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Computing Nonlinear Frequency Response Functions (FRFs) for
Systems with Iwan Joints
S. Iman Zare Estakhraji, Matthew S. Allen & Drithi Shetty
University of Wisconsin-Madison
International Modal Analysis Conference (IMAC XXXVIII), Houston, Texas
February 2020
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2
Nonlinearities due to joints are a major source of error in modeling complicated, built-up structures.
Micro- and Macro-slip in joints introduces nonlinearity
Nonlinearities due to joints are a major source of error in modeling complicated, built-up structures.
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Segalman’s 4-parameter Iwan model has proven effective at capturing the nonlinearity observed in joints in a variety of experimental studies.
[1] D. J. Segalman et al., “Handbook on Dynamics of Jointed Structures,” Sandia National Laboratories, Albuquerque, NM 87185, 2009.[2] B. Deaner et al., "Application of Viscous and Iwan Modal Damping Models to Experimental Measurements …," ASME JVA, vol. 137, p. 12, 2015[3] D. R. Roettgen and M. S. Allen, "Nonlinear characterization of a bolted, industrial structure using a modal framework," MSSP, vol. 84, pp. 152-170, 2017.
Slope = 1 + χ
micro-slip macro-slip
Log (Response Amplitude)
Lo
g (
Da
mp
ing
Ra
tio
)
FS
Slope = –1
no-slip
(material
damping)
Lo
g (
Dam
pin
g R
ati
o)
KT
FS
micro-slip macro-slip
Sti
ffn
ess
Log(Joint Force / Resp. Amplitude)
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The 4-parameter Iwan joint is straightforward to integrate, but no method exists for computing the nonlinear frequency response.
x(t, ϕ1)
x(t, ϕ2)
x(t, ϕ3)
F
x(t, ϕ4)
Iwan Model
p0,( )jz
𝐇 𝑇, 𝐳 =
𝐳(𝑇, 𝐳0) − 𝐳(0, 𝐳0) = 𝟎
ቤ𝜕𝐇
𝜕𝐳0 𝐳0,𝑇
ቤ𝜕𝐇
𝜕𝑇𝐳0,𝑇
𝐏𝑧T 𝑃𝑇
Δ𝐳0Δ𝑇
=−𝐇(𝑇, 𝐳0)
0
[1] M. Scheel et al., “Experimental Assessment of Polynomial Nonlinear State-Space and Nonlinear-Mode Models…,” MSSP, Submitted 2019.
?
ቤ𝜕𝐇
𝜕𝐳0 𝐳0,𝑇
NLFRF from [1]
• Iwan joints modeled with 30-100 sliders
• Discontinuous time histories
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Outline
• Computing Iwan slider states from history of reversal points.
• SDOF Case:• Reversals / Maximum
Displacement
• Implementation
• Extension to MDOF Systems
• Conclusions
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
Displacement
0
2
4
6
8
10
12
14
16
Jo
int
Nu
mb
er
For the time t=0.62732 T and =0.98543 n
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Iwan JointDetermining the positions of the Iwan sliders from the past
history of load reversals
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• An Iwan element is a one-dimensional, parallel arrangement of Jenkins elements.
• Each Jenkins element consists of a linear spring in series with a friction slider. -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
Displacement
0
2
4
6
8
10
12
14
16
Jo
int
Nu
mb
er
For the time t=0.62732 T and =0.98543 n
Top Surface
SliderSliding
Direction
Iwan Joint
Component A Component B
Bolt Assemblies
Joint Interface
x
y
z
1 2 3 4 5
Point s
Bottom Surface
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• Force-deflection relation for an element [1]:
• Nonlinear force in the joint [1]:
[1] Iwan, Wilfred D. "A distributed-element model for hysteresis and its steady-state dynamic response." (1966): 893-900.
: Strength of the element
: Distribution function
: Force of the element “i”
: Maximum displacement
Iwan Joint: Nonlinear Force
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Iwan Joint: Distribution function
• The maximum extension of the ith spring can be defined as [1]:
• Distribution function defined based on the power-law distribution [1]:
• The nonlinear force depends on the state of the sliders.
[1] Segalman, Daniel J. "A four-parameter Iwan model for lap-type joints." Journal of Applied Mechanics 72.5 (2005): 752-760.
0 2 4 6 8 10 12
Disp
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
Slid
er
Nu
mbe
r
Distribution function for 200 sliders
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Visualizing the State of the Sliders• The sliders don’t have
any internal dynamics.
• If the response is quasi-periodic, then the state of all of the sliders can be determined from xmax : the reversals (or the maximum displacement in past history).
