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Research ArticleNonlinear Resonance Analysis of Slender Portal Frames underBase Excitation

Luis Fernando Paullo Muntildeoz1 Paulo B Gonccedilalves1

Ricardo A M Silveira2 and Andreacutea Silva2

1Civil Engineering Department Pontifical Catholic University of Rio de Janeiro Rio de Janeiro RJ Brazil2Civil Engineering Department Federal University of Ouro Preto Ouro Preto MG Brazil

Correspondence should be addressed to Luis Fernando Paullo Munoz lfernandtecgrafpuc-riobr

Received 22 November 2016 Accepted 16 February 2017 Published 30 March 2017

Academic Editor Marcello Vanali

Copyright copy 2017 Luis Fernando Paullo Munoz et al 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

The dynamic nonlinear response and stability of slender structures in the main resonance regions are a topic of importance instructural analysis In complex problems the determination of the response in the frequency domain indirectly obtained throughanalyses in time domain can lead to huge computational effort in large systems In nonlinear cases the response in the frequencydomain becomes even more cumbersome because of the possibility of multiple solutions for certain forcing frequencies Thosesolutions can be stable and unstable in particular saddle-node bifurcation at the turning points along the resonance curves In thiswork an incremental technique for direct calculation of the nonlinear response in frequency domain of plane frames subjected tobase excitation is proposed The transformation of equations of motion to the frequency domain is made through the harmonicbalance method in conjunction with the Galerkin method The resulting system of nonlinear equations in terms of the modalamplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain thenonlinear resonance curves Suitable examples are presented and the influence of the frame geometric parameters and basemotionon the nonlinear resonance curves is investigated

1 Introduction

Lightweight framed structures are extensively used in engi-neering applications such as buildings industrial construc-tions off-shore platforms and aerospace structures In civilengineering framed metal structures such as gabled framesare efficient structural forms to withstand various loads suchas live snow wind earthquake and crane loads Linearmodels are usually used for the design of these structuresHowever as its slenderness increases the degree of geometricnonlinearity increases as well as the occurrence of instabilityphenomena Under dynamic loads slender frames may besubjected to unwanted (or unsafe) large vibration amplitudesand accelerations and dynamic jumps in the main resonanceregions due to hardening or softening frequency-amplituderelation and modal couplings Therefore an accurate non-linear dynamic analysis is required Recent earthquakeshave increased the interest in the development of reliable

analytical methods of assessing the dynamic response ofexisting structures and if necessary developing effectiveretrofit strategies So understanding the nonlinear dynamicsof framed structures under base excitation constitutes anessential step for their safe design

The seminal works by Koiter [1] and Roorda [2] haveshown the complex postbucking paths exhibited by slenderframes subsequently several authors have studied the non-linear behavior of reticulated structures liable to bucklingunder prescribed loading conditions A compilation of thesefirst results can be found in Bazant and Cedolin [3] whosummarize many topics about the stability of slender framesDesign prescriptions are found for example in Galambos[4] More recently Silvestre and Camotim [5] presented asemianalyticalmethod to analyze the geometrically nonlinearbehavior of plane frames including the influence of eventualbuckling mode interaction phenomena Later they presentedthe results concerning the elastic in-plane stability and

HindawiShock and VibrationVolume 2017 Article ID 5281237 21 pageshttpsdoiorg10115520175281237

2 Shock and Vibration

second-order behavior of unbraced single-bay pitched-roofsteel frames and proposed a methodology to design this typeof commonly used structure In particular they showed thatdue to the rafter slope the geometrically nonlinear behaviorsof orthogonal beam-and-column and pitched-roof frames arequalitatively different [6] The dynamics of such frames willbe investigated in this work More recently Galvao et al [7]assessed the effect of geometric imperfections and boundaryconditions on the postbuckling response and imperfection-sensitivity of L-frames

The dynamic response of slender frames in the mainresonance regions is an important topic in the analysis ofthese structures under time varying loads Several nonlinearphenomena have been studied in the last decades Simitses[8] studied the effect of static preloading and suddenlyapplied loads on the nonlinear vibrations and instabilities ofslender structures Chan andHo [9] conducted the nonlinearvibration analysis of steel frames with flexible connectionsMazzilli and Brasil [10] presented an analytical study ofthe nonlinear vibrations in a three-time redundant portalframe considering the effect of axial forces upon the naturalfrequencies The axial forces play an important role in tuningthe sway mode and the first symmetrical mode into a 1 2internal resonance Harmonic support excitations resonantwith the first symmetrical mode are then introduced and theamplitudes of nonlinear steady states are computed basedupon a multiple scales solution Later Soares and Mazzilli[11] described the implementation of a computer programto calculate nonlinear normal modes of structural systemsand generate individual modes of planar framed structuresexhibiting geometrically nonlinear behavior The procedurefollows the invariant manifold approach adapted to handleequations of motion of systems discretized by finite elementtechniques Chan and Chui [12] examined based on theirprevious works the nonlinear static and cyclic behavior ofsteel frames with semirigid connections McEwan et al [13]proposed a method for modeling the large deflection beamresponse involving multiple vibration modes while Ribeiro[14] analyzed the geometrically nonlinear vibrations of beamsand plane frames by the hierarchical finite element methodand examined the suitability of the proposed formulation fortime domain nonlinear analyses Da Silva et al [15] studiedthe nonlinear dynamics of a low-rise portal frame usingthe ANSYS finite element software The results show theinfluence of semirigid joints and geometrical nonlinearityon the steel frames dynamics More recently Galvao et al[16] investigated the effect of semirigid connections on thenonlinear vibrations of slender frames and obtained thenonlinear relation between the natural frequencies and staticpreloading Su and Cesnik [17] introduced a strain-basedgeometrically nonlinear beam formulation for structural andaeroelastic modeling and analysis of slender wings of veryflexible aircraft Solutions of different beam configurationsunder static loads and forced dynamic excitations are com-pared against ones from other geometrically nonlinear beamformulations Goncalves et al [18] published an experimentalanalysis of the nonlinear vibrations of a slendermetal columnunder self-weight Masoodi and Moghaddam [19] studiedthe nonlinear dynamics and natural frequencies of gabled

frames having flexible restraints and connections To controlunwanted nonlinear phenomena and vibrations of framesseveral control techniques are proposed in literature Forexample Palacios Felix et al [20] examined the nonlinearcontrol method based on the saturation phenomenon Theinteraction of a nonideal source with a portal frame leads tothe occurrence of interesting nonlinear phenomena duringthe passage through resonance when the nonideal excitationfrequency is near the portal frame natural frequency andwhen the nonideal system frequency is approximately twicethe controller frequency (two-to-one internal resonance)They also suggest the use of the portal frame for energyharvesting

The determination of a structurersquos response in the fre-quency domain characterized by the resonance curves andbifurcation diagrams is in some cases indirectly obtainedthrough a series of analyses in time domain varying step bystep a selected control parameter (see Parker and Chua [21])In this brute force approach for each control parameter valuethe equations of motion must be integrated long enough toreach the steady-state regime This leads to a huge computa-tional effort when analyzing structures with a large numberof degrees of freedom [16] In nonlinear cases the responsein the frequency domain becomes even more cumbersomebecause of the possibility of multiple solutions for certainforcing frequencies Those solutions can be stable and unsta-ble and bifurcations can appear in particular saddle-nodebifurcation at the turning points along the resonance curvesTo predict the nonlinear response in frequency domain somemethods are used such as the harmonic balance method(HBM) and other integral transformations Based on theHBM Nakhla and Valch [22] proposed a method for thesolution of nonlinear periodic networks avoiding the timedomain solution of the dynamic equations Lau et al [23]formulated an incremental variational process based on theHBM to study the nonlinear vibrations of elastic systemsFurther Cheung and Chen [24] used the formulation ofthe incremental harmonic balance method (IHB) to solvea system of differential equations with cubic nonlinearitywhich governs a wide range of engineering problems such aslarge-amplitude vibration of beams or plates An incrementalarc-length method combined with a cubic extrapolationtechnique is adopted to trace the response curves Chen et al[25] generalized this technique to use the incremental HBMin a finite element context being able to use the HBM inthe study of systems with several degrees of freedom Dueto its simple formulation and relatively easy implementationthe HBM is one of the most popular methods to study thenonlinear vibrations of structures [26] The algebraic systemof nonlinear equations which results from theHBM requiresa numerical procedure to obtain the solutions FrequentlyNewtonrsquos method is adopted to solve the nonlinear systemHowever in many cases the resonance curves exhibit limitpoints and in such cases the use of continuation methodsis necessary For example Ribeiro and Petyt [27] studied thefree and steady-state forced vibrations of thin plates Symboliccomputation is employed in the derivation of the modeland the equations of motion are solved by the Newton andcontinuation methods Von Groll and Ewins [28] described a

Shock and Vibration 3

numerical algorithm based on the harmonic balance methodto study rotorstator interaction dynamics under periodicexcitation The algorithm also calculates turning points andfollows solution branches via arc-length continuation Morerecently Ferreira and Serpa [29] described the applicationof the arc-length method to solve a system of nonlinearequations obtaining as a result the nonlinear frequencyresponse In a recent work Londono et al [30] used contin-uation methods in frequency domain to obtain the backbonecurves of several nonlinear systems while Renson et al [31]presented numerical methodologies for the computation ofnonlinear normal modes in mechanical system

