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NONLINEARITY IN MODAL AND VIBRATION TESTIN G

Norman F. Hunter, LANL, ESA-WR

International Modal Analysis ConferenceFebruary 3-7, 200 3Orlando, Florida

o Los AlamosN A TIONAL LABORATOR Y

Los Al amos Nati o nal Laborat o ry, an affir m ati ve action/equal opportu nity e mpl o yer, iso pe ra t ed by t he Un ive rsity o f Cali forni a for th e U .S.Department of Energy under contract W-7405- E NG-36 . By acceptance of this article, the publisher recognizes that the U .S . Governmentre ta ins a nonexc lus ive , roya l ty-f ree lic ense to p ublish or reproduce the p ubli s hed for m of thi s co n t ri buti o n, or to al low othe rs to d o so , fo r U. S .

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Nonlinearity in Modal and Vibratio nTesting

Norman Hunter

Engineering Sciences Divisio n

Weapon Response Group

Los Alamos National Laboratory

2/ 1/2003 Nonlinearity in Modal and VibrationTesting

Abstract

This set of slides describes some aspects of nonlinear

Vibration analysis thru use of analytical fromulas and

Examples from real or simulated test systems . TheSystems are drawn from a set of examples based on

Years of vibration testing experience . Both traditionaland new methods are used to describe nonlinear vibration .

2/ 1/2003 Nonlinearity in Modal and Vibration 2

Testing

What is Nonlinearity?

Linear Differential Equation : Mx" + Cx' + Kx = F

Nonlinear Differential Equation : Mx" + Cxi2 + K(x + ax 3= F

•Superposition Fails•Linear Frequency Response or Modal Model Does notAccurately Predict Measured Response .•Non-Gaussian Probability Density for Gaussian Excitation .•Coupling Between Frequencies .•FRF Changes as a Function of Excitation Type or Level .

2/1/2003 Nonlinearity in Modal and Vibration 3

Testing

Sources of Nonlinearity

• Joints (Microslip and Macroslip) . Hystersis .

• Large Deflections (Geometric Nonlinearities) .

• Material Nonlinearities (Foams, Viscoelasti cMaterials) .

• Friction .

• Loose Parts (Rattling) .

• Damage (Cracks) .

2/1/2003 Nonlinearity in Modal and Vibration 4Testing

How Nonlinearity Effects SystemResponse

• Time History

• Auto spectrum

• Frequency Response Function .

• Coherence

2/1/2003 Nonlinearity in Modal and Vibration

Testing

Nonlinear Effects on a Sinusoidal TimeHistory

Pu re Si nu sold and Slnusoid with Ha rm onic Distortio n

c~

E4

Time In S econ ds

25% Harmonic Distortion

Pure Sinus ol d and S inus ol d with Harmonic Distortio n

~

nE

100% Harmonic Distortion

For sinusoidal excitation harmonic distortion indicate sA nonlinearity somewhere in the system .


oi 02 e 3 e~ os n a ofTi me in Se co nd s

Nonlinear Effects on a Random Tim eHistory

Response of Linear Oscillato r

~ o.

CO

Q -0

Linear SDOF Oscillator

R esp on se of Polynomial Oscillato r

C7C

CO

Polynomial SDOF OscillatorAcce leration Response Acceleration Response .

2/ 1 /2003 Nonlinearity in M oda l and V i bra tionTesting

7

a i 02 oo a . o s as er ol asTime in Second s

I- ~J150 oi 02 o~ 04 as o a or o ~ a f i

Time i n Se c o n ds

Nonlinearity Effects on theAutospectrum

Autospectru m, Linear Ten De g re e of Free d om Syste m

10

16d

aE

ro

u 10

10

10 ~

i o io i o'F requency In Hz.

Autospectrum ofLinear TenDOFsystem.

A ut os pec trum, N onli near Ten Degree of Free d om Syste m

aE

V 10 "

10

10

io ,o

F re quency in H z

Increased HighAutospectrum of Frequency

NonLinear Response

TenDOF system .

2/ 1 /2003 Nonlinearity in Modal and Vibration 8Tes tin g

Nonlinear Effects on the Autospectrum

Linear Amplitude0 .5 _ Drive 'Po int. ~ . .