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Reversal Points / Maximum Displacement
• One moment after max:
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.39415 T and =0.99128 n
Joint is at a reversal point:
• The state of sliders can be specified:
✓ Green lines show
o Some of them are at
o Some of them are at Zero. (Never slipped)
• Reversal points can be used as a benchmark.
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• Slider #7 and #8 are still stuck at
-8 -6 -4 -2 0 2 4 6 8
Displacement
0
1
2
3
4
5
6
7
8
9
10
Jo
int
Nu
mb
er
For the time t=0.62851 T and =0.98729 n
State of Sliders
• Sliders #1~6 are sliding (red)
• Sliders #1~6 are following the joint by the distance of
(Green lines).
• Sliders #9 and 10 never slipped.
sliding direction
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• Force-deflection relation for an element [1]:
• Sum over the joints to compute the total force
• Total stiffness is the sum of the stiffnesses of the stuck springs.
[1] Iwan, Wilfred D. "A distributed-element model for hysteresis and its steady-state dynamic response." (1966): 893-900.
: Strength of the element
: Distribution function
: Force of the element “i”
: Maximum displacement
Knowing this, we can use the relations defining the Iwanjoint to compute the force at any instant.
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Modified Continuation Procedure
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Modified Continuation Procedure
• One additional state variable is added to the state vector:
• Shooting function:
• Convergence criteria:
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The Jacobians needed for the continuation algorithm are computed using finite differences.
• Correction (Newton-Raphson): • The step-size dilemma for sin(x) [1]:
[1] Mathur, Ravishankar. An analytical approach to computing step sizes for finite-difference derivatives. Diss. 2012.
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Application to an SDOF System
• Continuation state augmented with xmax (only one reversal considered)
• 100 sliders used to model the Iwanelement
MIwan Element
Klin
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SDOF System NLFRFs
0.156 0.1565 0.157 0.1575 0.158 0.1585 0.159 0.1595 0.16
Frequency (Hz)
100
101
102
103
|X|/
F (
s2/k
g)
Amplitude = 0.001N
Amplitude = 0.01N
Amplitude = 0.1N
Amplitude = 0.4N
Amplitude = 1N
Amplitude = 2N
Linear FRF
softening nonlinearity, increasing damping
Slider displacements for Amp=1 N show complicated, nonlinear
discontinuous response.
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Extension forMDOF Systems
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MDOF Algorithm
• Algorithm is similar, but there may be many reversals to keep track of.
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Results for a 3-DOF system with an IwanElement for different force levels
IwanM M M
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Conclusions
• While Iwan joints are complicated, the state of any Iwan joint can be determined algebraically if the displacement is known at enough reversal points.
• The Nonlinear Frequency Response (NLFRFs) can then be computed using continuation.
• With further development, this could be an effective tool for simulation or model updating using measurements.
Options for Computing NLFRFs
Time Integrate until Steady-
State
Shooting & Continuation
(this talk)
Harmonic Balance
(ongoing work)
~100 hours
10-15 minutes
~2-5 minutes
(computation times for SDOF system)
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Acknowledgements
• Iman Zare (who did most of the work!)
• The National Science Foundation• This material is based in part upon work supported by the
National Science Foundation under Grant Number CMMI-1561810. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.Iman Zare
all
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Results for a 3-DOF system with 3 IwanElements for different nonlinearities
0.1 0.15 0.2 0.25 0.3 0.35
Freq (Hz)
10-2
10-1
100
101
log
( M
ax (
x(t
)) )
Blue:Red:
MIwanElement
M M
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Iwan FRF
• A common numerical model for bolted or riveted joints is the Iwan model.
• In many applications it is desirable to predict the nonlinear Frequency Response Functions (FRFs) of a structure that contains joints.
• The steady-state response must be estimated over a range of frequencies.
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Continuation MethodA numerical method to obtain the FRF
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Continuation Procedure
• Numerical continuation can be used to obtain the FRFs of a nonlinear system.
• For a response to be considered steady-state, the displacement and velocity must be periodic.
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Numerical Integration
• The differential equation is solved using implicit integration methods: Newmark-Beta.
• The state variables are sent to the integration function as the initial conditions.
• All the history dependent variables should be specified as the initial conditions to start the integration.
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Continuation Procedure
• Prediction (pseudo-arclength continuation):
• Correction (Newton-Raphson):
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Iwan FRF: ChallengesThe implicit nature of the state variables makes it non-trivial to
use continuation to compute the frequency response using already established techniques such as the shooting method.
