Self-excited vibration control for axially fast excited beam by a time delay state feedback

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Self-excited vibration control for axially fast excited beam by a time delay state feedback

Mustapha Hamdi, Mohamed Belhaq

^*

Laboratory of Mechanics, University Hassan II-Aı¨n Chock, PB 5366 Maaˆrif, Casablanca, Morocco Accepted 18 February 2008

Abstract

This work examines the control of self-excited vibration of a simply-supported beam subjected to an axially high- frequency excitation. The investigation of the resonant cases are not considered in this paper. The control is imple- mented via a corrective position feedback with time delay. The objective of this control is to eliminate the undesirable self-excited vibrations with an appropriate choice of parameters. The issue of stability is also addressed in this paper.

Using the technique of direct partition of motion, the dynamic of discretized equations is separated into slow and fast components. The multiple scales method is then performed on the slow dynamic to obtain a slow ﬂow for the amplitude and phase. Analysis of this slow ﬂow provides analytical approximations locating regions in parameters space where undesirable self-excited vibration can be eliminated. A numerical study of these regions is performed on the original discretized system and compared to the analytical prediction showing a good agreement.

1. Introduction

In recent years, an increasing number of works has been devoted to study the influence of fast excitation on the slow dynamics of mechanical systems[1–3]. Such excitation can affect certain characteristic of systems such as equilibrium stability[4], linear stiffness[5], damping[6], natural frequencies[7]and stick-slip dynamics[8]. Fast excitation may also influence bifurcation of nonlinear systems such as symmetry breaking[9]and Hopf bifurcation[10].

Recent research focused attention on the eﬀect of axial high-frequency excitation on the slow behavior of elastic beams. Tcherniak[2]examined the inﬂuence of fast excitation on a simply supported beam and showed that the system’s dynamic properties may change. Jensen[11]performed a nonlinear buckling analysis of an elastic beam subjected to an axial static free and high-frequency excitation. It was shown that adding high-frequency excitation increases the buckling load and stable buckled equilibria may co-exist with the stabilized straight position.

Besides the aforementioned research on the effect of fast excitation on the slow dynamic of beams, there are also other works focusing mainly on the control of flexible beams. Olgac and Jalili[12]performed a modal analysis and studied stability of a flexible beam using a delayed resonator operated on the structure. Pinto and Goncßalves[13]studied the

doi:10.1016/j.chaos.2008.02.023

*Corresponding author.

E-mail address:mbelhaq@yahoo.fr(M. Belhaq).

Chaos, Solitons and Fractals 41 (2009) 521–532

www.elsevier.com/locate/chaos

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control of nonlinear oscillations of a shallow simply supported buckled beam subjected to transverse time-varying loads using bending moments distributed along the beam axis. Maccari[14]investigated vibration control near the primary resonance of cantilever beam with time delay. The control of chaotic motion was analyzed for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitation at the free end[15].

El-Bassiouny[16]examined vibrational control for the primary and 1:3 subharmonic resonances of a cantilever beam under state feedback control with a time delay. Nana Nbendjo and Woafo[17]considered the control with time delay to suppress chaotic vibrations of undamped buckled beam subjected to parametric excitation.

In this paper, we investigate the stability chart of the trivial equilibrium and the control of self-excited vibration of a simply-supported beam subjected to an axial high-frequency excitation. The control consists of a proportional position feedback with time delay. We focus attention on the interaction eﬀect of fast excitation and time delay on the stability chart and on the suppression of self-excited oscillation of the system. These undesirable vibrations may be induced, for instance, by an elastic support of the structure with a nonlinear damping.

This work relates to recent works dealing with the control of self-excited vibration in a delayed van der Pol pendulum subjected to high-frequency excitation[18,19].

The organization of the paper is as follows. In Section2, the equation of motion governing small vibrations about the straight beam position is given using Galerkin’s technique. Applying the method of direct partition of motion (DPM)[20], the dynamic of discretized equations is separated into fast components and the main autonomous equations governing the slow dynamic. The stability chart and the natural frequencies spectrum are analyzed for the single mode slow dynamic. In Section3, we use the multiple scales technique[21]on the slow dynamic to derive a slow ﬂow which is examined for locating regions in parameter space where undesirable self-excited vibration of the single mode beam can be inhibited. A numerical study is performed on the original discretized system and compared to analytical predictions for validation. Section4concludes the study.

