lbf
500 lbf
1000 lbf
5000 lbf
10000 lbf
Finite Element Model 2nd Mode
Modal ForceM
od
al E
ne
rgy
Dis
sip
ati
on
/Cy
cle
Discrete SimAnalytical ModelModal Sim
HowwelldoesthemodalIwanmodelpredicttheresponseofthestructure?
micro-slip
micro-slip
= -.99
Mode 2
102
103
104
105
106
670
672
674
676
678
680
682
684
686
688
1 lbf
5 lbf 10 lbf
50 lbf
100 lbf
500 lbf
1000 lbf
5000 lbf10000 lbf
Finite Element Model 3rd Mode
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Discrete SimAnalytical ModelModal Sim
102
103
104
105
106
10-6
10-4
10-2
100
102
104
1 lbf
5 lbf
10 lbf
50 lbf
100 lbf
500 lbf
1000 lbf
5000 lbf
10000 lbf
Finite Element Model 3rd Mode
Modal Force
Mo
da
l En
erg
y D
iss
ipa
tio
n/C
yc
le
Discrete SimAnalytical ModelModal Sim
HowwelldoesthemodalIwanmodelpredicttheresponseofthestructure?
macro-slipmicro-slip
macro-slipmicro-slip
Mode 3
17
TheresponsecanbesimulatedwithafewSDOFmodalsimulationsandcomparewellinthemicro‐slipregion.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5
0
51st Mode
Mod
al V
eloc
ity
Time (s)
Discrete Sim
Modal Sim
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5
0
52nd Mode
Mod
al V
eloc
ity
Time (s)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5
0
53rd Mode
Mod
al V
eloc
ity
Time (s)
Modal Responses
TheresponsecanbesimulatedwithafewSDOFmodalsimulationsandcomparewellinthemicro‐slipregion.
Physical Responses
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100
-50
0
50
100
150
200
250
Z-V
elo
city
(m
/s)
at M
idp
oin
t
Time (s)
FE Model with 50lbf Impact Force
Discrete Sim
Modal Sim
18
CanamodalIwanmodelbeusedtopredicttheresponseofactualexperimentaldata.
FE simulations: micro-slip region = Good Representation
macro-slip region = Fair Representation
Laboratory data?
Laboratorytestswereinitiallyconductedonabeamwithalinkattachedinthemiddle.
Nut
Link
Beam
Bolt
Washer
3.5"
20"
2" Same
dimensions as the FE model
Washers between link and beam were removed
The damping of this structure was found to be very small (joint carries little load) and hence a modified structure was studied instead.
19
Alternative:Two‐beamteststructure
Beam Suspension 2 strings
8 elastic bungees
Forcing Automatic hammer
Consistent force input
Thetwobeamteststructureshowsmeasurable,nonlineardamping.
Damping (%) at various torque levels: 10 in-lbs, 30 in-lbs, 50 in-lbs
1500.112640.166050.313
3560.267420.489000.572
9000.1617120.2974001.21
Percent Difference
(%)
50 in-lbs Torque, ζ
(%)
Percent Difference
(%)
30 in-lbs Torque, ζ
(%)
Percent Difference
(%)
10 in-lbs Torque, ζ
(%)
Elastic Mode #
20
MeasurementsweretakenwithascanninglaserDopplervibrometer.
Velocity Measurements:
Polytec OFV-534 Single Point LDV used as
reference.
Polytec PSV-400 Scanning Head used to
measure 70 points on the structure.
Comparisonprocedurefordiscreteandmodalsimulations.
Apply Impact Force Convert physicalresponse to modal
Laboratory Measurements:
Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
10
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Optimize Analytical Modelto frequency and energydissipation to each mode.