Stoykov and Ribeiro [32 33] investigated the geometri-cally nonlinear free vibrations of beams using a 119901-versionfinite element method The variation of the amplitude ofvibration with the frequency of vibration is determined andpresented in the form of backbone curves Coupling betweenmodes is investigated internal resonances are found andthe ensuing multimodal oscillations are described Formicaet al [34] present a computational framework to performparameter continuation of periodic solutions of nonlineardistributed-parameter systems represented by partial dif-ferential equations with time-dependent coefficients andexcitations As a case study they consider a nonlinear beamsubject to a harmonic excitation

An important design concern is the dynamics of slenderframes subjected to seismic loads In such cases the responseof the structure in frequency domain is important since thevulnerability of structure during an earthquake is related tothe relation between the natural vibration frequencies of thestructure and the frequency content of the seismic load asstudied in Paullo Munoz et al [35] who assess the frequencydomain response of slender frames under seismic excitationIn seismic events the ground acceleration magnitude is acrucial parameter since the effect of the excitation on thestructure is directly related to acceleration magnitude Mostrecorded events have a magnitude lower than 1119892 Howeverpeak acceleration greater than 1119892 has been registered Forexample an accelerationmagnitude of 17119892was registered forthe Los Angeles earthquake and a peak acceleration of 299119892was registered for the 2011 Tohoku earthquake [36] Douglas[37] presented a review of equations for the estimation ofpeak ground acceleration using strong-motion records whileKatsanos et al [38] reviewed alternative selection proceduresfor incorporating strong ground motion records within theframework of seismic design of structures There are manyother research areas where the influence of base accelerationon structural components is important including subsurfaceexplosions rotating machinery and launching vehicles

In this work an incremental technique for the directcalculation of the nonlinear dynamic response in frequencydomain of nonlinear plane frames discretized by the finite ele-ment method and subjected to a base excitation is proposedThe transformation of discretized equations of motion in thefinite element context to the frequency domain is accom-plished here through the classical harmonic balance method(HBM) For the nonlinear analysis a particular adaptationof the HBM-Galerkin methodology presented by Cheungand Chen [24] and generalized for the use in FEM context

by Chen et al [25] is proposed here The resulting systemof nonlinear equations in terms of the modal amplitudesand forcing frequency is solved by the Newton-Raphsonmethod together with an arc-length procedure to obtainthe nonlinear resonance curves Examples of commonlyused frame geometries are presented and the influence ofvertical and horizontal base motions on the frame nonlinearresponse as a function of the frame geometric parameters isanalyzed

2 Formulation

21 Equations of Motion For a structural system underthe action of harmonic forcing with a prescribed forcingfrequency Ω the nonlinear equations of motion can bewritten as

Mu (119905) + Cu (119905) + Fi (u (119905)) = 119865 cos (Ω119905) f (1)

where M C and Fi are the mass matrix damping matrixand the vector of nonlinear elastic forces respectively withFi being dependent on the nodal displacement vector u(119905)and on the geometric nonlinear formulation f is the vectorof external applied loads at the nodal points which gives theamount of forces at each DOF and 119865 and Ω are respectivelythe magnitude and frequency of the harmonic excitation

22 Harmonic Balance Method (HBM) for Linear AnalysisThe HBM is one of the most popular methods for thenonlinear analysis of dynamical systems In this method thedisplacement field is approximated by a finite Fourier seriesof the form

u (119905) = 119873sum119899=0

[A119899 cos (119899Ω119905) + B119899 sin (119899Ω119905)] (2)

where A119899 and B119899 are the modal amplitude vectors corre-sponding to the 119899th harmonic and 119873 is the number of termsconsidered in the approximation For the linear damped caseit is only necessary to retain the second term in (2) Thus thenodal displacement vector takes the form

u (119905) = A1 cos (Ω119905) + B1 sin (Ω119905) (3)

Introducing (3) into (1) results in

(ΩCA1 minus Ω2MB1) cos (Ωt)minus (Ω2MA1 + ΩCB1) sin (Ωt) = 119865f cos (Ωt) (4)

Taking 119865f = F0 and Fi = Ku(119905) where K is the linearstiffness matrix and applying the HBM to equation (4) thefollowing system is obtained

[ C minusΩM + K

minusΩ2M + K minusΩC] A1

B1 = F0

0 (5)

Equation (5) can be expressed in a more compact form as

K (Ω)D minus F = 0 (6)


where

K (Ω) = [ ΩC minusΩ2M + K

minusΩ2M + K minusΩC]

D = A1

B1

F = F00

(7)

As a result of the transformation of the system of equa-tions to the frequency domain and the use of continuationtechniques together with an arc-length constraint the forcingfrequency Ω and the modal amplitude vector D are theunknowns in (7) [29 39 40] Even in the linear case thetransformation of (1) to the frequency domain leads to anonlinear system of algebraic equations with the use ofnonlinear techniques such as the Newton-Raphson methodbeing necessary

23 HBM-Galerkin Methodology for Nonlinear Analysis Forthe nonlinear analysis considering that the steady-stateresponse is periodic it is convenient to use a periodicnondimensional time variable defined as

120591 = Ω119905 (8)

Substituting (8) into (1) the equation of motion can berewritten as

Ω2Mu (120591) + ΩCu (120591) + Fi (u (120591)) = 119865 cos (120591) f (9)

and each component of the displacement vector can beapproximated by

119906119894 (120591) = 119873sum119899=0

[119886119894119899 cos (120591) + 119887119894119899 sin (120591)] (10)

For a damped system with quadratic nonlinearity at leastthe first two terms of the Fourier series must be considered[39]

119906119894 (120591) = 1198861198940 + 1198871198941 cos (120591) + 1198881198941 sin (120591) (11)

For a damped system with only cubic nonlinearity it isnecessary to consider at least the following terms [39]

119906119894 (120591) = 1198861198941 cos (120591) + 1198871198941 sin (120591) (12)

Plane frames exhibit quadratic and cubic nonlinearitiesDefine

119906119894 = C (120591) d119894 (13)

with

C (120591)= ⟨1 sin (120591) cos (120591)⟩ d119894= ⟨1198861198940 1198871198941 1198881198941⟩T (14)

Consider the following relation

u (120591) = S (120591)D (15)

where

S (120591) =

[[[[[[[[[[[[[[

C (120591) 0 0 0 sdot sdot sdot 00 C (120591) 0 0 sdot sdot sdot 00 0 C (120591) 0 sdot sdot sdot 0

0 0 0 C (120591) sdot sdot sdot d 00 0 0 sdot sdot sdot 0 C (120591)

]]]]]]]]]]]]]]119873times3119873

D = ⟨d1 d2 d119873⟩T1times3119873

(16)

119873 is total number of degrees of freedom Introducing (15) into(9) the following matrix equation is obtained

Ω2MS (120591)D + ΩCS (120591)D + Fi (S (120591)D) = F0 cos (120591) (17)

Considering that the solution is periodic multiplyingboth sides of (17) by S(120591)T and integrating the resultingexpression over a period one obtains [40]

int21205870

S (120591)T [Ω2MS (120591)D + ΩCS (120591)D+ Fi (S (120591)D)] 119889120591 = int2120587

0S (120591)TF0 cos (120591) 119889120591

(18)

In a compact form (18) can be expressed as

Ω2MD + ΩCD + Fi (D) = F (19)

where

M = int21205870

S (120591)TMS (120591) sdot 119889120591C = int2120587

0S (120591)TCS (120591) sdot 119889120591

F = int21205870

S (120591)TF0 cos (120591) 119889120591Fi (D) = int2120587

0S (120591)TFi (S (120591) sdot D) 119889120591

(20)

Equation (19) is a nonlinear systemof algebraic equationsin which the unknowns are the forcing frequency Ω andthe modal amplitude vector D The internal force vector infrequency domain Fi(D) is obtained from the transformationof Fi(u(120591)) In this context an explicit expression for Fi(u(120591))is necessary The next step is the solution of the nonlinearsystem of algebraic equations


24 Solution of the Nonlinear System of Equations The trans-formation of the equations of motion from time to frequencydomain results in a nonlinear system of algebraic equationsboth for the linear and nonlinear cases as shown by (6) and(19) with the unknown variables being the forcing frequencyand the modal amplitudes In this section the technique tosolve the nonlinear system of equations must consider thepossibility of frequency and amplitude limit points In thiscontext (6) can be defined in a general form as

R (ΩD) = 0 (21)

Through an incremental analysis the dynamic equilib-rium in the 119894th step is given by

R (Ω119894D119894) = 0 (22)

The unknown variables in the 119894th step are obtained by theincremental analysis as

Ω119894 = Ω119894minus1 + ΔΩ119894D119894 = D119894minus1 + ΔD119894 (23)


R (Ω119894minus1 + ΔΩ119894D119894minus1 + ΔD119894) = 0 (24)