0.4 3~

~~0 . 3

0.2

0.10 0

0 2 2

0 4 0 4: ' --~

100

6 Location 100

' 6200 -200 LocationFrequency 300 8 Frequency 300 8

8 DOF System ,Linear 8 DOF System Rattling at Location Increased High Frequency

8 . Response


Nonlinear Effects on the FRF

F R F of I i nrar Tr n A n f Sv civ m

dv

o+

LL

Linear TenDOF System

, o

,o

v

C

6L

10~

1 0 ~

1 0 ~

~. . . . . ~• '~

•.•

~~• ~~

♦ ~

' ~~~~•

Frequency in Hz .

Linear TenDOF SystemNon-

FRf of Nonlinear Tendof

Increased Damping~ Additional Modes

Noisy FRF,More HighFrequencyResponse


Testing

10, 102

Frequency In Hz.

Basic Nonlinear Effects .

Time History - Harmonic Distortion . Non Gaussian response toRandom excitation .

Autospectrum- Broadens the Autospectrum By AddingFrequencies through frequency interactions . Often visible asIncreased values at frequencies above drive range .

Frequency Response Function- FRF peaks are lower, implyingIncreased damping. FRF is noisier.

Coherence - lower, especially at higher frequencies .


Testing

Locating Nonlinear Elements

If the nonlinear behavior generates high frequencies ,these may be greater in amplitude near thelocation of the nonlinearity .

2/ 1 /2003 Nonl ineari ty in Modal and Vibration 1 3

Tes t in g

.

Functions Designed to Quantify NonlinearBehavior

Amplitude Probability • Amplitude Probabilit yDensity .

Higher Order Spectra . .Dens ity.

B icoherence

• Time or AmplitudeDependent Models .

• Wavelets .

• State Space Models basedon Nonlinear DifferentialEquations .

. TriCoherence .


Testing

Probability Density Function

n nSample Filtered Random Time History

T,df (x) = f n in range x- dx to x+ dx

Probability that the magnitude of x at a randomly chosen time

lies between x -Ax and x + Ax limit as Ax 0X+ex

i°(x) = f pdf (x)dxx - dz

2/ 1 /2003 Nonlinearity in Modal and Vibration

Testing

Num ber Of Occurences ina mpli tude b and /to ta l

1 5

., . . . . ., .~ .. . . . .. v.~ v.v

Time in Seconds 0 0.5 1 1.5 2 2.5 3PDF

Probability Density

b

Pa<X<b = px xa

a < b

p(x) is the probability density of x .

P- 00 < X -< 00 --+ 00

p_ 00

)lx=1.O~ - --- -- - - -- --- -- -- -, -- -Area under the Probabili ty D ens ity is unity, whi ch is just away of saying that for sure the va lue of x is somewhere.


Testing

Moments of a Density

Mea n

StandardDeviatio n

Skewness

- 1 N .x =

I,

N j=1 ~.

1 NQ = ~ --x1~r_ 1 ~ = 1 ~

skewness = 1 N xl -z 3

IN j - 1 or

std =FLAII

ForGaussian

It

Q

zero

Kurtosis N x . -x 4 SO = F~96zero

kurtosis - 3N j -i or


Testing

.~ ~o .o>

o ~005

0,04

a~0 oz0 0,

~

2

2

,

.E■I■ 11,--

Histogram Estimate1,048,576 points ofrandom Noise .

y=randn(104 8576 , 1) ;

>> [n,xout]=hist(y,128) ;

>> bar(xout,n/ 1024)

>> xlabel('MAGNITUDE')

>> ylabel('PDF')

»

2/1/2003 Nonlinearity in Modal and Vibratio n

Testing

Estimating Probability Density

Histogram Estimate1,024 points ofrandom Noise.

1 8

3035 ~ -~ - - - - -

The Kernel Estimator

h k( x hX' l whe rel 1

h - window width .

n - number of data points .

i - point index

X , - Kernel Location

x - data point v alue.

Kernel K f k(x* - 1 .0

PP-

The kernel estimate at x is a scaledsummation of the points within thekernel, weighted by the kernelvalue at the point location .