Min f
Apply Impact Force Hilbert Transform
100
101
102
10124
125
126
127
128
129
Modal Joint F
(a)
Nat
ura
l F
req
uen
cy (
Hz
)
10
010
110
210
10-6
10-4
10-2
100
102
10
Modal Joint F
(b)
Mo
dal
En
erg
y D
issi
pat
ion
per
cyc
le
Compare Modal andDiscrete simulations
Modal Simulation:10
110
210
3
0.188
0.189
0.19
0.191
0.192
0.193
0.194
Modal Force
Nat
ura
l F
req
uen
cy (
Hz)
101
102
103
10-2
100
102
104
Modal Force
Mo
dal
En
erg
y D
iss
ipat
ion
/Cyc
le
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Mo
dal
Vel
oci
ty
Time (s)
Filtered
Average
0 20 400
0.5
1
Time (s)
Fo
rce
0 20 400
0.5
1
Time (s)
Fo
rce
21
Comparethestandarddeviationofthetrimmedsetversusuntrimmedset.
3.225172.481392.1Modal Hammer50
0.31015.30238.6450
0.0110.6160.3350
0.0050.2836.5250
0.0090.4420.8150
3.081139.341444.5Modal Hammer30
1.58558.24180.1430
0.1253.8452.8330
0.0190.5130.9230
0.0130.3824.1130
0.2136.10288.6410
0.0410.6886.4310
0.0250.2732.8210
0.0880.8020.2110
Standard Deviation of Trimmed (N)
Standard Deviation of Impact Force (N)
Mean Impact Force (N)
Hammer Level(1 lowest - 4 highest)
Torque(in-lbs)
Apply Impact Force
0 20 400
0.5
1
Time (s)F
orc
e
50 100 150 200 250 300 350 400 450 500 550 60010
-8
10-6
10-4
10-2
|Vel
oci
ty|
(m/s
)
Frequency (Hz)
All Measurements
Filtered
-0.2-0.1
00.1
0.2
-0.11-0.1-0.09
-0.8-0.6-0.4-0.200.20.4
1st Elastic Mode 120.9 Hz
-0.2-0.1
00.1
0.2
-0.11-0.1-0.09
-1
0
1
2nd Elastic Mode 214.8 Hz
-0.2-0.1
00.1
0.2
-0.11-0.1-0.09-1
0
1
3rd Elastic Mode 473.7 Hz
-0.2-0.1
00.1
0.2
-0.11-0.1-0.09
00.5
1
4th Elastic Mode 787.4 Hz
-0.2-0.1
00.1
0.2
-0.11-0.1-0.09
0.40.60.8
11.2
5th Elastic Mode 813.2 Hz
-0.2-0.1
00.1
0.2
-0.11-0.1-0.09
0.51
1.5
6th Elastic Mode 1050 Hz
Toisolatethemodes,measurementswerebandpassfiltered.
Convert physicalresponse to modal
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Mo
dal
Vel
oci
ty
Time (s)
Filtered
Average
Nextthemeasurementsweredividedbythemassnormalizedmodeshapevalue.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Mo
dal
Vel
oci
ty
Time (s)
Filtered
Average
22
ThemodalIwanmodelseemstohavedifficultycapturingtheenergydissipationandfrequency.
10-2
10-1
100
113
114
115
116
117
118
119
120
121
122
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Two Beam 1st Mode 30 in-lbs
Lab DataModal Iwan Model
10-2
10-1
100
10-12
10-10
10-8
10-6
Modal Force
Mo
da
l En
erg
y D
iss
ipa
tio
n/C
yc
le
Two Beam 1st Mode 30 in-lbs
Lab DataModal Iwan Model
Optimize Analytical Modelto frequency and energydissipation to each mode.
Min f
Howdoweaccountforthedampingassociatedwiththematerialandboundaryconditions?
Add a viscous damper to the modal Iwan model
M = 1
q
K∞
FS, KT, χ, β
C
23
ThemodalIwanmodelwithaviscousdampercapturesthelaboratorydataaccurately!
