Nonlinear Structural Dynamics: The Fundamentals Tutorial

Nonlinear Structural Dynamics:
The Fundamentals Tutorial
Douglas Adams
Distinguished Professor and Chair
Civil and Environmental Engineering
Professor of Mechanical Engineering
Vanderbilt University
International Modal Analysis Conference, Orlando, FL
February 3, 2014

2
Nonlinearity,friendorfoe?
It is fair to say that nonlinearity
is viewed as a detriment by
most designers and
experimentalists…
…perhaps for good reason.

3
Exhaust systems have a number of flexible elements
that facilitate resonant response but only for
increasing engine speed – how is this possible?
Exhaust system subject to
powertrain excitation
Decreasing
Increasing
f(t)
Nonlinearity–foe

4
Nonlinearity–friend
“If you hit a tuning fork twice as
hard it will ring twice as loud
but still at the same frequency.
That's a linear response. If you
hit a person twice as hard
they're unlikely just to shout
twice as loud. That property lets
you learn more about the
person than the tuning fork.”
Neil Gershenfeld, MIT
Credit: Wikipedia

5
Anexample: Findingdamageincompositesstructures
Suppose you want to find a small defect in a large
structure. How would you go about finding it?
Bond, R., Underwood, S., Cummins, J., and Adams, D. E., “Structural Health
Monitoring-Based Methodologies for Managing Uncertainty in Aircraft Structural Life
Assessment,” Structural Health Monitoring, submitted for review.

6
Youcouldtryvisuallyinspectingthematerial
Is this sandwich panel damaged?
Polypropylene honeycomb core sandwiched between
two fiberglass face sheets.
Core cracking Disbond

7
Youcouldalsotrylinearvibrationmeasurements
λ=c/f
But local changes can only be detected using high
frequency vibrations (100 kHz  1 in wavelength).
The problem with high frequency vibration is that it is
heavily damped (σn=-2πζfn) preventing the effects of
damage from being sensed in large structures.
c=2740 m/s

8
Nonlinearity,friendorfoe?
Is there a different [nonlinear] way?
Perform a mental exercise: push down on the material
where there is damage and then pull back up again.
Eric Dittman, Douglas E. Adams
mage
mage
5
4
0
eing
F ig. 10 Example of motion in a disbonded panel that leads
to bilinear stiffness
Bending stiffness of face sheet
Bending stiffness of face sheet
+ Stiffness of core

Damagemechanismsarenonlinearinnature
9

10
Anexperimenttostudythesemechanisms
Sandwich panel on vacuum surface to emulate
single degree of freedom system.4 Eric
F ig. 1 Testing Apparatus of Disbonded Aluminum Panels
This form of the equation once again allows for easy
Table 1 Primary Resonances
Damage Size in mm Prima
Undamaged 1161
25.0 467.5
63.5 359.5
101 296.0
Table 2 Testing Envelope of t
Damage in mm Testing Fre
25.0 467.50
233.75
155.80
63.5 359.50
179.75
119.80
101 296.0
148.0
Dittman, E., and Adams, D. E., “Identification of Cubic Nonlinearity in Disbonded
Aluminum Honeycomb Panels using Single Degree-of-Freedom Models”, Nonlinear
Dynamics, submitted for review .
M
K?

11
Sourcesofnonlinearbehaviorincompositematerialdamage
The source of quadratic stiffness nonlinearity is
obvious, but are their any cubic nonlinearities?Eric Dittman, Douglas E. Adams
Values by Damage
3
) R2
0.9828
0.9975
0.8672
Values by Damage
s3
/ m3
) R2
0.8645
0.9884
0.9930
the stiffness being
Table 8 Quadratic Stiffness and R-squared Values by Dam-
cubic nonlinearity.
3 Establishment of a SDOF nonlinear model
In prior work by Dittman and Adams [6], a disbonded
aluminum honeycomb panel exhibited a strong nonlin-
ear responsethat indicates a quadratic stiffness nonlin-
earity was present. The response data also contained
evidence of a cubic nonlinearity, although this nonlin-
earity was not identified by the previous SDOF models
that were proposed. Two new SDOF models are pro-
posed to explain the behaviors of the system:
ü + 2εµ ˙u + εκu2
+ ε2
N u3
+ ω2
0u = F cos(Ωt) (1)
ü + 2εµ ˙u + εκu2
+ ε2
N ˙u3
+ ω2
0u = F cos(Ωt) (2)