2. Equation of motion and slow dynamic

Consider a controlled simply-supported beam subjected to an axial high-frequency excitation having the form aX²cosðXtÞwhereais a non-dimensional amplitude of excitation andXis the excitation frequency (seeFig. 1). The control consists of a proportional position feedback with time delay. If we assume that the beam is mounted on an elastic support with nonlinear dampingFðw;wÞ_ of van der Pol type, the equation of motion governing small transverse deﬂectionswðx;tÞof the beam can be written in the dimensionless form as

€ wþ g

p²w_⁰⁰⁰⁰ ðabw²Þw_þ 1

p⁴w⁰⁰⁰⁰þkwðx;ssdÞ ¼ w⁰⁰aX²cosðXsÞ ð1Þ with the boundary conditions

w¼0 and w⁰¼0 at x¼0 ð2aÞ

w¼0 and w⁰¼0 at x¼1 ð2bÞ

wheregis the internal damping,aandbare the linear and nonlinear viscous damping,kands_drepresent the gain and time delay,aandXdenote the amplitude and frequency of the fast excitation ands¼x1tis the dimensionless time. The characteristic frequency corresponding to the first natural frequency of the structure isx1¼ ð^p_lÞ² ffiffiffiffi

EI m

q

, wherel,EIandm are the length, the constant bending stiﬀness and the mass per unity of length of the beam, respectively.

The transverse deﬂectionwðx;sÞcan be represented as

wðx;sÞ ¼X^N

k¼1

wkðsÞsinðkpxÞ ð3Þ

w(t- d, x)

Beam aΩ² cos(Ωt)

w x

τ F (w, w)

Fig. 1. Representation of the beam.

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With this assumption, equation of motion for the desired number of modes can be derived by substituting Eq.(3)into Eq.(1)and by performing the Galerkin’s method. This yields the set of equations

€

w_kþg_kw__k aw__kbX^N

m;n;l

K_kmnlw_mw_nw__l

!

þk⁴w_kþkw_kðssdÞ ¼w_ka_kX²cosðXsÞ ð4Þ

whereg_k¼gðk²pÞ² andK_kmnl¼R1

0 sinðkpxÞsinðmpxÞsinðnpxÞsinðlpxÞdxare thekth internal damping and the coeﬃ- cient of the modal interaction, respectively, andak¼ ðkpÞ²a.

2.1. Separation of motion and slow dynamic

Following[2], we apply the method of DPM to separate between the slow and fast dynamics of the individual oscillation mode. This separation provides the main autonomous equation governing the slow motion of the considered mode. We introduce two time scales: a fast timeT0¼Xs¼¹sand a slow time T1¼s, and we seek the solution w_kðT0;T1Þin the uniformly valid expansion to the order² having the form

w_kðT0;T1Þ ¼z_kðT1Þ þ/_1kðT0;T1Þ þ²/_2kðT0;T1Þ þOð³Þ ð5Þ The time derivatives is transformed according to d=ds¼¹D0þD1 and d²=ds²¼²D²₀þ2¹D0D1þD²₁, where D^j_i o^j=oT^j_i. Assume that the coeﬃcient a,bK_kmnl anda_kXare of the order of ⁰. The ﬁrst term of the expansion, zkðT1Þ, describes the slow main motion of the considered mode, whereas the/-terms stand for small overlays of fast motion. The fast component/_ik is considered to be small compared to the slow oscillation.

Inserting Eq.(5)into Eq.(4)and equating to zero the coeﬃcients of like powers of, we obtain to order¹:

D²₀/_1k¼g_1kðzk;T0Þ ð6Þ

where

g_1kðzk;T0Þ ¼zkakXcosðT0Þ ð7Þ

To order⁰, one obtains

D²₀/_2k¼ D²₁zk2D₀D1/_1kg_kðD1zkþD0/_1kÞ þ ðaðD1zkþD0/_1kÞ bX^N

m;n;l

K_kmnlzmznðD1zlþD0/_1lÞÞ k⁴zk

kz_kðT1s_dÞ þg_2kð/_1k;T0Þ RkðT0;T1Þ ð8Þ where

g_2kð/_1k;T0Þ ¼/_1kakXcosðT0Þ ð9Þ

The general solution to Eq.(6)is /_1kðT0;T1Þ ¼

Z Z

g_1kðzk;T0ÞdT0dT0þT0C1kðT1Þ þC2kðT1Þ ð10Þ whereC1k andC2kare arbitrary functions. To keep/_1k bounded asT0! 1,C1kmust vanish. Also, one can assume C2k¼0 because it has no contribution to the solution(5). Thus

