parameter and state estimation of shear buildings...

Research ArticleParameter and State Estimation of Shear BuildingsUsing Spline Interpolation and Linear Integral Filters

Antonio Concha 1 and Luis Alvarez-Icaza 2

1Facultad de Ingenierıa Mecanica y Electrica Universidad de Colima 28400 Coquimatlan COL Mexico2Instituto de Ingenierıa Universidad Nacional Autonoma de Mexico 04510 Coyoacan CDMX Mexico

Correspondence should be addressed to Luis Alvarez-Icaza alvarpumasiingenunammx

Received 25 February 2018 Accepted 18 April 2018 Published 19 July 2018

Academic Editor Daniele Baraldi

Copyright copy 2018 Antonio Concha and Luis Alvarez-Icaza This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A parameter identificationmethod and a high gain observer are proposed in order to identify themodel and to recover the state of aseismically excited shear building using acceleration responses of the ground and instrumented floors levels as well as the responsesat noninstrumented floors which are reconstructed by means of cubic spline shape functions The identification method can beimplemented online or offline and uses Linear Integral Filters whose bandwidthmust enclose the spectrum of a seismically excitedbuilding On the other hand the proposed state observer estimates the displacements and velocities of all the structure floors usingthe model estimated by the identification method The observer allows obtaining a fast response and reducing the state estimationerror while depending on a single gain The performance of the parameter and state estimators is verified through experimentscarried out on a five-story small scale building

1 Introduction

Acceleration responses of the building floor obtained whenit is excited through earthquakes or wind are employedfor parameter and state estimation of the structure or formonitoring its health Because of cost most of the buildingsare instrumented only at a few stories For this reasonthe estimation and health monitoring of a structure mustbe carried out using as data only the responses of theinstrumented stories

Parameter and state estimation of civil structures usingrecorded responses has attracted the attention of a largenumber of researchers around the world during the last fourdecades Some relevant references for not fully instrumentedstructures [1ndash15] are revised here Amini and Hedayati [1]propose a Sparse Component Analysis approach for modalidentification of structures where the number of sensors issmaller than the number of active modes References [2ndash5]employ the extended Kalman filter in order to estimate theparameters of a structure as well as its displacements andvelocities at noninstrumented floors This filter is obtained

by linearizing a nonlinear state equation that considers thebuilding parameters as states Nevertheless some poles ofthe linearized model may lie on the imaginary axis andas a consequence some parameter andor states may beunbounded On the other hand Zhou et al [6] recursivelyestimate the stiffness and damping ratios of a structureusing the first two derivatives of Log-Likelihood Measureand the knowledge of the building mass Mukhopadhyayet al [7] develop mode shape normalization and expansionapproaches that utilize the topology of the structuralmatricesfor estimating the mass and stiffness parameters of a buildingunder base excitation authors obtain the global identifia-bility requirements for their methodology and show thatfor estimating the structural parameters the number 119875 ofinstrumented floors should satisfy 119875 ge radic119899 and 119875 ge (1 +radic4119899 minus 3)2 if the floor masses at sensor locations are knownand if the total mass of the structure is known respectivelywhere 119899 is the number of floors of the building Yuan etal [8] estimate the mass and stiffness matrices of shearbuildings using a method depending on the first two ordersof modal data References [9ndash11] present OKID identification

HindawiShock and VibrationVolume 2018 Article ID 5206968 21 pageshttpsdoiorg10115520185206968

2 Shock and Vibration

techniques which determine the Markov parameters of aKalman observer in order to identity a building modelThesetechniques identify a discrete timemodel of the structure butif the sampling frequency of responses is too high in relationto the dominate frequencies of the structure then the poles ofthe estimated model lie close to the unit circle in the complexdomain whichmay produce parameter estimates statisticallyill-defined [16] Kaya el al [12] identify a structure usingthe Transfer Matrix formulation of the response where thehistories at noninstrumented floors are offline reconstructedusing the Mode Shape Based Estimation (MSBE) methodwhich assume that modal shapes can be approximated as alinear combination of the mode shapes of a shear beam anda bending beam It is worth mentioning that using MSBErequires solving two partial differential equations and theknowledge of the modal acceleration of the instrumentedfloors Hegde and Sinha [13] estimate the modal param-eters of a seismically excited torsionally coupled buildingusing limited number of responses modal parameters areextracted using an offline estimation methodology basedon the principles of the Natural Excitation Technique andthe Eigen Realization Algorithm authors use a cubic shapefunction for estimating the responses at noninstrumentedfloors On the other hand [14 15] carried out structuralidentification using ambient vibration measurements andoffline algorithms Huang [14] proposes a procedure foridentifying the dynamics characteristics of a shear buildingusing the multivariate ARVmodel whereas Chakraverty [15]estimates mass and stiffness parameters frommodal test dataand the Holzer criteria

This manuscript presents (1) an identification techniqueto identify the parameters of the model of a seismicallyexcited shear building using a limited number of responsesand (2) a high gain state observer that employs the identi-fied parameter in order to estimate the displacements andvelocities of both instrumented and noninstrumented floorsThe parameter and state estimators rely on the accelerationmeasurements of ground and some instrumented floors Theacceleration at noninstrumented floors is reconstructed bymeans of cubic shape spline functions that use the availablemeasurements The state observer gain is easily designed anddepends on a positive parameter that guarantees the stabilityand a fast response of the observer On the other handthe parameter estimator uses a parameterization with thefollowing features (1) assuming that building has Rayleighdamping (2) containing a vector whose entries are thestiffnessmass ratios of the building which are estimatedthrough a Least Squares algorithm and (3) employing LinearIntegral Filters (LIF) that attenuate low and high frequencynoise of the measured and reconstructed responses The LIFwere previously employed in Garrido and Concha [17] forparameter estimation of fully instrumented structures In thismanuscript the application of these filters is extended toestimate the model of buildings instrumented at only fewfloor levels Moreover by considering Rayleigh damping forthe structure the number of parameters of the proposedparameterization is the half of the one corresponding to theparameterization in [17] In comparison with techniques in[7ndash15] described above the proposed parameter estimator

allows attenuating low and high frequency noise of bothmeasured and reconstructed acceleration and can be imple-mented online at a low computational effort

It is worth mentioning that the cubic spline functionsemployed by the proposed parameter and state estimatorsare designed following the ideas in [18 19] Moreover thedesign of these functions does not require the knowledge ofthe modal shapes of the structure and they allow recursiveestimation of the responses at noninstrumented floors Tech-niques in [20ndash26] are also able to estimate the unmeasuredfloor level responses but unlike the cubic spline functionsthey are implemented offline and require an estimated build-ing model which may be unavailable

The paper has the following structure Sections 2 and 3present the shear building model and its parameterizationrespectively The cubic spline functions used for estimatingthe floor responses at noninstrumented floors are describedin Section 4 Section 5 shows the methodology for estimatingthe building parameters The proposed high gain observeris described in Section 6 Experiment results using theparameter and state estimators are shown in Section 7Finally Section 8 includes the conclusions of this paper

2 Mathematical Model of a Shear Building

The dynamics of a shear building with 119899 stories subjected toearthquake or basemotion is described through the followingmathematical model [27]

M (x (119905) + l119892 (119905)) + Cx (119905) + Kx (119905) = 0 (1)

x (119905) = [1199091 (119905) 119909119899 (119905)]119879 x (119905) = [1 (119905) 119899 (119905)]119879x (119905) = [1 (119905) 119899 (119905)]119879 l = [1 1 1]119879

(2)

xa (119905) = x (119905) + l119892 (119905) = [1198861 (119905) 119886119899 (119905)]119879 (3)

where variables 119909119894(119905) 119894(119905) and 119894(119905) are respectively thedisplacement velocity and acceleration of the 119894th floor whichare measured with respect to the basement Signal 119886119894(119905)represents the absolute acceleration at the 119894th floor and119892(119905) is the ground acceleration produced by earthquakeMoreover M and K are the mass and stiffness matricesrespectively which are defined as

M = [[[[[[[

1198981 0 sdot sdot sdot 00 1198982 sdot sdot sdot 0 d0 0 sdot sdot sdot 119898119899

]]]]]]]

K = [[[[[[[

1198961 + 1198962 minus1198962 sdot sdot sdot 0minus1198962 1198962 + 1198963 sdot sdot sdot 0 d0 0 sdot sdot sdot 119896119899]]]]]]]

(4)

Shock and Vibration 3

The entry 119896119894 (119894 = 1 2 119899) is the column lateral stiffnessbetween the 119894th and (119894 minus 1)th floors

The building has Rayleigh damping that is representedthrough the following matrix C

C = 1198860M + 1198861K (5)

Parameters 1198860 and 1198861 are constant and they are computed as[27]

12 [[[[1120596119894

1205961198941120596119895

120596119895

]]]][11988601198861] = [120577119894120577119895] (6)

where 120577V and 120596V V = 119894 119895 are the damping ratio and naturalfrequency of the Vth structural mode respectively

Remark 1 Computing the constants 1198860 and 1198861 in (6) requiresknowledge of the parameters 120596119894 120596119895 120577119894 and 120577119895 that corre-sponds to the 119894th and 119895th structural modes Natural frequen-cies 120596119894 and 120596119895 can be extracted by means of the Fourierspectra of the building responses produced by ambient orforce excitations On the other hand the parameters 120577119894 and120577119895 can be obtained with any of the following techniques

(i) Half-power bandwidth method [28] which estimatesthe damping ratios using the peaks of the responseFourier spectra

(ii) LogarithmicDecrementmethod which computes thedamping ratios in the time domain by means ofthe ratio between the amplitudes of any two closelyadjacent peaks of acceleration responses [29]

(iii) Enhanced Frequency Domain Decomposition meth-od that estimates the modal damping from the Singu-lar Value plots of recorded acceleration [30]

(iv) Using the recommended damping values shown inTable 1121 from [27] which depend on the type andcondition of the structure

The state space model corresponding to the differentialequation (1) is given by

(119905) = A120578 (119905) + B119892 (119905) (7)

A = [ Ontimesn IntimesnminusMminus1K minusMminus1C]

B = [Ontimes1minusl ](8)

120578 (119905) = [xT (119905) xT (119905)]119879= [1199091 (119905) 1199092 (119905) 119909119899 (119905) 1 (119905) 2 (119905) 119899 (119905)]119879 (9)

where Iptimesr and Optimesr are the identity and zero matrices withsize 119901 times 119903 respectively

The output equation of the state space model depends onthe absolute acceleration of the instrumented floors Let 119875 be

the number of instrumented floors and let 119894 = 1 2 119875 bethe 119894th instrumented floor numbered from bottom to top ofthe building Then the output equation is given by

y (119905) = Γxa (119905) = Γ [x (119905) + l119892 (119905)] = ΓD120578 (119905) D = [minusMminus1K minusMminus1C] (10)

where Γ isin 119877119875119909119899 is the localization matrix of the accelerome-ters whose entries are defined asΓ119894119895=

1 if the 119895th floor is the 119894th instrumented story0 otherwise

(11)

3 Model Parameterization

Assume first that all the stories are instrumented it meansthat Γ = I119899119909119899 then substituting (5) into (1) yields

x (119905) + l119892 (119905) = minusMminus1 [1198860M + 1198861K] x (119905) minusMminus1Kx (119905) (12)

Equation (12) can be rewritten as

xa (119905) + 1198860x (119905) = minus1198861Mminus1Kx (119905) minusMminus1Kx (119905) (13)

Define the following equalities

Mminus1Kx (119905) = u (119905) 120579Mminus1Kx (119905) = u (119905) 120579 (14)

where u(119905) is the time derivative of u(119905) and120579 = [ 11989611198981

11989621198981

11989621198982

sdot sdot sdot 119896119899119898119899

]119879 = [1205791 1205792 1205793 sdot sdot sdot 1205792119899minus1]119879u (119905)= minus[[[[[[[

1199091 (119905) 1199091 (119905) minus 1199092 (119905) 0 sdot sdot sdot 00 0 1199092 (119905) minus 1199091 (119905) sdot sdot sdot 0 0 0 0 sdot sdot sdot 119909119899 (119905) minus 119909119899minus1 (119905)]]]]]]]

(15)

Using equalities in (14) leads to the following expression

xa (119905) + 1198860x (119905) = [1198861u (119905) + u (119905)] 120579 (16)

In order to use only acceleration measurements andto attenuate low and high frequency measurement noise(16) is first derived three times with respect to time andsubsequently the resulting expression is integrated five timesover finite time intervals thus obtaining1198685 x(3)a (119905) + 1198860x(2) (119905) = 1198685 1198861u(2) (119905) + u(1) (119905) 120579 (17)