12
Sourcesofnonlinearbehaviorincompositematerialdamage
When the disbonded face sheet is free to vibrate,
what are the effects of the strains in the adhesive?Eric Dittman, Douglas E. Adams
F ig. 11 Simplified cross-section of a honeycomb panel show-
ing how the epoxy fillet binds the facesheet to the core
cubic nonlinearity.
3 Establishment of a SDOF nonlinear model
In prior work by Dittman and Adams [6], a disbonded
aluminum honeycomb panel exhibited a strong nonlin-
ear responsethat indicates a quadratic stiffness nonlin-
earity was present. The response data also contained
evidence of a cubic nonlinearity, although this nonlin-
earity was not identified by the previous SDOF models
that were proposed. Two new SDOF models are pro-
posed to explain the behaviors of the system:
ü + 2εµ ˙u + εκu2
+ ε2
N u3
+ ω2
0u = F cos(Ωt) (1)
ü + 2εµ ˙u + εκu2
+ ε2
N ˙u3
+ ω2
0u = F cos(Ωt) (2)

13
Polymers
P =s A, s = Ee
If P- then A-
this causes true stress s to increase,
which means e¯ and thus d ¯ and
this creates a "hardening" spring
Nonlinearstiffness

14
Acloserlookatlinearspringswiththephaseplane
We can use phase-plane diagrams as a means of
studying free and forced vibrations.
Displacement
Velocity
Rolling disk on an incline:
M +
ICM
a2
æ
è
ç
ö
ø
÷x + Kx = 0
x2
1/ K
+
x2
1/ Meff
= Constant
Phase plane shape:
Stiffness is constant in
linear system

15
Acloserlookatnonlinearspringswiththephaseplane
We generate the phase plane by conserving the
sum of the kinetic energy T(v) and potential
energy V(x):
1
2
mv2
+ V(x) = C
OR
v = ± 2C - 2V(x)
M +
ICM
a2
æ
è
ç
ö
ø
÷x + Kx +mx3
= 0

16
Phase plane diagrams reveal how stiffness (and
damping) characteristics change with amplitude.
- Which phase plane diagram contains nonlinearity?
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
position
velocity
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
position
velocity
M +
ICM
a2
æ
è
ç
ö
ø
÷x + K +mx2
( )x = 0
Whatistheeffectofnonlinearstiffnessonthefreeresponse?

17
Plug x(t)=Acos(t) into Duffing’s equation to find out!
-w2
AM cos wt( )+ AK cos wt( )= -mK Acos wt( )é
ë
ù
û
3
K -w2
Mé
ë
ù
ûAcos wt( )= -mKA3 3
4
cos wt( )+
1
4
cos 3wt( )
é
ë
ê
ù
û
ú
Mx+ Kx = -mKx3
If the stiffness [and damping] can change with
amplitude, then what happens to the modal
properties of the system? Use harmonic balance…
Howdoesthistranslateintochangesinnaturalfrequency?

18
9 9.5 10 10.5 11 11.5 12 12.5 13
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Frequency,  [rad/s]
Amplitude,A[m]
w
K
M
1+
3m
4
A2
æ
è
ç
ö
ø
÷
Backbone curve
Free oscillation frequency:
Shootingatamovingtarget–modulationofnaturalfrequency

19
σ in Hz
-20 -15 -10 -5 0 5 10 15 20
0
Figure 2.1. Amplitude of the response of a cubic stiffness EOM
excited at the primary resonance vs. detuning parameter. Coefficient
values are ω0 = 12, µ = 1, F = 5, and κ = 1.
σ in Hz
AmplitudeofResponseatPrimaryResonance
-20 -15 -10 -5 0 5 10 15 20
0
0.05
0.1
0.15
0.2
0.25
If the ωn changes as a function of the amplitude of
response, how does the frequency response change?
Howdoesthistranslateintochangesinforcedresponse?
Tracks the backbone

20
Frequencyresponsedistortion
Nonlinear systems respond at frequencies other than
the excitation frequency, this results in distortion.
Storer and Tomlinson, Mech. Systems and Signal Proc., 1993
 
 
     