/_1kðT0;T1Þ ¼ zkakXcosðT0Þ ð11Þ

wherezk¼zkðT1Þis considered as a constant in the fast timeT0.

To obtain an equation governing the slow dynamiczk, the term/_2kin Eq.(8)should be bounded. All terms of the right-hand sideRk are either 2p-periodic inT0or do not depend onT0. Hence,Rkcan be expressed in exact form as

RkðT0;T1Þ ¼ hRkðT0;T1Þi þHkðT0;T1Þ ð12Þ whereHk denoteT0-harmonic terms

HkðT0;T1Þ ¼X¹

j¼1

ðAjkðT1ÞsinðjT₀Þ þBjkðT1ÞcosðjT₀ÞÞ

withðAjk;B_jkÞdenoting the Fourier-coeﬃcients and whereh ideﬁnes theT₀-average operator hRkðT0;T1Þi ¼ 1

2p Z 2p

0

RkðT0;T1ÞdT₀ ð13Þ

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Inserting(12)into(8)one obtains

D²₀/_2k¼ hRkðT0;T1Þi þHkðT0;T1Þ ð14Þ

BecausehRkðT0;T1Þiis independent ofT0, Eq.(14)can be solved leading to /_2k¼1

2T²₀hGkðT0;T₁Þi þ Z Z

H_kðT0;T₁ÞdT₀dT₀þT₀C_3kðT1Þ þC_4kðT1Þ ð15Þ whereC3k andC4k are arbitrary functions of the slow timeT1. Using the same arguments as forC1k andC2k yields C3k¼C4k¼0. The integral term in Eq. (15) remains bounded as T0! 1 sinceHk is bounded periodic function, whereas the ﬁrst term is unbounded. Hence, for/_2k to be bounded for allT0>0, the conditionhRkðT0;T1Þi ¼0 is re- quired. With /_1k given by(11) andg_1k by a function periodic inT0 with zero mean,hg_1ki ¼0, one has h/_1ki ¼0, hD0/_1ki ¼0 andhD²₀/_1ki ¼0. Consequently, the equation for slow motionszk becomes

D²₁zkþg_kD1zk aD1zkbX^N

m;n;l

KkmnlzmznD1zl

!

þk⁴zkþkzkðT1sdÞ ¼ hg_2kð/_1k;T0Þi ð16Þ

Substituting the known function/_1k and averaging give the following main equation governing the slow components zkðT1ÞofwkðT0;T1Þ

D²₁zkþg_kD1zk aD₁zkbX^N

m;n;l

K_kmnlzmznD1zl

!

þ k⁴þðakXÞ² 2

!

zkþkz_kðT1s_dÞ ¼0 ð17Þ

Thus, the natural frequencies of the slow motion for the case of undamped undelayed oscillations is given by

x_k¼k²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þp⁴ðaXÞ²

2 s

ð18Þ

3. Stability chart

In this section, we study the eﬀect of time delay on the natural frequencies spectrum of the fast excited beam and the inﬂuence of high-frequency excitation on the stability chart of the trivial solution of the delayed slow motion (Eq.(17)).

The linear equation corresponding to Eq.(17)reads

€

zkþ ðg_kaÞz_kþx²_kz1þkzkðT1sdÞ ¼0 ð19Þ The stability analysis means the investigation of the trivial solution using the transcendental characteristic equation

s²þ ðg_kaÞsþx²_kþke^s^d^s¼0 ð20Þ

This equation possesses inﬁnitely many ﬁnite roots fork–0 andsd–0. To achieve stability, two dominant roots of Eq.

