Paper 88-2230 AIAAASMEASCEAHS 29th Structures Structural Dynamics and Materials Conf Williamsburg VA April 18-20 1988

DYN11MIC CEARACIERIZATION OF SlRUCIURES fJf PULSE PROBIN3 AW DECONVOUJIION middot

A s Carasso E Simiu

National Bureau of Standards Gaithersburg Maryland

Abstract

This paper outlines the mathematical and carputaticmal basis of a procedure for identifying the dynamic bebavior of linear structural systEllS from measurements of transient responses to specified pulses The dynamic behavior of such systems is fully characterized by dynamic Greens functioos which can be reconstructed by deccmvolutioo from response measurements The deconvolution involves the solution of an ill-posed integral equation and can be perforned by Caisidering an associated Cauchy prcblem for a partial differential equation in which the measured response is used to define the initial coodition The explosive grcwth of errors due to caitaminatjon of the response by noise is prevented by regularizing the prcblem so as to minintize a Tiklaiov functiCllal It is s00wn that infinitely divisible pulses which include as a particular case the inverse Gaussian pulse have properties which make it possible to perform cootinuous deconvolutioos and to ootain bourds on errors induced by noise The effectiveness of Uie proposed algorithm for reconstructing Greens functions is demonstrated in exanples involving dispersive and IlClldispersive matbers ~a dispersive structural network that may be representative of a large space structure i

ltrnenclature

wave prqiagation velocity (Fq coefficients in Fq 24 response to pulse excitatioo response contaminated by noise filtered response data bending stiffness dynamic Green e function Greens function with error due to noise shear stiffness Heaviside unit step functioo torsicmal stiffness length

m positive integer M bending mnent

~t) mass of lrubnoise

p(t) pulse s variable in Laplace transfonn darein t tine T torsional natent v shear force w deflectioo x space coordinate

29)

Research Mathematician Center for Aplied Mathetatics NBS FelltM Center for Building Technology

YI state variable vector

~ parameter defining noise distribution 6(t) Dirac delta function f positive coostant (Fq 13)

argurent in solution of wave equationmiddotr constant defining hysteretic danping

8 angular deformatiai due to torsion gt eigenvalues

constant (Fq 14)~ parameter in Fourier transform spsce pA mass of beam per unit length pI mass narent of inertia per unit length igt1J mass nunent of inertia about matber axis ti parameter in invetse Gaussian pulse T tine parameter (Fq 1)

angular defornation due to bending regularization parameter (Fq 15)

Introduction

It has been pointed out that qperating criteria governing autanated functions of large space structures such as directional orientation ~ figure main~e can be met only with distributed active caitrols bull For the controls to be effective accurate infornatian on the dynamic resmicrogtnse of the structure must be available This is also true of other flexible structures subjected to active controls (eg robot arms with noncollocated actuators and sensors2 ) The identification of dynamic prqierties is therefore essential in such applications

ltMing to the difficulties of earthbound dynamic testing and to the possibility that nechanical properties will change in flight as a result of envircnnental effects it is desirable to test the structure dynamically in service (on orbit) Dynamic testing can be perfomed by exciting the structure harmonically over a range of frequencies and xecovering the natural frequencies of vibration and the modal danping paraneters fran Uie analysis of the correspooding set of steady state tesmicrogtnses However under service corrlitions this metlcd might be impractical r it is desirable to extract the dynamic information befog sought in one fell swocp Since the dynamic Gteen s function fully characte1 izes dynamic behavior this can be done by exciting the structure wiU1 an apprqgtriate pulse and extracting the dynamic Greens function of interest firn1 the fOriespanding transient response

This paper sumerizes recent ~ current work performed at the National Bureau of Standards caxerning mathematical ~ carputational aspects of such a procedure Reconstructing the dynamic Greens function fratl the measured response to a

ou

01

Otamp

0 01 Ol 03 04 05 Tlflll (SllCOtOS)

Flg l Inverse Gaussian Pul se

The Laplace transform of Eq 7 is

l[p(rrt)]=exp(-rrsl2) ResgtO (8)

Therefore

p(01tl p(02t) = pcr1 + 02tl (9)

and we can write Eq 2 in the fonn

b(t) =p(crt) g(t)

= p(__ t) [p(__ t)J(m-l) g(t) (10) m m

pulse entails the solution of an ill-posed integral equatjcm in the presence of noise We first describe bdefly the mathematical basis of a~ algorithm for dgttaining such a solution To check its effectiveness we apply the algorithn to noisy synthetic response data for structures woose exact dynamic Greens functicms are knom These structures include a dispersive mecUum consisting of a Tfocshenko beam a nandispersive medium consisting of a sljder with elastic defometions due to shear strain and a net1JCgtrk consisting of torsianal metlgters and TiJl)shenko beams It is slnm that the prqiosed algor]thn results in reconstructed dynamic Greens functions that for practical purposes closely awraximate their exact coimterparts

We note that research is also being ccnlucted at the National Bureau of Standards on experimental aspects of the procedure described earlier Currently ~rk is proceeding on the develqnent of a spaceshyrated actuator capable of producing matharetically specified soort rrechanical pulses to be used for dynamic system identificaticn purposes

Inverse Gaussian PuJ se Prebes and Decaivoluticn of Respcmse

The excitation denoted by p(t) must be a srooth causal pulse of srort duration which - to an appr(l)riate scale -- satisfies the relation

ir)dT l (1)

We may refer to p(t) as a probability density function

The response to the pulse p(t) I

b(t 1 p(t - T )g(T)dT 0 ( t ( 00 (2) 0

will be in general a highly distorted image of the Greens function g(t) with the high frequency carponenls aroothed out (This will be illustrated by exanpl es presented in the sequel bull ) The practical problem is to reconstruct the dynamic Greens function g(t) from the measured ~ response ~(t) since the respcrise b(t) that ~uld have been recorded in the absence of noise is unknom Little is knom about such prdgtlans in general However when the p(t) is infinitely divisible (ie for each int~oger in p(t) is the m-fold COlNolution of another prdJability density defined on the positive axis) such prdgtlans have been intensively st~islt3 recently both matlanatically and carputatiooally bull bull

We first assume that the response is not ccntaminated by noise To dgttain g (t) fran Eq 2 we consider the Cauchy prdgtlem for the fellaring linear partial differential equation in x t

x gt 0 t gt 0 (3)

u(x0) =0 x gt 0

u(Ot = g(t) t gt o

the Fourier transform of the causal signal f (t) being defined as

l i g[f(t)] = f m = --- f(t)e-i~t dt (4)

i 2r 0 I i

Fourier transfonretion of the first of Eqs 3 yields an ordinary differential equation in u(x~ ) woose Solution by virtue of the third of Eqs 3 is

u(xU = ~p(~)]xg(0 (5)

We take the inverse Fourier transform of Fq 5 in which we set x = 1 The cawolution theoran then yields

I

u(lt) = [ p(t - r )g(r)dT (6)

Fran Fqs 2 aro 6 it follCMS that we can dgttain g Ctl by integrating Eq 3 fran x = l to x = 0 and using b(t) as initial data on x = l

we nor consider the irrportant particular case in which the kernel is the inverse Gausslan pulse

crH(t) p(crt) = ---- exp(-cr24t) (7)

14n3

) rrarent pulse equal to OOlp(crt Nm with rr 2 = 004 seconds is represented in Fig 1

1o1shy

The step by step marching procedure of the Cauchy prdgtlan conesparx3s to solving Eq 10 by successively resroving the narrdeg1 convolution factors p(am t) fran its right-hand side we refer to this procedure which allows the monitoring of the unaroothing process fran u(l t) = b(t) to u(O t) = gt) as continuous deconvolution

As indicated previously in practice the response data are caitaminated by noise so that

instead of Eq 2 we have I

bnCt) =fp(rt - T )ge(r)dr 0 lt t lt oo (11)

0 where ge(r) is an estinate of the Greens function as affected cy errors due to the unkn0wn noise

n(t) =li(t) - b(t) (12)

whose norm is asSllled to satisfy

jJct)dt5e 2 (13) 0

We also assume that there exists a positive constant such that

00

fos2lttldt ~ 2 (14)

The ratio

f = -- ltlt 1 (15)

micro

is the root mean square of the noise-to-signal ratio in li (t)

Laplace transformation of Eq 11 yields

exp(-usl2)Ge(s) = B(s) + N(s) (16)

where the notatian F(s) is used for the Laplace transform of f(t) On the other hand

exp(-usl2)G(s) = B(s) (17)

It can then be slnom cy using the Sclwrz inequality that

lGe(s) - G(sll 12Re s ------------ ~ ------1~1 INltsgtI (18)

IGe(sll

ie for large values of s the noise can result in enormous relative errors in Ge (s) bull Therefore using the measured reSpC8lse b (t) as initial data on

11 x = 1 can cause the solutwn to be drowned in spurious oscillations to the extent of rendering it useless for practical purposes This is illustrated in the following section for the case of the Tinoslienko bcani

It is therefore necessary to regularize the solutjC11 cy apprqiriatelY filtering the lt2tabull It is

4shom m that to dgttam the appraximat1on to g (t) middot that minintlzes the Tiklnlov fllllCtiaial associated wHh Eq llb7 it is necessary to replace the Fourier transform of the initial data an x 1 ~(0 cy the filtered data