 
       

ooooooo
oooo
ooooo
oo
oo
o
HHKH
H
HHKH
H
CjMK
H






53,,,,
0,,,
3,,
term(bilinear)tynonlineariquadraticnoistherebecause0,
1
1
5
1
2
5
4
1
3
13
2
21






  tj
o
o
eFyyKyCyM 
  2
1
     
  





tjnn
oooon
tj
oooo
tj
ooo
tj
oo
n
o
ooo
eFH
eFHeFHeFH
tytytytyty




,,,
,,,
)()()()()(
33
3
22
21
321

21
Frequency [Hz]
H1()H2(,)H3(,,)
We can calculate and plot so-called higher order
FRFs, for example, for the quarter car model.
H2
-Peaks at half the linear
resonance frequencies,
indicating quadratic nonlinearity
H3
-Peaks at one third the linear
resonance frequencies,
indicating cubic nonlinearity.
   
 n
n
n
X
nY
H



1
2 

Higher-orderfrequencyresponsefunctions

22
Arenonlinearvibrationmeasurementssensitivetodamage?
When the panel vibrates at two amplitudes, does
the modal stiffness change or is it constant?
According to the modal stiffness
changes leading to a change in frequency response.
w
K
M
1+
3m
4
A2
æ
è
ç
ö
ø
÷

Damage introduces
nonlinear properties
u
F
3D vibrometer used to collect
frequency response data from panel
Nonlinear behavior in multi-amplitude
frequency response data used to locate
damage
200 300 400 500 600 700 800
0
0.005
0.01
0.015
0.02
0.025
Frequency(Hz)
Magnitude(mm/s/N)
Damage indices are created based on
the magnitude and location of the
nonlinear behavior
23
Usingamplitude-dependentnonlinearitytodetectdamage

24
Testonvibratingfiberglasssandwichpanel
Actuator Location (Front
Side)
Disbond Damage (Front
Side)
Measurement Area (Front
Side)
Fiberglass Panel
Measurement Area
(Back Side)
Underwood, S., Plumlee, M., Adams, D., E., and Koester, D., “Structural Damage Detection in a
Sandwich Honeycomb Composite Rotor Blade Material Using Three-Dimensional Laser Velocity
Measurements,” 2009, Proceedings of the Annual Forum of the American Helicopter Society.

25
Nonlinearityduetolocalstiffnesschangewithamplitude
Difference between frequency response function
measurement at high and low amplitudes.
5
10
15
20 1
4
7
10
-2
-1
0
1
2
x 10
-3
Grid X
Grid Y
Amplitude(mm/s/N)

26
Takingacloserlookatacommonsourceofnonlinearity
ML2
q +MgLsinq = 0
sinMgL 

MgLsin qo +Dq( )= MgL Dq -
Dq3
3!
+
Dq5
5!
-
æ
è
ç
ö
ø
÷@ MgLDq
qo
8 Eric Dittman, Douglas E. Adams
Table 8 Quadratic Stiffness and R-squared Values by Dam-
age Size
Damage (mm) Quadratic Stiffness (N/ m2
) R2
25 1,480*106
0.9510
63.5 460*106
0.9992
101 89.9*106
0.9125
ing nonlinearity. While previous work was able to sug-
gest the existence of a quadratic stiffness nonlinearity
and a cubic nonlinearity in a disbonded aluminum hon-
eycomb panel, the data and analysis results in this pa-
per have confirmed the quadratic stiffness nonlinearity
as well as determined that the cubic nonlinearity is a
stiffness nonlinearity. The quadratic stiffness nonlinear-
ity can also be described as a bi-linear stiffness. This
arises from the change in stiffness seen on the facesheet
as it rises off the honeycomb core and then descends
back to the core and presses into it, as seen in Figure
10.
This action creates two distinct stiffness regimes,
which can also be modeled as a quadratic stiffness. This
behavior is most noticeable when the system is excited
at one half the primary frequency. When the system is
excited at the primary frequency, the response ampli-
F ig. 11 Simplified cross-section of a honeycomb panel show-
ing how the epoxy fillet binds the facesheet to the core
fit is not as good as the damage size increases. There is
also a scaling factor that has not been accounted for in
this determination of κ, which is ε. From a comparison
of the values of κ, ε is around 0.5.
The physical behavior behind of the cubic stiffness
is not yet isolated. Since it is a cubic phenomenon, the
physical mechanism resists the displacement of the sys-
tem equally against the positive and negative displace-
ment. With the data fit strengthening as the damage
grows larger, it is thought that the epoxy may play a
role. As the damage grows, more of the epoxy fillets
that bind the facesheet to the core, seen in Figure 11
become exposed to the vibratory input. It is thought
that these exposed fillets provide the basis of the cubic
stiffness.
7 Summary
A single degree-of-freedom model has been identified
that dictates the motion of a disbonded aluminum hon-
eycomb panel when excited at the primary resonance
and thesuperharmonic frequencies. Using vibration data
coefficients for the equation of motion were calculated
using a least-squares method. The high R-squared val-
ueson thecoefficientsgivestrong confidencein both the
model and the coefficients. The disbond exhibits both
quadratic and cubic stiffness. These damage modes are
most easily seen when the panel is excited at one-half
and one-third the primary resonance, respectively. The
quadratic stiffness arises from the two different stiff-