Magnitude


Testing

Kernel Density Estimate, Linear ScaleGaussian Random Noise

0 . 4

0.3 5

0 .3

0 .25

0 . 2

0 . 1 5

0 . 1

0 . 0 5

C

0 .4

0 .35

0 . 3

0 .25

0 .2

0. 15

0 . 1

0.06

2~ 0 PO INT S

MA G N ITUOE

nts


Testing

,a ' .,o, 1

,0- 4

Gaussian Probability Density, Log Scale

io-,

~10-5 -4 -3 -2 -l O 1 ? 3 4 S

Ke rnel Density Estimate plotted on a Logarithmic Scale forGaussian Random Noise .,o" . . _ -- . - - - . - -. . - -- ,

„„ >76 Points


Testing

~_ . ~ . . ~

Estimated Probability Density, Linear Single Degree of Freedom SystemDriven by Gaussain Nois e

o~Ske ixness=0 .0078

03,

Var- =0 .0170 .25

Kurtosis=-0 .1165 ozVariance Kd=0 .035 O1 5 !,

o, -

0 os0L

,o-,o ,o,o1 0

10 ' .a

1 O ' •°

1 O °O

n is Gaussian,is estimate d)nse .

Estimated Probability Density, Linear and Log Scales, Response of linear Single Degreeof Freedom System .

2/1/2003 Nonlinearity in Modal and Vibration 22Testin g

-4 -2 O 2 4 B

ANALOG NONLINEAR OSCILLATOR

1 0'

10

~~ 10zw0~J 102c o

co0

d 10 3

10 4

1a- s-2

P DF o f A SC I N

Kurtos is= 1 .98 std=+/- 0 . 11 9

Cres t= 2 . 999

:?/ l /2003

1 01

10

>- 1 O-~

Fnzw0

~ 102J

0cl : 10-3d

1 0 4

1 05

PDF of ASCOUT

Kurtos is=3 . 09 std=+/- 0 .2 939

Crest= 4.61Nonlinearity in Modal and Vibratio n

Testing2 3

MAGN I TUDE

Skewness=-0 .2026 std=+/- 0 .05 1

-1 0 1 2MAGNITUDE

Skewness= -0 . 000 1 5 std=+/- 0 .0014

Higher Order SpectraSecond OrderSpectra

HigherOrderSpectra

Hxx((o) = X (w)X *t(o)

Hxy w )=Y(w)X*(w)

Power Spectrum

Cross Spectrum

H(cvl,`0 2, W1 + w2)= X

(0)JX

(0J2

)X

(cvl + (0 2 ) Autobispectrum

Y( W1 l~2 9 . . .C~N) Cj 1 +(. j2 + . . .+ ~N1 =XrilX(~21X(~31 . . .X(~1 + ~2 + . . . ~N~ 1

(0j1 ~ 1 ~ 1 ~ )

Autonspectrum

Higher Order Spectra have a number of variants depending on theCombination of X ( Input ) and Y(response) terms on the Right handside of the equations .


Testing

Higher Order Spectra

IC164coherence,

Y2 (CO l'C-0 2'Co

1 + 2 J

[E(X(w1 )x(w2)Y ( Col + C0 2

))]2

[E(X(w i)x(w2))]2[ ~ 1 2

E(Y 12 ~IJ

0 .0s y2(cvl'cv2'cvi + 2 )

S 1 .0

Unity for complete quadratic dependence , zero for no quadratic dependence

2 \

[E(X(w1 ~X (`~'~ }X( .̀~'~ ~X ( 3 .̀~' ,))] 2

Y ~1 ' ~t' ~~ 3~~ 1 2

E

(

x

('C0j

X ~

C0j

X

(C0j)

][E(X(3w

i

))

]

2

0 .0 s y2 (cvI

, cv, , cv, , 3cv' ) S 1 . 0

2/ 1 /2003 Nonlinearity in Modal and Vibration 25Testing

A Simple Example of Hi gh er Order Spectra

105

~'a

105CF)ca

1p1o

101510~

S 1 =sin(2*pi*20*t)

S2=sin(2*pi*30*t)

2/ 1 /2003

101

10

Frequency

Sout=sl .*s2

Nonlinearity in Modal and Vibration

Testing

103

26

PSD Green Two Sines , Red Product

Bicoherence of the Product of Two Sinusoid s

Bicoherence of Sine Produc t

0 . 5 1 . . .~•~•~~~~' ~

0600

200

Frequency F 2

2/ 1 /2003

0 0

PeaksCorrespond toQuadraticDependence .