- 1Plto1 2 ~ltn (19) ibntlt~gt = -----------------shy

lPltU 12 + laquo212irl

The choice of w in Eq 19 is determined cy information on the approximate noise level inherent in the measurerents being performed and an information of experimental or analytic origin oo the awroximate bull time dependence of the dynamic Greens function Nlmerieal experinentation using interactive graphicsmiddot is then quite effective in locating the cptimal value ofw middot

the calculation of error botmds Greens functions being reccmstructed (seeS8 -f~r details)

pplicaticns

The purpose of the applications presented in this section is to daronstrate the effectiveness of the prcposed algorithm for reccmstructing dynamic Greens functions fran n6isy reSpC81Ses to infinitely divisible pulses We calfine ourselves in this paper to inverse Gaussian pulses (Eq 7) whkh as seen in Fig 1 are tmimodal A rich varie~ of infinitely divisible pulses can be constructed which can be useful for roodeling distortions superirrposed erall bscillations deviatjons fran a prescribed shape and other perturbations due to the instrumentation pplications of the theory of multimodal pulses to the structural dynantlcs jdentifkation prdgtlan will be presented in a forthcaning paper

The noisy response b (t) is constructed as follows We first calcula~e the dynamic Greens function g(t) for the member or system being considered We then use Eq 2 to calculate the response b(t) The response bll(t) is created synthetically by adding noise nit) to b(t) For any given t the noise (perturbation) n(t) is a randan mimer drawn fran a tmiform distribution in the range plusmn ti b (t) For exanple if ti = 001 we refer to the perturbation as bulla noisebull

Given the noisy response bn (t) we reconstruct the dynamic Greens function cy using the algoritbn based on the solution of the regularized Cauchy problem associated ~ith Eq 11 Details of the algoritlmi are given in bull middot

Sinply SuflXrted Tinoshenko Beambull

Jynamic Greens Function We seek the dynamic Greens flllCtion for the slcpe at x = t of the Timoshenko beam with the loading and support ccoUtions s~ in Fig 2 The equations of motion of the beam are

for the dvnamic

av a2w aM a2w -- pA --- -- - v PI --- (20) ax at2

I

ax at2

with the constitutive relations

awClitM=EI--- V=GAa--+it (21) ax ax

The state variable vector is defined as

(22)

Fourier transfonration of Eqs 20-21 yields

om ~

A7h9 ~ x bull

t I The prcperty of infinite divisibility allows Fig 2 Sirrply SUJpgtrted Tinoshenko Beam

THEOR1TICApound RESPONSI TD DELTA IUNCTION llDlltNT liiGVLARIZlD RtCONSTRUCTION 01 DlpoundTA HSPONSI

OOt

DDI

D

-001

-ooz

-003

-004

(a)

0 D5 I 15

Tlllt StcDNDS)

INVERSE CAUSSIAN RESPONSt

D03

ooz

aor

0

-0DI

-oot

-003 (b)

-D04 0 D5 I 15 z

rtllf (SICONDS)

DIRECT RECONSTRCCTION or DELTA RESPONSt

001 shy

THfORITICAL RtsPDNSt ro STIP IUNCTIDN 1101INT

002

001

-oor

-aoz

-001

OSl ADDED NOSI 01ttA =00)

(c) -00-1 - -------r--middot-r----middot- --------~----

a 05 I 1s

Till SECONDS)

0001

0

-0001

-0001

-0003

-0004

-ooos

-0006

-0001 (d)

-0008

-0009 0 os t6

T111 (SICONDS)

DOt

001

0

-D01

-001

-003 (e)061 ADDID NOSf OllECA=tU-3)

0 0$ I 15 fllf SICDNDS)

ERlDR IN llECONSTRUCTION or DELTA RESPONSE

0002 ~-----------------

0001

0

-0001

-0002

-0003 (f)(DSl ADDED NOSl 01ECA=ZSE-3)

-0004 ---shy0 0S I I 5

TlllE (SECONDS)

o rN RrcalisrliiicriON o srrr usPoNsl

DO

ooos ~

0 -ooos

-001

(0Sl ADDED NOISl OMECAZSE-3) (g)

------middot----- -------r--middotmiddot 0 os I 5

0001

D

-0001

-ooot

-0003

-D004

-0005

-D006

-D007 (DSI ADDfD NDSf OlllCAbulll51-1)

-D008 (h)

-0009 D 05 I 15

rtlll (SECONDS)

Fig 3 Results Cigttained for Tinoshenko Beam

-----middot middot-middot---middotmiddot

0 1 0 -lGA8

0 0 1EI 0aY fy~ (23)~ =

0 -pI~2 0 1

pAE2 0 0 0

The general solution for v islO

V= A1cosh(gt1xtl + ~sinh(gt 1xtl + A3cos(gt2xO

(24)

(plusmn gt-1 andplusmn igt2 are the eigenvalues of the matrix in Eq 231 note tllat gt 1 and gt 2 are functions of ~ bull ) The general solutions for the other carponents of the state vector follow ilmediately fran the Fourier transforms of Eqs 20 and 21

The boundary cooditions are

w = o M= o018lttgt Nn at x = 0 (25)

WOJ M=O at x bullI

The slqie at x = a is

2a = --=-=-- tA1gt1 coshgt1 + ~gt-126inhgt1 ax ssl pAE2t2

- ~gt22cosgt2 + A4gt22singt2J (26)

It follows fran the boundary caiditioos that

(27)

gt2singt2 +A3 ---shy

gt-1

l A3 = - ~ -----shy

gt2 tangt2

pAE2t3 ~ = ----------------shy

gt-1 (gt12 + gt-22)EI

The dynamic Gzeen s function being sought is the inverse Fourier transfollll of Fq 26

Calculations were performed assuming the following values for the ~ters of the beam p A = 0425 tgm pl = 278 x 10- kgm EI = 0555(1 + irJ sgnEl Ntf GAs =0711(1 + ilsgnE) N t = Y m = 001 (Trese values were calculated in in an attatpt to dgttain an equivalent Tilloshenko rotating beam whose response would match the measu1ed response of a flexible rdgtot amf) The calculated dynamic Greens functiw is slrMn in Fig 3a for 0 lt t lt 2 s It is noted that the Tilrcgtshenko beam whose motion is governed by a fourth 01-der partial differential equaticn is a dispersive medium

Numerical Reconstructiw of Dynamic Greens Fmction Fran Ngtisy Response The slcpe at x - f due to an invetse Gaussian pulse with = 02 sl2 (Fig 1) acting at x = 0 is ootained fran Fq 2 and

is shJwn in Fig 3b The response is seen to be a distorted image of the dynamic Greens function (Fig 3a) with high frequency caiponents sroothed out we add 05 noise to the response at each of 720 equidistant paints in the interval [0 2) s

Direct reconstruction of the Greens function flan noisy data is slnm in Fig 3c tbte the strwg arrplification typical of ill-posed prcblens of the srall anamt of noise in the data The regulazized reconstruction with = 25 x 10-3 shlwn in Fig 3e is a great improvement over the dizect reconstruction Note blwever that shazp features in the early part of the response have been rounded by tiie reconstzuction algoriU111 Figute 3f slns the errors in tiie reconstruction (ie tlie differences between the original and the reconstructed Gzeen El

function) bull The largest errors occur in the early part of the signal and axe due to the minute rounding nentianed above

Recall that dynamic Greens fuIJctions are used to calculate the respCXJSe to physically realizable (ie SIOOth) loadings by convolutiw as in Eq 2 The errors in the reconstnicted Gzeens function will cause errors in the calculated response However owing to the avezaging action of the integration process these errors will be considerably smaller than those affecting the reconstructed Greens function To illustrate the favorable effect of integration we consider the slope of the beam at x = f due to the step mioont OOlH(t) Figures 3d and 3h shM the step response calculated by convolution of the step function with the kernel calsisting of the original and the reconstructed Greens function respectively The agreement bettNeen the two responses is much closer than is the case for the corresparling Greens functions This is also seen in Fig 3g which slt1s the ezrors in the xegularized step response T1ie naximum absolute error occurs near the middle of the inte1val and is of the order of 01 of the corresponding ordinate of the response

Mechanical Slider With Oefomations Due To Shear

~-Dynamic Green middots Function The equations of

lllOtion and the constitutive relatioos for this maiber depicted in Fig 4 are

av a2w aw v (28)--- = pA ----

ax at2 ax

t

x ow

Fig 4 Mechanical Slider

ou

01

Otamp

0 01 Ol 03 04 05 Tlflll (SllCOtOS)

Flg l Inverse Gaussian Pul se

The Laplace transform of Eq 7 is

l[p(rrt)]=exp(-rrsl2) ResgtO (8)

Therefore

p(01tl p(02t) = pcr1 + 02tl (9)

and we can write Eq 2 in the fonn

b(t) =p(crt) g(t)

= p(__ t) [p(__ t)J(m-l) g(t) (10) m m

pulse entails the solution of an ill-posed integral equatjcm in the presence of noise We first describe bdefly the mathematical basis of a~ algorithm for dgttaining such a solution To check its effectiveness we apply the algorithn to noisy synthetic response data for structures woose exact dynamic Greens functicms are knom These structures include a dispersive mecUum consisting of a Tfocshenko beam a nandispersive medium consisting of a sljder with elastic defometions due to shear strain and a net1JCgtrk consisting of torsianal metlgters and TiJl)shenko beams It is slnm that the prqiosed algor]thn results in reconstructed dynamic Greens functions that for practical purposes closely awraximate their exact coimterparts

We note that research is also being ccnlucted at the National Bureau of Standards on experimental aspects of the procedure described earlier Currently ~rk is proceeding on the develqnent of a spaceshyrated actuator capable of producing matharetically specified soort rrechanical pulses to be used for dynamic system identificaticn purposes