27
Nonlinear systems detune their own resonance as
they respond with different amplitudes.
Pendulum resonates and slowly falls out of resonance
Effectsofgeometricnonlinearityonfreeresponse

28
he following form:
¨u + 2εµ ˙u + εκu3
+ ω2
0u = εF cos(Ωt)
hod of Multiple Time Scales introduces Tn asthenth
timescale.
the other times scales are orders of ε so that T1 = εt, etc. It is
ponse of u could be rewritten as u = u0 + εu1 and that the diff
ritten asoperatorsof theform d
dt
= D0 + εD1, where Dn is thede
sion with respect to Tn. By keeping only order ε0
and ε1
terms, E
tten as:
ε0
: D2
u + ω2
u = 0
sintroducesTn asthenth
timescale. Starting
orders of ε so that T1 = εt, etc. It is assumed
en as u = u0 + εu1 and that the differentials
orm d
dt
= D0 + εD1, whereDn isthederivative
keeping only order ε0
and ε1
terms, Equation
(2.5)
1u0 − 2µD0u0 − κu3
0 + F cos(Ωt). (2.6)
¨u + 2εµ ˙u + εκu3
+ ω2
0u = ε
TheMethod of Multiple Time Scales introduces
with T0 = t, the other times scales are orders of ε s
that the response of u could be rewritten as u = u
could berewritten asoperatorsof theform d
dt
= D0
of theexpression with respect to Tn. By keeping onl
(2.4) is rewritten as:
ε0
: D2
0u0 + ω2
0u0 = 0
ε1
: D2
0u1 + ω2
0u1 = − 2D0D1u0 − 2µD0u
Solving Equation (2.5) for u0 yields:
˙u + εκu3
+ ω2
0u = εF cos(Ωt) (2.4)
e Scales introduces Tn asthenth
timescale. Starting
es are orders of ε so that T1 = εt, etc. It is assumed
rewritten as u = u0 + εu1 and that the differentials
of theform d
dt
= D0 + εD1, whereDn isthederivative
Tn. By keeping only order ε0
and ε1
terms, Equation
0 (2.5)
− 2D0D1u0 − 2µD0u0 − κu3
0 + F cos(Ωt). (2.6)
Method of MultipleTime Scales introduces Tn as the nth
time
= t, the other times scales are orders of ε so that T1 = εt, etc
e response of u could be rewritten as u = u0 + εu1 and that t
erewritten asoperatorsof theform d
dt
= D0 + εD1, where Dn i
xpression with respect to Tn. By keeping only order ε0
and ε1
t
rewritten as:
ε0
: D2
0u0 + ω2
0u0 = 0
ε1
: D2
0u1 + ω2
0u1 = − 2D0D1u0 − 2µD0u0 − κu3
0 + F cos(Ω
Equation (2.5) for u0 yields:
n
t, the other times scales are orders of ε so that T1 = εt, etc. It is assu
esponse of u could be rewritten as u = u0 + εu1 and that the differe
ewritten as operators of theform d
dt
= D0 + εD1, where Dn is thederiv
ression with respect to Tn. By keeping only order ε0
and ε1
terms, Equ
written as:
ε0
: D2
0u0 + ω2
0u0 = 0
ε1
: D2
0u1 + ω2
0u1 = − 2D0D1u0 − 2µD0u0 − κu3
0 + F cos(Ωt).
quation (2.5) for u0 yields:
u0 = A(T1)eıω0T0
+ ¯A(T1)e− ıω0T0
is the complex conjugate of A.
uning parameter, σ, is introduced here. σ, with a value
o understand how theresponseamplitudeand phasing ch
y changes in the neighborhood of a known value. Upon
etuning parameter Ω = ω0 + εσ into Equation (2.6), thef
ed:
D2
0u1 + ω2
0u1 = (− 2ıω0A′
− 2µıω0A − 3κA2 ¯A +
1
2
FeıσT
− κA3
e3ıω0T0
+ c.c
ng γ = σT1 − β, the equations are written with the detuning parameter,
y as follows:
a′
= − µa +
F
2ω0
sin(γ) (2.1
aγ′
= σa −
3κ
8ω0
a3
+
F
2ω0
cos(γ). (2.1
se equations are examined in the steady state to determine the response
tation force. Instead of solving for a in terms of σ, the equation is solved f
ms of a, the response amplitude:
σ =
3κ
8ω0
a2
±
1
a
F 2
4ω0
− µ2a2. (2.1
sponse with respect to the detuning parameter σ is shown in Figures 2.1 a
Now,let’saddaforce–Methodofmultipletimescales
Credit: E. Dittman, 2013