(20)should be placed on the imaginary axis at the desired resonant frequency, while other roots remain in the stable left-half of the complex plane. The imaginary characteristic roots ares¼ ixc, wherex_cis the resonance frequency and i¼ ffiffiffiffiffiffiffi

p1

. The subscript c implies the crossing of the root loci on the imaginary axis. Substitutings¼ ixc into Eq.(20)and solving for the control parameterskcandsdcyields

kc¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððx²_kx²_cÞ²þ ððg_kaÞxcÞ²Þ q

ð21Þ and

sdc¼ 1

x_c arctan ðg_kaÞxc

x²_kx²_c

þ2ð‘1Þp

; ‘¼1;2;3;. . . ð22Þ

where‘corresponds to the‘th branch.

The natural undamped undelayed frequencies, Eq. (18), and the undamped delayed frequencies x_ck ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k⁴ð1þp⁴^ðaXÞ₂²Þ k² q

of the slow motion are shown inFig. 2for the ﬁrst three modes versus fast excitation force intensity. The critical value of feedback gain must satisfy the conditionk6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k⁴ð1þp⁴^ðaXÞ₂²Þ q

for eachkth mode. This ﬁgure shows the inﬂuence of time delay on the natural frequencies. It can be seen that each natural frequency spectrum is

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splitted into two branches which come closer to the undelayed natural spectrum (dashed line) for large excitation force intensity. The splitting is more visible for the ﬁrst frequency. This means that the beam may have, for the same excitation, two harmonic vibrations in the vicinity of the natural undelayed frequencies.

The stability chart is presented inFig. 3for ﬁxedg₁a¼0:067,a1¼0:02 andX¼100. The dashed region corresponds to the stability domain of the trivial equilibrium of the mode beam under consideration. Above this region the trivial equilibrium is unstable.

InFig. 4, we show for the two ﬁrst lobes ofFig. 3, the eﬀect of the high-frequency excitationXon the stability chart.

It is seen that the eﬀect of fast excitation increases the stability domain, shifts the lobes left and increases the peaks maxima.Figs. 5 and 6 illustrate time traces of the slow componentz1ðsÞfor two operating points S and U selected in the stable and unstable zones inFig. 3, respectively.

4. Self-excited vibration and control

In this section, we use the method of multiple scales[21]to derive the slow ﬂow of slow motion corresponding to Eq.

(17). This slow ﬂow is analyzed to obtain analytical approximation of self-excited vibrations. Introducing a non-dimensional bookeeping parameterland scaling parametersg₁¼lg~₁,a¼l~a,bK1111¼b₁¼lb~₁, andk¼l~k, Eq.(17)reads

€

z1þg~₁z_1 ð~ab~₁z²₁Þz_1þx²₁z1þ~kz1ðssdÞ ¼0 ð23Þ whereðÞ ¼_ _ds^d andx²₁¼1þ^ða¹₂^XÞ²

.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 5 10 15

Excitation, aΩ ωk andωc±

ωc1+ ω1 ωc2− ωc±

⋅⋅⋅⋅⋅⋅ ωk

Fig. 2. The ﬁrst three undelayed natural frequencies (dashed line) and the corresponding delayed frequencies (solid line) vs. fast excitation force intensity;k¼0:8.

0 2 4 6 8 10 12 14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time delay, τdc Feedback gain, λc

•

S

U

Fig. 3. Stability chart of the trivial solution of Eq.(19);g1a¼0:067,a1¼0:02,X¼100.

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0 1 2 3 4 5 6 7 8 9 10 0

0.5 1 1.5 2 2.5 3 3.5

Time delay, τdc Feedback gain, λc

Ω=50 Ω=100

Ω=150

Fig. 4. Eﬀect of varyingXon the stability chart;g₁a¼0:067,a1¼0:02.

0 5 10 15 20 25 30

−0.08

−0.06

−0.04

−0.02 0 0.02 0.04 0.06

Time,τ z1(τ)

Fig. 5. The time trace of slow componentz1ðsÞresponse corresponding to point S inFig. 3;kc¼0:5;sdc¼2:4.

0 10 20 30 40 50 60 70 80 90 100

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

Time,τ z1(τ)

Fig. 6. The time trace of slow componentz1ðsÞresponse corresponding to point U inFig. 3;kc¼1;sdc¼6:6.