Inverse Gaussian PuJ se Prebes and Decaivoluticn of Respcmse

The excitation denoted by p(t) must be a srooth causal pulse of srort duration which - to an appr(l)riate scale -- satisfies the relation

ir)dT l (1)

We may refer to p(t) as a probability density function

The response to the pulse p(t) I

b(t 1 p(t - T )g(T)dT 0 ( t ( 00 (2) 0

will be in general a highly distorted image of the Greens function g(t) with the high frequency carponenls aroothed out (This will be illustrated by exanpl es presented in the sequel bull ) The practical problem is to reconstruct the dynamic Greens function g(t) from the measured ~ response ~(t) since the respcrise b(t) that ~uld have been recorded in the absence of noise is unknom Little is knom about such prdgtlans in general However when the p(t) is infinitely divisible (ie for each int~oger in p(t) is the m-fold COlNolution of another prdJability density defined on the positive axis) such prdgtlans have been intensively st~islt3 recently both matlanatically and carputatiooally bull bull

We first assume that the response is not ccntaminated by noise To dgttain g (t) fran Eq 2 we consider the Cauchy prdgtlem for the fellaring linear partial differential equation in x t

x gt 0 t gt 0 (3)

u(x0) =0 x gt 0

u(Ot = g(t) t gt o

the Fourier transform of the causal signal f (t) being defined as

l i g[f(t)] = f m = --- f(t)e-i~t dt (4)

i 2r 0 I i

Fourier transfonretion of the first of Eqs 3 yields an ordinary differential equation in u(x~ ) woose Solution by virtue of the third of Eqs 3 is

u(xU = ~p(~)]xg(0 (5)

We take the inverse Fourier transform of Fq 5 in which we set x = 1 The cawolution theoran then yields

I

u(lt) = [ p(t - r )g(r)dT (6)

Fran Fqs 2 aro 6 it follCMS that we can dgttain g Ctl by integrating Eq 3 fran x = l to x = 0 and using b(t) as initial data on x = l

we nor consider the irrportant particular case in which the kernel is the inverse Gausslan pulse

crH(t) p(crt) = ---- exp(-cr24t) (7)

14n3

) rrarent pulse equal to OOlp(crt Nm with rr 2 = 004 seconds is represented in Fig 1

1o1shy

The step by step marching procedure of the Cauchy prdgtlan conesparx3s to solving Eq 10 by successively resroving the narrdeg1 convolution factors p(am t) fran its right-hand side we refer to this procedure which allows the monitoring of the unaroothing process fran u(l t) = b(t) to u(O t) = gt) as continuous deconvolution

As indicated previously in practice the response data are caitaminated by noise so that

instead of Eq 2 we have I

bnCt) =fp(rt - T )ge(r)dr 0 lt t lt oo (11)

0 where ge(r) is an estinate of the Greens function as affected cy errors due to the unkn0wn noise

n(t) =li(t) - b(t) (12)

whose norm is asSllled to satisfy

jJct)dt5e 2 (13) 0

We also assume that there exists a positive constant such that

00

fos2lttldt ~ 2 (14)

The ratio

f = -- ltlt 1 (15)

micro

is the root mean square of the noise-to-signal ratio in li (t)

Laplace transformation of Eq 11 yields

exp(-usl2)Ge(s) = B(s) + N(s) (16)

where the notatian F(s) is used for the Laplace transform of f(t) On the other hand

exp(-usl2)G(s) = B(s) (17)

It can then be slnom cy using the Sclwrz inequality that

lGe(s) - G(sll 12Re s ------------ ~ ------1~1 INltsgtI (18)

IGe(sll

ie for large values of s the noise can result in enormous relative errors in Ge (s) bull Therefore using the measured reSpC8lse b (t) as initial data on

11 x = 1 can cause the solutwn to be drowned in spurious oscillations to the extent of rendering it useless for practical purposes This is illustrated in the following section for the case of the Tinoslienko bcani

It is therefore necessary to regularize the solutjC11 cy apprqiriatelY filtering the lt2tabull It is

4shom m that to dgttam the appraximat1on to g (t) middot that minintlzes the Tiklnlov fllllCtiaial associated wHh Eq llb7 it is necessary to replace the Fourier transform of the initial data an x 1 ~(0 cy the filtered data

- 1Plto1 2 ~ltn (19) ibntlt~gt = -----------------shy

lPltU 12 + laquo212irl

The choice of w in Eq 19 is determined cy information on the approximate noise level inherent in the measurerents being performed and an information of experimental or analytic origin oo the awroximate bull time dependence of the dynamic Greens function Nlmerieal experinentation using interactive graphicsmiddot is then quite effective in locating the cptimal value ofw middot

the calculation of error botmds Greens functions being reccmstructed (seeS8 -f~r details)

pplicaticns

The purpose of the applications presented in this section is to daronstrate the effectiveness of the prcposed algorithm for reccmstructing dynamic Greens functions fran n6isy reSpC81Ses to infinitely divisible pulses We calfine ourselves in this paper to inverse Gaussian pulses (Eq 7) whkh as seen in Fig 1 are tmimodal A rich varie~ of infinitely divisible pulses can be constructed which can be useful for roodeling distortions superirrposed erall bscillations deviatjons fran a prescribed shape and other perturbations due to the instrumentation pplications of the theory of multimodal pulses to the structural dynantlcs jdentifkation prdgtlan will be presented in a forthcaning paper

The noisy response b (t) is constructed as follows We first calcula~e the dynamic Greens function g(t) for the member or system being considered We then use Eq 2 to calculate the response b(t) The response bll(t) is created synthetically by adding noise nit) to b(t) For any given t the noise (perturbation) n(t) is a randan mimer drawn fran a tmiform distribution in the range plusmn ti b (t) For exanple if ti = 001 we refer to the perturbation as bulla noisebull

Given the noisy response bn (t) we reconstruct the dynamic Greens function cy using the algoritbn based on the solution of the regularized Cauchy problem associated ~ith Eq 11 Details of the algoritlmi are given in bull middot

Sinply SuflXrted Tinoshenko Beambull

Jynamic Greens Function We seek the dynamic Greens flllCtion for the slcpe at x = t of the Timoshenko beam with the loading and support ccoUtions s~ in Fig 2 The equations of motion of the beam are

for the dvnamic

av a2w aM a2w -- pA --- -- - v PI --- (20) ax at2

I

ax at2

with the constitutive relations

awClitM=EI--- V=GAa--+it (21) ax ax

The state variable vector is defined as

(22)

Fourier transfonration of Eqs 20-21 yields

om ~

A7h9 ~ x bull

t I The prcperty of infinite divisibility allows Fig 2 Sirrply SUJpgtrted Tinoshenko Beam

THEOR1TICApound RESPONSI TD DELTA IUNCTION llDlltNT liiGVLARIZlD RtCONSTRUCTION 01 DlpoundTA HSPONSI

OOt

DDI

D

-001

-ooz

-003

-004

(a)

0 D5 I 15

Tlllt StcDNDS)

INVERSE CAUSSIAN RESPONSt

D03

ooz

aor

0

-0DI

-oot

-003 (b)

-D04 0 D5 I 15 z

rtllf (SICONDS)

DIRECT RECONSTRCCTION or DELTA RESPONSt

001 shy

THfORITICAL RtsPDNSt ro STIP IUNCTIDN 1101INT

002

001

-oor

-aoz

-001

OSl ADDED NOSI 01ttA =00)

(c) -00-1 - -------r--middot-r----middot- --------~----

a 05 I 1s

Till SECONDS)

0001

0

-0001

-0001

-0003

-0004

-ooos

-0006

-0001 (d)

-0008

-0009 0 os t6

T111 (SICONDS)

DOt

001

0

-D01

-001

-003 (e)061 ADDID NOSf OllECA=tU-3)

0 0$ I 15 fllf SICDNDS)

ERlDR IN llECONSTRUCTION or DELTA RESPONSE

0002 ~-----------------

0001

0

-0001

-0002

-0003 (f)(DSl ADDED NOSl 01ECA=ZSE-3)

-0004 ---shy0 0S I I 5

TlllE (SECONDS)

o rN RrcalisrliiicriON o srrr usPoNsl

DO

ooos ~

0 -ooos

-001

(0Sl ADDED NOISl OMECAZSE-3) (g)

------middot----- -------r--middotmiddot 0 os I 5

0001

D

-0001

-ooot

-0003

-D004

-0005

-D006

-D007 (DSI ADDfD NDSf OlllCAbulll51-1)

-D008 (h)

-0009 D 05 I 15

rtlll (SECONDS)

Fig 3 Results Cigttained for Tinoshenko Beam

-----middot middot-middot---middotmiddot

0 1 0 -lGA8

0 0 1EI 0aY fy~ (23)~ =

0 -pI~2 0 1

pAE2 0 0 0

The general solution for v islO

V= A1cosh(gt1xtl + ~sinh(gt 1xtl + A3cos(gt2xO

(24)

(plusmn gt-1 andplusmn igt2 are the eigenvalues of the matrix in Eq 231 note tllat gt 1 and gt 2 are functions of ~ bull ) The general solutions for the other carponents of the state vector follow ilmediately fran the Fourier transforms of Eqs 20 and 21

The boundary cooditions are

w = o M= o018lttgt Nn at x = 0 (25)

WOJ M=O at x bullI

The slqie at x = a is

2a = --=-=-- tA1gt1 coshgt1 + ~gt-126inhgt1 ax ssl pAE2t2

- ~gt22cosgt2 + A4gt22singt2J (26)