29
Failureofthesuperpositionprinciple
written as:
ε0
: D2
0u0 + ω2
0u0 = 0
ε1
: D2
0u1 + ω2
0u1 = − 2D0D1u0 − 2µD0u0 − κu3
0 + F cos(Ωt).
quation (2.5) for u0 yields:
u0 = A(T1)eıω0T0
+ ¯A(T1)e− ıω0T0
s the complex conjugate of A.
ning parameter, σ, is introduced here. σ, with a value in Hz, allow
o understand how theresponseamplitudeand phasing changeasthefo
changes in the neighborhood of a known value. Upon substitution

30
values are ω0 = 12, µ = 1, F = 5, and κ = 1.
σ in Hz
AmplitudeofResponseatPrimaryResonance
-20 -15 -10 -5 0 5 10 15 20
0
0.05
0.1
0.15
0.2
0.25
equations are examined in the steady state to determine the response to
ion force. Instead of solving for a in terms of σ, the equation is solved for
s of a, the response amplitude:
σ =
3κ
8ω0
a2
±
1
a
F 2
4ω0
− µ2a2. (2.14)
onse with respect to the detuning parameter σ is shown in Figures 2.1 and
sponse exhibits a backbone curve, which indicates that the maximum am-
the response moves away from the primary resonance with increasing force
Shiftingresonance,multipleamplitudesofresponse

31
A linear system is the sum of its parts:
If y = f x( )
then f ax1 + bx2( ) = f ax1( )+ f by2( )
= af x1( )+bf x2( )
Linearsuperpositionprinciple

32
In the linear
equation of
motion for a
pendulum we
expect to see
one steady
state response
regardless of
the initial
conditions.
Not-So-Simple Pendulum (cont.)
3.7
The steady state response of this linearized pendulum with damping at the
excitation frequency is UNIQUE – it does not depend on what the initial
conditions are. Also note there is only one frequency in the response after
the transient decays.
0 1000 2000 3000 4000 5000 6000 7000
-1
-0.5
0
0.5
1
TIME[s]
THETA[rad]
0 1000 2000 3000 4000 5000 6000 7000
-1
-0.5
0
0.5
1
TIME[s]
THETA[rad]
t
q
t
q
Same amplitude
for different ICs
ML2
q +MgLq =t(t)
attraction for the alternative responses. This type of
primary resonance because it occurs near the linea
0 1000
-1
-0.5
0
0.5
1
THETA[rad]
0 1000
-1
-0.5
0
0.5
1
THETA[rad]
q
q
0.8 0.9 1 1.1 1.2 1.3
10
-2
10
-1
10
0
10
1
FREQUENCY[rad/s]
MAGNITUDE[rad] 0.8 0.9 1 1.1 1.2 1.3
10
-2
10
-1
10
0
10
1
FREQUENCY[rad/s]
MAGNITUDE[rad]
w
oq
oq
Linear
Nonlinear
Nonlinear Vibration –D. E. Adams, 7/21/04
Whatweexpecttoseein[linear]pendulum