The superindex (119894) denotes the 119894th time derivative of thecorresponding signal and 1198685sdot is a Linear Integral Filter(LIF) The general definition a LIF is given by119868120572 120588 (119905)

= 1120575120572minus1 int119905

119905minus120575int1205911

1205911minus120575sdot sdot sdot int120591120572minus1

120591120572minus1minus120575120588 (120591120572) 119889120591120572 sdot sdot sdot 1198891205911 (18)


where 120572 is the number of integrations and 120575 is the timeintegration window length defined as120575 = ℏ119879119904 (19)

where ℏ gt 0 and 119879119904 is the sampling period of 120588The Laplace transform of 119868120572120588(119905) in (18) is given by [17]

L [119868120572 120588 (119905)] = 120575119866120572 (119904)L [120588 (119905)] (20)

119866120572 (119904) = (1 minus 119890minus120575119904120575119904 )120572 1003816100381610038161003816119866120572 (119895120596)1003816100381610038161003816 = 100381610038161003816100381610038161003816100381610038161003816 sin (120587120596120596119888)(120587120596120596119888)

100381610038161003816100381610038161003816100381610038161003816120572

120596119888 = 2120587120575 119891119888 = 1120575

(21)

Terms 120596119888 and 119891119888 determine the bandwidth of the 120572th-orderlow-pass filter 119866120572(119904) in rads and Hz respectively

Equation (17) is equivalent to the following parameteri-zation which will be employed for parameter identificationpurposes

120594 (119905) = 120601 (119905) 120579 (22)

120594 (119905) = 11205753 11986823sum

119895=0

(3119895) (minus1)119895 xa (119905 minus 119895120575)+ 11988601205752 1198683

2sum119895=0

(2119895) (minus1)119895 x (119905 minus 119895120575)(23)

120601 (119905) = 11988611205752 11986832sum

119895=0

(2119895) (minus1)119895 u (119905 minus 119895120575)+ 11205751198684

1sum119895=0

(1119895) (minus1)119895 u (119905 minus 119895120575)(24)

It is important tomention that variables 120594(119905) and 120601(119905) canbe written in the Laplace domain as

L [120594 (119905)] = H3 (119904)L [xa (119905)] + 1198860H2 (119904)L [x (119905)]L [120601 (119905)] = 1198861H2 (119904)L [u (119905)] +H1 (119904)L [u (119905)] (25)

whereH1 (119904) = 1205751199041198665 (119904) H2 (119904) = 12057511990421198665 (119904) H3 (119904) = 12057511990431198665 (119904)

(26)

Remark 2 The filters H119894(119904) 119894 = 1 2 3 are band pass filtersand they are designed to encompass the frequency band ofthe seismically excited building and to attenuate low and highfrequency measurement noise

Table 1 Boundary conditions for a shear building

Boundary Absolute acceleration119886(119905) = (119905) + 119892(119905)ℎ = 0 119886(0 119905) = 119892(119905)1015840119886(0 119905) = 0ℎ = 119867 10158401015840119886 (119867 119905) = 0101584010158401015840119886 (119867 119905) = 04 Estimation of the Responses atNoninstrumented Floor Levels

In practice most of the buildings are instrumented in onlysome floors it means that not all signals 1198861(119905) 119886119899(119905)which appear in variables x(119905) xa(119905) and u(119905) of parameter-ization (22) are available In order to implement x(119905) xa(119905)and u(119905) the acceleration types at noninstrumented floorsare reconstructed by means of cubic spline shape functionswhich are described in this section

Consider a building with height 119867 and 119899 stories thatis instrumented at its basement and at 119875 floors as shownin Figure 1 Moreover let ℎ0 = 0 and 1199030(119905) = 119892(119905)be the height and acceleration response at the basementrespectively Similarly terms ℎ119901 and 119903119901(119905) 119901 = 1 2 119875 arethe height and absolute acceleration at the 119901th instrumentedfloor respectively where 119903119901(119905) isin xa(119905)

Let 119886(ℎ 119905) be the absolute acceleration of the buildingat height ℎ Using (3) yields 119886(0 119905) = 119892(119905) and 119886(119867 119905) =119899119886(119905) In addition let ℎlowast be the height of the 119908th non-instrumented floor that is located within the subintervalΔ119901 = ℎ119901+1 minus ℎ119901 delimited by two instrumented floors withheights ℎ119901 and ℎ119901+1 where ℎ119901+1 gt ℎ119901 The response at thisnoninstrumented floor is given by 119886(ℎlowast 119905) or equivalently119886(ℎlowast 119905) = 119886119908(119905) according to definition (3) An estimate119909119886(ℎlowast 119905) = 119909119886119908(119905) of 119886(ℎlowast 119905) = 119908119886(119905) is computed throughthe following cubic spline shape function

119909119886119908 (119905) = 119909119886 (ℎlowast 119905)= 119886119901 (119905) + 119887119901 (119905) [ℎlowast minus ℎ119901] + 119889119901 [ℎlowast minus ℎ119901]2+ 119890119901 (119905) [ℎlowast minus ℎ119901]3 ℎlowast isin [ℎ119901 ℎ119901+1]

(27)

where 119886119901(119905) 119887119901(119905) 119889119901(119905) and 119890119901(119905) are the coefficients of the119901th cubic polynomial which are computed at every samplinginstant from continuity of the spline function from responses119903119901(119905) 119901 = 0 1 2 119875 at the instrumented floors andfrom the boundary conditions of the absolute accelerationof the building These conditions assume that the buildingbehaves as a cantilever and they are shown in Table 1 wheresuperscripts 1015840 10158401015840 and 101584010158401015840 indicate the first second and thirdderivative with respect to the spatial variable ℎ AppendicesA and B present the cubic spline function obtained if theacceleration response at the top floor is available or notrespectively

Once absolute acceleration 119909119886119908(119905) of the 119908th noninstru-mented floor has been obtained it is possible to estimate its


ℎP = H

ℎPminus1

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

Δℎ2

Δℎ1

Δℎ0

(a) Building instrumented at the top floor

H

ℎPminus1

ℎP

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

ΔℎP

Δℎ2

Δℎ1

Δℎ0

(b) Building noninstrumented at the top floor

Figure 1 Building with height119867 instrumented in 119875 floor levels

relative acceleration 119909119908(119905) through the expression 119909119908(119905) =119909119886119908(119905) minus 119892(119905) Then the unavailable responses 119908(119905) and119886119908(119905) in variables x(119905) xa(119905) and u(119905) are replaced by theirestimates 119909119908(119905) and 119909119886119908(119905) respectively From now on vari-ables constructed with recorded and reconstructed responsesare denoted as xa(119905) x(119905) and u(119905) which are given by

x (119905) = x (119905) + 120576 (119905)xa (119905) = xa (119905) + 120576a (119905)u (119905) = u (119905) + 120576u (119905) (28)

where terms 120576(119905) 120576119886(119905) and 120576u(119905) depend on the error of the

reconstructed responses and on the noise of the recordedresponses

5 Parameter Estimation of the Building

Substituting signals x(119905) x119886(119905) and u(119905) given in (28) into(22)ndash(24) yields

(119905) = (119905) 120579 + 1198685 [120582 (119905)] (29)

where variables (119905) and (119905) have the same structure as 120594(119905)and 120601(119905) respectively However (119905) and (119905) contain theterms xa(119905) x(119905) and u(119905) instead of xa(119905) x(119905) and u(119905) Onthe other hand vector 120582(119905) depends on 120576 120576a and 120576u TheLaplace transform of 120582(119905) is given by

Λ (119904) = L [120582 (119905)]= 1199043L [120576a] + 1199042 (1198860L [120576] minus 1198861L [120576u] 120579)minus 119904L [120576u] 120579

(30)

Employing (20) and (30) allows obtaining the nextLaplace transform of 1198685[120582(119905)]

L [1198685 [120582 (119905)]] = H3 (119904)L [120576a]+H2 (119904) (1198860L [120576] minus 1198861L [120576u] 120579)minusH1 (119904)L [120576u] 120579(31)

whereH119894(119904) 119894 = 1 2 3 were previously defined in (26)Expression (29) can be rewritten as

(119896119879119904) = (119896119879119904) 120579 + 1198685 [120582 (119896119879119904)] (32)

where 119905 = 119896119879119904 119896 = 0 1 2 are the sampling instants ofsignals (119905) and (119905) Omitting 119879119904 in (32) leads to

(119896) = (119896) 120579 + 1198685 [120582 (119896)] (33)

In order to estimate the parameter vector 120579 in (33) theLeast Squares (LS) algorithm is employed which is definedas [31]

= P (119873) 119873sum119896=0

119879 (119896) (119896)

P (119873) = [ 119873sum119896=0

119879 (119896) (119896)]minus1 (34)

where P is the covariance matrix119873 is the number of samplesof and Note that vector can be computed only if matrixP exists

On the other hand 120579 can also be recursively identifiedsince the responses at noninstrumented floor levels can be


computed at every sampling period 119879119904 The recursive versionof the LS denoted as RLS is given by

(119896) = (119896 minus 1) + L (119896) 120598 (119896)L (119896)= P (119896 minus 1) 119879 (119896) [120573I119899times119899 + (119896)P (119896 minus 1) (119896)119879]minus1

P (119896) = [P (119896 minus 1) minus L (119896) (119896)P (119896 minus 1)]120573120598 (119896) = (119896) minus (119896) (119896)

(35)

where120573 is the forgetting factor such that 0 lt 120573 le 1 moreover120598(119896) is the output estimation error

Proposition 3 Suppose that input signal 119892(119905) is persistentlyexciting at least of order b = 2119899 minus 1 where b is the numberof parameters of 120579 to be estimated then vector norm of theparameter estimation error defined as = 120579minus is boundedMoreover the smaller the perturbation term 1198685[120582(119896119879119904)] in (33)the smaller the value of Proof It is given in books [31 32]

51 Estimation of Modal Parameters Once that vector hasbeen estimated with the offline or the online estimationmethodology it is possible to identify the natural frequenciesand modal damping factors of the building To this end thefollowing matrix A which is an estimate of matrixA given in(8) is constructed

A = [ Ontimesn IntimesnminusMminus1K minusMminus1C] (36)

Mminus1K = [[[[[[[[

1205791 + 1205792 minus1205792 sdot sdot sdot 0minus1205793 1205793 + 1205794 sdot sdot sdot 0 d0 0 sdot sdot sdot 1205792119899minus1

]]]]]]]](37)

Mminus1C = 1198860Intimesn + 1198861Mminus1K (38)

where parameters 120579119894 119894 = 1 2 2119899 minus 1 are the entries ofvector Note that (38) is deduced from (5)

The eigenvalues of the matrix A in (36) are given by

1205821198941 = minus120590119894 + 119895120603119894 = minus120577119894119894 + 119895119894radic1 minus 1205772

1205821198942 = minus120590119894 minus 119895120603119894 = minus120577119894119894 minus 119895119894radic1 minus 1205772 (39)

where 120590119894 = 120577119894119894 and 120603119894 = 119894radic1 minus 1205772 Moreover 119894 and 120577119894119894 = 1 2 119899 are respectively the estimates of the natural

frequency 120596119894 and damping factor 120577119894 corresponding to the 119894thmode of building model (1)

From (39) the following equations for computing theparameters 119894 and 120577119894 are obtained119894 = radic1205902119894 + 1206032

119894 120577119894 = 120590119894119894

119894 = 1 2 119899

(40)

52 Estimation of Matrices119872 119870 and 119862 Assume that mass1198981 of the building is known then matrix C and the entries ofmatrices M and K are given by1 = 11205791119894 = 119894minus1120579(2119894minus2)

119894 = 119894120579(2119894minus1) 119894 = 2 3 119899C = 1198860M + 1198861K

(41)

A similar procedure can be carried out if another floor massis known instead of1198981

6 High Gain State Observer

The proposed high gain state observer employs the buildingmodel estimated by the LS method This observer estimatesthe complete state of a building instrumented at only fewfloorlevels and it is given by120578 (119905) = A (119905) + B119892 (119905) + L (119905) (119905) = xa (119905) minus yo (119905) (42)

xa (119905) = xa (119905) + 120576a (119905) = D120578 (119905) + 120576a (119905) yo (119905) = D (119905) (43)

= [xT xT]119879 = [1199091 1199092 119909119899 1199091 1199092 119909119899]119879 (44)

where is an estimate of 120578 L is the observer gain matrixvariable xa is established in (28) term yo is the absoluteacceleration estimated by the observer matrix A is presentedin (36) and matrix D which is an estimate of D in (10) isdefined as