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Using the method of multiple scales, a first order uniform expansion of the solution to Eq.(23)is sought in the form z1ðs;lÞ ¼z11ðT1;T2Þ þlz12ðT1;T2Þ þOðl²Þ ð24Þ where the independent time scales are defined asT1¼sandT2¼ls. It follows that the derivatives with respect tos become_ds^d¼D1þlD2 and_ds^d²2¼D²₁þ2lD1D2þl²D2, whereD^j_n¼o^j=oT^j_n. Substituting Eq.(24) into Eq.(23), using the notationD^j_n¼o^j=oT^j_n and equating coefficients of like powers ofl, leads to

Orderl⁰:

D²₁z11þx²₁z11¼0 ð25Þ

Orderl¹:

D²₁z12þx²₁z12¼ 2D1D2z11g~₁D1z11þ ð~ab~₁z²₁₁ÞD1z11~kz11ðT1sdÞ ð26Þ The solutions to the orderl⁰is of the complex form

z11¼AðT2Þexpðix1T1Þ þAðT2Þexpðix1T1Þ ð27Þ

whereAandAare complex conjugate functions. This solution is substituted into Eq.(26)to obtain

D²₁z12þx²₁z12¼ ½2iD1D2Aix₁g~₁Aþ ð~ab~₁AAÞix1A~kAe^ix¹^s^de^ix¹^T¹þccþNST ð28Þ where symbol cc denotes the parts of the complex conjugate of the function at right-hand side of Eq.(28), and NST represent parts that do not produce secular terms. Eliminating the terms that produce secular terms from Eq.(28), we have

2ix1D2Aix₁g~₁Aþ ð~ab~₁AAÞix1A~kAexpðix1s_dÞ ¼0 ð29Þ The functionAmay be expressed in the polar form

A¼1

2RexpðihÞ ð30Þ

whereRandhare real functions with respect toT2. Substituting Eq.(30)into Eq.(29)and separating the real and imaginary parts, yield

R⁰þ1

2g~₁R 1 2~aR1

8b~₁R³

~k

2x₁Rsinðx1sdÞ ¼0 ð31aÞ

h⁰ ~k 2x1

Rcosðx1sdÞ ¼0 ð31bÞ

0 50 100 150 200 250

0 0.5 1 1.5 2 2.5 3 3.5

Time delay, τ d

Excitation frequency,Ω

λ=0.2 λ=0.25

SO

no SO

Fig. 7. Curves delimiting the existence region of self-oscillation, conditions(35) and (36); SO: self-oscillation.

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where the prime denotes a derivative with respect toT2. An equilibrium point in Eqs.(31)corresponds to a periodic motion in the slow dynamic (23)and toquasi periodic vibration in the original system(4). Equilibria are obtained by settingR⁰¼h⁰¼0 in Eqs.(31). This leads to

R¼0;R¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4

b₁ ag₁þ k x1

sinx₁s_d s

ð32Þ The condition for the solutionR¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4

b₁ðag₁þ_x^k

1sinx1sdÞ

q to be real is

ag₁þ k

x₁ sinx1sdP0 ð33Þ

By settingx1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ^ða¹₂^XÞ² q

, Eq.(33)becomes

80 100 120 140 160 180 200 220

0 0.5 1 1.5 2 2.5

Time delay, τd

Excitation frequency,Ω

Analytical result Numerical result λ=0.2

SO

no SO no SO

Fig. 8. Comparison between analytical results based on conditions(35) and (36)and numerical integration of Eq. (23), SO: self- oscillation.

50 100 150 200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Feedback gain, λ

Excitation frequency, Ω Analytical result Numerical result

τ_d=1

τd=0.8

SO no SO

Fig. 9. Comparison between analytical results (Eqs.(35) and (36)) and numerical integration of Eq.(23), SO: self-oscillation.

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sin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þða1XÞ²

2 s

sd

0

@

1

APg₁a

k

2 s

ð34Þ

The inequality(34)provides the two following conditions corresponding to the birth of limit cycle

sd> 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ^ða¹₂^XÞ²

q arcsin g₁a

k

2 0 s

@

1

A ð35Þ

80 100 120 140 160 180 200

0 0.5 1 0 0.5 1 1.5 2 2.5

Excitation frequency, Ω Feedback gain,

λ Time delay, τd

no SO

SO

no SO

Fig. 10. Surface curve locating region of suppression of self-excited oscillation. The white section lying in the plane k¼0:2 corresponds to the curve given inFig. 8.