It follows fran the boundary caiditioos that

(27)

gt2singt2 +A3 ---shy

gt-1

l A3 = - ~ -----shy

gt2 tangt2

pAE2t3 ~ = ----------------shy

gt-1 (gt12 + gt-22)EI

The dynamic Gzeen s function being sought is the inverse Fourier transfollll of Fq 26

Calculations were performed assuming the following values for the ~ters of the beam p A = 0425 tgm pl = 278 x 10- kgm EI = 0555(1 + irJ sgnEl Ntf GAs =0711(1 + ilsgnE) N t = Y m = 001 (Trese values were calculated in in an attatpt to dgttain an equivalent Tilloshenko rotating beam whose response would match the measu1ed response of a flexible rdgtot amf) The calculated dynamic Greens functiw is slrMn in Fig 3a for 0 lt t lt 2 s It is noted that the Tilrcgtshenko beam whose motion is governed by a fourth 01-der partial differential equaticn is a dispersive medium

Numerical Reconstructiw of Dynamic Greens Fmction Fran Ngtisy Response The slcpe at x - f due to an invetse Gaussian pulse with = 02 sl2 (Fig 1) acting at x = 0 is ootained fran Fq 2 and

is shJwn in Fig 3b The response is seen to be a distorted image of the dynamic Greens function (Fig 3a) with high frequency caiponents sroothed out we add 05 noise to the response at each of 720 equidistant paints in the interval [0 2) s

Direct reconstruction of the Greens function flan noisy data is slnm in Fig 3c tbte the strwg arrplification typical of ill-posed prcblens of the srall anamt of noise in the data The regulazized reconstruction with = 25 x 10-3 shlwn in Fig 3e is a great improvement over the dizect reconstruction Note blwever that shazp features in the early part of the response have been rounded by tiie reconstzuction algoriU111 Figute 3f slns the errors in tiie reconstruction (ie tlie differences between the original and the reconstructed Gzeen El

function) bull The largest errors occur in the early part of the signal and axe due to the minute rounding nentianed above

Recall that dynamic Greens fuIJctions are used to calculate the respCXJSe to physically realizable (ie SIOOth) loadings by convolutiw as in Eq 2 The errors in the reconstnicted Gzeens function will cause errors in the calculated response However owing to the avezaging action of the integration process these errors will be considerably smaller than those affecting the reconstructed Greens function To illustrate the favorable effect of integration we consider the slope of the beam at x = f due to the step mioont OOlH(t) Figures 3d and 3h shM the step response calculated by convolution of the step function with the kernel calsisting of the original and the reconstructed Greens function respectively The agreement bettNeen the two responses is much closer than is the case for the corresparling Greens functions This is also seen in Fig 3g which slt1s the ezrors in the xegularized step response T1ie naximum absolute error occurs near the middle of the inte1val and is of the order of 01 of the corresponding ordinate of the response

Mechanical Slider With Oefomations Due To Shear

~-Dynamic Green middots Function The equations of

lllOtion and the constitutive relatioos for this maiber depicted in Fig 4 are

av a2w aw v (28)--- = pA ----

ax at2 ax

t

x ow

Fig 4 Mechanical Slider

instead of Eq 2 we have I

bnCt) =fp(rt - T )ge(r)dr 0 lt t lt oo (11)

0 where ge(r) is an estinate of the Greens function as affected cy errors due to the unkn0wn noise

n(t) =li(t) - b(t) (12)

whose norm is asSllled to satisfy

jJct)dt5e 2 (13) 0

We also assume that there exists a positive constant such that

00

fos2lttldt ~ 2 (14)

The ratio

f = -- ltlt 1 (15)

micro

is the root mean square of the noise-to-signal ratio in li (t)

Laplace transformation of Eq 11 yields

exp(-usl2)Ge(s) = B(s) + N(s) (16)

where the notatian F(s) is used for the Laplace transform of f(t) On the other hand

exp(-usl2)G(s) = B(s) (17)

It can then be slnom cy using the Sclwrz inequality that

lGe(s) - G(sll 12Re s ------------ ~ ------1~1 INltsgtI (18)

IGe(sll

ie for large values of s the noise can result in enormous relative errors in Ge (s) bull Therefore using the measured reSpC8lse b (t) as initial data on

11 x = 1 can cause the solutwn to be drowned in spurious oscillations to the extent of rendering it useless for practical purposes This is illustrated in the following section for the case of the Tinoslienko bcani

It is therefore necessary to regularize the solutjC11 cy apprqiriatelY filtering the lt2tabull It is

4shom m that to dgttam the appraximat1on to g (t) middot that minintlzes the Tiklnlov fllllCtiaial associated wHh Eq llb7 it is necessary to replace the Fourier transform of the initial data an x 1 ~(0 cy the filtered data

- 1Plto1 2 ~ltn (19) ibntlt~gt = -----------------shy

lPltU 12 + laquo212irl

The choice of w in Eq 19 is determined cy information on the approximate noise level inherent in the measurerents being performed and an information of experimental or analytic origin oo the awroximate bull time dependence of the dynamic Greens function Nlmerieal experinentation using interactive graphicsmiddot is then quite effective in locating the cptimal value ofw middot

the calculation of error botmds Greens functions being reccmstructed (seeS8 -f~r details)

pplicaticns

The purpose of the applications presented in this section is to daronstrate the effectiveness of the prcposed algorithm for reccmstructing dynamic Greens functions fran n6isy reSpC81Ses to infinitely divisible pulses We calfine ourselves in this paper to inverse Gaussian pulses (Eq 7) whkh as seen in Fig 1 are tmimodal A rich varie~ of infinitely divisible pulses can be constructed which can be useful for roodeling distortions superirrposed erall bscillations deviatjons fran a prescribed shape and other perturbations due to the instrumentation pplications of the theory of multimodal pulses to the structural dynantlcs jdentifkation prdgtlan will be presented in a forthcaning paper

The noisy response b (t) is constructed as follows We first calcula~e the dynamic Greens function g(t) for the member or system being considered We then use Eq 2 to calculate the response b(t) The response bll(t) is created synthetically by adding noise nit) to b(t) For any given t the noise (perturbation) n(t) is a randan mimer drawn fran a tmiform distribution in the range plusmn ti b (t) For exanple if ti = 001 we refer to the perturbation as bulla noisebull

Given the noisy response bn (t) we reconstruct the dynamic Greens function cy using the algoritbn based on the solution of the regularized Cauchy problem associated ~ith Eq 11 Details of the algoritlmi are given in bull middot

Sinply SuflXrted Tinoshenko Beambull

Jynamic Greens Function We seek the dynamic Greens flllCtion for the slcpe at x = t of the Timoshenko beam with the loading and support ccoUtions s~ in Fig 2 The equations of motion of the beam are

for the dvnamic

av a2w aM a2w -- pA --- -- - v PI --- (20) ax at2

I

ax at2

with the constitutive relations

awClitM=EI--- V=GAa--+it (21) ax ax

The state variable vector is defined as

(22)

Fourier transfonration of Eqs 20-21 yields

om ~

A7h9 ~ x bull

t I The prcperty of infinite divisibility allows Fig 2 Sirrply SUJpgtrted Tinoshenko Beam

THEOR1TICApound RESPONSI TD DELTA IUNCTION llDlltNT liiGVLARIZlD RtCONSTRUCTION 01 DlpoundTA HSPONSI

OOt

DDI

D

-001

-ooz

-003

-004

(a)

0 D5 I 15

Tlllt StcDNDS)

INVERSE CAUSSIAN RESPONSt

D03

ooz

aor

0

-0DI

-oot

-003 (b)

-D04 0 D5 I 15 z

rtllf (SICONDS)

DIRECT RECONSTRCCTION or DELTA RESPONSt

001 shy

THfORITICAL RtsPDNSt ro STIP IUNCTIDN 1101INT

002

001

-oor

-aoz

-001

OSl ADDED NOSI 01ttA =00)

(c) -00-1 - -------r--middot-r----middot- --------~----

a 05 I 1s

Till SECONDS)

0001

0

-0001

-0001

-0003

-0004

-ooos

-0006

-0001 (d)

-0008

-0009 0 os t6

T111 (SICONDS)

DOt

001

0

-D01

-001

-003 (e)061 ADDID NOSf OllECA=tU-3)

0 0$ I 15 fllf SICDNDS)

ERlDR IN llECONSTRUCTION or DELTA RESPONSE

0002 ~-----------------

0001

0

-0001

-0002

-0003 (f)(DSl ADDED NOSl 01ECA=ZSE-3)

-0004 ---shy0 0S I I 5

TlllE (SECONDS)

o rN RrcalisrliiicriON o srrr usPoNsl

DO

ooos ~

0 -ooos

-001

(0Sl ADDED NOISl OMECAZSE-3) (g)

------middot----- -------r--middotmiddot 0 os I 5

0001

D

-0001

-ooot

-0003

-D004

-0005

-D006

-D007 (DSI ADDfD NDSf OlllCAbulll51-1)

-D008 (h)

-0009 D 05 I 15

rtlll (SECONDS)

Fig 3 Results Cigttained for Tinoshenko Beam

-----middot middot-middot---middotmiddot

0 1 0 -lGA8

0 0 1EI 0aY fy~ (23)~ =

0 -pI~2 0 1

pAE2 0 0 0

The general solution for v islO

V= A1cosh(gt1xtl + ~sinh(gt 1xtl + A3cos(gt2xO

(24)