10-2
10-1
100
113
114
115
116
117
118
119
120
121
122
Modal Force
Na
tura
l Fre
qu
en
cy
(H
z)
Two Beam 1st Mode 30 in-lbs
Lab DataModal Iwan ModelModal Iwan & VD Model
10-2
10-1
100
10-12
10-10
10-8
10-6
Modal ForceM
od
al E
ne
rgy
Dis
sip
ati
on
/Cy
cle
Two Beam 1st Mode 30 in-lbs
Lab DataModal Iwan ModelModal Iwan & VD ModelVD Model
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.01
-0.005
0
0.005
0.01
0.015
Ve
loc
ity (m
/s) a
t M
idp
oin
t
Time (s)
Filtered DataSimulated Model
TheresponsecanthenbesimulatedasasuperpositionofnonlinearSDOFsystems.
24
Theseconceptsalsoholdforrealistichardware
Front and rear catalytic converters assembled together with required assembly torque and exhaust manifold gasket.
System hung freely suspended by bungee cords to complete a roving hammer test.
0 50 100 150 200 250 300 350 400 450 500
10-1
100
Frequency [Hz]
Co
mp
osi
te F
RF
[m/s
2 ]
Composite FRF for Coupled Catalyic Converter System
Combined Composite FRF
Composite FRF 101X
Composite FRF 101Z
LinearExperimentalModalAnalysis
Force levels kept low to estimate linear damping.
Bending0.0045348.686
Torsion0.0044262.715
Localized Heat Shield Mode0.0004247.384
Localized Heat Shield Mode0.0005243.413
Bending0.0043175.422
Bending0.0030113.701
Deflection TypeDamping
Ratio
Natural Frequency
[Hz]
Modal Index
25
10-4
10-3
10-2
10-1
100
10-2.6
10-2.5
10-2.4
10-2.3
10-2.2
Damping vs. Velocity Amplitude
Amplitude (m/s)
Dam
pin
g R
atio
Strike 204z1
Strike 304z1
Strike 304x1
Strike 204z2
Strike 304z2
Strike 304x2
Strike 204z3
Strike 304z3
Strike 304x3
ModalIwanmodelaccuratelycapturesthedampingversusamplitudeforvariousinputpoints(variouscombinationsofthedifferentmodalamplitudes)!
Because the quasi-modal framework is valid, we can reduce the number of tests and test/model comparisons dramatically!
Preview:ModalIwanparameterscan(*)beusedtodeducediscretejointparameters.
Measurements of modal response in micro-slip regime used to deduce discrete joint properties. [Segalman, Allen, Eriten & Hoppman, ASME-IDETC 2015]
26
Conclusions
Modal Iwan models can represent real structures quite accurately! Academic Model
micro-slip region = Excellent Representation
macro-slip region = Excellent Representation
FE Model micro-slip region = Good Representation
macro-slip region = Fair Representation
Laboratory Data micro-slip region = Excellent Representation
macro-slip region = Not Obtained
Simulation Work:
Michael Starr, Daniel Segalman, Michael Guthrie, Brandon Deaner, Robert Lacayo
Lab Work:
Jill Blecke, Matthew Allen, Hartono Sumali, Michael Starr, Randy Mayes, Brandon Zwink, Patrick Hunter, Dan Segalman, Steve Myatt
Some of the work described was conducted at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. Special thanks to…
Acknowledgements
Univ. Wisconsin Research Group:
Former: Brandon Deaner, Robert Kuether, Hamid Ardeh, Dan Roettgen, David Ehrhardt, Current: Robert Lacayo, Kurt Hoppman, Melih Eriten, …
B. Deaner, M. S. Allen, M. J. Starr, D. J. Segalman, and H. Sumali, "Application of Viscous and Iwan Modal Damping Models to Experimental Measurements From Bolted Structures," ASME Journal of Vibrations and Acoustics, vol. 137, p. 12, 2015.