D = [Mminus1K Mminus1C] (45)

with Mminus1K and Mminus1C shown in (37) and (38) respectivelyDefine the state estimation error (119905) as follows (119905) = 120578 (119905) minus (119905) = [x (119905)

x (119905)] minus [x (119905)x (119905)] = [x (119905)x (119905)]= [1199091 1199092 119909119899 1199091 1199092 119909119899]119879

(46)


Then the state estimation error dynamics 120578(119905) is given by120578 (119905) = (119905) minus 120578 (119905)= (A minus LD) 120578 (119905) + (A minus LD) (119905) minus L120576a (119905)= (A minus LD) 120578 (119905) + (A minus LD) (120578 (119905) minus (119905))minus L120576a (119905) = (A minus LD) (119905) + Ξ (119905)

(47)

= Alowast (119905) + Ξ (119905) (48)

with

Alowast = (A minus LD) Ξ (119905) = (A minus LD) 120578 (119905) minus L120576a (119905) (49)

A = A minus AD = D minus D

(50)

Proposition 4 Let L be the gain of the state observer definedas

L = [Ontimesn minus120574Intimesn]119879 (51)

(1) If 120574 satisfies0 le 120574 le 120581 = min [9848581 9848582 984858119899] 984858119894 = 11205772119894 minus 1 119894 = 1 2 119899

(52)

where 120577119894 is presented in (40) and min[sdot] denotes thesmallest value of the set [sdot] then one has the following(a) The eigenvalues of matrixAlowast in (48) are given by120582lowast1198941 = minus120590lowast119894 + 119895120603lowast

119894 120582lowast1198942 = minus120590lowast119894 minus 119895120603lowast119894

(53)

120590lowast119894 = 120577lowast119894 120596lowast119894

120603lowast119894 = 120596lowast

119894radic1 minus 120577lowast2 (54)

120596lowast119894 = 119894radic1 + 120574120577lowast119894 = 120577119894radic1 + 120574 (55)

where 119894 = 1 2 119899 and 119894 and 120577119894 are thenatural frequency and damping factor of the 119894thmode of the estimated building model which arecomputed using (40)

(b) Increasing the value of gain 120574 in (51)-(52) allowsthe real part of the eigenvalues of matrixAlowast to bemore negative

(c) Norm of the estimation error is bounded andwhen 119905 997888rarr infin it satisfies

1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn

11 + 120574 IntimesnminusMminus1K minusMminus1C]]

Mminus1K = Mminus1K minus Mminus1KMminus1C = Mminus1C minus Mminus1C

120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)

where M is a matrix whose columns are theeigenvectors of matrix Alowast

(2) If 120574 997888rarr infin then the eigenvalues of119860lowast approach 120582lowast1198941 =minus119894(2120577119894) and 120582lowast1198942 = minusinfin 119894 = 1 2 119899Proof See Appendix C

61 Attenuation of the State Estimation Error The norm(119905) of the state estimation error can be reduced as follows

(i) Increase 120574 in order to decrease E and to obtain a fastresponse for the observer Based on our experiencegood results in the state estimation are obtained usinga gain 120574 between 12058110 and 120581 where 120581 was defined in(52)

(ii) Reduce Mminus1K and Mminus1C which allows decreas-ing E According to Proposition 3 the smallerthe term 1198685[120582] in (33) is the smaller the parametricerror norms Mminus1K and Mminus1C are Signal 1198685[120582] isfiltered through the filters H1(119904) H2(119904) and H3(119904)shown in (31) They are designed to include thebandwidth of the building responses and to filtermeasurement noise which in turn attenuates the term1198685[120582] and norms Mminus1K and Mminus1C

(iii) Instrument the building at regular intervals over itsheight which permits reducing 120576a [19 33] Thesereferences also show that increasing the total numberof instrumented floors decreases the value of 120576a

(iv) Attenuate themeasurement noise corrupting the stateobserver To achieve this vector xa(119905) in (28) isreplaced by the following filtered vector xaf(119905) invariable of the state observer (42)xaf (119905) = L

minus1 [F1 (119904)] lowast xa (119905) (58)

where lowast is the convolution operator and F1(119904) is afourth-order low-pass Butterworth filter whose cut-off frequency is appropriately chosen


Figure 2 Experimental structure

Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)

Figure 3 Excitation of the experimental structure

7 Experimental Results

Figure 2 depicts a five-story small scale building with dimen-sions 65times55times1755 cmwhich is used during the experimentsThe structure is mounted over a shake table with Parkerlinear motors 406T03LXR and it is built with aluminumwith exception of a column of each floor which is madefrom brass During the experiments the structure is excitedwith the North-South component of the Mexico City 1985earthquake which is fitted in amplitude to be in agreementwith the structure and shown in Figure 3The responses of theshake table and floors are measured through Analog Devicesaccelerometers model ADXL203 placed at every floor andat the base Moreover the absolute position of each floorsis obtained by means of Micro-Epsilon laser sensors modeloptoNCDT 1302 These sensors are only used for comparingthe displacements and velocities of the floors with theirestimates provided by the high gain state observer Filteringthe response of the laser sensors with the filter (75120587)2119904(119904 +75120587)2 produces the nominal velocity of the building floorsFilter F1(119904) in (58) has a cut-off frequency of 100Hz Theparameter identification algorithm and the state observer areimplemented inMatlab-Simulink Data acquisition is carriedout through twoNational Instruments PCI-6221 cards whichcommunicate with a personal computer by means of the

Simulink Desktop Real-Time toolbox It is worth mentioningthat a sampling period of 1ms is used during the experiments

The first two natural frequencies of the small scale build-ing are 1205961 = 1107 and 1205962 = 318 rads which were obtainedfrom frequency response experiments It is considered thatthe first two modes of the structure have a damping factor of1 ie 1205771 = 1205772 = 001 based on experimental data and sinceit is slightly damped During experiments we have observedthat 1205771 and 1205772 vary between 07 and 16 and this variationdepends on the excitation signal In order to compute thecoefficients 1198860 and 1198861 of the Rayleigh damping we decidedto fix both damping ratios 1205771 and 1205772 to 1 since this valueis between 07 and 16 These natural frequencies anddamping factors allow obtaining parameters 1198861 = 01642 and1198862 = 00005 using (6) which are employed for computing thevariables and of parameterization (33)

In order to determine the effectiveness of the cubic splinefunction in the response reconstruction of the119908th noninstru-mented floor the following function E119908 is computed whichwas previously defined in [19]

E119908 = radicsum119875119896=0 [119886119908 (119896) minus 119909119886119908 (119896)]2sum119873

119896=0 2119886119908 (119896) (59)

where 119908119886 and 119909119908119886 are the nominal and estimated absoluteacceleration of the 119908th building floor respectively Valueof function E119908 depends on the number of instrumentedfloors and on their distribution along the building heightLet Π119875

119895 119895 = 1 2 120580 120580 = (119899[119875(119899 minus 119875)]) be all possibledistributions of 119875 instrumented floors of a building with 119899stories Each distribution produces a set of errorsE119908 and themean value of these errors is denoted as E It is consideredthat the smaller the value ofE the better the performance ofthe cubic spline functions

On the other hand let 119894 be the identification error inpercentage () of the 119894th natural frequency 119894 correspondingto estimated building model Performance index 119864 in (60)measures the quality of the identified model by taking intoaccount all errors 119894 ie

119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)

The smaller the value of 119864 the better the quality of themodel Since function 119864 depends on the integration period120575 of the LIF it is useful to compute the following terms119864lowast

= min120575isin[120575min120575max]

119864 (120575) 120575lowast = argmin120575isin[120575min120575max]

119864 (120575) 119891lowast119888 = 1120575lowast

(61)

The performance of the high gain state observer is alsoexamined To this end function 119864x(120575) in (62) which alsodepends on 120575 is computed


Table 2 Results obtained when all floors are instrumentedΠ5119895 Instrumented floors Non-instrumented floors E 119864lowast

120575lowast [s] 119891lowast119888 [Hz] 119864∘

x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818

Table 3 Results with four instrumented floorsΠ4119895 119895 Instrumented floors Non-instrumented floors E 119864lowast


x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083

Table 4 Results with three instrumented floorsΠ3119895 119895 Instrumented floors Non-instrumented floors E 119864lowast


x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 3 5 2 4 048 517 0046 2174 1388 0054 18522 0 2 4 5 1 3 068 521 0048 2083 1198 005 203 0 2 3 5 1 4 063 608 0046 2174 1463 0056 17864 0 1 2 5 3 4 053 855 0054 1852 254 0048 20835 0 1 3 4 2 5 061 903 005 20 2627 0046 21746 0 1 2 4 3 5 055 998 005 20 2183 0044 22737 0 1 4 5 2 3 074 1098 0052 1923 1871 0038 26328 0 2 3 4 1 5 078 1315 007 1429 2588 005 209 0 3 4 5 1 2 098 1457 006 1667 1271 0064 156310 0 1 2 3 4 5 096 ndash ndash ndash ndash ndash ndash

119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)

The better the reconstruction of the state the smaller thevalue of 119864x(120575) The following expressions depending on theminimum of function 119864x(120575) are also computed during theexperiments 119864∘

x = min120575isin[120575min120575max]

119864x (120575) 120575∘ = argmin120575isin[120575min120575max]

119864x (120575) 119891∘119888 = 1120575∘

(63)

The goal of the experiments presented in the next subsec-tion is to obtain and compare the value of the performanceindexes defined above when the small scale building isinstrumented from one to five floors A comparison ofmeasured and estimated values of the state will also beincluded In all the experiments the value of the observer gain120574 is fixed to 120574 = 100071 Results with Five Instrumented Floors This case repre-sents the fully instrumented case where no cubic splinesfunctions are used and it is included for comparison pur-poses Table 2 shows the values of 119864lowast

120575lowast 119864∘x and 120575∘ for

the unique distribution of five instrumented floors which isnamed Π5

1 Moreover Figures 4(a) and 4(b) show respec-tively the variation of the performance indexes 119864(120575) and119864x(120575) where 120575 isin [001 012] Note that 120575lowast = 120575∘ indicatingthat the estimated model with the best quality does notproduce the smallest value of 119864x(120575) The reason is that term120576a in (43) that depends only on the measurement noise inthis case with full instrumentation of the structure affects theperformance of state observer which provokes the fact thatthe identified model with the best quality does not originatethe smallest value 119864∘

x From Figures 4(a) and 4(b) it ispossible to see that 119864(120575) varies within [196 2947] whereasfunction 119864x(120575) takes values within the interval [637 1484]72 Results with Four Instrumented Floors Table 3 presentsthe parameters E 119864lowast

120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are

computed using the distributions Π4119895 119895 = 1 2 5

Furthermore Figures 5(a) and 5(b) depict respectively thefunctions 119864 and 119864x with respect to 120575 From these figuresit is possible to observe that values of 119864(120575) produced bydistributions Π4

119896 119895 = 1 2 4 for 120575 isin [002 005] aresimilar Moreover functions 119864x(120575) corresponding to thesedistributions have a small variation for 120575 isin [005 01] Onthe other hand the smallest value of E is obtained withdistributionΠ4

3 but this layout does not produce the smallestvalues of 119864lowast

and 119864∘x Note that the largest values of 119864(120575) and119864x(120575) for 120575 isin [002 006] are computed with distribution Π4

5ie where the top floor is not instrumented

73 Results with Three Instrumented Floors Table 4 presentsthe values of E 119864lowast

120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 for distributions


0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575

Figure 4 Functions 119864(120575) and 119864x(120575) computed when all the floors are instrumented

0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575

Figure 5 Functions 119864(120575) and 119864x(120575) computed with four instrumented floors

Π3119895 119895 = 1 2 9 of three instrumented floors It is

not possible to identify the building model using layoutΠ310 since the reconstructed absolute acceleration types at

noninstrumented floors produce a regressor vector suchthat covariance matrices P in (34) and (35) do not existOn the other side Figures 6 and 7 show respectively thefunctions 119864 and 119864x with respect to 120575 Note that distributionsΠ3

119895 119895 = 1 2 3 which are uniform or approximately uniformalong the building height allow obtaining small values of 119864lowast

and 119864∘

x which are close to the ones computed with layoutsΠ4119895 119895 = 1 2 3 of four instrumented floors On the other

hand distributions Π3119896 119896 = 7 8 9 that are not uniform over

the building height produce the largest values of E and 119864lowast

however a small value of 119864∘x is computed with Π3

9

Figure 8 depicts the absolute acceleration types 1199091198862 and1199091198864 and the two noninstrumented floors which are recon-structed by the cubic shape function in the configurationΠ3