(plusmn gt-1 andplusmn igt2 are the eigenvalues of the matrix in Eq 231 note tllat gt 1 and gt 2 are functions of ~ bull ) The general solutions for the other carponents of the state vector follow ilmediately fran the Fourier transforms of Eqs 20 and 21

The boundary cooditions are

w = o M= o018lttgt Nn at x = 0 (25)

WOJ M=O at x bullI

The slqie at x = a is

2a = --=-=-- tA1gt1 coshgt1 + ~gt-126inhgt1 ax ssl pAE2t2

- ~gt22cosgt2 + A4gt22singt2J (26)

It follows fran the boundary caiditioos that

(27)

gt2singt2 +A3 ---shy

gt-1

l A3 = - ~ -----shy

gt2 tangt2

pAE2t3 ~ = ----------------shy

gt-1 (gt12 + gt-22)EI

The dynamic Gzeen s function being sought is the inverse Fourier transfollll of Fq 26

Calculations were performed assuming the following values for the ~ters of the beam p A = 0425 tgm pl = 278 x 10- kgm EI = 0555(1 + irJ sgnEl Ntf GAs =0711(1 + ilsgnE) N t = Y m = 001 (Trese values were calculated in in an attatpt to dgttain an equivalent Tilloshenko rotating beam whose response would match the measu1ed response of a flexible rdgtot amf) The calculated dynamic Greens functiw is slrMn in Fig 3a for 0 lt t lt 2 s It is noted that the Tilrcgtshenko beam whose motion is governed by a fourth 01-der partial differential equaticn is a dispersive medium

Numerical Reconstructiw of Dynamic Greens Fmction Fran Ngtisy Response The slcpe at x - f due to an invetse Gaussian pulse with = 02 sl2 (Fig 1) acting at x = 0 is ootained fran Fq 2 and

is shJwn in Fig 3b The response is seen to be a distorted image of the dynamic Greens function (Fig 3a) with high frequency caiponents sroothed out we add 05 noise to the response at each of 720 equidistant paints in the interval [0 2) s

Direct reconstruction of the Greens function flan noisy data is slnm in Fig 3c tbte the strwg arrplification typical of ill-posed prcblens of the srall anamt of noise in the data The regulazized reconstruction with = 25 x 10-3 shlwn in Fig 3e is a great improvement over the dizect reconstruction Note blwever that shazp features in the early part of the response have been rounded by tiie reconstzuction algoriU111 Figute 3f slns the errors in tiie reconstruction (ie tlie differences between the original and the reconstructed Gzeen El

function) bull The largest errors occur in the early part of the signal and axe due to the minute rounding nentianed above

Recall that dynamic Greens fuIJctions are used to calculate the respCXJSe to physically realizable (ie SIOOth) loadings by convolutiw as in Eq 2 The errors in the reconstnicted Gzeens function will cause errors in the calculated response However owing to the avezaging action of the integration process these errors will be considerably smaller than those affecting the reconstructed Greens function To illustrate the favorable effect of integration we consider the slope of the beam at x = f due to the step mioont OOlH(t) Figures 3d and 3h shM the step response calculated by convolution of the step function with the kernel calsisting of the original and the reconstructed Greens function respectively The agreement bettNeen the two responses is much closer than is the case for the corresparling Greens functions This is also seen in Fig 3g which slt1s the ezrors in the xegularized step response T1ie naximum absolute error occurs near the middle of the inte1val and is of the order of 01 of the corresponding ordinate of the response

Mechanical Slider With Oefomations Due To Shear

~-Dynamic Green middots Function The equations of

lllOtion and the constitutive relatioos for this maiber depicted in Fig 4 are

av a2w aw v (28)--- = pA ----

ax at2 ax

t

x ow

Fig 4 Mechanical Slider

THEOR1TICApound RESPONSI TD DELTA IUNCTION llDlltNT liiGVLARIZlD RtCONSTRUCTION 01 DlpoundTA HSPONSI

OOt

DDI

D

-001

-ooz

-003

-004

(a)

0 D5 I 15

Tlllt StcDNDS)

INVERSE CAUSSIAN RESPONSt

D03

ooz

aor

0

-0DI

-oot

-003 (b)

-D04 0 D5 I 15 z

rtllf (SICONDS)

DIRECT RECONSTRCCTION or DELTA RESPONSt

001 shy

THfORITICAL RtsPDNSt ro STIP IUNCTIDN 1101INT

002

001

-oor

-aoz

-001

OSl ADDED NOSI 01ttA =00)

(c) -00-1 - -------r--middot-r----middot- --------~----

a 05 I 1s

Till SECONDS)

0001

0

-0001

-0001

-0003

-0004

-ooos

-0006

-0001 (d)

-0008

-0009 0 os t6

T111 (SICONDS)

DOt

001

0

-D01

-001

-003 (e)061 ADDID NOSf OllECA=tU-3)

0 0$ I 15 fllf SICDNDS)

ERlDR IN llECONSTRUCTION or DELTA RESPONSE

0002 ~-----------------

0001

0

-0001

-0002

-0003 (f)(DSl ADDED NOSl 01ECA=ZSE-3)

-0004 ---shy0 0S I I 5

TlllE (SECONDS)

o rN RrcalisrliiicriON o srrr usPoNsl

DO

ooos ~

0 -ooos

-001

(0Sl ADDED NOISl OMECAZSE-3) (g)

------middot----- -------r--middotmiddot 0 os I 5

0001

D

-0001

-ooot

-0003

-D004

-0005

-D006

-D007 (DSI ADDfD NDSf OlllCAbulll51-1)

-D008 (h)

-0009 D 05 I 15

rtlll (SECONDS)

Fig 3 Results Cigttained for Tinoshenko Beam

-----middot middot-middot---middotmiddot

0 1 0 -lGA8

0 0 1EI 0aY fy~ (23)~ =

0 -pI~2 0 1

pAE2 0 0 0

The general solution for v islO

V= A1cosh(gt1xtl + ~sinh(gt 1xtl + A3cos(gt2xO

(24)

(plusmn gt-1 andplusmn igt2 are the eigenvalues of the matrix in Eq 231 note tllat gt 1 and gt 2 are functions of ~ bull ) The general solutions for the other carponents of the state vector follow ilmediately fran the Fourier transforms of Eqs 20 and 21

The boundary cooditions are

w = o M= o018lttgt Nn at x = 0 (25)

WOJ M=O at x bullI

The slqie at x = a is

2a = --=-=-- tA1gt1 coshgt1 + ~gt-126inhgt1 ax ssl pAE2t2

- ~gt22cosgt2 + A4gt22singt2J (26)

It follows fran the boundary caiditioos that

(27)

gt2singt2 +A3 ---shy

gt-1

l A3 = - ~ -----shy

gt2 tangt2

pAE2t3 ~ = ----------------shy

gt-1 (gt12 + gt-22)EI

The dynamic Gzeen s function being sought is the inverse Fourier transfollll of Fq 26

Calculations were performed assuming the following values for the ~ters of the beam p A = 0425 tgm pl = 278 x 10- kgm EI = 0555(1 + irJ sgnEl Ntf GAs =0711(1 + ilsgnE) N t = Y m = 001 (Trese values were calculated in in an attatpt to dgttain an equivalent Tilloshenko rotating beam whose response would match the measu1ed response of a flexible rdgtot amf) The calculated dynamic Greens functiw is slrMn in Fig 3a for 0 lt t lt 2 s It is noted that the Tilrcgtshenko beam whose motion is governed by a fourth 01-der partial differential equaticn is a dispersive medium

Numerical Reconstructiw of Dynamic Greens Fmction Fran Ngtisy Response The slcpe at x - f due to an invetse Gaussian pulse with = 02 sl2 (Fig 1) acting at x = 0 is ootained fran Fq 2 and

is shJwn in Fig 3b The response is seen to be a distorted image of the dynamic Greens function (Fig 3a) with high frequency caiponents sroothed out we add 05 noise to the response at each of 720 equidistant paints in the interval [0 2) s

Direct reconstruction of the Greens function flan noisy data is slnm in Fig 3c tbte the strwg arrplification typical of ill-posed prcblens of the srall anamt of noise in the data The regulazized reconstruction with = 25 x 10-3 shlwn in Fig 3e is a great improvement over the dizect reconstruction Note blwever that shazp features in the early part of the response have been rounded by tiie reconstzuction algoriU111 Figute 3f slns the errors in tiie reconstruction (ie tlie differences between the original and the reconstructed Gzeen El

function) bull The largest errors occur in the early part of the signal and axe due to the minute rounding nentianed above

Recall that dynamic Greens fuIJctions are used to calculate the respCXJSe to physically realizable (ie SIOOth) loadings by convolutiw as in Eq 2 The errors in the reconstnicted Gzeens function will cause errors in the calculated response However owing to the avezaging action of the integration process these errors will be considerably smaller than those affecting the reconstructed Greens function To illustrate the favorable effect of integration we consider the slope of the beam at x = f due to the step mioont OOlH(t) Figures 3d and 3h shM the step response calculated by convolution of the step function with the kernel calsisting of the original and the reconstructed Greens function respectively The agreement bettNeen the two responses is much closer than is the case for the corresparling Greens functions This is also seen in Fig 3g which slt1s the ezrors in the xegularized step response T1ie naximum absolute error occurs near the middle of the inte1val and is of the order of 01 of the corresponding ordinate of the response