The frequency and amplitude increments ΔΩ119894 and ΔA119894can be calculated iteratively as

ΔΩ119896119894 = ΔΩ119896minus1119894 + 120575Ω119896119894 ΔD119896119894 = ΔD119896minus1119894 + 120575D119896119894

(25)

where 120575Ω119896119894 and 120575D119896119894 are the correctors which can be obtainedthrough the first variation of (24) as

120575R119896= 120597R119896minus1120597D119894 120575D119896119894 + 120597R119896minus1120597Ω119894 120575Ω119896119894 (26)

or in a more compact form as

Km119896minus1120575D119896119894 + 120575Ω119896119894 f119896minus1 = 120575R119896 (27)

To solve (27) an additional constraint equation is neces-sary In this work a spherical arc-length constraint is used[41 42]

ΔD119879119894 ΔD119894 + (ΔΩ119896119894 )2 f119896minus1Tf119896minus1 minus Δ1198972119894 = 0 (28)

From (24) (25) (27) and (28) it is possible to obtain theiterative frequency corrector as

120575Ω119896119894 =

plusmn

Δ1198972119894radicvTv+f119896minus1Tf119896minus1

for 119896 = 1

minus r119896119879v119896

v119896Tv119896for 119896 gt 1

(29)

where

v119896 = (Km119896minus1)minus1 f119896minus1r119896 = (Km119896minus1)minus1 R119896minus1

(30)

The signal of the first frequency corrector also calledfrequency predictor can be obtained using the positive workcriterion as

sign (120575Ω1119894 ) = sign((v119896)T sdot f119896minus1) (31)

For linear systems Km and f can be defined by thefollowing expressions

Km = [ ΩC minusΩ2M + K


f = [ C minus2120596Mminus2120596M minusC ]D

(32)

Analogously for the nonlinear case Km and f can beobtained from the following relations

Km = Ω2M + ΩC + 120597Fi (D)120597D

f = (2ΩM + C) sdot D(33)

25 Geometric Nonlinear Consideration An explicit expres-sion for Fi(u(120591)) is necessary Considering an incrementalanalysis the nodal displacement vector can be expressed as

u (119905119894) = u (119905119894minus1) + Δu (119905) (34)

Analogously the internal forces vector can be calculated as

Fi (119905119894) = Fi (119905119894minus1) + ΔFi (119905) (35)

Fi can be derived once the kinematic relations andconstitutive law are defined These relations depend on theadopted nonlinear strain measure formulation [7 16] Inthis paper only the geometric nonlinearity is consideredIn this context the internal force increment vector can beapproximated as

ΔFint = (KL + KNL) Δu (36)

where KL is the linear elastic stiffness matrix and KNLis the nonlinear stiffness matrix Considering a linearizedformulation the nonlinear stiffness matrix for an Euler-Bernoulli beam-column plane element in a local system ofreference can be defined as [7 16]


KNL =

[[[[[[[[[[[[[[[[[[[[[[[[

119875119871 1198721 + 11987221198712 0 minus 119875119871 minus 1198721 + 11987221198712 01198721 + 11987221198712 61198755119871 11987510 minus 1198721 + 11987221198712 minus 61198755119871 11987510

0 11987510 11987511987115 0 minus 11987510 minus 11987511987130minus 119875119871 minus 1198721 + 11987221198712 0 119875119871 1198721 + 11987221198712 0

minus 1198721 + 11987221198712 minus 61198755119871 minus 11987510 1198721 + 11987221198712 61198755119871 minus 119875100 11987510 minus 11987511987130 0 minus 11987510 11987511987115

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

(37)

where 119875 is the average internal axial force and 1198721 and 1198722are the internal bending moments in the initial and the endnodes respectively For a prismatic bean-column elementthe internal forces can be expressed as a function of the axialand transversal displacements respectively 119906(119909) and V(119909) as

119875 = 119864119860 ( 120597119906 (119909)120597119909 + 1119871 int1198710

( 120597V (119909)120597119909 )2 sdot 119889119909)119872 = 119864119868 ( 1205972V (119909)1205971199092 )

(38)

where 119864 is Youngrsquos modulus 119860 is the cross-sectional areaand 119868 is the moment of inertia The displacement fieldscan be expressed by interpolation of nodal displacementcomponents as

119906 (119909) = 2sum119894=1

119867119894 (119909) 119906119894

V (119909) = 6sum119894=3

119867119894 (119909) 119906119894(39)

where 119867119894(119909) is the shape function used for the finite elementdiscretization Analogous to (15) the increment of the nodaldisplacement vector can by calculated as

Δu (119905) = S (120591) ΔD (40)

Solving (15) (35) (36) and (40) the nodal internal forcevector Fi at the 119894th increment can be obtained as

Fi119894 = KLS (120591)D + Fi119894minus1 + KNLS (120591) ΔD (41)

Finally the nodal internal force vector in frequencydomain defined in (20) can be calculated as

Fi119894 (D) = KLD119894 + KNLΔD + Fi119894minus1 (D) (42)

where

KL = int21205870

S (120591)119879KLS (120591) 119889120591KNL = int2120587

0S (120591)119879KNLS (120591) 119889120591

Fi119894minus1 (D) = int21205870

S (120591)119879FNL119894minus1119889120591(43)

In linear case Fi(D) = KLD and the resultant nonlinearsystem of equations defined by (19) is the same as the systemobtained through the classical HBM (see (5))

26 Equivalent Rotation Matrix In order to assemble theglobal system of equations the matrices M C KL and KNLmust be defined in global coordinates The matrices M Cand KL are the same for any reference system since theyare constant On the other hand KNL defined by (37) isformulated in a local reference frame Thus its rotation tothe global reference system is necessary It is defined in globalreference frame by

KNL = int21205870

S (120591)119879T119879KNLTS (120591) 119889120591 (44)

where T is a rotation matrix The rotation matrix for aplane beam-column element can be found in [15] In termsof the computational implementation it is not convenientto rotate the stiffness matrix KNL before frequency domaintransformation since the elements of KNL have large expres-sion which depend on nodal displacement even in the localreference frame So it is convenient to rotate KNL after thetransformation of the equations to the frequency domainusing an equivalent rotation matrix In this case it takes theform

KNL = T1015840119879int21205870

S (120591)119879KNLS (120591) 119889120591T1015840 (45)


Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)

Figure 1 Tower model with concentrated mass at top and elastoplastic support

0

005

01

015

02

025

Am

plitu

de n

orm

(m)

Time dom int k = 1011 kNmradTime dom int k = infiniteHBM k = 1011 kNmradHBM k = infinite

500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)

Figure 2 Variation of themaximumdisplacement at the top of the tower as a function of the forcing frequencyΩ obtained with the harmonicbalance method (HBM) and with time domain integration 119860119909 = 08119892

m = 10kgkl = 5Nm

k = kl + knlu(t)2

f(t) = 04m middot cos (휔t)

u(t)

m

Figure 3 1-DOF model with cubic nonlinear stiffness under harmonic excitation

where T1015840 is the equivalent rotation matrix which satisfies thefollowing condition

T1015840119879S (120591)119879 = S (120591)119879T119879 (46)

Taking into account the block-diagonal characteristic of S(120591)and since T is an orthogonal matrix it is possible to find

an explicit expression for T1015840 as a function of T through thefollowing expression

T1015840 = [[[[[

t101584011 t10158401NGL d

t1015840NGL1 t1015840NGLNGL

]]]]]

(47)


Time domain simulationTime domain simulationProposed methodProposed method

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3

Figure 4 Nonlinear resonance curves Maximum amplitude as a function of the forcing frequency

1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm

bf = 10 cmIxx = 684e minus 12m4

A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H

Figure 5 Pitched-roof frame model and four first natural vibration modes

where NGL is the number of degrees of freedom and t1015840119894119895 is asubmatrix given by

t1015840119894119895 = T119894119895I3 (48)

where I3 is an identity matrix of order three

3 Numerical Examples

31 Linear Analysis Validation As a first example a simpletower model with a concentrated mass at the top and anelastic rotational support at the base is studied to validate thelinear formulation The finite element model is composed of10 beam-column Euler-Bernoulli elements The geometricaland material properties are shown in Figure 1 The tower isunder the action of a horizontal harmonic base displacement

Figure 2 shows the variation of the maximum displace-ment at the top of the tower as a function of the forcing

frequency Ω (resonance curve) for a forcing magnitude 119860119909 =08119892 where 119892 is the acceleration of gravity considering twovalues of the elastic support stiffness 119896119903 The results obtainedwith the present formulation and continuation scheme agreewith the results obtained from a time domain analysis Thisshows the coherence of the result obtained with the proposedmethod

32 Nonlinear Analysis Validation As a second example asimple mass-spring system with cubic nonlinearity underharmonic excitation is studied to validate the nonlinearformulation The schematic representation of the system andrelevant properties are shown in Figure 3 In this exampledamping is not considered

Figure 4 shows the nonlinear resonance curves obtainedthrough the present HBM-Galerkin methodology and timedomain simulations for positive (hardening) and negative(softening) cubic nonlinearity (119896nl = plusmn10Nm3) The results