100 200 ---

Frequency F1

400500

Blue lineindicates aplaneof symmetry

---~-~-~-~~~ -

~` -- -

10 20 30 40 50 60 70 80 90 1 QO

[E(X `~, )4602MC91 +~z))]2

Y21~1+ 2 2[E(X(w)X(w))]2[E(Y(wi + ~2 l

)

]

Nonline arity in Modal and VibrationTesting

The 45 degree line isa line of constantf1 +f2 .

100

90

80

70

60 ,

50 ~- ~

4

0 30

2010 0

27

BICOHERENCE EST IMATED FORINDEPENDENT SIGNALS.

0 . 1

0 .06

0150

Gaussian Random Noise, 153 blocks, 128 Points Each . Bicoherenc e

Estimate .

Bicoherence, Random Noise, 153, 128 pt . Block s

These levels are based on averaging out the

. . . effects of unrelated signals , more blocks ofaveraging means a lower level .

80


Testing

rrequency r1 0 p Frequency F1

STATISTICAL ERROR IN THE BICOHERENCE

0 .K' . . . . . .

0 . 1 ~ . • ' ,

O . Ofi . . ~ ~ . • .

O . Ofi

. • I0.04

0 . 02

] .. . .. . .

1517

100 1 5C

100

OA25 . ~ ~

0 .02 . . ~ . .

0 . 015 ~ . .

0 0 1

0 005 ~ •

0 .. - `150

150

50

0 0

75 Blocks of 256 Points each .

50

387 Bloc ks of 256 Points ea c h .


Testing

Bicoherence, Acceleration of Mass 1 0

Tendof Sys tem

o, . . . . - ~ . ._ . . ._ . . . _aoe . . . . . --- ~- - .006 . ~ . . . . . . . .004

002 ._. ~~ . _ _ . .-400 _ - ~ - -

250~-

200

150150

50 6O

o, ~ ~ . . - . . . _ . . . . . - : .o 043 ~

_ ~ . . . . ~0.06

o 04

002o

zso ~zoo

so

200

250

250

200

linear

bilinear2/1/2003 Nonlinearity in Modal and Vibration 30

Testing

Bicoherence Example, QuadraticallyRelated Signals

08eeo .02

. r. . . .. .. .

7.1 h To

~r.~3DO

p

T=!L 0 : 32 767] /1 000 ;

x =1).OS*randn(32767,1)+(sin(2*pi*50*t) .*exp(-t / 10))' ;

y=0 .07*randn(32767,1)+x . ^2 ;

[bc,ff,rndout] =bicoc 1(x,y,1000,512) ;

me .sh(ff,bc)

Cantour(ff,bc,20); 31

Methods Reviewed

40 Auto spectrum

40 Frequency Response Functio n

* Coherence

• Amplitude Probability Density (Skewness,Kurtosis) .

• Bicoherence, Tricoherence (to be illustrated) .


Review Some Systems

« Linear Ten Degree of Freedom Oscillato r

N Bilinear Ten Degree of Freedom Oscillator .lip Eight Degree of Freedom System (Linear and

.nonlinear) .

2/ 1/2003 Nonlinearity in Modal and Vibration 33Testing

M10K9,10 C9,1 0

M9K89 C89

M 2K 12 C12MI

Ten Degree of Freedom System

Linear Case : k45 = k ref

Bilinear Case: x4 - xs < 0

k45 = 4 .Ok ref

x4 - xs > 0

k45 = 0 .25 kref

Loss of Stiffn ess in Te n si o n


Ten Degree of Freedom Linear System

x 10-6s~

6

4

2

0

-2

-4

-6

I ! ~ ,, II~ Jl ~b I ,

J } +I j i

I.I' ~

; M

-8 L0 1 2 3 4 5 6 7 8 9

Acce leration Respon se of Mass 1 0


Testing

Tendof Linear

E

io

10

Frequen cy1 0

Frequency Response FunctionForce to Mass 10

cd

0U

Coherence Function, Force toMass 10 .