0 10 20 30 40 50 60 70 80 90 100

−4

−3

−2

−1 0 1 2 3 4

Time, τ w1(τ) and z1(τ)

A

72 72.5 73 73.5 74 74.5 75 75.5

−4

−3

−2

−1 0 1 2 3 4

Time, τ w1(τ) and z1(τ)

w₁(τ)

z₁(τ)

a b

Fig. 11. (a) Numerical solution for the full single mode motionw1ðsÞand its slow componentz1ðsÞ. (b) Enlargement of fragment A;

X¼100,k¼0:6,sd¼0:5.

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0 20 40 60 80 100 120 140 160 180 200

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

Time, τ z1(τ)

τd=0.25

0 20 40 60 80 100 120 140 160 180 200

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

Time, τ z1(τ)

τd=1

0 20 40 60 80 100 120 140 160 180 200

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

Time, τ z1(τ)

τd=2

Fig. 12. Numerical time histories for the slow motionz1ðsÞfor diﬀerent values ofsdinFig. 8;X¼100,k¼0:2.

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and

sd< 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ^ða¹₂^XÞ²

q parcsin g₁a

k

2 s 0

@

1 A 0

@

1

A ð36Þ

Using Eq.(31b)and the argument of the cosine in Eq.(27)leads to the frequency of the self-excited oscillation x¼x1 k

2x₁Rcosx1sd ð37Þ

A condition for the existence of a limit cycle is guaranteed because this frequency is positive for allsd.

In what follows we fix the parametersa1¼0:02,g₁¼0:197,a¼0:13 andb₁¼0:2. InFig. 7, we show for different values ofkthe conditions(35) and (36)locating regions where self-excited oscillations are absent. These conditions are valid when the condition^g¹_kâ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ^ða¹₂^XÞ² q

61 is held. These curves suggest that for a ﬁxed value of the delay amplitude, there exists a critical valueX_crfor which the system my have self-excited vibration. This critical valueX_cras well as the correspondingsd;crare obtained from Eqs.(33)–(36)and written as

Xcr¼ 1 a₁

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 k²

ðg₁aÞ²1 v !

uu

t ð38Þ

and

sd;cr¼g₁a k

p

2 ð39Þ

In the case ofFig. 7andk¼0:25, these critical values areX_cr¼252:6 ands_d;cr¼0:42.

FromFig. 7, we can see that the region where self-oscillations can be eliminated increased by increasing the delay amplitudek.Fig. 9shows the effect of the time delay on the suppression region. Comparisons between analytical results and numerical integration of Eq.(23)usingdde23[22]are shown inFigs. 8 and 9. Note that the conditions(35) and (36) are not affected by the nonlinear damping coefficientb₁at the leading order.

Fig. 10illustrates the surface curve in 3-dimensional (X,k,s_d) parameter space delimiting the region where limit cycle can be eliminated. The white section lying in the plane k¼0:2 corresponds to the curve given in Fig. 8. In Fig. 11the numerical time histories of the full motionw1ðsÞ, Eq.(4), and of the slow dynamicz1ðtÞ, with fragment A enlarged inFig. 11b are compared showing a good agreement. Finally, Fig. 12shows time histories of the slow dynamicz₁ðsÞwhen we move along the lineX¼100 inFig. 8for the valuess_d¼0:25,s_d¼1 ands_d¼2.