0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

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0

00005

0001

00015

0002

00025

|Dy|

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0

00005

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00015

0002

00025

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|Dy|

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0025

003

0035

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0003

|Dy|

Figure 6 Variation of the norm of the horizontal and vertical vibration amplitudes at the roof top as a function of the excitation frequencyfor a horizontal base excitation and three values of the roof height 119867 119860119909 = 096119892

obtained with the present HBM-Galerkin scheme and withtime domain simulations are very close validating the non-linear formulation It can be also observed that the algorithmis able to bypass the limit point associated with the saddle-node bifurcation As expected the time domain analysisis not able to trace the unstable branches of resonancecurves

33 Frequency Domain Analysis of Slender Frames

331 Pitched-Roof Frame A pitched-roof frame fixed at thebase and with a constant cross section is studied to assessthe behavior of this type of commonly used structure underhorizontal and vertical base excitation The geometry andmaterial properties of the frame are shown in Figure 5 as well


H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|

Figure 7 Variation of the norm of the vertical vibration amplitude at the roof top as a function of the excitation frequency for a vertical baseexcitation and three values of the roof height 119867 119860119910 = 096119892

Table 1 Four first natural frequencies of the pitched-roof frame

119867 (m) Natural vibration frequency (rads)1st mode 2nd mode 3rd mode 4th mode

00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208

as the configuration of the first four natural vibration modeswith the first and third modes being antisymmetric and thesecond and fourth modes being symmetric The influence ofthe roof height on the response is also investigated Table 1shows the first four natural frequencies for the three valuesof 119867 considered in the numerical analysis The vibration fre-quencies decrease with119867 as the frame slenderness increasesThe structure is modeled with twenty bean-column elements8 elements of the same size for the columns and 12 elementsfor the roof

Figure 6 shows the variation of the norm of the horizontaland vertical components of the displacement at top of the

frame 119863119909 and 119863119910 respectively as a function of the forcingfrequency Ω considering both linear (blue) and nonlinear(red) FE formulations for three values of the height of theroof pitches (119867 = 0m 3m and 6m) and considering ahorizontal excitation magnitude 119860119909 = 096119892 The horizontalbase motion excites the first and the third modes only whichcorrespond to predominantly lateral deformation modesThe nonlinear effect is present in the first resonance regionwhere a hardening behavior is observed The first peak andconsequently the nonlinearity increase with increasing roofheight with the nonlinear maximum vibration amplitudebeing slightly lower than the linear one After the first peakthe nonlinear and linear responses are practically coincidentThe vertical displacement 119863119910 is smaller than the verticalcomponent and is overestimated in a linear analysis Figure 7displays the response for a vertical excitation magnitude119860119910 = 096119892 The vertical base motion excites the secondand fourth modes these modes correspond to symmetricvibration modes with predominantly vertical motion Forvertical base excitation the resonance peak of the verticalmotion decreases when the roof height increases and the


H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

Figure 8 Variation of the norm of the vertical and horizontal vibration amplitudes at the roof top as a function of the excitation frequencyfor simultaneous vertical and horizontal excitation and three values of the roof height 119867 119860119909 = 08119892 and 119860119910 = 0667119860119909

effect of nonlinearity is negligible In this case due to thesymmetry of the displacement field the lateral motion of thetop node is zero

Figure 8 shows the norm of the horizontal and verticaldisplacement as a function of the forcing frequency for theframe excited in both the horizontal and vertical directions

This is a typical excitation in seismic analysis where bothcomponents are usually present Following the suggestionof some seismic code [43] the intensity of the vertical basemotion is adopted as 067 times the intensity of the horizontalbase motion (119860119910 = 066119860119909) In this case all vibrationmodes are excitedThe influence of geometric nonlinearity is


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))

Figure 9 Arched-roof frame model and four first natural vibration modes

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

Figure 10 Influence of the roof height on the nonlinear response of the arched-roof frame under horizontal excitation 119860119909 = 096119892

evident in the two first resonance regions where a hardeningbehavior leading to possible dynamic jumps is observed Asthe roof height increases themagnitude of119863119909 correspondingto the first resonant peak increases while the magnitude ofsecond peak decreases On the other hand the magnitude

of vertical displacement 119863119910 is reduced in the two firstresonance regions with the increase of the roof height Thesenonlinear resonance curves exhibit amplitude and frequencylimit points This indicates the existence of fold (saddle-node) bifurcations which can only be mapped by the use of


0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

Figure 11 Influence of the roof height on the nonlinear response of the arched-roof frame under vertical excitation 119860119910 = 096119892

continuation techniquesThe solutions between the two limitpoints are unstable

332 Arched-Roof Frame Now an arched-roof framefixed atthe base and with a constant cross section is studied to assessthe behavior of this typical structural form under horizontaland vertical base excitationThe structure has the same spancolumn height and cross section as the pitched roof as wellas the same material properties as shown in Figure 9 toenable comparisons of the results An incomplete circulararch with height 119867 is considered Figure 9 also shows theconfiguration of the first four vibration modes Table 2 showsthe first four natural frequencies for three values of the archheight 119867 As in the previous example the first and thirdmodes are antisymmetric while the second and fourthmodesare symmetric

Figures 10 and 11 show the variation of the norm ofthe vertical and horizontal displacements at top of the archas a function of the forcing frequency for horizontal andvertical harmonic excitation respectively and consideringboth linear (blue) and nonlinear (red) formulations For the

Table 2 Natural vibration frequencies of the arched-roof frame


10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341

horizontal base excitation the first and the third modes areexcited and the nonlinear effect is only noticed in the firstpeak leading to a hardening behavior and decrease of themaximum lateral displacement As in the previous examplethe first peak increases with the increase of the roof heightFor vertical base excitation only the second and fourthmodes are excited and the nonlinear effect is slightly notedin the first peak only where a softening behavior is observedIn this case the first peak amplitude decreases when the roofheight increases Comparing the results of the pitched roofand arched roof for 119867 = 3m one can observe that the archedroof is more flexible leading to higher vibration amplitudes


H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|


Figure 12 shows the variation of the maximum horizontal(119863119909) and vertical displacements (119863119910) at the top of archconsidering both horizontal and vertical base excitationswith119860119910 = 0667119860119909 and 119860119909 = 08119892 In this case all vibrationmodes are excited The influence of the geometric nonlin-earity is evident in the two first resonance regions where

a hardening behavior leading to possible dynamic jumps isobserved As the roof height increases the magnitude of119863119909 corresponding to the first resonant peak increases whilethe magnitude of second peak decreases The magnitude ofvertical displacement 119863119910 decreases in the two first resonanceregions with the increase of the roof height


0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Figure 13 Influence of the roof height on the nonlinear response of the frame under horizontal and vertical excitation 119860119909 = 08119892 and 119860119910 =0667119860119909

333 Pitched-Roof Framewith a Long Span Now the pitchedroof is again analyzed considering a span length 119871 = 160mincreasing thus the framersquos slenderness The other propertiesof the system are the same as those shown in Figure 5 Table 3shows the first four natural frequencies for the three values of

the roof height Compared with the results in Table 1 a strongdecrease in the natural frequencies is observed

Figure 13 shows the resonance curves of the horizontal(119860119909) and vertical (119860119910) displacement at the top of the frameconsidering vertical and horizontal harmonic base excitation


LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

Figure 14 Influence of increasing magnitude 119860119909 on the nonlinear response of the frame under horizontal excitation Norm of the horizontaldisplacement at top of pitched-roof as a function of the forcing frequency 119871 = 10 and 119867 = 6m

acting simultaneously with amplitude of the horizontal accel-eration of 119860119909 = 08119892 and vertical acceleration of 119860119910 =066119860119909 The nonlinear effects are visible in the two firstresonance regions similar to the frame with 119871 = 10mHowever the two first natural frequencies are close especiallyfor 119867 = 0m leading to modal interaction between thetwo first modes and a large excitation region where large-amplitude vibrations occur Compared with the results inFigure 6 for 119871 = 10m a marked increase in the lateraland vertical displacements is observedThe vertical displace-ments are particularly large around the second vibrationfrequency

334 Strong Base Motion Now the effect of strong basemotion on the nonlinear dynamics of the pitched-roofframe (Section 331) arched-roof (Section 332) frame andpitched-roof frame with long span (Section 333) is assessedFor this a parametric study considering the variation of theground acceleration magnitude is presented A maximumacceleration peak of 36119892 for a single direction excitationand of 30119892 for horizontal and vertical excitations actingsimultaneously is adopted

Figure 14 shows the variation of the maximum horizontaldisplacement at top of the pitched-roof frame (Figure 5)as a function of the forcing frequency for increasing base


LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|

Figure 15 Influence of increasing base motion 119860119909 on the nonlinear response of the frame under vertical excitation Norm of the verticaldisplacement at top of pitched roof as a function of the forcing frequency 119871 = 10m and 119867 = 6m

Table 3 Natural vibration frequency of pitched-roof frame

119867 (m) Natural frequency of vibration (rads)1st mode 2nd mode 3rd mode 4th mode

00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947

excitation magnitude A sharp increase in the vibrationamplitude and the hardening nonlinearity of the responsein the first resonance region is observed For strong basemotions the effect of the nonlinearity in the vicinity of thethird vibration frequency begins to appear being also of thehardening type The second mode is not excited