C ohere nce, Li n e ar Te n D e gre e o f F reed o m S y stem

.~_ . - . .' O


10 1 0

Fre quency

L inear Ten Degree of F reed o m System

1 d

id

41 0

010~CL

1 d

id

-1

Ten Degree of Freedom Linear System Mass 5 Response

Skew ness= 0 . 011 1Std=0.044

Kurtosis= 0 .0477Std=0.044


0 1Response Amplitude X 1

ps

Tendof Linear

Bicoherence,Tendof, No Bilinear Spring

0 . 08 . . . .~~~~' :

0 . 06 •

0 . 04

0 . 0244

0

:

600

200

Frequency F2 0 0

400

200 300

100Frequency F1

9

450

400 ~

350

300

250

200

-

150-

100 L Q; - o

50

p 0 100 200 300 400

2/l/2003 Nonlinearity in Modal and Vibration 38Testing

Imaginary Part of Ten Degree of Freedom System TransferFunctions .

x 10-5 Imaginary Part of Ten DOF System Transfer Function s

1

1 0

Locatior

Log of Frequency in H z .

2 . 5

Testing

1 ~ .~0 .5

R=radius .Shorter radiusMeans increase dDamping . ~

0 . 90=ang;le , increasin gAngle means increasing 0 .8

Frequency. 0.7~

~ 0 . 6~Q 0 . 5z

0 0 . 4

0 . 3

0 .2

0 . 1

n

Ten Dof Linear

LINEAR TEN DOF SYSTEM

xk

xx

x

x

r

0

~t

0 0 .2 0 .4 0 .6 0 .8 1REAL PART


Testing

Acceleration Time History ResponseIODOF Bilinear Oscillator

x10 6

4

2

Cu

r. .~~ 0

E

-2

4

Acceleration Response of Mass 10 , Bilinear System


Testing

60 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 . 9 1

Time in Seco nds

Tendof Bilinear

Frequency Response Funct ion of Mass 10, Bili n ear Syste m

v

aE

0 .

0 .

o ,

l , 0 .

c

CUo .

L0

U

o .

o .

0 .

a

Frequenc y

1 Transfer Function Coherence


10' 10 10 3

Frequency

Coherence Functi on of Ma s s 10 , B ilinear System

Autospectrum, Linear and Bilinear Tendof Syste m

-3o

-35

-40

-45

5050

150

200

300 0

Linear System

Distinct humpat location .~-5

-35

-40 -- - - _ ,~ .

-45 ; , \\

-5050

100

150 - - - -

10 200 86

250 4

300 0 2

Bilinear System


Testing

10

Tendof Bilinear Bicoherence, mass 5 response

Bicoherence,Tendof, Bilinear Spring

0 .1 . . ~ . . .

0 .0$

0 .06

0 .04

, . . . . . . . . • ~ ,

. . . ..- .~:.~ ~ .~~4~ ~~~

0.02: ' •. ~~' ~ ~\a 'R\~

600

400

200

Fre quency F2 0 p

~•~~~' 400 500300

200100

Frequency F1

300

250

200

150

1 00

50


00 50 100 150 200

1BILINEAR TEN DOF SYSTE M

0 . 9

0 . 8

0 .7

0 .6

0 - !577(D 0 .42

0 . 3

0 . 2

0 .1

0 L

0 0 . 2 o_4 0.6 0.8 1REAL PART


Testing

Local Mode Shapes-Linear and Bilinear Ten DOF System

1 .5

1

0 .5

0

-0 . 5

-1

0 2

2 .5

2

1 .5

1

0 .5

0

-0 .5

- 1

-1 .5

-20


4 6 8 1 ( 2 4 6 8 10

Eight Degree-of-Freedom System

s=^ '. ¢ Lr• ~n' • ~'

• •~ ~ ' • }.sw ~~~

.

_ ~ • • . w •.

y~

Chuck Farrar , Bill Baker , Scott Doebling Los Alamos ,2/1/2003 ESA-EA Nonlinearity in Modal and Vibration 47

Testing

R721 u7a, d7 a

R721 u7a

~

f' R721d7a

102

10 '

100

10 - '10' 102


Testing

Kurtosis as a function of Bumper Locatio n

12 ---~-~

--------- . . . . . . - - - - ----~,~10 - ~ ~~/ --- ----- --~- ;~- - - ,~--- ~--- . . .--- - .-------- - --

.s --- ~ .