1It is shown that these responses are close to their nominalvalue On the other side Figure 9 shows the entries 120579119894 119894 =1 2 9 of the estimated vector provided by the RLS

and corresponding to layout Π31 Note that parameters 120579119894119894 = 1 2 9 converge to a neighborhood around a constant

value in approximately 5 s

74 Results with Two Instrumented Floors Values of E 119864lowast120575lowast 119891lowast

119888 119864∘x 120575∘ and 119891∘

119888 for distributions Π2119895 119895 = 1 2 7

are shown in Table 5 Furthermore Figures 10 and 11 presentrespectively the performance indexes 119864 and 119864x computedby varying 120575 Layout Π2

1 with which the building is instru-mented at approximately regular intervals originates thesmallest value of 119864lowast

and small values of E and 119864∘x among

the ones shown in Table 5 On the other hand distributionsΠ25 and Π2

6 that also are approximately uniform along thebuilding height produce large values of 119864lowast

but small valuesof E and 119864∘

x It is worth mentioning that configurations Π2119895 119895 = 7 8 9 10 which are not uniform generate large values of

E Note also that value of 119864lowast computed with the identified

building model corresponding to Π27 is large Moreover it is

not possible to identify the model of the structure throughlayouts Π2

119895 119895 = 8 9 10Figures 12(a) and 12(b) present the displacements 1199095

estimated by the proposed state observer using layouts


0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9

Figure 6 Function 119864(120575) calculated with three instrumented floors

0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9

Figure 7 Function 119864x(120575) computed with three instrumented floors

Π41 and Π2

1 respectively Distribution Π41 was included for

comparison purposes to show a degradation in the quality ofthe displacement reconstruction when usingΠ2

1 In additionFigure 13 compares the velocities 1199095 obtained using thesesamedistributionsNote that the quality of the reconstructionof 1199095 is even worse than that of 1199095 when using Π2

1Note that in the distributionΠ4

1 only the acceleration 3119886is estimated by means of the cubic spline shape functionwhereas in the distribution Π2

1 three acceleration types 11198863119886 and 5119886 are reconstructed Figures 14(a) and 14(b) depictthe acceleration 1199093119886 obtained with the distributions Π4

1 andΠ21 respectively It is shown that both estimates are close to

the nominal acceleration 3119886 On the other hand Figure 15presents the other two estimated acceleration types 1199091119886 and1199095119886 corresponding to the distribution Π2

1 By comparingFigures 14(b) 15(a) and 15(b) it is concluded that theacceleration that is best reconstructed in the distribution Π2

1is 311988675 Results with One Instrumented Floor It is not possible toidentify the model of the building with all distributions of

one instrumented floor since the reconstructed accelerationresponses generate a regressor vector with which thecovariance matrices P in (34) and (35) do not exist

76 Parameter Estimates Obtained with Two Three Fourand Five Instrumented Floors Define (Π119875

119895 120575lowast) as the 119895thdistribution of 119875 instrumented floors with its correspondingintegration period 120575lowast of the LIF that produces 119864lowast

Table 6presents the estimated parameters 120579119894 119894 = 1 2 9 bythe proposed technique with (Π2

1 120575lowast) (Π31 120575lowast) (Π4

1 120575lowast) and(Π51 120575lowast) Moreover this Table shows parameters 119903 120577119903 119903

and 119903 119903 = 1 2 5 computed using those layouts Itis possible to observe that the error in all the estimatednatural frequencies 119903 and damping factors 120577119903 is less than4 Note that for (Π5

1 120575lowast) (Π41 120575lowast) (Π3

1 120575lowast) and (Π21 120575lowast)

the error of the parameter estimates 120579119894 119894 = 1 2 9 isless than 10 20 25 and 73 respectively Moreoverfor all these distributions the maximum error obtained byestimating floor mases and stiffness parameters decreases asthe number of instrumented floors increases For example


Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a

(a) Comparison between 1199092119886 and 1199092119886

Time (s)0 1 2 3 4 5 6 7 8 9 10

minus5minus4minus3minus2minus1

01234

Acce

lera

tion

(mＭ

2)

x4ax4a

(b) Comparison between 1199094119886 and 1199094119886

Figure 8 Reconstructed signals 1199092119886 and 1199094119886 corresponding to the distribution Π31

500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9

Figure 9 Estimated vector produced by the RLS and corresponding to the configuration Π31

0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7

Figure 10 Function 119864(120575) computed with two instrumented floors


0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9

Figure 11 Function 119864x(120575) computed with two instrumented floors

Disp

lace

men

t (m

)

minus004minus003minus002minus001

0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5

(a) State estimate 1199095 corresponding to the distribution Π41

Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5

(b) State estimate 1199095 corresponding to the distribution Π21

Figure 12 Estimates produced by the state observer in the distributions Π41 and Π2

1

the floor masses identified with layouts (Π31 120575lowast) and (Π4

1 120575lowast)have an error less than 18 and 8 respectively

77 Summary of Experimental Results and Discussion Bycomparing the results shown in Sections 71ndash76 the followingpoints are concluded

(i) The term 119891lowast = 1120575lowast tends to decrease as the numberof instrumented floors is reduced Based on this factand in our experience the parameter 120575 should beselected so that 120596119888 = 2120587120575 takes a value withinthe interval 120596119888 isin [18120596maxradic119875119899 28120596maxradic119875119899]where 120596max is the maximum natural frequency of thestructure

(ii) Parameter 119891∘ does not depend on the number ofinstrumented floors During experiments term 119891∘

usually takes values close to 20Hz with either com-plete or reduced instrumentation

(iii) The quality of the estimated model mainly dependson the sensor location over the structure When the

building is instrumented at regular intervals overits height then usually (1) the cubic spline shapefunction yields good results in the reconstruction ofthe unknown responses and (2) the parameter andstate estimators have good performance

(iv) In distributions of sensors that are not uniformalong the building height the cubic spline functionproduces responses with large errors and thereforethe quality of the estimated model is not guaranteedIn some cases it is not even possible to identifythe model and state of the structure using thoseresponses

(v) As expected increasing the number of instrumentedfloors allows decreasing the error of the responsescomputed by the cubic spline function This occurswith an increase in the cost of instrumentationAccording to our experience when the ratio 119875119899 isaround 04ndash06 good results in terms of the perfor-mance indexes 119864(120575) and 119864x(120575) and the reproductionof displacements and velocities can be obtained


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a

(a) Acceleration 1199093119886 corresponding to the distribution Π41


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a

(b) Acceleration 1199093119886 corresponding to the distribution Π21

Figure 14 Acceleration 1199093119886 obtained with the distributions Π41 and Π2

1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a

(a) Acceleration 1199091119886 obtained with the distribution Π21

minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a

(b) Acceleration 1199095119886 obtained with the distribution Π21

Figure 15 Acceleration 1199091119886 and acceleration 1199095119886 corresponding to the distributions Π41 and Π2

1

Values of 119875119899 smaller than 04 may yield numericalproblems as shown in the experimental results for thecase of one instrumented floor

Remark 5 Themethods described in [1 3 6ndash15] report goodresults in the parameter estimation of buildings with ratios

119875119899 up to 05 033 05 043 05 033 025 05 018 05 05and 05 respectively Note that the ratio 119875119899 = 05 is themost common among them Moreover all of these methodsoperate offline with exception of the techniques presented in[3 6] and most of them require transforming the buildingmodel from continuous one to its discrete time counterpart


Table 5 Results with two instrumented floorsΠ2119895119895 Instrumented floors Non-instrumented floors E 119864lowast


x 120575∘ [s] 119891∘119888 [Hz]

1 0 2 4 1 3 5 101 540 0054 1852 207 0042 23812 0 1 4 2 3 5 114 1246 0074 1352 2842 0048 20833 0 1 5 2 3 4 123 1468 0084 1190 2929 004 254 0 3 4 1 2 5 128 1835 006 1667 2537 006 16675 0 2 5 1 3 4 099 1905 0058 1724 2383 0046 21746 0 3 5 1 2 4 114 339 0064 1563 1612 0048 20837 0 4 5 1 2 3 147 3854 006 1667 2271 005 208 0 2 3 1 4 5 141 ndash ndash - ndash ndash ndash9 0 1 3 2 4 5 139 ndash ndash - ndash ndash ndash10 0 1 2 3 4 5 224 ndash ndash ndash ndash ndash ndash

Table 6 Parameter estimates corresponding to the distributions Π21 Π3

1 Π41 and Π5

1 of two three four and five floors

Nominalvalue Unit Estimate(Π5

1 120575lowast) Error

Estimates(Π41 120575lowast) Error

Estimates(Π3



1205791 = 17083 1s2 16480 353 16290 464 16448 372 29405 72131205792 = 11443 1s2 11344 086 11055 339 11218 196 16081 40531205793 = 13409 1s2 13191 163 12629 582 14621 904 11044 17641205794 = 13409 1s2 13377 024 13432 017 15043 1219 12303 8251205795 = 13409 1s2 12575 622 15268 1387 12765 480 11507 14191205796 = 13409 1s2 13078 247 15987 1923 14661 933 11006 17921205797 = 13409 1s2 14413 749 13366 032 16689 2446 13927 3861205798 = 13409 1s2 14566 863 12991 311 13449 030 9244 31061205799 = 13409 1s2 13461 038 13177 173 10650 2057 11440 14681205961 = 1107 rads 1090 151 1086 188 1081 232 1136 2661205962 = 3180 rads 3164 049 3187 025 3166 042 3107 2271205963 = 4886 rads 4846 081 4915 059 4751 276 4821 1331205964 = 6163 rads 6213 081 6080 135 6277 185 6232 1131205965 = 7014 rads 7014 0 7193 255 7237 318 7275 3721205771 = 1 1 0 1 0 1 0 099 11205772 = 1 1 0 1 0 1 0 099 11205773 = 131 13 076 131 0 128 229 13 0761205774 = 157 158 064 155 127 16 191 159 1271205775 = 175 175 0 179 229 18 286 181 3431198981 = 1078 Kg 1078 0 1078 0 1078 0 1078 01198982 = 920 Kg 927 077 944 257 827 1009 1570 70621198983 = 920 Kg 986 720 830 977 975 595 1678 82431198984 = 920 Kg 895 273 993 793 856 692 1326 44161198985 = 920 Kg 968 526 979 642 1081 1753 1072 16481198961 = 18415 Nm 17766 353 17560 464 17731 372 31699 72141198962 = 12336 Nm 12229 087 11917 339 12093 197 17335 40531198963 = 12336 Nm 12402 087 12675 275 12443 087 19312 56551198964 = 12336 Nm 12898 456 13272 757 14291 1585 18471 49731198965 = 12336 Nm 13035 566 12900 457 11516 665 12260 062

Although the ratio 119875119899 = 04 corresponding to the proposedmethod is larger than the ratios119875119899 obtained with the offlinetechniques described in [9 10 12] the proposed techniquecan be implemented online and can detect parameter changesproduced by an anomaly in the structure due to a weakcomponent or failure of elements Moreover unlike themethodology in [3] the proposed method does not needmeasurements of displacements and velocities of the floors

which are more difficult to obtain than floor accelerationmeasurements

8 Conclusions

This paper presented a parameter estimation method anda high gain state observer that respectively identify themodel and state of a shear building using acceleration


measurements from only few floor levels and the responsesat noninstrumented floors which are computed throughspline cubic shape functions The building model obtainedby the LS parameter estimation method is employed bythe state observer The performance of both the observerand the identification method was verified in a five-storyexperimental structure using one two three four and fiveinstrumented floorsThe experimental results give the follow-ing conclusions (1) it is not possible to estimate the modeland state of the structure with only one instrumented floor(2) in general the parameter and state estimators producegood results if the layout of recording sensors is uniformor approximately uniform along the height of the building(3) increasing the number of instrumented floors improvesthe quality of the estimated model and reduces the stateestimation error (4) if the structure is not instrumented atregular or approximately regular intervals over its heightthen a good quality of the identified model is not guaranteedEven in this case it may be not possible to identify the modeland state of the structure (5) a reasonable trade-off betweenestimators performance and the number of instrumentedfloors occurs when three floors are instrumented It is worthmentioning that instrumenting two floors also yields goodresults as long as the distribution is regular

Appendix

A Cubic Spline Shape Function ObtainedUsing the Response at ℎ=119867

Let a building with height 119867 119899 stories be instrumentedat its basement and at 119875 floors and be divided into 119875subintervals where each subinterval is delimited by twoinstrumented floors as shown in Figure 1 The absoluteacceleration at height ℎ within the 119901th subinterval [ℎ119901 ℎ119901+1]119901 = 0 1 2 119875minus1 is estimated through the following cubicshape spline function