Figure 15 shows the frame response due to verticalharmonic acceleration considering 119860119909 = 18119892 and 119860119909 =36119892 The effect of the geometric nonlinearity is observedin the resonance regions associated with the second andfourth vibration modes exhibiting softening behavior Thesoftening behavior is due to quadratic nonlinearities and isconnected with the interaction between axial and transver-sal beam displacements In such cases the linear analy-sis underestimates the maximum vibration response Thesoftening effect due to negative quadratic nonlinearity canbe observed in other slender structures under compres-sive loads like shallow arches and arises from the inter-action between in-plane compressive forces and bending[25]

Figure 16 shows the horizontal and vertical componentsof vibration as function of forcing frequency at top of

pitched-roof frame (119871 = 10m and 119867 = 6m) consideringharmonic horizontal and vertical base excitations actingsimultaneously and considering 119860119910 = 0667119860119909 and twovalues of the acceleration amplitude 119860119909 = 15119892 and 119860119909 =30119892 The nonlinear effect is present in all resonance regionsfor the two displacement components A strong differencebetween the linear and nonlinear responses is particularlyobserved in the first resonance region

Finally Figure 17 shows the horizontal and verticaldisplacement components at top of pitched-roof frame as afunction of forcing frequency for the frame with 119871 = 16mand 119867 = 6m considering harmonic horizontal and verticalbase excitations acting simultaneously with 119860119910 = 0667119860119909for two values of 119860119909 15119892 and 30119892 Again the strong basemotion leads to a highly nonlinear response

In all cases analyzed here the resonance curves for the lin-ear and nonlinear cases were easily obtained by the proposednumerical methodology which can be easily applied to anystructural system discretized by the FEM

4 Conclusions

In this work an incremental technique for the direct cal-culation of the nonlinear resonance curves of plane framesdiscretized by the finite element method and subjected to abase excitation is proposedThe transformation of discretizedequations of motion in the finite element method context tothe frequency domain is accomplished here through the clas-sical harmonic balance method together with the GalerkinmethodThe resulting system of nonlinear equations in terms


LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|

Figure 16 Horizontal and vertical displacements at top of the frame as a function of the forcing frequency due to simultaneous horizontaland vertical forcing 119871 = 10m and 119867 = 6m

of the modal amplitudes and forcing frequency is solved bythe Newton-Raphson method together with an arc-lengthprocedure to obtain the nonlinear resonance curves Theformulation of the proposed method is validated comparingthe present results with time domain simulations showingcoherence and precisionThe algorithm is able tomap regionswith coexisting stable and unstable solutions A pitched-roof frame and an arched-roof frame under horizontal andvertical base excitation with varying roof height and spansare studied to illustrate the influence of the nonlinearityon the four first resonance regions Horizontal base motion

excites only the odd modes while vertical base motionexcites only the evenmodes Under simultaneous vertical andhorizontal excitation all modes are excited The roof heighthas a strong influence on the resonant peaks and degree ofnonlinearityThe results also show the influence of increasingbase excitation on the nonlinear behavior Depending onthe excitation magnitude hardening or softening behaviorcan be observed The proposed method is shown to be auseful method for the analysis of slender frame structures infrequency domain since it is able to obtain the resonancecurves with precision and small computational effort This


LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|

Figure 17 Horizontal and vertical displacements at the top of the frame as a function of the forcing frequency due to simultaneous horizontaland vertical forcing 119871 = 16m and 119867 = 6m

can be used as a preliminary design tool for frames underseismic and other types of base excitation leading to a saferdesign

Conflicts of Interest

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

References

[1] W T Koiter ldquoPost-buckling analysis of simple two-bar framerdquoin Recent Progress in Applied Mechanics B Broberg J Hult andF Niordson Eds pp 337ndash354 1967

[2] J Roorda The instability of imperfect elastic structures [PhDthesis] University College London London UK 1965

[3] Z Bazant and L Cedolin Stability of Structures Oxford Univer-sity Press Oxford UK 1991

[4] T V Galambos Guide to Stability Design Criteria for MetalStructures John Wiley amp Sons 1998

[5] N Silvestre and D Camotim ldquoAsymptotic-numerical methodto analyze the postbuckling behavior imperfection-sensitivityand mode interaction in framesrdquo Journal of EngineeringMechanics vol 131 no 6 pp 617ndash632 2005

[6] N Silvestre and D Camotim ldquoElastic buckling and second-order behaviour of pitched-roof steel framesrdquo Journal of Con-structional Steel Research vol 63 no 6 pp 804ndash818 2007


[7] A S Galvao P B Goncalves and R A M Silveira ldquoPost-buckling behavior and imperfection sensitivity of L-framesrdquoInternational Journal of Structural Stability and Dynamics vol5 no 1 pp 19ndash35 2005

[8] G J Simitses ldquoEffect of static preloading on the dynamicstability of structuresrdquoAIAA journal vol 21 no 8 pp 1174ndash11801983

[9] S L Chan and G W M Ho ldquoNonlinear vibration analysis ofsteel frames with semirigid connectionsrdquo Journal of StructuralEngineering vol 120 no 4 pp 1075ndash1087 1994

[10] C E NMazzilli andRM L R F Brasil ldquoEffect of static loadingon the nonlinear vibrations of a three-time redundant portalframe analytical and numerical studiesrdquo Nonlinear Dynamicsvol 8 no 3 pp 347ndash366 1995

[11] M E S Soares and C E N Mazzilli ldquoNonlinear normal modesof planar frames discretized by the finite element methodrdquoComputers and Structures vol 77 no 5 pp 485ndash493 2000

[12] S L Chan and P T Chui Eds Non-Linear Static and CyclicAnalysis of Steel Frames with Semi-Rigid Connections Elsevier2000

[13] M I McEwan J R Wright J E Cooper and A Y T LeungldquoA combined modalfinite element analysis technique for thedynamic response of a non-linear beam to harmonic excitationrdquoJournal of Sound and Vibration vol 243 no 4 pp 601ndash6242001

[14] P Ribeiro ldquoHierarchical finite element analyses of geometricallynon-linear vibration of beams and plane framesrdquo Journal ofSound and Vibration vol 246 no 2 pp 225ndash244 2001

[15] J G S Da Silva L R O De Lima P D S Vellasco S DeAndrade and R A De Castro ldquoNonlinear dynamic analysisof steel portal frames with semi-rigid connectionsrdquo EngineeringStructures vol 30 no 9 pp 2566ndash2579 2008

[16] A SGalvao A RD Silva RAM Silveira andP BGoncalvesldquoNonlinear dynamic behavior and instability of slender frameswith semi-rigid connectionsrdquo International Journal of Mechan-ical Sciences vol 52 no 12 pp 1547ndash1562 2010

[17] W Su and C E S Cesnik ldquoStrain-based geometrically non-linear beam formulation for modeling very flexible aircraftrdquoInternational Journal of Solids and Structures vol 48 no 16-17pp 2349ndash2360 2011

[18] P B Goncalves D L B R Jurjo C Magluta and N RoitmanldquoExperimental investigation of the large amplitude vibrations ofa thin-walled column under self-weightrdquo Structural Engineeringand Mechanics vol 46 no 6 pp 869ndash886 2013

[19] A R Masoodi and S H Moghaddam ldquoNonlinear dynamicanalysis and natural frequencies of gabled frame having flexiblerestraints and connectionsrdquo KSCE Journal of Civil Engineeringvol 19 no 6 pp 1819ndash1824 2015

[20] J L Palacios Felix J M Balthazar and R M L R F Brasil ldquoOnsaturation control of a non-ideal vibrating portal frame founda-tion type shear-buildingrdquo Journal of Vibration and Control vol11 no 1 pp 121ndash136 2005

[21] T S Parker and L O Chua Practical Numerical Algorithms forChaotic Systems Springer New York NY USA 1989

[22] M S Nakhla and J Vlach ldquoA piecewise harmonic balancetechnique for determination of periodic response of nonlinearsystemsrdquo IEEE Transactions on Circuits and Systems vol 23 no2 pp 85ndash91 1976

[23] S L Lau Y K Cheung and S Y Wu ldquoIncremental harmonicbalance method with multiple time scales for aperiodic vibra-tion of nonlinear systemsrdquo ASME Journal of Applied Mechanicsvol 50 no 4 pp 871ndash876 1983

[24] Y K Cheung and S H Chen ldquoApplication of the incrementalharmonic balance method to cubic non-linearity systemsrdquoJournal of Sound and Vibration vol 140 no 2 pp 273ndash2861990

[25] S H Chen Y K Cheung and H X Xing ldquoNonlinear vibrationof plane structures by finite element and incremental harmonicbalancemethodrdquoNonlinear Dynamics vol 26 no 1 pp 87ndash1042001

[26] A Grolet and F Thouverez ldquoComputing multiple periodicsolutions of nonlinear vibration problems using the harmonicbalance method and Groebner basesrdquo Mechanical Systems andSignal Processing vol 52-53 no 1 pp 529ndash547 2015