..__ - - - . .-- -,- ---- ---~

- -- - --- -- -- . . -- ~=~~ -:_ ----- .-- ~- ---~ A

7 ---- -- - --- - : - - -----

6 <- . . . . . ----------- - r

s . . . . . . . ---~----

4 ---- ._ . '~~ . .

3 k : . . . . . ~ - .-------- ~i`~ . _=- -

2 ~ _ .------- ; -- >------ - --------- - ---------- - ~ --/

1 1 ~ ,- _ I _ ~ I r,

1 2 3 4 5 6

'--~---_~

-------- ;-------

-- - . -- . . . . . . . . . . . .

--------------~-- . - -- ~- - ----~--. -- . -- - - - - - - - - - - -.

-- - ---C - -- .---~-- --- ---_

7,7 8

Bumper at 1 -2 ,Cases 11 -1 2

Bumper at 4-5,cases 9-10

Bumper at 7-8 ,

Cases 7-8

No Bumper

cases

1-6


1

u_0 1a-

1

1

LL0 1IL

1

1 1-30 -20 -10 0 10 20 30 -30 -20 -10 0 1 0

Magnitude Magnitude

No Bumper Present Bumper Present

Skewness-0.121 Std=0.158 Skewness=0.768 Std=0 .161

Kurtosis= 0.277 Std=0.084 Kurtosis= 0.644 Std=0 .11 22/1/2003 Nonlinearity in Modal and Vibration

Testing

20 30

50

_1 R721d7a Mass 8 PDF1 "

_~ R721 u7a Mass 8 PDF1^

8D OF Mode Shape 1 vs . TimeNo Bumber

Shape 1 R721U7a

2

~ . ~ . .

0

. ' .-1 .

. . ' , ~ .

-2

1 . ~ .

20 .54

0 6

Time in Seconds Lacation


8DOF Mode Shape 1 vs . TimeBumber

Shape 1 R721D7a

3

0

-31 .~

0 .5

0 g 6

Time in Seconds

4

Loc a tio n

2D

2/ l /2003 Nonlinearity in Modal and Vibration 52Testing

1 ~ _ . . ~ .

-5-,o

-, 50

100

200

300 0

5

0 -5

-, o

100

200

2/ 1 /2003 2300 0

46

Autospectra, 8 Dof System

No Bumper

5 .

0

-5

8

0

100

200 E

300 0

Bumper at 7-8

No Bumber

86

4 Nonlinearity in Modal and Vibration

Testing

53

8

. l-~~

1 00

0

-5

-10

-1 50

88

Nonlinear , Bumber atLocation 4 -5

300 0

Linear or Nonlinear?

Bumper Location ?


Autospectra, 8 Dof System

Frequency Response Function 8DOF System

Imaginary PartTransfer Functions, BDOF, R72 1 U7A

1 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :. . . . . . . .

:. . . . . . .

:. . . . . . . ,

. . ~ . . . . . . . .~ .100

. . . . . . . . . . . . .

50

0

-50

-100

-15010

. ._ . ~ . . .

. . . . . . .. ~ ., . . .. . . . . . . . . . . . . . . . . . . . .••~r • '

. . •• ~ F ~

, . . . . ~ ti • ~ ' a~,

, ~ . ~ • e~~~` . . . . . . . .~t; - . ' i , ~ • ' .

. . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . , . . . .V, -. . . .. . . . . . . . . . ` . . .

. . .

Location00 10 20 30

Frequencv

40 50 60 70


Testing

Comparative Frequency Response Functions, 8 Dof System .

BumperBumper at .4-5410 at 1-2 ,~ .51 2a0

_2

-5

-4

-10 100

100 150

8

8 250 42006 2

250 4 300 p

300 0

,o , - ~ Bumper,o _ 5at 7-8

5 No 'Bumper

H 0

-5

-10 -10ioo 1

00150

8 150200 6

250 4 200 82 6

2/ 1 /2003 300 ° Nonlinearity in Modal and Vi Sb°rati 2 4 56Testing 300 0

8DOF BUMBER FrequencyResponse Function vs . Time

Shape 1 R721 D7 a

10

a

-10

1 . 5

" . . ~ . ~~: ;~~~",r~~.• . .. . . .,. .. ..~° - s:t

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