5. Conclusions

In this paper, we have investigated the linear stability chart and self-excited vibration of a parametrically driven and self-excited hinged-clamped beam under time delay control consisting of a proportional position feedback. Using the Galarkin’s technique, the single mode model governing small vibrations of the beam near the stationary trivial solution is provided. First, we have applied the method of DPM to obtain the main autonomous equation governing the slow averaged motions. The spectrum of natural frequencies for the beam and the stability chart of the trivial solution are analyzed in the presence of high-frequency excitation and time delay state feedback. We have shown that in presence of time delay, the spectrum of natural frequencies splits into two branches lying below and above the undelayed natural spectrum. The width between the two frequency branches, which depend on the gain delay, are more visible for the first natural frequency spectrum. By increasing the excitation, the width decreases and the branches approach the undelayed spectrum. This result suggests that in the delayed case, the beam may have two harmonic vibrations close to the undelayed natural frequency. Furthermore, it was seen that high-frequency excitation has a significant effect on the stability chart, that is, as the excitation frequency increases, so does the stability domain, the lobes shift left and the peaks maxima increment. This suggests that for large excitation frequency, stability may be achieved for large gain delay and for moderate time delay. In a second part, we have performed the method of multiple scales on the slow dynamic to obtain a slow flow for the amplitude and phase. Analysis of this slow flow provides analytical approximations locating regions in parameters space where undesirable self-excited vibration can be inhibited. Numerical study of this region was performed on the original discretized system and compared to analytical predictions showing a every good agreement. It was shown that the region where self-excited vibrations are eliminated increases by increasing the delay amplitude. For fixed values of delay amplitude, self-oscillation persists for appropriate large frequency excitation and time delay.

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References

[1] Chelomei VN. Mechanical paradoxes caused by vibrations. Soviet Phys Doklady 1983;28:387–90.

[2] Tcherniak D. The inﬂuence of fast excitation on a continuous system. J Sound Vib 1999;227(2):343–60.

[3] Schmitt JM, Bayly PV. Bifurcation in the mean angle of a horizontally shaken pendulum: analysis and experiment. Nonlinear Dyn 1998;15:1–14.

[4] Thomsen JJ, Tcherniak DM. Chelomei’s pendulum explained. Proc Royal Soc Lond 2001;A457:1889–913.

[5] Jensen JS, Tcherniak DM. Stiﬀning eﬀects of high-frequency excitation: experiments for an axially loaded beam. J Appl Mech 2000;67(2):397–730.

[6] Hansen MH. Eﬀect of high-frequency excitation on natural frequencies of spinning discs. J Sound Vib 2000;243(4):577–89.

[7] Chatterjee S, Singha TK, Karmakar SK. Non-trivial eﬀect of fast vibration on the dynamics of a class of nonlinearly damped mechanical systems. J Sound Vib 2003;260(4):711–30.

[8] Thomsen JJ. Using fast vibrations to quench friction-induced oscillations. J Sound Vib 1999;228(5):1079–102.

[9] Mann BP, Koplow MA. Symmetry breaking bifurcations of parametrically excited pendulum. Nonlinear Dyn 2006;4:427–37.

[10] Bourkha R, Belhaq M. Eﬀect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos, Solitons &

Fractals 2007;34:621–7.

[11] Jensen JS. Buckling of an elastic beam with added high-frequency excitation. Int J Non-linear Mech 2000;35:217–27.

[12] Olgac N, Jalili N. Modal analysis of ﬂexible beams with delayed resonator vibration absorber: theory and experiment. J Sound Vib 1998;218(2):307–31.

[13] Pinto OC, Goncßalves PB. Active non-linear control of buckling and vibrations of a ﬂexible buckled beam. Chaos, Solitons &

Fractals 2002;14:227–39.

[14] Maccari A. Vibration control for the primary resonance of a cantilever beam by a time delay state feedback. J Sound Vib 2003;259(2):241–51.

[15] Zhang W. Choatic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Chaos, Solitons & Fractals 2005;26:731–45.

[16] El-Bassiouny AF. Vibrational control of cantilever beam with time delay state feedback. Z Naturforsch 2006;61a:629–40.

[17] Nana Nbendjo BR, Woafo P. Active control with delay of horseshoes chaos using piezoelectric absorber on a buckled beam under parametric excitation. Chaos, Solitons & Fractals 2007;32:73–9.

[18] Belhaq M, Sah S. Horizontal fast excitation in delayed van der Pol oscillator. Comm Nonlinear Sci Num Simulat 2008;13:1706–13.

[19] Sah S, Belhaq M. Eﬀect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator.

Chaos, Solitons & Fractals,2008;37:1489–96.

[20] Blekhman II. Vibrational mechanics-nonlinear dynamic eﬀects, general approach, application. Singapore: World Scientiﬁc; 2000.

[21] Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley; 1979.

[22] Shampine LF, Thompson S. Solving delay diﬀerential equations with dde23. PDF available on-line athttp://www.radford.edu/

~thompson/webddes/tutorial.pdf, 2000.