[27] P Ribeiro and M Petyt ldquoGeometrical non-linear steady stateforced periodic vibration of plates part I model and conver-gence studiesrdquo Journal of Sound and Vibration vol 226 no 5pp 955ndash983 1999

[28] G Von Groll and D J Ewins ldquoThe harmonic balance methodwith arc-length continuation in rotorstator contact problemsrdquoJournal of Sound andVibration vol 241 no 2 pp 223ndash233 2001

[29] J V Ferreira and A L Serpa ldquoApplication of the arc-lengthmethod in nonlinear frequency responserdquo Journal of Sound andVibration vol 284 no 1-2 pp 133ndash149 2005

[30] J M Londono S A Neild and J E Cooper ldquoIdentification ofbackbone curves of nonlinear systems from resonance decayresponsesrdquo Journal of Sound and Vibration vol 348 pp 224ndash238 2015

[31] L Renson G Kerschen and B Cochelin ldquoNumerical compu-tation of nonlinear normal modes in mechanical engineeringrdquoJournal of Sound and Vibration vol 364 pp 177ndash206 2016

[32] S Stoykov and P Ribeiro ldquoNonlinear free vibrations of beamsin space due to internal resonancerdquo Journal of Sound andVibration vol 330 no 18-19 pp 4574ndash4595 2011

[33] S Stoykov and P Ribeiro ldquoStability of nonlinear periodicvibrations of 3D beamsrdquoNonlinear Dynamics vol 66 no 3 pp335ndash353 2011

[34] G Formica A Arena W Lacarbonara and H DankowiczldquoCoupling FEM with parameter continuation for analysis ofbifurcations of periodic responses in nonlinear structuresrdquoJournal of Computational and Nonlinear Dynamics vol 8 no2 Article ID 021013 2013

[35] L F Paullo Munoz P B Goncalves R A M Silveira and A RD Silva ldquoNon-linear dynamic analysis of plane frame structuresunder seismic load in frequency domainrdquo in Proceedings of theInternational Conference on Engineering Vibration LjubljanaSlovenia September 2016

[36] httpsenwikipediaorgwikiPeak ground acceleration[37] J Douglas ldquoEarthquake ground motion estimation using

strong-motion records a review of equations for the estimationof peak ground acceleration and response spectral ordinatesrdquoEarth-Science Reviews vol 61 no 1 pp 43ndash104 2003

[38] E I Katsanos A G Sextos and G D Manolis ldquoSelection ofearthquake ground motion records a state-of-the-art reviewfrom a structural engineering perspectiverdquo Soil Dynamics andEarthquake Engineering vol 30 no 4 pp 157ndash169 2010

[39] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons 2008

[40] R Bouc ldquoSur la methode de Galerkin-Urabe pour les systemesdifferentiels periodiquesrdquo International Journal of Non-LinearMechanics vol 7 no 2 pp 175ndash188 1972

[41] M A Crisfield ldquoA fast incrementaliterative solution procedurethat handles lsquosnap-throughrsquordquo Computers and Structures vol 13no 1ndash3 pp 55ndash62 1981


[42] E Ramm ldquoStrategies for tracing the non-linear response nearlimit-pointsrdquo in Nonlinear Finite Element Analysis in StructuralMechanics W Wunderlich E Stein and K J Bathe Eds pp63ndash89 Springer Berlin Germany 1981

[43] V C Ministerio RNE E030 Norma de Diseno SismorresistenteReglamento Nacional de Construcciones Lima Peru 2007

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



second-order behavior of unbraced single-bay pitched-roofsteel frames and proposed a methodology to design this typeof commonly used structure In particular they showed thatdue to the rafter slope the geometrically nonlinear behaviorsof orthogonal beam-and-column and pitched-roof frames arequalitatively different [6] The dynamics of such frames willbe investigated in this work More recently Galvao et al [7]assessed the effect of geometric imperfections and boundaryconditions on the postbuckling response and imperfection-sensitivity of L-frames

The dynamic response of slender frames in the mainresonance regions is an important topic in the analysis ofthese structures under time varying loads Several nonlinearphenomena have been studied in the last decades Simitses[8] studied the effect of static preloading and suddenlyapplied loads on the nonlinear vibrations and instabilities ofslender structures Chan andHo [9] conducted the nonlinearvibration analysis of steel frames with flexible connectionsMazzilli and Brasil [10] presented an analytical study ofthe nonlinear vibrations in a three-time redundant portalframe considering the effect of axial forces upon the naturalfrequencies The axial forces play an important role in tuningthe sway mode and the first symmetrical mode into a 1 2internal resonance Harmonic support excitations resonantwith the first symmetrical mode are then introduced and theamplitudes of nonlinear steady states are computed basedupon a multiple scales solution Later Soares and Mazzilli[11] described the implementation of a computer programto calculate nonlinear normal modes of structural systemsand generate individual modes of planar framed structuresexhibiting geometrically nonlinear behavior The procedurefollows the invariant manifold approach adapted to handleequations of motion of systems discretized by finite elementtechniques Chan and Chui [12] examined based on theirprevious works the nonlinear static and cyclic behavior ofsteel frames with semirigid connections McEwan et al [13]proposed a method for modeling the large deflection beamresponse involving multiple vibration modes while Ribeiro[14] analyzed the geometrically nonlinear vibrations of beamsand plane frames by the hierarchical finite element methodand examined the suitability of the proposed formulation fortime domain nonlinear analyses Da Silva et al [15] studiedthe nonlinear dynamics of a low-rise portal frame usingthe ANSYS finite element software The results show theinfluence of semirigid joints and geometrical nonlinearityon the steel frames dynamics More recently Galvao et al[16] investigated the effect of semirigid connections on thenonlinear vibrations of slender frames and obtained thenonlinear relation between the natural frequencies and staticpreloading Su and Cesnik [17] introduced a strain-basedgeometrically nonlinear beam formulation for structural andaeroelastic modeling and analysis of slender wings of veryflexible aircraft Solutions of different beam configurationsunder static loads and forced dynamic excitations are com-pared against ones from other geometrically nonlinear beamformulations Goncalves et al [18] published an experimentalanalysis of the nonlinear vibrations of a slendermetal columnunder self-weight Masoodi and Moghaddam [19] studiedthe nonlinear dynamics and natural frequencies of gabled

frames having flexible restraints and connections To controlunwanted nonlinear phenomena and vibrations of framesseveral control techniques are proposed in literature Forexample Palacios Felix et al [20] examined the nonlinearcontrol method based on the saturation phenomenon Theinteraction of a nonideal source with a portal frame leads tothe occurrence of interesting nonlinear phenomena duringthe passage through resonance when the nonideal excitationfrequency is near the portal frame natural frequency andwhen the nonideal system frequency is approximately twicethe controller frequency (two-to-one internal resonance)They also suggest the use of the portal frame for energyharvesting

The determination of a structurersquos response in the fre-quency domain characterized by the resonance curves andbifurcation diagrams is in some cases indirectly obtainedthrough a series of analyses in time domain varying step bystep a selected control parameter (see Parker and Chua [21])In this brute force approach for each control parameter valuethe equations of motion must be integrated long enough toreach the steady-state regime This leads to a huge computa-tional effort when analyzing structures with a large numberof degrees of freedom [16] In nonlinear cases the responsein the frequency domain becomes even more cumbersomebecause of the possibility of multiple solutions for certainforcing frequencies Those solutions can be stable and unsta-ble and bifurcations can appear in particular saddle-nodebifurcation at the turning points along the resonance curvesTo predict the nonlinear response in frequency domain somemethods are used such as the harmonic balance method(HBM) and other integral transformations Based on theHBM Nakhla and Valch [22] proposed a method for thesolution of nonlinear periodic networks avoiding the timedomain solution of the dynamic equations Lau et al [23]formulated an incremental variational process based on theHBM to study the nonlinear vibrations of elastic systemsFurther Cheung and Chen [24] used the formulation ofthe incremental harmonic balance method (IHB) to solvea system of differential equations with cubic nonlinearitywhich governs a wide range of engineering problems such aslarge-amplitude vibration of beams or plates An incrementalarc-length method combined with a cubic extrapolationtechnique is adopted to trace the response curves Chen et al[25] generalized this technique to use the incremental HBMin a finite element context being able to use the HBM inthe study of systems with several degrees of freedom Dueto its simple formulation and relatively easy implementationthe HBM is one of the most popular methods to study thenonlinear vibrations of structures [26] The algebraic systemof nonlinear equations which results from theHBM requiresa numerical procedure to obtain the solutions FrequentlyNewtonrsquos method is adopted to solve the nonlinear systemHowever in many cases the resonance curves exhibit limitpoints and in such cases the use of continuation methodsis necessary For example Ribeiro and Petyt [27] studied thefree and steady-state forced vibrations of thin plates Symboliccomputation is employed in the derivation of the modeland the equations of motion are solved by the Newton andcontinuation methods Von Groll and Ewins [28] described a







2 Formulation


Mu (119905) + Cu (119905) + Fi (u (119905)) = 119865 cos (Ω119905) f (1)



u (119905) = 119873sum119899=0

[A119899 cos (119899Ω119905) + B119899 sin (119899Ω119905)] (2)


u (119905) = A1 cos (Ω119905) + B1 sin (Ω119905) (3)