33
In the actual
equation of
motion for a
nonlinear
pendulum we see
two steady state
responses when
we are just
beneath the linear
resonance.
ML2
q +MgLsinq =t(t)
Not-So-Simple Pendulum (cont.)
steady state response of this nonlinear system is NOT UNIQUE – it
s depend on what the initial conditions are; there are domains of
action for the alternative responses. This type of response is called a
mary resonance because it occurs near the linear resonance.
0 1000 2000 3000 4000 5000 6000 7000
-1
-0.5
0
0.5
1
TIME[s]
THETA[rad]
0 1000 2000 3000 4000 5000 6000 7000
-1
-0.5
0
0.5
1
TIME[s]
THETA[rad]
t
q
t
q
1 1.1 1.2 1.3
FREQUENCY[rad/s]
1 1.1 1.2 1.3
FREQUENCY[rad/s] w
Linear
Nonlinear
0 1000
-1
-0.5
0
THET
0 1000
-1
-0.5
0
0.5
1
THETA[rad]
q
0.8 0.9 1 1.1 1.2 1.3
10
-2
10
-1
FREQUENCY[rad/s]
MAGNIT
0.8 0.9 1 1.1 1.2 1.3
10
-2
10
-1
10
0
10
1
FREQUENCY[rad/s]
MAGNITUDE[rad]
w
oq
Linear
Nonlinear
Nonlinear VibrationD. E. Adams, 7/21/04
Whatweactuallyseeina[nonlinear]pendulum

34
Thin aircraft panels can experience sonic fatigue,
which is exacerbated by large deflections – why?
Credit: S.M. Spottswood, AFRL
y
K
Due to bending only
Stiffness
y
With stretching of neutral axis
Stiffness
Whymightthistypeofsteadystatesolutionposeaproblem?

35
Response at
mid point
Response
Excitation
Credit:
Increasing
amplitude

36
In a nonlinear system, we even see new
resonances because of the interaction of the two
components of the response (free and forced).
ML2
q +MgLsinq =t(t)
6000 6020 6040 6060 6080 6100
-0.1
-0.05
0
0.05
0.1
TIME [s]
THETA[rad]
6000 6020 6040 6060 6080 6100
-1
-0.5
0
0.5
1
TIME [s]
THETA[rad]
t

t

0 1 2 3 4 5 6
10
-2
10
-1
10
0
10
1
10
2
10
3
Frequency [rad/s]
MAGH()[rad/N-m]

A
o
Superharmonic
resonance
Nonlinearresonance–there’smore…super-harmonics

37
In a nonlinear system, we even see new
resonances because of the interaction of the two
components of the response (free and forced).
ML2
q +MgLsinq =t(t)
2400 2410 2420 2430 2440 2450
-1
-0.5
0
0.5
1
TIME [s]
THETA[rad]
2400 2410 2420 2430 2440 2450
-0.02
-0.01
0
0.01
0.02
TIME [s]
THETA[rad]
t

t

0 1 2 3 4 5 6
10
-2
10
-1
10
0
10
1
10
2
10
3
Frequency [rad/s]
MAGH()[rad/N-m]

A
o

Subharmonic
resonance
Nonlinearresonance–sub-harmonicresonance

38
Carbon Reinforced Fiberglass
• Young’s Modulus : 4.75~4.95 GPa
• Failure strength : 29.6~57.5GPa
Howcanthislackofsuperpositionbebeneficial?

39
Windenergytestbed
Chen
Fleeter
Kim, S., Adams, D. E., Sohn, H., Rodriguez Rivera, G., Vitek, J., Carr, S., and Grama, A., “Validation of Vibro-
Acoustic Modulation of wind turbine blades for structural health monitoring using operational vibration as
a pumping signal,” 2013, Proc of the 9th Intl. Workshop on Structural Health Monitoring, Palo Alto, CA .

40
Smartrotor
• 24 circuit slipring (12 chns)
• Installed in turbine nacelle
• Attached with a flexible
coupling
DC Triaxial
Accelerometers and
Macro Fiber Composite
Actuators
Anemometer
Optical
Tachometer
Shaft and Bearing
Assembly
Slip
Ring

Usinglackofsuperpositionfordetectionofcracking
Sensor
Actuator
crack
Structural vibration due to
wind turbine operation
Probing signal
from actuator
Sidebands
from crack
Pumping signal from shear &
bending loads
41

Quadratic damping
Linear damping
42
Most damping mechanisms are nonlinear but we
describe them using equivalent viscous damping.
Fluid drag damping:
0 KyyyyM  
Effective damping
Let’snotforgetaboutdampingnonlinearities

43
Oscillations take place in the vocal folds due to air
flow allowing humans to vocalize – how?
Human vocal system
Close-up of fundamental
voice production system
Credit: Lucero, TEMA Tend. Mat. Apl. Comput., 2005
Animportantclassofnonlineardampers

44
Oscillations took place in the Tacoma Narrows Bridge
in 1940 due to a similar phenomena.
Koughan, J., The University of Texas at Austin, 1996
Astructuralexample–themostinfamousoscillation