119909119886 (ℎ) = 119886119901 + 119887119901 (ℎ minus ℎ119901) + 119889119901 (ℎ minus ℎ119901)2+ 119890119901 (ℎ minus ℎ119901)3 ℎ isin [ℎ119901 ℎ119901+1] (A1)

For the sake of simplicity the argument 119905 of signal 119909119886(ℎ 119905) andparameters 119886119901(119905) 119887119901(119905) 119888119901(119905) 119886119901(119905) has been omitted in thisequation

Coefficients 119886119901 119887119901 119889119901 and 119890119901 are computed by assumingthat the 119875 cubic polynomials in (A1) have continuous firstand second derivatives with respect to the spatial variableℎ This assumption allows obtaining the following equations[34]

Δℎ119901 = ℎ119901+1 minus ℎ119901 119901 = 0 1 2 119875 minus 1 (A2)

119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)

119887119901 = 1Δℎ119901 (119886119901+1 minus 119886119901) minus Δℎ1199013 (2119889119901 + 119889119901+1) 119901 = 0 1 2 119875 minus 1 (A4)

119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)

Δℎ119901minus1119889119901minus1 + 2 (Δℎ119901minus1 + Δℎ119901) 119889119901 + Δℎ119901119889119901+1= 3Δℎ119901 (119903119901+1 minus 119903119901) minus 3Δℎ119901minus1 (119903119901 minus 119903119901minus1) 119901 = 1 2 119875 minus 1

(A6)

First the unknown parameters 119889119901119875119901=0 of the system oflinear equations (A6) are computed Then the constants119887119901119875minus1119901=0 and 119890119901119875minus1119901=0 are obtained using equations (A4) and(A5) respectively In order to solve the system of equations(A6) two extra equations are needed which are deducedfrom the boundary conditions 1015840119886(0) and 10158401015840119886 (119867) Taking intoaccount these conditions produces

1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)

Substituting 1015840119886(0) = 0 and 10158401015840119886 (119867) = 0 given in Table 1into (A7) yields

1198870 = 0 (A8)

119889119875 = 0 (A9)

Substituting 1198870 = 0 into (A4) and using (A3) lead to

1Δℎ0 (1199031 minus 1199030) minus Δℎ03 (21198890 + 1198891) = 0 (A10)

Expressions (A6) (A9) and (A10) can be rewritten inmatrix form as follows

SD = V (A11)

S

=[[[[[[[[[[

2Δℎ0 Δℎ0 0 0 sdot sdot sdot 0Δℎ0 2 (Δℎ0 + Δℎ1) Δℎ1 0 sdot sdot sdot 00 Δℎ1 2 (Δℎ1 + Δℎ2) Δℎ2 sdot sdot sdot 0 d0 0 0 0 sdot sdot sdot 1

]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[

3Δℎ0 (1199031 minus 1199030)3Δℎ1 (1199032 minus 1199031) minus 3Δℎ0 (1199031 minus 1199030)3Δℎ2 (1199033 minus 1199032) minus 3Δℎ1 (1199032 minus 1199031)0

]]]]]]]]]]]]]

(A13)

From (A11) the matrixD is computed as follows

D = Sminus1V (A14)

where the inverse matrix Sminus1 exists since S is a diagonallydominant matrix Once matrixD is obtained the parameters119887119901119875minus1119901=0 and 119890119901119875minus1119901=0 are computed using (A4) and (A5)

B Cubic Spline Shape Function DeducedWhen the Response at ℎ=119867 Is Not Available

In this case the building is divided into 119875 + 1 subintervals asshown in Figure 1(b) where the subinterval Δℎ119875 is given byΔℎ119875 = 119867 minus ℎ119875 (B1)

The absolute acceleration at height ℎwithin the 119901th subinter-val is computed with (A1) where 119901 = 0 1 2 119875

Since the cubic polynomials corresponding to the subin-tervals Δℎ119875minus1 and Δℎ119875 have the same spatial derivative atℎ = ℎ119875 the next equality is obtained [34]1015840119886 (119867) = 119887119875 = 119887119875minus1 + 2119889119875minus1Δℎ119875minus1 + 3119890119875minus1Δ2

119875minus1 (B2)

Moreover the cubic spline function has the followingsecond and third derivatives with respect to ℎ at the boundary119867 10158401015840119886 (119867) = 2119889119875 + 6119890119875Δ119875 (B3)

101584010158401015840119886 (119867) = 6119890119875 (B4)

Substituting the boundary conditions 10158401015840119886 (119867) = 101584010158401015840119886 (119867) =0 given in Table 1 into (B3) and (B4) yields119889119875 = 0 (B5)

119890119875 = 0 (B6)

Moreover the boundary condition 1015840119886(0) = 0 in Table 1produces expression (A10)

Coefficients 119889119901119875119901=0 are computed by solving the set ofequations (A6) (A10) and (B5) Subsequently the param-eters 119887119901119875119901=0 and 119890119901119875119901=0 are obtained using (A4) (B2) and(A5) (B6) respectively

C Proof of Proposition 4

Proof Substituting the gain L = [Ontimesn minus120574Intimesn]119879 into term(A minus LD) of (47) produces120578 (119905) = Alowast (119905) + Ξ (119905) (C1)

Alowast = (A minus LD)= [ Ontimesn Intimesnminus (1 + 120574) Mminus1K minus (1 + 120574) Mminus1C

] (C2)

where matrices A and D are defined in (36) and (45)respectively

Using expression (119905) in (46) permits writing the homo-geneous part of (C1) as follows

x (119905) + (1 + 120574) Mminus1Cx (119905) + (1 + 120574) Mminus1Kx (119905)= Ontimes1

(C3)


Mminus1C = Mminus1CMminus1K = Mminus1K

(C4)

Using these definitions it is possible to write (C3) as1(1 + 120574)Mx (119905) + Cx (119905) + Kx = Ontimes1 (C5)

Modal analysis allows expressing (C5) as the following 119899uncoupled differential equations1(1 + 120574)119894∙ 119902119894 (119905) + 119888119894∙ 119902119894 (119905) + 119894∙119902119894 (119905) = 0

119894 = 1 2 119899 (C6)

where 119902119894(119905) denotes modal coordinate and parameters 119894∙119888119894∙ and 119894∙ are estimates of the modal mass damping andstiffness which are defined as

x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)

where 119894 = 1 2 119899 variable 120595i is the natural mode vectorcorresponding to the 119894th natural frequency 119894 (40) of theestimated building model ie

[K minus 2119894 M]120595i = Ontimes1 119894 = radic 119894∙119894∙

119894 = 1 2 119899 (C8)


Equation (C6) is equivalent to the following expression

119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)

where 119894 = 1 2 119899 and 120577119894 are the modal damping ratios ofthe identified building model

Using equalities in (55) it is possible to write (C9) as

119902119894 + 2120577lowast119894 120596lowast119894 119902119894 + 120596lowast2

119894 119902119894 = 0 (C10)

The characteristic equation corresponding to the 119894thdifferential equation in (C10) is given by


119894 = 0 119894 = 1 2 119899 (C11)

The roots 120582lowast1198941 and 120582lowast1198942 119894 = 1 2 119899 of (C11) or equivalentlythe eigenvalues of Alowast in (C2) are given by expressions(53)ndash(55) Therefore point (1)(a) of Proposition 4 has beenproved

On the other hand substituting equalities in (55) into thecharacteristic equation (C11) gives

1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)

Equation (C12) can be rewritten as

1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)

According to the root locus technique the two roots 120582lowast1198941and 120582lowast1198942 in (53)ndash(55) of each characteristic equation in (C13)traverse the path shown in Figure 16 It is possible to see thatroot locus has a break-in point where the two roots 120582lowast1198941 and120582lowast1198942 return to the real axis Let 984858119894 be the value of the gain 120574in which the two roots 120582lowast1198941 and 120582lowast1198942 of the 119894th characteristicequation become realThis gain 984858119894 is computed by consideringas zero the imaginary part of 120582lowast1198941 and 120582lowast1198942 ie

120596lowast119894radic1 minus 120577lowast2 = 0 997904rArr120577lowast2 = 1 997904rArr(1 + 120574) 1205772 = 1 997904rArr

984858119894 = 11205772119894 minus 1(C14)

Thus increasing 120574 from 0 to 984858119894 allows moving the real partof the roots 120582lowast1198941 and 120582lowast1198942 further and further away from theleft side of the real axis Therefore if 120574 satisfies 0 le 120574 le 120581 =min[9848581 9848582 984858119899] then the real part of the roots 120582lowast1198941 and 120582lowast1198942

Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i

Figure 16 Root locus for the 119894th characteristic equation in (C13)

119894 = 1 2 119899 becomes more and more negative as the gain 120574varies from zero to 120581 Thus point (1)(b) of Proposition 4 hasbeen proved

From Figure 16 it is possible to see that roots 120582lowast1198941 and120582lowast1198942 of the 119894th characteristic equation converge to the zerosofZ119894(119904) as 120574 997888rarr infin which are given by

1199111198941 = minus 1198942120577119894 1199111198942 = minusinfin 119894 = 1 2 119899(C15)

Therefore point (2) of Proposition 4 has been provedIn order to prove point (1)(c) the solution of (48) is used

which is given by

(119905) = eAlowastt (0) + int119905

0eAlowast(tminus120591)Ξ (120591) 119889120591 (C16)

where eAlowastt is the state transition matrix of system (48)Since all the eigenvalues 120582lowast1198941 and 120582lowast1198942 119894 = 1 2 119899 ofAlowast

are different the matrix eAlowastt can be rewritten as

eAlowastt = MeDlowasttMminus1 (C17)

where columns ofmatrixM are the eigenvectors correspond-ing to the eigenvalues 120582lowast1198941 and 120582lowast1198942 119894 = 1 2 119899 moreovermatrix eDlowastt is given by

eDlowastt = [[[[[[[[

119890120582lowast11119905 0 sdot sdot sdot 00 119890120582lowast12119905 sdot sdot sdot 0 d0 0 sdot sdot sdot 119890120582lowast1198992119905]]]]]]]]= eRlowastteIlowastt (C18)

= [[[[[[[[

119890minus120590lowast1 119905 0 sdot sdot sdot 00 119890minus120590lowast1 119905 sdot sdot sdot 0 d0 0 sdot sdot sdot 119890minus120590lowast119899 119905]]]]]]]]

[[[[[[[[

119890119895120603lowast1 119905 0 sdot sdot sdot 00 119890minus119895120603lowast1 119905 sdot sdot sdot 0 d0 0 sdot sdot sdot 119890minus119895120603lowast119899]]]]]]]](C19)

Substituting the equality eDlowastt = eRlowastteIlowastt (C18) into (C17)yields

eAlowastt = MeRlowastteIlowasttMminus1 (C20)


Matrix eAlowastt (C20) satisfies

10038171003817100381710038171003817eAlowastt10038171003817100381710038171003817 = 10038171003817100381710038171003817MeRlowastteIlowasttMminus110038171003817100381710038171003817 le 120583119890minus120590lowast119898119905120590lowast119898 = min1le119894le119899

120590lowast119894 (C21)

120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 10038171003817100381710038171003817eRlowastt10038171003817100381710038171003817 le 119890minus120590lowast11989811990510038171003817100381710038171003817eIlowastt10038171003817100381710038171003817 = 1(C22)

where sdot = sdot 119901 is the induced matrix norm and 119901 can be1 2 orinfin

The following inequality follows from (C21)

10038171003817100381710038171003817eAlowastt (0)10038171003817100381710038171003817 le 120583 1003817100381710038171003817 (0)1003817100381710038171003817 119890minus120590lowast119898119905 (C23)

Applying the matrix norm on both sides of (C16) andusing inequalities (C21) and (C23) produce

1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)

Substituting the gain L (51) into Ξ(119905) defined in (49) leadsto

Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn

11 + 120574 IntimesnminusMminus1K minusMminus1C]]100381710038171003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817120578 (119905)1003817100381710038171003817

+ 120574 100381710038171003817100381710038171003817[Ontimesn minusIntimesn]119879100381710038171003817100381710038171003817 1003817100381710038171003817120576a (119905)1003817100381710038171003817 100381710038171003817100381710038171003817[Ontimesn minusIntimesn]119879100381710038171003817100381710038171003817 = 1(C25)

where Mminus1K and Mminus1C are defined in (57)Substituting inequality (C25) into (C24) and taking the

limit of (119905) as 119905 997888rarr infin yield (56) Thus finally the point(1)(c) of Proposition 4 has been proved