[ C minusΩM + K


B1 = F0

0 (5)




where



D = A1

B1

F = F00

(7)



120591 = Ω119905 (8)


Ω2Mu (120591) + ΩCu (120591) + Fi (u (120591)) = 119865 cos (120591) f (9)


119906119894 (120591) = 119873sum119899=0

[119886119894119899 cos (120591) + 119887119894119899 sin (120591)] (10)


119906119894 (120591) = 1198861198940 + 1198871198941 cos (120591) + 1198881198941 sin (120591) (11)


119906119894 (120591) = 1198861198941 cos (120591) + 1198871198941 sin (120591) (12)


119906119894 = C (120591) d119894 (13)

with

C (120591)= ⟨1 sin (120591) cos (120591)⟩ d119894= ⟨1198861198940 1198871198941 1198881198941⟩T (14)


u (120591) = S (120591)D (15)

where

S (120591) =

[[[[[[[[[[[[[[



]]]]]]]]]]]]]]119873times3119873

D = ⟨d1 d2 d119873⟩T1times3119873

(16)




int21205870


0S (120591)TF0 cos (120591) 119889120591

(18)


Ω2MD + ΩCD + Fi (D) = F (19)

where

M = int21205870

S (120591)TMS (120591) sdot 119889120591C = int2120587

0S (120591)TCS (120591) sdot 119889120591

F = int21205870

S (120591)TF0 cos (120591) 119889120591Fi (D) = int2120587

0S (120591)TFi (S (120591) sdot D) 119889120591

(20)




R (ΩD) = 0 (21)


R (Ω119894D119894) = 0 (22)







(25)








120575Ω119896119894 =

plusmn


for 119896 = 1

minus r119896119879v119896

v119896Tv119896for 119896 gt 1

(29)

where


(30)







(32)


Km = Ω2M + ΩC + 120597Fi (D)120597D

f = (2ΩM + C) sdot D(33)


u (119905119894) = u (119905119894minus1) + Δu (119905) (34)


Fi (119905119894) = Fi (119905119894minus1) + ΔFi (119905) (35)





KNL =

[[[[[[[[[[[[[[[[[[[[[[[[




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

(37)


119875 = 119864119860 ( 120597119906 (119909)120597119909 + 1119871 int1198710

( 120597V (119909)120597119909 )2 sdot 119889119909)119872 = 119864119868 ( 1205972V (119909)1205971199092 )

(38)


119906 (119909) = 2sum119894=1

119867119894 (119909) 119906119894

V (119909) = 6sum119894=3

119867119894 (119909) 119906119894(39)


Δu (119905) = S (120591) ΔD (40)





where

KL = int21205870

S (120591)119879KLS (120591) 119889120591KNL = int2120587

0S (120591)119879KNLS (120591) 119889120591

Fi119894minus1 (D) = int21205870

S (120591)119879FNL119894minus1119889120591(43)



KNL = int21205870

S (120591)119879T119879KNLTS (120591) 119889120591 (44)


KNL = T1015840119879int21205870

S (120591)119879KNLS (120591) 119889120591T1015840 (45)


Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)


0

005

01

015

02

025

Am

plitu

de n

orm

(m)


500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)


m = 10kgkl = 5Nm

k = kl + knlu(t)2


u(t)

m



T1015840119879S (120591)119879 = S (120591)119879T119879 (46)



T1015840 = [[[[[

t101584011 t10158401NGL d


]]]]]

(47)



0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















2 Formulation


Mu (119905) + Cu (119905) + Fi (u (119905)) = 119865 cos (Ω119905) f (1)



u (119905) = 119873sum119899=0

[A119899 cos (119899Ω119905) + B119899 sin (119899Ω119905)] (2)


u (119905) = A1 cos (Ω119905) + B1 sin (Ω119905) (3)




[ C minusΩM + K


B1 = F0

0 (5)




where



D = A1

B1

F = F00

(7)



120591 = Ω119905 (8)


Ω2Mu (120591) + ΩCu (120591) + Fi (u (120591)) = 119865 cos (120591) f (9)


119906119894 (120591) = 119873sum119899=0

[119886119894119899 cos (120591) + 119887119894119899 sin (120591)] (10)


119906119894 (120591) = 1198861198940 + 1198871198941 cos (120591) + 1198881198941 sin (120591) (11)


119906119894 (120591) = 1198861198941 cos (120591) + 1198871198941 sin (120591) (12)


119906119894 = C (120591) d119894 (13)

with

C (120591)= ⟨1 sin (120591) cos (120591)⟩ d119894= ⟨1198861198940 1198871198941 1198881198941⟩T (14)


u (120591) = S (120591)D (15)

where

S (120591) =

[[[[[[[[[[[[[[



]]]]]]]]]]]]]]119873times3119873

D = ⟨d1 d2 d119873⟩T1times3119873

(16)




int21205870


0S (120591)TF0 cos (120591) 119889120591

(18)


Ω2MD + ΩCD + Fi (D) = F (19)

where

M = int21205870

S (120591)TMS (120591) sdot 119889120591C = int2120587

0S (120591)TCS (120591) sdot 119889120591

F = int21205870

S (120591)TF0 cos (120591) 119889120591Fi (D) = int2120587

0S (120591)TFi (S (120591) sdot D) 119889120591

(20)




R (ΩD) = 0 (21)


R (Ω119894D119894) = 0 (22)







(25)








120575Ω119896119894 =

plusmn


for 119896 = 1

minus r119896119879v119896

v119896Tv119896for 119896 gt 1

(29)

where


(30)







(32)


Km = Ω2M + ΩC + 120597Fi (D)120597D

f = (2ΩM + C) sdot D(33)


u (119905119894) = u (119905119894minus1) + Δu (119905) (34)


Fi (119905119894) = Fi (119905119894minus1) + ΔFi (119905) (35)





KNL =

[[[[[[[[[[[[[[[[[[[[[[[[




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

(37)


119875 = 119864119860 ( 120597119906 (119909)120597119909 + 1119871 int1198710

( 120597V (119909)120597119909 )2 sdot 119889119909)119872 = 119864119868 ( 1205972V (119909)1205971199092 )

(38)


119906 (119909) = 2sum119894=1

119867119894 (119909) 119906119894

V (119909) = 6sum119894=3

119867119894 (119909) 119906119894(39)


Δu (119905) = S (120591) ΔD (40)





where

KL = int21205870

S (120591)119879KLS (120591) 119889120591KNL = int2120587

0S (120591)119879KNLS (120591) 119889120591

Fi119894minus1 (D) = int21205870

S (120591)119879FNL119894minus1119889120591(43)



KNL = int21205870

S (120591)119879T119879KNLTS (120591) 119889120591 (44)


KNL = T1015840119879int21205870

S (120591)119879KNLS (120591) 119889120591T1015840 (45)


Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)


0

005

01

015

02

025

Am

plitu

de n

orm

(m)


500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)


m = 10kgkl = 5Nm

k = kl + knlu(t)2


u(t)

m



T1015840119879S (120591)119879 = S (120591)119879T119879 (46)



T1015840 = [[[[[

t101584011 t10158401NGL d


]]]]]

(47)



0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where



D = A1

B1

F = F00

(7)



120591 = Ω119905 (8)


Ω2Mu (120591) + ΩCu (120591) + Fi (u (120591)) = 119865 cos (120591) f (9)


119906119894 (120591) = 119873sum119899=0

[119886119894119899 cos (120591) + 119887119894119899 sin (120591)] (10)


119906119894 (120591) = 1198861198940 + 1198871198941 cos (120591) + 1198881198941 sin (120591) (11)


119906119894 (120591) = 1198861198941 cos (120591) + 1198871198941 sin (120591) (12)


119906119894 = C (120591) d119894 (13)

with

C (120591)= ⟨1 sin (120591) cos (120591)⟩ d119894= ⟨1198861198940 1198871198941 1198881198941⟩T (14)


u (120591) = S (120591)D (15)

where

S (120591) =

[[[[[[[[[[[[[[



]]]]]]]]]]]]]]119873times3119873

D = ⟨d1 d2 d119873⟩T1times3119873

(16)




int21205870


0S (120591)TF0 cos (120591) 119889120591

(18)


Ω2MD + ΩCD + Fi (D) = F (19)

where

M = int21205870

S (120591)TMS (120591) sdot 119889120591C = int2120587

0S (120591)TCS (120591) sdot 119889120591

F = int21205870

S (120591)TF0 cos (120591) 119889120591Fi (D) = int2120587

0S (120591)TFi (S (120591) sdot D) 119889120591

(20)




R (ΩD) = 0 (21)


R (Ω119894D119894) = 0 (22)







(25)








120575Ω119896119894 =

plusmn


for 119896 = 1

minus r119896119879v119896

v119896Tv119896for 119896 gt 1

(29)

where


(30)







(32)


Km = Ω2M + ΩC + 120597Fi (D)120597D

f = (2ΩM + C) sdot D(33)


u (119905119894) = u (119905119894minus1) + Δu (119905) (34)