B. J. Deaner, M. S. Allen, M. J. Starr, and D. J. Segalman, “Application of Viscous and Iwan Modal Damping Models to Experimental Measurements from Bolted Structures," presented at the International Design Engineering Technical Conferences, Portland, Oregon USA, 2013.
M. S. Allen, R. J. Kuether, B. Deaner, and M. W. Sracic, "A Numerical Continuation Method to Compute Nonlinear Normal Modes Using Modal Reduction," presented at the 53rd Structures, Structural Dynamics, and Materials Conference (SDM), Honolulu, Hawaii, 2012.
27
0 5 10 15 20 25 30 35 40 45 50
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
Time (ms)
Res
po
nse
Response
Zero Pts
AlternativeT/FAnalysis:ZeroedEarly‐timeFFT
Nonlinearity is assumed to be active at high amplitudes and inactive at lower amplitudes
The response then becomes more linear as more of the initial nonlinear response is nullified.
Impulse responses with initial segments of varying length set to zero are compared in the frequency domain.
The nonzero portion of each impulse response begins at a point in which the response is near zero.
High Amplitude Nonlinear
Low Amplitude Linear
Excitation
0 5 10 15 20 25 30 35 40 45 50
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
Time (ms)
Res
po
nse
Response
Zero Pts
AlternativeT/FAnalysis:ZeroedEarly‐timeFFT
Nonlinearity is assumed to be active at high amplitudes and inactive at lower amplitudes
The response then becomes more linear as more of the initial nonlinear response is nullified.
Impulse responses with initial segments of varying length set to zero are compared in the frequency domain.
The nonzero portion of each impulse response begins at a point in which the response is near zero.
High Amplitude Nonlinear
Low Amplitude Linear
Excitation
1340 1360 1380 1400 1420 1440
105
106
Frequency (Hz)
Mag
nit
ud
e
NLDetect: FFT of Time Response - Truncated at zero points)
0
13.0927.04
42.18
58.35
82.28 107
129.6
28
Beam:ZEFFTs
500 1000 1500 2000 2500 3000 3500 4000
104
105
106
107
Frequency (Hz)
Mag
nit
ud
eNLDetect: FFT of Time Response - Truncated at zero points)
0
13.0927.04
42.18
58.35
82.28 107
129.6
1400 and 2750 Hz modes clearly soften as the amplitude increases.
FEA Mode 1: 644 Hz
FEA Mode 3: 1581 Hz
FEA Mode 6: 3035 Hz
1340 1360 1380 1400 1420 1440
105
106
Frequency (Hz)
Mag
nit
ud
e
NLDetect: FFT of Time Response - Truncated at zero points)
0
13.0927.04
42.18
58.35
82.28 107
129.6All of these lines come from one response with different parts erased!
Beam:BENDandIBEND
IBEND: Somewhat
nonlinear before 70ms,
Significantly nonlinear before 50ms.
1385 1390 1395 1400 1405 1410 1415 1420
106
107
Frequency (Hz)
Ma
gn
itu
de
NLDetect: FFT of Time Response - Truncated at zero points)
114
Fit to 114
130
Extrap from 114 to 129.6
71
Extrap from 114 to 70.95
30.6
Extrap from 114 to 30.58
2.66
Extrap from 114 to 2.655
0 20 40 60 80 100 120 1400
0.2
0.4
0.6
0.8
1
1.2
1.4
time (ms)
No
rmal
ized
In
teg
ral
Normalized Integral of Difference FRF over Frequency
0.2
0.4
0.6
0.8
1
1.2Integral
fit time
29
“LinearBeam”
Results on previous slides for 200 lbf peak force
This shows the ZEFFTs for a 40 lbf peak force
3 Hz shift observed from high to low amplitude.
1400 1405 1410 1415 1420
105
106
Frequency (Hz)
Mag
nit
ud
eNLDetect: FFT of Time Response - Truncated at zero points)
011.9926.3443.2759.8181.05117.7