Mechanical Slider With Oefomations Due To Shear

~-Dynamic Green middots Function The equations of

lllOtion and the constitutive relatioos for this maiber depicted in Fig 4 are

av a2w aw v (28)--- = pA ----

ax at2 ax

t

x ow

Fig 4 Mechanical Slider

0 1 0 -lGA8

0 0 1EI 0aY fy~ (23)~ =

0 -pI~2 0 1

pAE2 0 0 0

The general solution for v islO

V= A1cosh(gt1xtl + ~sinh(gt 1xtl + A3cos(gt2xO

(24)

(plusmn gt-1 andplusmn igt2 are the eigenvalues of the matrix in Eq 231 note tllat gt 1 and gt 2 are functions of ~ bull ) The general solutions for the other carponents of the state vector follow ilmediately fran the Fourier transforms of Eqs 20 and 21

The boundary cooditions are

w = o M= o018lttgt Nn at x = 0 (25)

WOJ M=O at x bullI

The slqie at x = a is

2a = --=-=-- tA1gt1 coshgt1 + ~gt-126inhgt1 ax ssl pAE2t2

- ~gt22cosgt2 + A4gt22singt2J (26)

It follows fran the boundary caiditioos that

(27)

gt2singt2 +A3 ---shy

gt-1

l A3 = - ~ -----shy

gt2 tangt2

pAE2t3 ~ = ----------------shy

gt-1 (gt12 + gt-22)EI

The dynamic Gzeen s function being sought is the inverse Fourier transfollll of Fq 26

Calculations were performed assuming the following values for the ~ters of the beam p A = 0425 tgm pl = 278 x 10- kgm EI = 0555(1 + irJ sgnEl Ntf GAs =0711(1 + ilsgnE) N t = Y m = 001 (Trese values were calculated in in an attatpt to dgttain an equivalent Tilloshenko rotating beam whose response would match the measu1ed response of a flexible rdgtot amf) The calculated dynamic Greens functiw is slrMn in Fig 3a for 0 lt t lt 2 s It is noted that the Tilrcgtshenko beam whose motion is governed by a fourth 01-der partial differential equaticn is a dispersive medium

Numerical Reconstructiw of Dynamic Greens Fmction Fran Ngtisy Response The slcpe at x - f due to an invetse Gaussian pulse with = 02 sl2 (Fig 1) acting at x = 0 is ootained fran Fq 2 and

is shJwn in Fig 3b The response is seen to be a distorted image of the dynamic Greens function (Fig 3a) with high frequency caiponents sroothed out we add 05 noise to the response at each of 720 equidistant paints in the interval [0 2) s

Direct reconstruction of the Greens function flan noisy data is slnm in Fig 3c tbte the strwg arrplification typical of ill-posed prcblens of the srall anamt of noise in the data The regulazized reconstruction with = 25 x 10-3 shlwn in Fig 3e is a great improvement over the dizect reconstruction Note blwever that shazp features in the early part of the response have been rounded by tiie reconstzuction algoriU111 Figute 3f slns the errors in tiie reconstruction (ie tlie differences between the original and the reconstructed Gzeen El

function) bull The largest errors occur in the early part of the signal and axe due to the minute rounding nentianed above

Recall that dynamic Greens fuIJctions are used to calculate the respCXJSe to physically realizable (ie SIOOth) loadings by convolutiw as in Eq 2 The errors in the reconstnicted Gzeens function will cause errors in the calculated response However owing to the avezaging action of the integration process these errors will be considerably smaller than those affecting the reconstructed Greens function To illustrate the favorable effect of integration we consider the slope of the beam at x = f due to the step mioont OOlH(t) Figures 3d and 3h shM the step response calculated by convolution of the step function with the kernel calsisting of the original and the reconstructed Greens function respectively The agreement bettNeen the two responses is much closer than is the case for the corresparling Greens functions This is also seen in Fig 3g which slt1s the ezrors in the xegularized step response T1ie naximum absolute error occurs near the middle of the inte1val and is of the order of 01 of the corresponding ordinate of the response

Mechanical Slider With Oefomations Due To Shear

~-Dynamic Green middots Function The equations of

lllOtion and the constitutive relatioos for this maiber depicted in Fig 4 are

av a2w aw v (28)--- = pA ----

ax at2 ax

t

x ow

Fig 4 Mechanical Slider

bull bull

10

i

-10

i I

-10 gI tltbullgt

so that

a2w a2w2---- = a ---- (29)

at2 middota

GAaa2 =---shy

p A

The boundary canditions are

aw V=O (or -- = 0) for x = 0 (30)

ax a2w

Il1o ---- = - 0 (t) - v for x = e at2

Note that the notion of the slider is governed bymiddot the wave equation Unlike the Tinoshenko beam the slider is therefore a nondispersive nedium The general solution of Fq 29 is

w =f(at - x) + F(at + x) (31

The first boundary candition yields

w = f(at - x) + f(at + x (32)

lb dgttain the functiaial fonn of f ( n [the argument r nay be put equal to at - x or at + x as required]

we note that the second of Fqsbull30 is a coMensed fonn of the boundary caidi tion

Il1o a2w aw ---- ---- = - a2 ---- for x = r (33

p A at2 ax

and the initial conditions 1

w = 0 for t = 0 and 0 lt x lt (34) middot

aw 1 lim --- shy for x = e

t=+O 0 t

As shJwn for an entirely analogous prdgtlem in12 the first of Fqs ~4 inllies

aw aw --=O and ---=O for t = Or 0 lt x lt t (35) ax (lt

Fquations 35 and the first of Fqs 34 for x c O yield

1

(36)fen = o

lb dgttain the expression Of f( l) for the interval t lt t lt 3e we use Fq 33 which pro1Tides the caitinuing equation the second of Fqs 34 and the condition that there is no sudden change in the di spl acanent at x =t The result is

l PA f en = - ----- c1 - exp c - --- c r - t gtn lt31 gt middot

apA Ille

The expressions of fen for successive intervals (2k - lHlt t lt (2k + llL where k = 2 3 bullbullbull aremiddot dgttained recUIsively using the caitinuing equation middot the condi Uon of caitinui ty of the velocity at x = middot l and the candi tian that there is no sudden change

in the displacement at x = l The dynamic Greens function of the acceleration at x = t n derived Eran Fq 32 in which the apprcpriate expressions for f (I) were used is shJwn in Fig 6a for ~ = 0 225 kg pA = 0425 kgro and GA8 = 0711 N l1ie Greens functions for the velocity and displacerent at x = tn are shJwn in Figs 6b and 6c respectively

Numerical Reconstruction of lynamic Greens Function From Noisy Response The acceleration respca1se at x =1 11 due to an inverse Gaussian pulse with u = 05 sl12 acting at x = l is sln-m in Fig S The respci1se is seen to be a highly distorted image of the correspcmding Green s function (Fig 6a) Next 1 noise was added to this response The

Fig 5 kceleraticn Due to Inverse Gaussian Pulse

dynamic Greens function reconstructed fran the noisy response to the pulle using the regularization parameter w = 10 x 10- is shown in Fig 6d The reccvstructed Greens function has less sharp peaks than its exact COmterpart and it is also affected by spurious high-frequency carpCllents Ngtte that the accuracy of the reconstruction would have been lnproved had a sharper pulse been used

The reconstructed Greens function of the velocity at x =t n was dgttained by integrating the reconstructed Greens function of the acceleration with respect to tine The result is shown in Fig 6e The agrearent with the exact Greens function of the velocity (Fig 6b) is nruch hetter than for the acceJ eratjons1 as indicated earlier integration considerably reduces errors due to deconvolution in the presence of noise Integration of the recoostructed velocity (ie integrating the reconstructed acceleration twice) results in a reconstructed Greens function of the displacesrent (Fig 6f) that is virtually indistinguishable fran the corresponding exact Greens function (Fig 6c)

Structural Network With Torsional Members and Tinoshenko Beams

The hipothetical space structure of Fig7 corudsts of four identical torsiaial irerbers (1-2 1 -2 2shy3 2-3) and two TiJioshenko beams (2-2 and 3-3 The mcment cannecticns between the torsiaial irerbers and the Tinoshenko beams are rigid The network and its inqiulsive loadings are symretrical about the line joining the roidpoints of the beams

Exact Acceleration

1000

500

Nbulle o

-500

-1000 0 2 3 4

-

---

l I I

tltsl

Cal

Exact Velocity

10

5

fO 0 E

-5

-10 0 2 3 4

tltsl

(bl

Exact Displacement

1

0

-1

-2

-3

-4 m -5

-6

-7 -8 -9

0 1 2 3 4 tltsl

ltcl

Reconstructed Acceleration

1000

500

- ----

-1 l I

N-E fl)

0

-500

-1000 5

10

5

fl) 0E

-5

-10 5

0

-1

-2

-3 m -4

-5

-6

-7 -a -a

5

0 1 2 3 4 tltsl

(di

Reconstructed Velocity

0 1 2 3 4 t(S)

Cegt Reconstructed Displacement

0 2 3 4 tltsl

(fl

Fig 6 Results Obtained for Mechanical Slider

5

5

5

shy shy

3

=~=pound==-1middot-3--amp-shy

Fig 7 Structural Network ltl = 195 m)

The equation of JlOtion and the ccmstitutive equation for the torsiooal metbers are

ClT Tae (38)--- = --shyax ax JG1

From the first of Eqs 38 it follows that the general solution for the Fourier transfonn of the rotation of in each torsianal metber is of the fonn