45
Oscillations take place in reed vales in reciprocating
compressors for similar reasons. Let’s look closer.
Piston in cylinder with
reed valve to manifold
Suction valve
Cylinder wall
Suction
manifold
Discharge
manifold
Discharge valve
Control volume
Piston
Amechanicalexampleinvolvingnonlineardamping
Park, Bilal and Adams, ASME JVA, 2008

46
Steady state response of the reed valve.
Acloserlookatthisnonlineardampingmechanism

47
A simple nonlinear damping law, in the Van der Pohl
oscillator, can be used to replicate this behavior.
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-500
-400
-300
-200
-100
0
100
200
THETA [deg]
a*(theta.
2
-b)[N-m-s]

Nonlinear
Damping
Coefficient
+ POS
- NEG
Nonlineardampingrestoringforce

48
Equation of motion for Van der Pohl oscillator:
Energy change per cycle (for a periodic motion) = 0:
q +c q2
-b( )q +wn
2
q = 0
˙q ´c q2
-b( )˙q0
2p
wn
ò dt = 0
Assume q = Acoswnt
cA2
wn
2
A2
cos2
wnt - b( )0
2p
wn
ò sin2
wntdt = 0

49
cA2
wn
2
A2 1
2
+
cos2wnt
2
é
ëê
ù
ûú´
1
2
-
sin2wnt
2
é
ëê
ù
ûú-
b
2
+
bsin2wnt
2
æ
è
ç
ö
ø
÷0
2p
wn
ò dt = 0
cA2
wn
2 A2
4
-
b
2
æ
è
ç
ö
ø
÷0
2p
wn
ò dt = 0
pcwn A2 A2
2
- b
é
ë
ê
ù
û
ú = 0
A = 0,A = 2b
So we conclude that oscillations take place with an amplitude of:
And these oscillations are stable if:
c > 0
A = 2b

50
Reference textbooks
Nayfeh and Mook, “Nonlinear Oscillations,” Wiley, 1979.
Davis, “Introduction to Nonlinear Differential and Integral Equations,”
Dover, 1998.
There are a few big-picture things to remember when
approaching analytical work in nonlinear systems:
- Nearly all methods build upon the response of the
corresponding linear system, i.e., nonlinearity = 0
- All methods require only trigonometry and solution of linear
differential equations (sometimes integration too)
- Must define the order of magnitude of nonlinear terms
relative to linear terms (damping, forces)
- Must solve for possible responses and then check the
stability of those responses (route to steady state)
Quickasideonnonlinearanalyticalmethods

LADSS – What You Need to Know…NLs, July 19-20, 2010

52
Nonlinearities are common in:
- Some materials
Plasticity, hysteresis
- Flexible structures
Geometry
- Structures with joints and fasteners
Friction and geometry
- Structures with preload with buckling
Multiple equilibria
- Structures that interact with fluids
Boundary condition
- Every structure (over large operating range)
aM

F

 E
Material properties
Physics (forcing)
Kinematics (motions)
Sourcesofnonlinearity

53
Nonlinearity is more difficult to understand
- We use linear modes to understand linear systems; we can
use normal forms to understand most nonlinearities.
- Algebra, trig, and linear differential equations are needed.
Nonlinearity is only important 5% of the time
- Squeak & rattle and brake squeal only occur for certain
operating conditions, but lead to serious warranty problems.
- Small changes in analytical or experimental initial conditions
can lead to major qualitative changes in dynamics.
Nonlinearity can be treated like noise in our data
- Chaos was believed to be random noise until 1961 but has
since been shown to be a nonlinear phenomenon.
- When systems are nonlinear, noise will never be Gaussian in
nature and we need to be extra careful about estimation.
2 1 1
cos cos2
2 2
t t 
Nonlinearmyths(Jackson,1989)

54
Nonlinearity is only important for large amplitudes
- The effects of Coulomb (dry) friction due to rubbing are most
apparent for small amplitudes of response.
- Nonlinear resonance phenomena are often quite nonlinear for
small response amplitudes (jump bifurcations).
- A common practice in experimental dynamics is to add dither
signals to the excitation to reduce the effects of nonlinearities.
Nonlinear analysis is pointless because no closed-
form solutions are available
- Qualitative solutions are available and are often more useful
for design purposes than closed-form solutions.
- Approximate methods to first-order accuracy are readily
available and accessible.
Refer to Jackson (1989) for other myths in nonlinear dynamics.
Nonlinearmyths(Jackson,1989)