Nomenclature

LIF Linear Integral FilterLS Least SquaresRLS Recursive Least Squares119899 isin N Number of floors of a buildingM isin R119899119909119899 Mass matrix of the building kgC isin R119899119909119899 Damping matrix of the building

NsdotsmK isin R119899119909119899 Stiffness matrix of the building Nmx(119905) isin R1198991199091 Displacement vector of the floors

relative to the building basement mx(119905) isin R1198991199091 Velocity vector of the floors relative

to the building basement msx(119905) isin R1198991199091 Acceleration vector of the floors

relative to the building basementms2119892(119905) isin R Ground acceleration produced by anearthquake ms2

l isin R1198991199091 Vector of onesxa(119905) isin R1198991199091 Absolute acceleration vector of the

floors ms21198860 isin R Rayleigh damping coefficient 1s1198861 isin R Rayleigh damping coefficient s120596119894 isin R+ Natural frequency of the 119894th moderads120577119894 isin R+ Damping ratio of the 119894th mode

A isin R21198991199092119899 State matrix of a seismically excitedbuilding

B isin R21198991199091 Input vector of a seismically excitedbuilding

D isin R1198991199092119899 Output matrix of a seismicallyexcited building

120578(119905) isin R21198991199091 State of a seismically excited building119875 isin N Number of instrumented floorsΓ isin R119875119909119899 Localization matrix of accelerometersu(119905) isin R119899119909(2119899minus1) Matrix composed of acceleration of

floors relative to the buildingbasement ms2

120579 isin R(2119899minus1)1199091 Stiffnessmass vector to be identifiedN(kgsdotm) Superscript used in estimatedparameters or signals119868120572120588(119905) LIF representing 120572 integrals overfinite time intervals of 120588(119905)120575 isin R+ Time integration window length ofthe LIF s119879119904 isin R+ Sampling period s

L[sdot] Laplace operator120601(119905) isin R119899119909(2119899minus1) Regressor vector120594(119905) isin R1198991199091 Output of the parameterization

model119866120572(119904) 120572th-order LIF120596119888 isin R+ Cut-off frequency of the filter 119866120572(119904)rads119891119888 isin R+ Cut-off frequency of the filter 119866120572(119904)Hz

H119894(119904) 119894 = 1 2 3 Band pass filters


119867 isin R+ Building heightℎ119901 isin R+ Height of the 119901th instrumented floor119903119901 isin R Absolute acceleration of the 119901thinstrumented floor ms2(ℎ 119905) isin R Absolute acceleration at height ℎ ofthe building ms2119908119886(119905) isin R Absolute acceleration at the 119908thnoninstrumented floor ms2ℎlowast isin R+ Height at the 119908th noninstrumentedfloorΔ119901 isin R+ Subinterval delimited by twoinstrumented floors119886119901 119887119901 119889119901 119890119901 Coefficients of the 119901th cubicpolynomial

120598(119905) isin R1198991199091 Error between estimated andnominal relative acceleration ms2

120598a(119905) isin R1198991199091 Error between estimated andnominal absolute acceleration ms2

120598u(119905) isin R119899119909(2119899minus1) Matrix composed of errors betweenestimated and nominal relativeacceleration ms2

120582(119905) isin R1198991199091 Signal depending on the accelerationestimation errors119873 isin N Number of samples used by the LS

P isin R(2119899minus1)119909(2119899minus1) Covariance matrix of the LS or RLSL isin R1198991199092119899 State observer gain matrix120574 isin R+ Multiplying factor of the state

observer gain matrixΠ119875119895 119895th distribution of 119875 instrumented

floorsE119908 isin R+ Metric to determine the effectiveness

in the reconstruction of the 119908thnoninstrumented floor

E isin R+ Mean value of a set of errorsE119908 of adistribution of instrumented floors119894 isin R+ Identification error in percentage ofthe 119894th estimated natural frequency119864(120575) isin R+ Performance index that measures thequality of the identified model119864lowast

isin R+ Minimum value of 119864(120575)120575lowast isin R+ Time integration window length thatproduces the minimum value of119864(120575) s119891lowast

119888 isin R+ The reciprocal of 120575lowast Hz119864x(120575) isin R+ Performance of the high gain stateobserver119864∘

x isin R+ Minimum value of function 119864x(120575)120575∘ isin R+ Time integration window length thatproduces the minimum value of119864x(120575) s119891∘

119888 isin R+ The reciprocal of 120575∘ Hz

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Antonio Concha acknowledges Programa para el Desar-rollo Profesional Docente (PRODEP-SEP) for supportingthis research Luis Alvarez-Icaza acknowledges support fromUNAM-PAPIIT Grant IN104218

References

[1] F Amini andYHedayati ldquoUnderdetermined blindmodal iden-tification of structures by earthquake and ambient vibrationmeasurements via sparse component analysisrdquo Journal of Soundand Vibration vol 366 pp 117ndash132 2016

[2] M Hoshiya and E Saito ldquoStructural identification by extendedKalman filterrdquo Journal of Engineering Mechanics vol 110 no 12pp 1757ndash1770 1984

[3] C G Koh L M See and T Balendra ldquoEstimation of structuralparameters in time domain A substructure approachrdquo Earth-quake Engineering amp Structural Dynamics vol 20 no 8 pp787ndash801 1991

[4] A Corigliano and S Mariani ldquoParameter identification inexplicit structural dynamics Performance of the extendedKalman filterrdquo Computer Methods Applied Mechanics and Engi-neering vol 193 no 36-38 pp 3807ndash3835 2004

[5] S J Ghosh D Roy and C S Manohar ldquoNew forms of extendedKalman filter via transversal linearization and applications tostructural system identificationrdquo Computer Methods AppliedMechanics and Engineering vol 196 no 49-52 pp 5063ndash50832007

[6] J Zhou A Mita and L Mei ldquoAn improved bayesian structuralidentification using the first two derivatives of log-likelihoodmeasurerdquo Journal of Structures vol 2015 pp 1ndash9 2015

[7] S Mukhopadhyay H Lus and R Betti ldquoStructural identifica-tion with incomplete instrumentation and global identifiabilityrequirements under base excitationrdquo Structural Control andHealth Monitoring vol 22 no 7 pp 1024ndash1047 2015

[8] P Yuan Z Wu and X Ma ldquoEstimated mass and stiffnessmatrices of shear building from modal test datardquo EarthquakeEngineering amp Structural Dynamics vol 27 no 5 pp 415ndash4211998

[9] J-N Juang M Phan L G Horta and R W LongmanldquoIdentification of observerKalman filter Markov parameterstheory and experimentsrdquo Journal of Guidance Control andDynamics vol 16 no 2 pp 320ndash329 1993

[10] H Lus R Betti and R W Longman ldquoIdentification of linearstructural systems using earthquake-induced vibration datardquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1449ndash1467 1999

[11] F Vicario M Q Phan R Betti and R W Longman ldquoOutput-only observerKalman filter identification (O3KID)rdquo StructuralControl and Health Monitoring vol 22 no 5 pp 847ndash872 2015

[12] Y Kaya S Kocakaplan and E Safak ldquoSystem identification andmodel calibration of multi-story buildings through estimationof vibration time histories at non-instrumented floorsrdquo Bulletinof Earthquake Engineering vol 13 no 11 pp 3301ndash3323 2015


[13] G Hegde and R Sinha ldquoMethod of modal identification oftorsionally-coupled buildings using earthquake responsesrdquo inProceedings of the The 14th World Conference on EarthquakeEngineering pp 1ndash8 2008

[14] C S Huang ldquoStructural identification from ambient vibrationmeasurement using the multivariate AR modelrdquo Journal ofSound and Vibration vol 241 no 3 pp 337ndash359 2001

[15] S Chakraverty ldquoIdentification of structural parameters ofmultistorey shear buildings from modal datardquo EarthquakeEngineering amp Structural Dynamics vol 34 no 6 pp 543ndash5542005

[16] H Garnier and L Wang Identification of Continuous-timeModels from SampledData Springer-Verlag LondonUK 2008

[17] R Garrido and A Concha ldquoEstimation of the parameters ofstructures using acceleration measurementsrdquo in Proceedings ofthe Universite Libre de Bruxelles pp 1791ndash1796 Belgium July2012

[18] M P Limongelli ldquoOptimal location of sensors for reconstruc-tion of seismic responses through spline function interpola-tionrdquo Earthquake Engineering amp Structural Dynamics vol 32no 7 pp 1055ndash1074 2003

[19] M P Limongelli ldquoPerformance evaluation of instrumentedbuildingsrdquo ISET Journal of Earthquake Technology vol 42 no2-3 pp 47ndash61 2005

[20] F E Udwadia ldquoMethodology for optimum sensor locationsfor parameter identification in dynamic systemsrdquo Journal ofEngineering Mechanics vol 120 no 2 pp 368ndash390 1994

[21] E Heredia-Zavoni and L Esteva ldquoOptimal instrumentationof uncertain structural systems subject to earthquake groundmotionsrdquo Earthquake Engineering amp Structural Dynamics vol27 no 4 pp 343ndash362 1998

[22] M M Abdullah A Richardson and J Hanif ldquoPlacement ofsensorsactuators on civil structures using genetic algorithmsrdquoEarthquake Engineering amp Structural Dynamics vol 30 no 8pp 1167ndash1184 2001

[23] J Li and S S Law ldquoSubstructural response reconstruction inwavelet domainrdquo Journal of Applied Mechanics vol 78 no 4Article ID 041010 10 pages 2011

[24] S S Law J Li and Y Ding ldquoStructural response reconstructionwith transmissibility concept in frequency domainrdquoMechanicalSystems and Signal Processing vol 25 no 3 pp 952ndash968 2011

[25] J He X Guan and Y Liu ldquoStructural response reconstructionbased on empirical mode decomposition in time domainrdquoMechanical Systems and Signal Processing vol 28 pp 348ndash3662012

[26] Z Wan S Li Q Huang and T Wang ldquoStructural responsereconstruction based on the modal superposition method inthe presence of closely spaced modesrdquoMechanical Systems andSignal Processing vol 42 no 1-2 pp 14ndash30 2014

[27] A K Chopra Dynamics of Structures Theory and Applicationsto Earthquake Engineering Prentice Hall Englewood Cliffs NJUSA 2001

[28] G A Papagiannopoulos and G D Hatzigeorgiou ldquoOn the useof the half-power bandwidth method to estimate damping inbuilding structuresrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 7 pp 1075ndash1079 2011

[29] J Butterworth J H Lee and B Davidson ldquoExperimentaldetermination ofmodal damping from full scale testingrdquo inPro-ceedings of the 13th world conference on earthquake engineeringvol 310 Vancouver Canada 2004

[30] A Devin and P Fanning ldquoImpact of nonstructural componentson modal response and structural dampingrdquo in Topics on theDynamics of Civil Structures vol 1 pp 415ndash421 Springer NewYork NY USA 2012

[31] T Soderstrom and P Stoica System Identification Prentice HallNew York NY USA 1989

[32] P Ioannou and B Fidan Adaptive Control Tutorial vol 11SIAM Philadelphia USA 2006

[33] R K Goel ldquoMode-based procedure to interpolate strongmotion records of instrumented buildingsrdquo ISET Journal ofEarthquake Technology vol 45 no 3-4 pp 97ndash113 2008

[34] R L Burden and J D Faires ldquoAnalisis numericordquo ThomsonLearning 2002

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom


techniques which determine the Markov parameters of aKalman observer in order to identity a building modelThesetechniques identify a discrete timemodel of the structure butif the sampling frequency of responses is too high in relationto the dominate frequencies of the structure then the poles ofthe estimated model lie close to the unit circle in the complexdomain whichmay produce parameter estimates statisticallyill-defined [16] Kaya el al [12] identify a structure usingthe Transfer Matrix formulation of the response where thehistories at noninstrumented floors are offline reconstructedusing the Mode Shape Based Estimation (MSBE) methodwhich assume that modal shapes can be approximated as alinear combination of the mode shapes of a shear beam anda bending beam It is worth mentioning that using MSBErequires solving two partial differential equations and theknowledge of the modal acceleration of the instrumentedfloors Hegde and Sinha [13] estimate the modal param-eters of a seismically excited torsionally coupled buildingusing limited number of responses modal parameters areextracted using an offline estimation methodology basedon the principles of the Natural Excitation Technique andthe Eigen Realization Algorithm authors use a cubic shapefunction for estimating the responses at noninstrumentedfloors On the other hand [14 15] carried out structuralidentification using ambient vibration measurements andoffline algorithms Huang [14] proposes a procedure foridentifying the dynamics characteristics of a shear buildingusing the multivariate ARVmodel whereas Chakraverty [15]estimates mass and stiffness parameters frommodal test dataand the Holzer criteria