- - AX - f gtx9(x) = 6(0)cos( ---) + T(O) ----sin( -) (39)

e JGi e lgtlJ

) c--- 3112u (40) JG1

A shnHar relation is dgttained for T(x) fran the second of Eqs 38 The general solution for the Fourier transfonns of the shear force in each beam is of the sarre fonn as Eq 24 The general solutions for the Fourier transfonns of the other CCJlil011el1ts of the state vector (-w and Ml follow fran Fq 24 and the Fourier transforms of Fqs 20 and 21

There are altogether 16 tJllknain coefficients (two for each torsional marber and four for each beam) The 16 requisite boundary ccmditicns are the follCMing torques at joints 1 and 1 are equal to unity ie to the Fourier transform of tle applied torques (2 conditions) sum of torsional and bending norents at joints 2 2 3 3 are zero (4 conditiaus)J shear forces at the errls of the beams are zero since torsional rnexnbers are assumed nol to be capable of transnitting shear force (4 caiditions) 1 rotations of collinear torsional nerbers at joints 2 and 2 bull are equal (2 condi tiais) 1 rotations of torsional marbers and slcpes of beams at joints 2 2 3 3 are equal (4 cooditions) ltlte that it is possible In this case to take advantage of symretry which allows the reduction of the nuniler of UJJknams to twelve Once the tmknown coefficients are

detennined as functions of the frequency parameter the Greens function being sooght is cigttained by inverse Fourier tiansfonnation of the quantity of interestmiddot (eg rotaticn slcpe norent)

Calculations were perfonned for the follCMing values of the network parameters (seell pp 1~-20) pA = 239 kg~J pI = 118 kg m EI= 177x10 (1 + isgnE l N Jtf t = 195 m = 0001 The Greens functicri for the slcpe at joint 3 of beam 3-3 bull due to the norents inpulses of Fig 7 is shJwn in Fig ea The disturbances consisting of the moment jnpulses at joints 1 end 1 prcpagate to joint 3 partly through the beams 3-3 and 2-2 which are dispersive That is the varicus frequency catJOleiltB Of the disturbances which were originally synthesized in a sharp signal travel at different speeds end arrive at joint 3 at different titres The dispersion Of the higher freQUEICY catJOleiltS is evident in Fig Ba A similar dispersion occurs in the case of the Greens functicn shJwn in Fig 3a The higher the danping the faster the high frequency carpxients are smiothed out This is seE1l by carparing the Greens functiais for the slcpe of the beam at joint 2 calculated by assuning the hysteretic datrping to be 01 an the cne hand (Fig Sa) and 1 on the other (Fig 9) Finally note the contrast beoieen the Greens functiais of Figs 3a Ba and 9 which are typical of dispersive media and the Greens functicns for the naidispersive nechanical slider (Figs 4a b c)

Nurrerical Recanstructicn of Dynamic Greens Function Fran Ngtisy Response Figure Sb shools the response cbtained by caivolving the Greens function of n~middot Ba and an inverse Gaussian pulse with 02 s bull The drastic smoothing out of the sharp features of the Greens function is evident The noisy respaise was ccnstructed by adding lbull noise to the response of Fig Sb

Exact Acceleration

1000

500

Nbulle o

-500

-1000 0 2 3 4

-

---

l I I

tltsl

Cal

Exact Velocity

10

5

fO 0 E

-5

-10 0 2 3 4

tltsl

(bl

Exact Displacement

1

0

-1

-2

-3

-4 m -5

-6

-7 -8 -9

0 1 2 3 4 tltsl

ltcl

Reconstructed Acceleration

1000

500

- ----

-1 l I

N-E fl)

0

-500

-1000 5

10

5

fl) 0E

-5

-10 5

0

-1

-2

-3 m -4

-5

-6

-7 -a -a

5

0 1 2 3 4 tltsl

(di

Reconstructed Velocity

0 1 2 3 4 t(S)

Cegt Reconstructed Displacement

0 2 3 4 tltsl

(fl

Fig 6 Results Obtained for Mechanical Slider

5

5

5

shy shy

3

=~=pound==-1middot-3--amp-shy

Fig 7 Structural Network ltl = 195 m)

The equation of JlOtion and the ccmstitutive equation for the torsiooal metbers are

ClT Tae (38)--- = --shyax ax JG1

From the first of Eqs 38 it follows that the general solution for the Fourier transfonn of the rotation of in each torsianal metber is of the fonn

- - AX - f gtx9(x) = 6(0)cos( ---) + T(O) ----sin( -) (39)

e JGi e lgtlJ

) c--- 3112u (40) JG1

A shnHar relation is dgttained for T(x) fran the second of Eqs 38 The general solution for the Fourier transfonns of the shear force in each beam is of the sarre fonn as Eq 24 The general solutions for the Fourier transfonns of the other CCJlil011el1ts of the state vector (-w and Ml follow fran Fq 24 and the Fourier transforms of Fqs 20 and 21

There are altogether 16 tJllknain coefficients (two for each torsional marber and four for each beam) The 16 requisite boundary ccmditicns are the follCMing torques at joints 1 and 1 are equal to unity ie to the Fourier transform of tle applied torques (2 conditions) sum of torsional and bending norents at joints 2 2 3 3 are zero (4 conditiaus)J shear forces at the errls of the beams are zero since torsional rnexnbers are assumed nol to be capable of transnitting shear force (4 caiditions) 1 rotations of collinear torsional nerbers at joints 2 and 2 bull are equal (2 condi tiais) 1 rotations of torsional marbers and slcpes of beams at joints 2 2 3 3 are equal (4 cooditions) ltlte that it is possible In this case to take advantage of symretry which allows the reduction of the nuniler of UJJknams to twelve Once the tmknown coefficients are

detennined as functions of the frequency parameter the Greens function being sooght is cigttained by inverse Fourier tiansfonnation of the quantity of interestmiddot (eg rotaticn slcpe norent)

Calculations were perfonned for the follCMing values of the network parameters (seell pp 1~-20) pA = 239 kg~J pI = 118 kg m EI= 177x10 (1 + isgnE l N Jtf t = 195 m = 0001 The Greens functicri for the slcpe at joint 3 of beam 3-3 bull due to the norents inpulses of Fig 7 is shJwn in Fig ea The disturbances consisting of the moment jnpulses at joints 1 end 1 prcpagate to joint 3 partly through the beams 3-3 and 2-2 which are dispersive That is the varicus frequency catJOleiltB Of the disturbances which were originally synthesized in a sharp signal travel at different speeds end arrive at joint 3 at different titres The dispersion Of the higher freQUEICY catJOleiltS is evident in Fig Ba A similar dispersion occurs in the case of the Greens functicn shJwn in Fig 3a The higher the danping the faster the high frequency carpxients are smiothed out This is seE1l by carparing the Greens functiais for the slcpe of the beam at joint 2 calculated by assuning the hysteretic datrping to be 01 an the cne hand (Fig Sa) and 1 on the other (Fig 9) Finally note the contrast beoieen the Greens functiais of Figs 3a Ba and 9 which are typical of dispersive media and the Greens functicns for the naidispersive nechanical slider (Figs 4a b c)

Nurrerical Recanstructicn of Dynamic Greens Function Fran Ngtisy Response Figure Sb shools the response cbtained by caivolving the Greens function of n~middot Ba and an inverse Gaussian pulse with 02 s bull The drastic smoothing out of the sharp features of the Greens function is evident The noisy respaise was ccnstructed by adding lbull noise to the response of Fig Sb

3

=~=pound==-1middot-3--amp-shy

Fig 7 Structural Network ltl = 195 m)

The equation of JlOtion and the ccmstitutive equation for the torsiooal metbers are

ClT Tae (38)--- = --shyax ax JG1

From the first of Eqs 38 it follows that the general solution for the Fourier transfonn of the rotation of in each torsianal metber is of the fonn

- - AX - f gtx9(x) = 6(0)cos( ---) + T(O) ----sin( -) (39)

e JGi e lgtlJ

) c--- 3112u (40) JG1

A shnHar relation is dgttained for T(x) fran the second of Eqs 38 The general solution for the Fourier transfonns of the shear force in each beam is of the sarre fonn as Eq 24 The general solutions for the Fourier transfonns of the other CCJlil011el1ts of the state vector (-w and Ml follow fran Fq 24 and the Fourier transforms of Fqs 20 and 21

There are altogether 16 tJllknain coefficients (two for each torsional marber and four for each beam) The 16 requisite boundary ccmditicns are the follCMing torques at joints 1 and 1 are equal to unity ie to the Fourier transform of tle applied torques (2 conditions) sum of torsional and bending norents at joints 2 2 3 3 are zero (4 conditiaus)J shear forces at the errls of the beams are zero since torsional rnexnbers are assumed nol to be capable of transnitting shear force (4 caiditions) 1 rotations of collinear torsional nerbers at joints 2 and 2 bull are equal (2 condi tiais) 1 rotations of torsional marbers and slcpes of beams at joints 2 2 3 3 are equal (4 cooditions) ltlte that it is possible In this case to take advantage of symretry which allows the reduction of the nuniler of UJJknams to twelve Once the tmknown coefficients are

detennined as functions of the frequency parameter the Greens function being sooght is cigttained by inverse Fourier tiansfonnation of the quantity of interestmiddot (eg rotaticn slcpe norent)