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State of sliders
• It specifies if a slider is slipping or not:
• For the case of steady-state response, the state of sliders should be periodic.
• The state of sliders should be considered as the initial condition as well.
The non-linear force of Iwan joint
is history dependent.
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Reversal Points
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.28377 T and =0.99128 n
• Before max: • At max:
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.39413 T and =0.99128 n
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Reversal Points
• At max:
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.39413 T and =0.99128 n
• One moment after max:
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.39415 T and =0.99128 n
Joint is at a reversal point
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Algorithm
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Algorithm
Never slipped
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.58353 T and =0.99128 n
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Slipping
Algorithm
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Algorithm
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Finite Difference
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Finite Difference
• The step-size dilemma for sin(x) [1]:
[1] Mathur, Ravishankar. An analytical approach to computing step sizes for finite-difference derivatives. Diss. 2012.
• Finite difference is used:
• Step-size makes the differences:
o Number of iteration
o Next predictions
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Step-Size Algorithm
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Results
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SDOF System
• Just one reversal point is considered, and it seems that is enough.
• 100 sliders are used.
• Different Load cases is considered.
• For MDOF system more reversal points should be considered.
MIwan Element
Klin
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SDOF System
0.156 0.1565 0.157 0.1575 0.158 0.1585 0.159 0.1595 0.16
Frequency (Hz)
100
101
102
103|X
|/F
(s
2/k
g)
Amplitude = 0.001N
Amplitude = 0.01N
Amplitude = 0.1N
Amplitude = 0.4N
Amplitude = 1N
Amplitude = 2N
Linear FRF
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SDOF System
0.156 0.1565 0.157 0.1575 0.158 0.1585 0.159 0.1595 0.16
Frequency (Hz)
100
101
102
103
|X|/
F (
s2/k
g)
Amplitude = 0.001N
Amplitude = 0.01N
Amplitude = 0.1N
Amplitude = 0.4N
Amplitude = 1N
Amplitude = 2N
Linear FRF
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Preiodicity
1 2 3 4 5 6 7 8 9
t/T
1
2
3
4
5
Perc
ent
Err
or
10-6
Error of initial displacement
1 2 3 4 5 6 7 8 9
t/T
-2
-1
0
Pe
rcen
t E
rro
r
10-4
Error of initial velocity
0.0685 0.069 0.0695 0.07 0.0705 0.071 0.0715 0.072 0.0725 0.073
Freq(Hz)
2
4
6
8
10
12
Ma
x(x
(t))
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MDOF SystemThe challenges and a need to a new algorithm
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MDOF System
• More harmonics because of more non-linearity in the system.
• More than two reversal points, all of them should be considered.
• The number of reversal points may change at each iteration. That causes the size of Jacobians to be Dynamic.
0 0.142857 0.285714 0.428571 0.571429 0.714286 0.857143 1
t/T
-10
-8
-6
-4
-2
0
2
4
6
8
Dis
p
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MDOF Algorithm
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MDOF Algorithm : Part I
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MDOF Algorithm : Part II
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MDOF Algorithm : Part III
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MDOF Algorithm : Part IV • It continues until we get the state of sliders are periodic.
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0.1 0.15 0.2 0.25 0.3 0.35
Freq (Hz)
10-2
10-1
100
101
log
( M
ax (
x(t
)) )
= - 0.5
= -0.8
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Appendix
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Iwan FRF: Challenges
• The implicit nature of the state variables makes it non-trivial to use continuation to compute the frequency response using already established techniques such as the shooting method.
• A novel method to numerically compute the non-linear FRFs of a system with an Iwan element,
• The reversal points over the response period is included as a state variable.
• The shooting method is modified to account for the added state variable.
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Reversal Points
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.39413 T and =0.99128 n
• One moment after max:
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Reversal Points
• When and the 1th to 7th
sliders are sliding, the 8th slider is stuck at some initial location. The 9th and 10th sliders are stuck at equilibrium.
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement
0
2
4
6
8
10
12
Jo
int
Nu
mbe
r
For the time t=0.28377 T and =0.99128 n
• When and the 1th to 7th
sliders are sliding, the 8th slider is stuck at some initial location. The 9th and 10th sliders are stuck at equilibrium.
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Iwan_FRF/MHB/3DOF/my-try-captured-all-weeks.fig
Iwan_FRF/MHB/3DOF/my-try-captured-all-the-peak-max(xdt).fig
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KT
FS
micro-slipmacro-slip
Sti
ffn
ess
Joint Force / Resp. Amplitude