This manuscript presents (1) an identification techniqueto identify the parameters of the model of a seismicallyexcited shear building using a limited number of responsesand (2) a high gain state observer that employs the identi-fied parameter in order to estimate the displacements andvelocities of both instrumented and noninstrumented floorsThe parameter and state estimators rely on the accelerationmeasurements of ground and some instrumented floors Theacceleration at noninstrumented floors is reconstructed bymeans of cubic shape spline functions that use the availablemeasurements The state observer gain is easily designed anddepends on a positive parameter that guarantees the stabilityand a fast response of the observer On the other handthe parameter estimator uses a parameterization with thefollowing features (1) assuming that building has Rayleighdamping (2) containing a vector whose entries are thestiffnessmass ratios of the building which are estimatedthrough a Least Squares algorithm and (3) employing LinearIntegral Filters (LIF) that attenuate low and high frequencynoise of the measured and reconstructed responses The LIFwere previously employed in Garrido and Concha [17] forparameter estimation of fully instrumented structures In thismanuscript the application of these filters is extended toestimate the model of buildings instrumented at only fewfloor levels Moreover by considering Rayleigh damping forthe structure the number of parameters of the proposedparameterization is the half of the one corresponding to theparameterization in [17] In comparison with techniques in[7ndash15] described above the proposed parameter estimator

allows attenuating low and high frequency noise of bothmeasured and reconstructed acceleration and can be imple-mented online at a low computational effort

It is worth mentioning that the cubic spline functionsemployed by the proposed parameter and state estimatorsare designed following the ideas in [18 19] Moreover thedesign of these functions does not require the knowledge ofthe modal shapes of the structure and they allow recursiveestimation of the responses at noninstrumented floors Tech-niques in [20ndash26] are also able to estimate the unmeasuredfloor level responses but unlike the cubic spline functionsthey are implemented offline and require an estimated build-ing model which may be unavailable

The paper has the following structure Sections 2 and 3present the shear building model and its parameterizationrespectively The cubic spline functions used for estimatingthe floor responses at noninstrumented floors are describedin Section 4 Section 5 shows the methodology for estimatingthe building parameters The proposed high gain observeris described in Section 6 Experiment results using theparameter and state estimators are shown in Section 7Finally Section 8 includes the conclusions of this paper

2 Mathematical Model of a Shear Building

The dynamics of a shear building with 119899 stories subjected toearthquake or basemotion is described through the followingmathematical model [27]

M (x (119905) + l119892 (119905)) + Cx (119905) + Kx (119905) = 0 (1)

x (119905) = [1199091 (119905) 119909119899 (119905)]119879 x (119905) = [1 (119905) 119899 (119905)]119879x (119905) = [1 (119905) 119899 (119905)]119879 l = [1 1 1]119879

(2)

xa (119905) = x (119905) + l119892 (119905) = [1198861 (119905) 119886119899 (119905)]119879 (3)

where variables 119909119894(119905) 119894(119905) and 119894(119905) are respectively thedisplacement velocity and acceleration of the 119894th floor whichare measured with respect to the basement Signal 119886119894(119905)represents the absolute acceleration at the 119894th floor and119892(119905) is the ground acceleration produced by earthquakeMoreover M and K are the mass and stiffness matricesrespectively which are defined as

M = [[[[[[[

1198981 0 sdot sdot sdot 00 1198982 sdot sdot sdot 0 d0 0 sdot sdot sdot 119898119899

]]]]]]]

K = [[[[[[[

1198961 + 1198962 minus1198962 sdot sdot sdot 0minus1198962 1198962 + 1198963 sdot sdot sdot 0 d0 0 sdot sdot sdot 119896119899]]]]]]]

(4)




C = 1198860M + 1198861K (5)


12 [[[[1120596119894

1205961198941120596119895

120596119895

]]]][11988601198861] = [120577119894120577119895] (6)








(119905) = A120578 (119905) + B119892 (119905) (7)



120578 (119905) = [xT (119905) xT (119905)]119879= [1199091 (119905) 1199092 (119905) 119909119899 (119905) 1 (119905) 2 (119905) 119899 (119905)]119879 (9)







(11)









11989621198981

11989621198982

sdot sdot sdot 119896119899119898119899



(15)


xa (119905) + 1198860x (119905) = [1198861u (119905) + u (119905)] 120579 (16)



= 1120575120572minus1 int119905

119905minus120575int1205911






L [119868120572 120588 (119905)] = 120575119866120572 (119904)L [120588 (119905)] (20)

119866120572 (119904) = (1 minus 119890minus120575119904120575119904 )120572 1003816100381610038161003816119866120572 (119895120596)1003816100381610038161003816 = 100381610038161003816100381610038161003816100381610038161003816 sin (120587120596120596119888)(120587120596120596119888)

100381610038161003816100381610038161003816100381610038161003816120572

120596119888 = 2120587120575 119891119888 = 1120575

(21)



120594 (119905) = 120601 (119905) 120579 (22)

120594 (119905) = 11205753 11986823sum

119895=0

(3119895) (minus1)119895 xa (119905 minus 119895120575)+ 11988601205752 1198683

2sum119895=0

(2119895) (minus1)119895 x (119905 minus 119895120575)(23)

120601 (119905) = 11988611205752 11986832sum

119895=0

(2119895) (minus1)119895 u (119905 minus 119895120575)+ 11205751198684

1sum119895=0

(1119895) (minus1)119895 u (119905 minus 119895120575)(24)



whereH1 (119904) = 1205751199041198665 (119904) H2 (119904) = 12057511990421198665 (119904) H3 (119904) = 12057511990431198665 (119904)

(26)








(27)




ℎP = H

ℎPminus1

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

Δℎ2

Δℎ1

Δℎ0


H

ℎPminus1

ℎP

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

ΔℎP

Δℎ2

Δℎ1

Δℎ0




x (119905) = x (119905) + 120576 (119905)xa (119905) = xa (119905) + 120576a (119905)u (119905) = u (119905) + 120576u (119905) (28)





(119905) = (119905) 120579 + 1198685 [120582 (119905)] (29)



(30)




(119896119879119904) = (119896119879119904) 120579 + 1198685 [120582 (119896119879119904)] (32)


(119896) = (119896) 120579 + 1198685 [120582 (119896)] (33)


= P (119873) 119873sum119896=0

119879 (119896) (119896)

P (119873) = [ 119873sum119896=0

119879 (119896) (119896)]minus1 (34)







(35)





Mminus1K = [[[[[[[[


]]]]]]]](37)









119894 120577119894 = 120590119894119894

119894 = 1 2 119899

(40)


119894 = 119894120579(2119894minus1) 119894 = 2 3 119899C = 1198860M + 1198861K

(41)




xa (119905) = xa (119905) + 120576a (119905) = D120578 (119905) + 120576a (119905) yo (119905) = D (119905) (43)

= [xT xT]119879 = [1199091 1199092 119909119899 1199091 1199092 119909119899]119879 (44)




x (119905)] minus [x (119905)x (119905)] = [x (119905)x (119905)]= [1199091 1199092 119909119899 1199091 1199092 119909119899]119879

(46)



(47)

= Alowast (119905) + Ξ (119905) (48)

with



(50)




(52)



(53)


120603lowast119894 = 120596lowast






1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn



120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)








minus1 [F1 (119904)] lowast xa (119905) (58)




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





C = 1198860M + 1198861K (5)


12 [[[[1120596119894

1205961198941120596119895

120596119895

]]]][11988601198861] = [120577119894120577119895] (6)








(119905) = A120578 (119905) + B119892 (119905) (7)



120578 (119905) = [xT (119905) xT (119905)]119879= [1199091 (119905) 1199092 (119905) 119909119899 (119905) 1 (119905) 2 (119905) 119899 (119905)]119879 (9)







(11)









11989621198981

11989621198982

sdot sdot sdot 119896119899119898119899



(15)


xa (119905) + 1198860x (119905) = [1198861u (119905) + u (119905)] 120579 (16)



= 1120575120572minus1 int119905

119905minus120575int1205911






L [119868120572 120588 (119905)] = 120575119866120572 (119904)L [120588 (119905)] (20)

119866120572 (119904) = (1 minus 119890minus120575119904120575119904 )120572 1003816100381610038161003816119866120572 (119895120596)1003816100381610038161003816 = 100381610038161003816100381610038161003816100381610038161003816 sin (120587120596120596119888)(120587120596120596119888)

100381610038161003816100381610038161003816100381610038161003816120572

120596119888 = 2120587120575 119891119888 = 1120575

(21)



120594 (119905) = 120601 (119905) 120579 (22)

120594 (119905) = 11205753 11986823sum

119895=0

(3119895) (minus1)119895 xa (119905 minus 119895120575)+ 11988601205752 1198683

2sum119895=0

(2119895) (minus1)119895 x (119905 minus 119895120575)(23)

120601 (119905) = 11988611205752 11986832sum

119895=0

(2119895) (minus1)119895 u (119905 minus 119895120575)+ 11205751198684

1sum119895=0

(1119895) (minus1)119895 u (119905 minus 119895120575)(24)



whereH1 (119904) = 1205751199041198665 (119904) H2 (119904) = 12057511990421198665 (119904) H3 (119904) = 12057511990431198665 (119904)

(26)








(27)




ℎP = H

ℎPminus1

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

Δℎ2

Δℎ1

Δℎ0


H

ℎPminus1

ℎP

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

ΔℎP

Δℎ2

Δℎ1

Δℎ0




x (119905) = x (119905) + 120576 (119905)xa (119905) = xa (119905) + 120576a (119905)u (119905) = u (119905) + 120576u (119905) (28)





(119905) = (119905) 120579 + 1198685 [120582 (119905)] (29)



(30)




(119896119879119904) = (119896119879119904) 120579 + 1198685 [120582 (119896119879119904)] (32)


(119896) = (119896) 120579 + 1198685 [120582 (119896)] (33)


= P (119873) 119873sum119896=0

119879 (119896) (119896)

P (119873) = [ 119873sum119896=0

119879 (119896) (119896)]minus1 (34)







(35)





Mminus1K = [[[[[[[[


]]]]]]]](37)









119894 120577119894 = 120590119894119894

119894 = 1 2 119899

(40)


119894 = 119894120579(2119894minus1) 119894 = 2 3 119899C = 1198860M + 1198861K

(41)




xa (119905) = xa (119905) + 120576a (119905) = D120578 (119905) + 120576a (119905) yo (119905) = D (119905) (43)

= [xT xT]119879 = [1199091 1199092 119909119899 1199091 1199092 119909119899]119879 (44)




x (119905)] minus [x (119905)x (119905)] = [x (119905)x (119905)]= [1199091 1199092 119909119899 1199091 1199092 119909119899]119879

(46)



(47)

= Alowast (119905) + Ξ (119905) (48)

with



(50)




(52)



(53)


120603lowast119894 = 120596lowast






1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn



120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)








minus1 [F1 (119904)] lowast xa (119905) (58)




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





L [119868120572 120588 (119905)] = 120575119866120572 (119904)L [120588 (119905)] (20)

119866120572 (119904) = (1 minus 119890minus120575119904120575119904 )120572 1003816100381610038161003816119866120572 (119895120596)1003816100381610038161003816 = 100381610038161003816100381610038161003816100381610038161003816 sin (120587120596120596119888)(120587120596120596119888)

100381610038161003816100381610038161003816100381610038161003816120572

120596119888 = 2120587120575 119891119888 = 1120575

(21)



120594 (119905) = 120601 (119905) 120579 (22)

120594 (119905) = 11205753 11986823sum

119895=0

(3119895) (minus1)119895 xa (119905 minus 119895120575)+ 11988601205752 1198683

2sum119895=0

(2119895) (minus1)119895 x (119905 minus 119895120575)(23)

120601 (119905) = 11988611205752 11986832sum

119895=0

(2119895) (minus1)119895 u (119905 minus 119895120575)+ 11205751198684

1sum119895=0

(1119895) (minus1)119895 u (119905 minus 119895120575)(24)



whereH1 (119904) = 1205751199041198665 (119904) H2 (119904) = 12057511990421198665 (119904) H3 (119904) = 12057511990431198665 (119904)

(26)








(27)