Fi (119905119894) = Fi (119905119894minus1) + ΔFi (119905) (35)





KNL =

[[[[[[[[[[[[[[[[[[[[[[[[




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

(37)


119875 = 119864119860 ( 120597119906 (119909)120597119909 + 1119871 int1198710

( 120597V (119909)120597119909 )2 sdot 119889119909)119872 = 119864119868 ( 1205972V (119909)1205971199092 )

(38)


119906 (119909) = 2sum119894=1

119867119894 (119909) 119906119894

V (119909) = 6sum119894=3

119867119894 (119909) 119906119894(39)


Δu (119905) = S (120591) ΔD (40)





where

KL = int21205870

S (120591)119879KLS (120591) 119889120591KNL = int2120587

0S (120591)119879KNLS (120591) 119889120591

Fi119894minus1 (D) = int21205870

S (120591)119879FNL119894minus1119889120591(43)



KNL = int21205870

S (120591)119879T119879KNLTS (120591) 119889120591 (44)


KNL = T1015840119879int21205870

S (120591)119879KNLS (120591) 119889120591T1015840 (45)


Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)


0

005

01

015

02

025

Am

plitu

de n

orm

(m)


500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)


m = 10kgkl = 5Nm

k = kl + knlu(t)2


u(t)

m



T1015840119879S (120591)119879 = S (120591)119879T119879 (46)



T1015840 = [[[[[

t101584011 t10158401NGL d


]]]]]

(47)



0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











R (ΩD) = 0 (21)


R (Ω119894D119894) = 0 (22)







(25)








120575Ω119896119894 =

plusmn


for 119896 = 1

minus r119896119879v119896

v119896Tv119896for 119896 gt 1

(29)

where


(30)







(32)


Km = Ω2M + ΩC + 120597Fi (D)120597D

f = (2ΩM + C) sdot D(33)


u (119905119894) = u (119905119894minus1) + Δu (119905) (34)


Fi (119905119894) = Fi (119905119894minus1) + ΔFi (119905) (35)





KNL =

[[[[[[[[[[[[[[[[[[[[[[[[




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

(37)


119875 = 119864119860 ( 120597119906 (119909)120597119909 + 1119871 int1198710

( 120597V (119909)120597119909 )2 sdot 119889119909)119872 = 119864119868 ( 1205972V (119909)1205971199092 )

(38)


119906 (119909) = 2sum119894=1

119867119894 (119909) 119906119894

V (119909) = 6sum119894=3

119867119894 (119909) 119906119894(39)


Δu (119905) = S (120591) ΔD (40)





where

KL = int21205870

S (120591)119879KLS (120591) 119889120591KNL = int2120587

0S (120591)119879KNLS (120591) 119889120591

Fi119894minus1 (D) = int21205870

S (120591)119879FNL119894minus1119889120591(43)



KNL = int21205870

S (120591)119879T119879KNLTS (120591) 119889120591 (44)


KNL = T1015840119879int21205870

S (120591)119879KNLS (120591) 119889120591T1015840 (45)


Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)


0

005

01

015

02

025

Am

plitu

de n

orm

(m)


500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)


m = 10kgkl = 5Nm

k = kl + knlu(t)2


u(t)

m



T1015840119879S (120591)119879 = S (120591)119879T119879 (46)



T1015840 = [[[[[

t101584011 t10158401NGL d


]]]]]

(47)



0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










KNL =

[[[[[[[[[[[[[[[[[[[[[[[[




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

(37)


119875 = 119864119860 ( 120597119906 (119909)120597119909 + 1119871 int1198710

( 120597V (119909)120597119909 )2 sdot 119889119909)119872 = 119864119868 ( 1205972V (119909)1205971199092 )

(38)


119906 (119909) = 2sum119894=1

119867119894 (119909) 119906119894

V (119909) = 6sum119894=3

119867119894 (119909) 119906119894(39)


Δu (119905) = S (120591) ΔD (40)





where

KL = int21205870

S (120591)119879KLS (120591) 119889120591KNL = int2120587

0S (120591)119879KNLS (120591) 119889120591

Fi119894minus1 (D) = int21205870

S (120591)119879FNL119894minus1119889120591(43)



KNL = int21205870

S (120591)119879T119879KNLTS (120591) 119889120591 (44)


KNL = T1015840119879int21205870

S (120591)119879KNLS (120591) 119889120591T1015840 (45)


Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)


0

005

01

015

02

025

Am

plitu

de n

orm

(m)


500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)


m = 10kgkl = 5Nm

k = kl + knlu(t)2


u(t)

m



T1015840119879S (120591)119879 = S (120591)119879T119879 (46)



T1015840 = [[[[[

t101584011 t10158401NGL d


]]]]]

(47)



0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ma

kr

E = 310GPa

A = 62831m2

I = 3952m4

H = 70m

휌 = 240000 kgm3

Ma = 15000000 kg

C = 05M

HEI

= Acos (Ωt)ug(t)


0

005

01

015

02

025

Am

plitu

de n

orm

(m)


500 1000 1500 2000 2500 3000 3500 4000000Ω (rads)


m = 10kgkl = 5Nm

k = kl + knlu(t)2


u(t)

m



T1015840119879S (120591)119879 = S (120591)119879T119879 (46)



T1015840 = [[[[[

t101584011 t10158401NGL d


]]]]]

(47)



0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

0

5

10

15

20

25

30

Am

plitu

de n

orm

(m)

10 20 30 400Ω (rads)

10 20 30 400Ω (rads)

kNL = 10Nm3

kNL = minus10Nm3


1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H



t1015840119894119895 = T119894119895I3 (48)









0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0005

001

0015

002

0025

003

0035

000 5000 10000 15000 20000

H = 0m

H = 3m

H = 6m

H = 0m

H = 3m

H = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ω (rads)

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

|Dy|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|

0

0005

001

0015

002

0025

003

0035

|Dx|

0

0005

001

0015

002

0025

003

0035

|Dx|

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

0

00005

0001

00015

0002

00025

0003

|Dy|






H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










H = 6m

H = 3m

LinearNonlinear

LinearNonlinear

LinearNonlinear

H = 0m

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

0000

0005

0010

0015

0020|D

y|

0000

0005

0010

0015

0020

|Dy|




00 2182 4897 13054 16130 1961 4714 10823 1598060 1540 3642 8321 9208





H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










H = 0m

H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

H = 3m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

H = 0m

LinearNonlinear

5000 10000 15000000Ω (rads)

0000

0002

0004

0006

0008

0010

0012

|Dy|

0000

0005

0010

0015

0020

0025

0030

|Dx|






1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1st mode

2nd mode

3rd mode

4th mode

E = 210GPa휌 = 785 tonm3

d = 15 cm


A = 173e minus 6m2

tw

tfbf

100m

60m

y

y

d xx

H R =10

sin (2 arc tg(H10))


H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)





0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0000

0005

0010

0015

0020

|Dy|

H = 1m H = 3m

H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000 20000000Ω (rads)

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|

5000 10000 15000 20000000Ω (rads)

0000

0005

0010

0015

0020

|Dy|







10 2102 5173 12260 1670530 1781 4851 9777 1516050 1320 4079 7402 11341



H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










H = 1m H = 1m

H = 3mH = 3m

H = 5m H = 5m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000휔 (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

5000 10000 15000000Ω (rads)

0000

0005

0010

0015

0020

0025

|Dx|

0000

0005

0010

0015

0020

0025

0030

|Dx|

0000

0005

0010

0015

0020

0025

|Dx|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|

0000

0002

0004

0006

0008

0010

|Dy|





0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

0000

0010

0020

0030

0040

0050

0060

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

0000

0003

0006

0009

0012

0015

0018

|Dy|

H = 0mH = 0m

H = 3m H = 3m

H = 6mH = 6m

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear






LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 048g Ax = 096g

Ax = 18g

Ax = 36g

0

0005

001

0015

002

0025

003

0035

|Dx|

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

002

004

006

008

01

012

014|D

x|

0

001

002

003

004

005

006

007

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dx|

2000 4000 6000 8000 10000000Ω (rads)






LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LinearNonlinear

LinearNonlinear

Ay = 18gAy = 36g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

|Dy|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

|Dy|




00 1660 2109 6068 1173530 1552 2558 5803 919560 1293 2482 5024 5947







4 Conclusions



LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

Ax = 15g

Ay = 10gAx = 30g

Ay = 20g

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

0

0001

0002

0003

0004

0005

0006

0007

0008

0009

001

|Dy|

0

001

002

003

004

005

006|D

x|

0

002

004

006

008

01

012

|Dx|

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002|D

y|





LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LinearNonlinear

LinearNonlinear

LinearNonlinear

LinearNonlinear

0000

0050

0100

0150

0200

0250

0000

0020

0040

0060

0080

0100

0120

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

2000 4000 6000 8000 10000000Ω (rads)

|Dx|

|Dx|

Ax = 30g

Ay = 20g

Ax = 15g

Ay = 10g

Ax = 15g

Ay = 10g

Ax = 30g

Ay = 20g

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

|Dy|

0000

0005

0010

0015

0020

0025

0030

0035|D

y|





References














































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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