55
Reference textbook
Worden and Tomlinson, “Nonlinearity in Structural Dynamics,”
Taylor & Francis, 2001
There are a few big-picture things to remember when
approaching experimental work in nonlinear systems:
- Record time histories, not spectra
- Do not average the data upfront, be careful of noise
- Sample quite a bit faster than you think you need to
- Remember that everything from the stiffness to the steady-
state response can depend on amplitude and initial
conditions
- Harmonic (or slow swept sine) testing is generally better
than random testing due to distortion
- Must sufficiently observe (spatially) nonlinear elements.
Experimentalguidelinesinnonlinearsystems

56
Acceleration response of
diaphragm with NL stiffness
Aliasing of 3rd order
harmonic
Must ensure that sampling frequency is high enough
to observe both the excitation frequency and the
distortion, i.e., nonlinear harmonics, to avoid bias.
Samplefastenough

57
Donotoverlooktheobvious(removetimefromyourplots)
Restoring Force Technique (or Force State Mapping)
Masri and Caughey, Journal of Applied Mechanics, 1979.
Haroon, Adams, and Luk, ASME JVA, 2005.
M2
x2
= F x1
- x2
,x1
- x2( )
x2
x1
xb

58
     3 1 3
cos cos 3 cos
4 4
o o ot t t      tt oo  2cos
2
1
2
1
cos2

Ordinary and partial coherence measurements are a
means of identifying sources of correlated noise.
Bendat, “Nonlinear Systems Techniques,” Wiley, 1998.
Donotoverlooktheobvious(usespectralcoherence)

59
Timedatahasalottooffer,e.g.,freeresponsesteprelaxation
Hilbert transforms can be used to identify modulation due
to nonlinearity as a function of free response amplitude.
Feldman, Mechanical Systems and Signal Proc., 1994
   
 1
h
y
y t y t d
t


 


    H
2 2
( ) ( ) ( )
( )
( ) arctan
( )
h
h
A t y t y t
y t
t
y t

 
 
  
 
03
 KyKyyCyM 

60
Acknowledgethenon-stationarynatureofnonlinearsystems
Time-frequency maps (spectrograms, wavelets, etc.)
are also used to visualize non-stationary signals from
the free and forced response.
     



  
deytwtY j
sp ,
x2
x1
xb

61
Frequencyresponseconceptsarestilluseful
When nonlinear feedback is considered, a new set of
FRF-like expressions can be derived that account for
nonlinearity in the feedback loop.
Adams and Allemang, Journal of Sound and Vibration, 1999.
Spottswood and Allemang, Journal of Sound and Vibration, 2006.
X (w){ }= HL
(w)é
ë
ù
û m1
(w)
é
ëê HL
(w)é
ë
ù
û Bn1{ }
mNn
(w) HL
(w)é
ë
ù
û BnNn
{ }ù
û ´
F(w){ }
-Xn1
(w)
-XnNn
(w)
é
ë
ê
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
ú
Nonlinear generating
functions
Nonlinear components
NL
0 1 2 3 4 5 6
10
-2
10
-1
10
0
10
1
10
2
10
3
Frequency [rad/s]
MAGH()[rad/N-m]
- Capable of MIMO NL modeling without
need for large number of excitations but
must sufficiently observe spatial response

62
Butbecarefulabouthowyoutreat“noise”indata
Which technique you use to estimate linear and
nonlinear parameters determines the types of noise
that can be accommodated.
+
knÁ ×( )
2
X F
A w( ) +
Reverse path
Method (Bendat)
- Noise on force
NIFO
Method (Adams)
- Noise on response
+
knÁ ×( )
2
F X
H w( )-
Haroon, M. and Adams, D., E., “A Modified H2 Algorithm for Improved Frequency
Response Function and Nonlinear Parameter Estimation,” 2008, JSV.

63
There are many other noteworthy topics and articles
for your consideration, including but not limited to:
Staszweski and Chance, Wavelets, IMAC, 1997.
Vinh and Liu, Modal analysis, IMAC, 1989.
Wright and Hammond, FRF estimators, IMAC, 1991.
Bedrossan and Rice, Volterra series, IEEE, 1971.
Strogatz, Nonlinear dynamics, Addison-Wesley, 1994.
And many many more…
Someotherresourcestoconsider

Nonlinear Structural Dynamics: The Fundamentals Tutorial

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