ℎP = H

ℎPminus1

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

Δℎ2

Δℎ1

Δℎ0


H

ℎPminus1

ℎP

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

ΔℎP

Δℎ2

Δℎ1

Δℎ0




x (119905) = x (119905) + 120576 (119905)xa (119905) = xa (119905) + 120576a (119905)u (119905) = u (119905) + 120576u (119905) (28)





(119905) = (119905) 120579 + 1198685 [120582 (119905)] (29)



(30)




(119896119879119904) = (119896119879119904) 120579 + 1198685 [120582 (119896119879119904)] (32)


(119896) = (119896) 120579 + 1198685 [120582 (119896)] (33)


= P (119873) 119873sum119896=0

119879 (119896) (119896)

P (119873) = [ 119873sum119896=0

119879 (119896) (119896)]minus1 (34)







(35)





Mminus1K = [[[[[[[[


]]]]]]]](37)









119894 120577119894 = 120590119894119894

119894 = 1 2 119899

(40)


119894 = 119894120579(2119894minus1) 119894 = 2 3 119899C = 1198860M + 1198861K

(41)




xa (119905) = xa (119905) + 120576a (119905) = D120578 (119905) + 120576a (119905) yo (119905) = D (119905) (43)

= [xT xT]119879 = [1199091 1199092 119909119899 1199091 1199092 119909119899]119879 (44)




x (119905)] minus [x (119905)x (119905)] = [x (119905)x (119905)]= [1199091 1199092 119909119899 1199091 1199092 119909119899]119879

(46)



(47)

= Alowast (119905) + Ξ (119905) (48)

with



(50)




(52)



(53)


120603lowast119894 = 120596lowast






1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn



120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)








minus1 [F1 (119904)] lowast xa (119905) (58)




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ℎP = H

ℎPminus1

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

Δℎ2

Δℎ1

Δℎ0


H

ℎPminus1

ℎP

ℎ3

ℎ2

ℎ1

ℎ0 = 0

rP

rPminus1

r3

r2

r1

r0

ΔℎPminus1

ΔℎP

Δℎ2

Δℎ1

Δℎ0




x (119905) = x (119905) + 120576 (119905)xa (119905) = xa (119905) + 120576a (119905)u (119905) = u (119905) + 120576u (119905) (28)





(119905) = (119905) 120579 + 1198685 [120582 (119905)] (29)



(30)




(119896119879119904) = (119896119879119904) 120579 + 1198685 [120582 (119896119879119904)] (32)


(119896) = (119896) 120579 + 1198685 [120582 (119896)] (33)


= P (119873) 119873sum119896=0

119879 (119896) (119896)

P (119873) = [ 119873sum119896=0

119879 (119896) (119896)]minus1 (34)







(35)





Mminus1K = [[[[[[[[


]]]]]]]](37)









119894 120577119894 = 120590119894119894

119894 = 1 2 119899

(40)


119894 = 119894120579(2119894minus1) 119894 = 2 3 119899C = 1198860M + 1198861K

(41)




xa (119905) = xa (119905) + 120576a (119905) = D120578 (119905) + 120576a (119905) yo (119905) = D (119905) (43)

= [xT xT]119879 = [1199091 1199092 119909119899 1199091 1199092 119909119899]119879 (44)




x (119905)] minus [x (119905)x (119905)] = [x (119905)x (119905)]= [1199091 1199092 119909119899 1199091 1199092 119909119899]119879

(46)



(47)

= Alowast (119905) + Ξ (119905) (48)

with



(50)




(52)



(53)


120603lowast119894 = 120596lowast






1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn



120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)








minus1 [F1 (119904)] lowast xa (119905) (58)




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






(35)





Mminus1K = [[[[[[[[


]]]]]]]](37)









119894 120577119894 = 120590119894119894

119894 = 1 2 119899

(40)


119894 = 119894120579(2119894minus1) 119894 = 2 3 119899C = 1198860M + 1198861K

(41)




xa (119905) = xa (119905) + 120576a (119905) = D120578 (119905) + 120576a (119905) yo (119905) = D (119905) (43)

= [xT xT]119879 = [1199091 1199092 119909119899 1199091 1199092 119909119899]119879 (44)




x (119905)] minus [x (119905)x (119905)] = [x (119905)x (119905)]= [1199091 1199092 119909119899 1199091 1199092 119909119899]119879

(46)



(47)

= Alowast (119905) + Ξ (119905) (48)

with



(50)




(52)



(53)


120603lowast119894 = 120596lowast






1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn



120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)








minus1 [F1 (119904)] lowast xa (119905) (58)




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




(47)

= Alowast (119905) + Ξ (119905) (48)

with



(50)




(52)



(53)


120603lowast119894 = 120596lowast






1003817100381710038171003817 (119905)1003817100381710038171003817 le 120583120577∙∙

E 1003817100381710038171003817120578 (119905)1003817100381710038171003817 + 120583120574 1003817100381710038171003817120576a (119905)1003817100381710038171003817(1 + 120574) 120577∙∙

(56)

E = [[Ontimesn



120583 = M 10038171003817100381710038171003817Mminus110038171003817100381710038171003817 120577∙∙ = min1le119894le119899

120577119894119894

(57)








minus1 [F1 (119904)] lowast xa (119905) (58)




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Time (s)

minus15

minus1

minus05

0

05

1

18

0 25 5 75 10 125 15 175 20

xg

(mＭ

2)








119896=0 2119886119908 (119896) (59)




119864 (120575) = radic 5sum119894=1

2119894

119894 () = 1003816100381610038161003816119894 minus 1205961198941003816100381610038161003816120596119894

times 100 (60)


= min120575isin[120575min120575max]


119864 (120575) 119891lowast119888 = 1120575lowast

(61)





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 3 4 5 ndash ndash 196 0035 2857 637 0055 1818



x 120575∘ [s] 119891∘119888 [Hz]

1 0 1 2 4 5 3 018 350 004 25 1164 0048 20832 0 1 3 4 5 2 027 359 0038 2632 945 0058 17243 0 1 2 3 5 4 016 418 004 25 1338 0048 20834 0 2 3 4 5 1 045 579 0044 2273 750 0062 16135 0 1 2 3 4 5 032 1232 0068 1471 2607 0048 2083



x 120575∘ [s] 119891∘119888 [Hz]


119864x (120575) = 5sum119894=1

[10038171003817100381710038171199091198941003817100381710038171003817 + 1003817100381710038171003817100381711990911989410038171003817100381710038171003817] 10038171003817100381710038171199091198941003817100381710038171003817 = radic 119873sum

119895=1

1199092119894 1003817100381710038171003817100381711990911989410038171003817100381710038171003817 = radic 119873sum

119895=1

1199092119894(62)


x = min120575isin[120575min120575max]


119864x (120575) 119891∘119888 = 1120575∘

(63)


120575lowast 119864∘x and 120575∘ for




120575lowast 119891lowast119888 119864∘

x 120575∘ and 119891∘119888 which are








120575lowast 119891lowast119888 119864∘



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

5

10

15

20

25

30E

001 002 004 006 008 01 012

(s)

(a) 119864(120575) versus 120575

0

25

5

75

10

125

15

001 002 004 006 008 01 012

(s)

E x

(b) 119864x(120575) versus 120575


0

5

10

15

20

25

30

E

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

(a) 119864(120575) versus 120575

0

10

20

30

40

(s)002 004 006 008 01 012

Π41

Π42 Π4

5

Π44

Π43

E x

(b) 119864x(120575) versus 120575






and 119864∘





9






119888 119864∘x 120575∘ and 119891∘















0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

10

20

30

40

50E

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)002 004 006 008 01 014012

E

Π36

Π37 Π3

9

Π38

(b) Π3119896 119896 = 6 7 8 9


0

10

20

30

40

(s)002 004 006 008 01 014012

Π31

Π32 Π3

5

Π34

Π33

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

(s)002 004 006 008 01 014012

Π36

Π37 Π3

9

Π38

E x

(b) Π3119896 119896 = 6 7 8 9


Π41 and Π2















1 120575lowast) (Π31 120575lowast) (Π4



1 120575lowast) (Π41 120575lowast) (Π3




Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Time (s)0 1 2 3 4 5 6 7 8 9 10

minus3

minus2

minus1

0

1

225

Acce

lera

tion

(mＭ

2)

x2ax2a


Time (s)0 1 2 3 4 5 6 7 8 9 10


01234

Acce

lera

tion

(mＭ

2)

x4ax4a



500

750

1000

1250

1500

1750

2000

N(

kgmiddotm

)

Time (s)0 1 2 3 4 5 6 7 8 9 10

12

45

3

(a) 120579119894 119894 = 1 2 3 4 5

500

750

1000

1250

1500

1750

2000N

(kg

middotm)

Time (s)0 1 2 3 4 5 6 7 8 9 10

67

89

(b) 120579119894 119894 = 6 7 8 9


0

10

20

30

40

50

(s)004 006 008 01 014012

E

Π21

Π22 Π2

5

Π24

Π23

(a) Π2119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 005 006 007 009008

E

Π26

Π27

(b) Π2119896 119896 = 6 7



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

10

20

30

40

50

(s)004 006 008 01 014012

Π21

Π22 Π2

5

Π24

Π23

E x

(a) Π3119896 119896 = 1 2 3 4 5

0

10

20

30

40

50

(s)004 006 008 01 014012

Π26

Π27

E x

(b) Π3119896 119896 = 6 7 8 9


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Disp

lace

men

t (m

)


0001002003

0045

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1












Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5


Velo

city

(ms

)


00102030405

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5

x5



1


012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



012

35

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x3a

x3a



1

minus2

minus1

0

1

2

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x1a

x1a


minus7minus6

minus4

minus2

0

2

45

Acce

lera

tion

(mＭ

2)

Time (s)0 1 2 3 4 5 6 7 8 9 10

x5a

x5a



1







x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





x 120575∘ [s] 119891∘119888 [Hz]



1 Π41 and Π5





Estimates(Π3






8 Conclusions




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Appendix







119886119901 = 119903119901 119901 = 0 1 2 119875 (A3)


119890119901 = 119889119901+1 minus 1198891199013Δℎ119901 119901 = 0 1 2 119875 minus 1 (A5)


(A6)


1015840119886 (0) = 119887010158401015840119886 (119867) = 2119889119875 (A7)


1198870 = 0 (A8)

119889119875 = 0 (A9)




SD = V (A11)

S

=[[[[[[[[[[


]]]]]]]]]](A12)


D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



D =[[[[[[[[[[

119889011988911198892119889119875

]]]]]]]]]]

V =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(A13)


D = Sminus1V (A14)






119875minus1 (B2)


101584010158401015840119886 (119867) = 6119890119875 (B4)


119890119875 = 0 (B6)






] (C2)




(C3)



(C4)



119894 = 1 2 119899 (C6)


x (119905) = 119899sum119903=1

120595i119902119894 (119905) 119894∙ = 120595i119879M120595i119888119894∙ = 120595i119879C120595i119894∙ = 120595i119879K120595i

(C7)



119894 = 1 2 119899 (C8)



119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119902119894 + 2 (1 + 120574) 120577119894119894 119902119894 + (1 + 120574) 2119894 119902119894 = 0

120577119894 = 119888119894∙(2119894∙119894) (C9)




119894 119902119894 = 0 (C10)



119894 = 0 119894 = 1 2 119899 (C11)



1199042 + 2 (1 + 120574) 120577119894119894119904 + (1 + 120574) 2119894 = 0 119894 = 1 2 119899 (C12)


1 + 120574Z119894 (119904)Q119894 (119904) = 1 + 120574P119894 (119904) = 0P119894 (119904) = Z119894 (119904)

Q119894 (119904) = 2120577119894119894 (119904 + 1198942120577119894)1199042 + 2120577119894119894119904 + 2119894

(C13)



984858119894 = 11205772119894 minus 1(C14)


Imaginaryaxis

Realaxis

Break-inpoint

minusi

2i






which is given by

(119905) = eAlowastt (0) + int119905








= [[[[[[[[


[[[[[[[[







120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





120590lowast119894 (C21)






1003817100381710038171003817 (119905)1003817100381710038171003817 le 119890minus(1+120574)120577∙∙120583[1003817100381710038171003817 (0)1003817100381710038171003817 minus Ξ (119905)(1 + 120574) 120577∙∙

]+ 120583 Ξ (119905)(1 + 120574) 120577∙∙120590lowast119898 = (1 + 120574) 120577∙∙ = (1 + 120574)min

1le119894le119899120577119894119894

(C24)


Ξ (119905) le (1 + 120574) 100381710038171003817100381710038171003817100381710038171003817100381710038171003817[[Ontimesn





Nomenclature
































Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

















Data Availability




Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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