Calculations were perfonned for the follCMing values of the network parameters (seell pp 1~-20) pA = 239 kg~J pI = 118 kg m EI= 177x10 (1 + isgnE l N Jtf t = 195 m = 0001 The Greens functicri for the slcpe at joint 3 of beam 3-3 bull due to the norents inpulses of Fig 7 is shJwn in Fig ea The disturbances consisting of the moment jnpulses at joints 1 end 1 prcpagate to joint 3 partly through the beams 3-3 and 2-2 which are dispersive That is the varicus frequency catJOleiltB Of the disturbances which were originally synthesized in a sharp signal travel at different speeds end arrive at joint 3 at different titres The dispersion Of the higher freQUEICY catJOleiltS is evident in Fig Ba A similar dispersion occurs in the case of the Greens functicn shJwn in Fig 3a The higher the danping the faster the high frequency carpxients are smiothed out This is seE1l by carparing the Greens functiais for the slcpe of the beam at joint 2 calculated by assuning the hysteretic datrping to be 01 an the cne hand (Fig Sa) and 1 on the other (Fig 9) Finally note the contrast beoieen the Greens functiais of Figs 3a Ba and 9 which are typical of dispersive media and the Greens functicns for the naidispersive nechanical slider (Figs 4a b c)

Nurrerical Recanstructicn of Dynamic Greens Function Fran Ngtisy Response Figure Sb shools the response cbtained by caivolving the Greens function of n~middot Ba and an inverse Gaussian pulse with 02 s bull The drastic smoothing out of the sharp features of the Greens function is evident The noisy respaise was ccnstructed by adding lbull noise to the response of Fig Sb

2

1

2

UJ bull15E-2 1 noise

0 llC

w D

9 CJ

-1

omiddot llC w 0 +----- D

9 CJ

-1

002

0 05 to tltagt (b)

famp

Exact Step Response

20 0 05 10 t(amp)

15

(e)

Reconstructed Step Response

20

001

in

~ 0 llC

w D

9 -001 CJ

-+---- in 0 llC

w D

CJ

-002

0 005 10 teal

20

(C) (f)

Exact Greens Function Reconstructed Greens Function

20 1bull15E-3 1 noise

10

in 0 llC

w D

9 CJ

-1

0 05 10 15 20 0 05 10 15 20 tltal tisgt

ltagt (d)

Response to Inverse Gaussian Pulse Reconstructed Greens Function

0 +--~0 --J

001

w bull15E-3 1 noise

Q+---shy

9 -001

-002

Fig 8 Results Obtained for Structural Network ~ = 0001

--~-- --- middot--- __middotmiddot --middotmiddot -middot ~

Slrle lurel Nbulllugtoflr

05

~

0

-05

_ 0 05 15 I

Tiii (bull)

Fig 9 Exact Greens Function I = 001

The regularized recaistruction of ~ dynamic Greens functian using w 175 x 10- is slxlwn in Fig Bd Mich of the detailed structure of the Greens functicn has been presexved al though a few discrepancies can be detected However the errors inherent in Fig Bd are of little consequence when the reconstructed Greens function is used as a tool for predicting the response to a givenmiddot smooth fllllction f(t) Indeed as pointed out earlier Ellall pointwise errors tend to be averaged out by the convolutioo process As an exanple the teconstructed step response (Fig Sf) cbtained by cawoluticn of the unit step function with the reconstructed Greens function of Fig Sd is virtually indistinguishable fran the exact step response (Fig middot Sc) It is noted that if the errors in the reconstructed Greens functions are large -- as maymiddot be the case in an unregularized (direct) reconstructidn - the convolution process may not be sufficiently effective in averagilig them out

To illustrate the inportance of selecting the cptimal value of the regularizing pararreter we sl~ in Fig 8e a reconstruction in which w = 15 x 10- bull Because this value is too large it causes the srooUtlng out of legi tilnate high-frequency carpooents of the signal

For the case = 001 Fig 10 slxMs a view of 12 successive steps (see Eq 10) in the reconstruction process that proceeds fran middotthe response to the pulse (first trace in foregrourrl) to the Greens function being sought (last trace in background) bull such diagrams intelfgtreted in the light of error bouros developed in can be helpful in attempts to determine whether certain high-frequency ccnpcmmts of a reconstructed Greens functionbelong legitimately to the signal or are spurious additions due to noise

Jdditional details on various iispects of the 10rk presented here aze available in bull

ainclusions

A mathematical and computational capability exists that gtOUld enable the reliable experinental determination of dynamic Greens functions for structures The method is based on prqiagating specific types of pulses leading to a tractable clecaivolution prtblem in the presence of noise

1cknCMledgments

This work was spcnsored by the Directorate of 1erospace Sciences Air Force Office of Scientific Research Dr Anttony Ilt lltros served as the project monitor~ The Minerals Management Service us Departnient of the Interior (Charles E Smith Research Program Manager) also provided partial support

Fig 10 View of Successive Partial Deconvolutions for Structural Network I = 001

Slrle lurel Nbulllugtoflr

05

~

0

-05

_ 0 05 15 I

Tiii (bull)

Fig 9 Exact Greens Function I = 001

The regularized recaistruction of ~ dynamic Greens functian using w 175 x 10- is slxlwn in Fig Bd Mich of the detailed structure of the Greens functicn has been presexved al though a few discrepancies can be detected However the errors inherent in Fig Bd are of little consequence when the reconstructed Greens function is used as a tool for predicting the response to a givenmiddot smooth fllllction f(t) Indeed as pointed out earlier Ellall pointwise errors tend to be averaged out by the convolutioo process As an exanple the teconstructed step response (Fig Sf) cbtained by cawoluticn of the unit step function with the reconstructed Greens function of Fig Sd is virtually indistinguishable fran the exact step response (Fig middot Sc) It is noted that if the errors in the reconstructed Greens functions are large -- as maymiddot be the case in an unregularized (direct) reconstructidn - the convolution process may not be sufficiently effective in averagilig them out

To illustrate the inportance of selecting the cptimal value of the regularizing pararreter we sl~ in Fig 8e a reconstruction in which w = 15 x 10- bull Because this value is too large it causes the srooUtlng out of legi tilnate high-frequency carpooents of the signal

For the case = 001 Fig 10 slxMs a view of 12 successive steps (see Eq 10) in the reconstruction process that proceeds fran middotthe response to the pulse (first trace in foregrourrl) to the Greens function being sought (last trace in background) bull such diagrams intelfgtreted in the light of error bouros developed in can be helpful in attempts to determine whether certain high-frequency ccnpcmmts of a reconstructed Greens functionbelong legitimately to the signal or are spurious additions due to noise

Jdditional details on various iispects of the 10rk presented here aze available in bull

ainclusions

A mathematical and computational capability exists that gtOUld enable the reliable experinental determination of dynamic Greens functions for structures The method is based on prqiagating specific types of pulses leading to a tractable clecaivolution prtblem in the presence of noise

1cknCMledgments

This work was spcnsored by the Directorate of 1erospace Sciences Air Force Office of Scientific Research Dr Anttony Ilt lltros served as the project monitor~ The Minerals Management Service us Departnient of the Interior (Charles E Smith Research Program Manager) also provided partial support

Fig 10 View of Successive Partial Deconvolutions for Structural Network I = 001

1

References

1 Ashley H Sane Considerations on Farthbound i Dynamic Testing of Large Space Structures Paper 86-0908 27th Structures Structural Dynarnjcs and Materjals Conference Part 2 San Antonio Texas May 19- 21 1986 roerican Institute of Aerrnautics and Astronautics New York NY pp 362-373

2 Cannon R H Jr and Scllnitz E Initial Experiments on the End-Point Control of a Flexible One-Link Rcbot The International Journal of Rcbotics Research Vol 3 No 3 Fall 1984 pp 62-75

3 Crawley E F am de Luis J Use of Piezoeleetric Actuators as Elements of Intelligent Systans AIAA Journal Vol 25 Oct 1987 pp 1373-1385

4 Carasso A s and Hsu N N~ Prdgte Wavefonns and Deconvolution in the Experimental Dcterminatjon of Elastic Greens FUnctions SIAM Journal oo llppl led Matlanatics Vol 45 No 3 June 1985 pp 369-382

S Carasso A s Infinitely Divisible Pulses Ccrltinuoos Deconvolution and the Characted zation of Linear Time Invariant SystEllS SIAM Journal m Applied Matlanatics Vol 47 No 4 Aug 1987 pp 892-927

6 FranklIn J N On Tikhmov s Method for IllPosed PrdlJEllS bull Matlanatics of Catputatfon Vol 28 1974 pp 889-907

7 Franklin J J Minimum Prlnciples for Ill shyPosed PrdlJ ens SIAM Journal m Matleretical

_Analysis Vol 8 1978 pp 638-650

8 Carasso A s and Hsu N N L00 ErTors in i Partial Deconvolutim of the Inverse Gaussian Pulse Sl1M Journal on Awl ied Matlenatics Vol 45 No 6 Dec 1985 pp 1029-1038

9 Burty W C and Rubinstein M F Dynamics of Structures Prentice-Hall Engletood Cliffs NJ 1964p 88

10 Pestel E c and Leckie F A Matrix Methods in Elastanechanics MGraw-Hill New York 1963 p 133

11 von Flotow A H Distmbance Prcpagatiai in Structural Networks Control of Large Space Structures Ph D Dissertatiai Departznent of Aeronautics atld Astronautics Stanford University June 1984

12 love A E H A Treatise on the Matlenatkal TflEOry Of EJastlcity 4th ed Dover New York 1944 p 435

13 Carasso A s and Sirniu E Identification of Dynamic Greens Functions in Structural Networks paper sulmitted for possible publication to the AIAA Journal

shy

• Dynamic characterization of structures by pulse probing and deconvcution

• Dynamic characterization of structures by pulse probing and deconvcution