Horizontal fast excitation in delayed van der Pol oscillator

Horizontal fast excitation in delayed van der Pol oscillator

Mohamed Belhaq ^* , Si Mohamed Sah

Laboratory of Mechanics, University Hassan II-Aı¨n Chock, PB 5366 Maaˆrif, Casablanca, Morocco Received 16 February 2007; received in revised form 17 February 2007; accepted 19 February 2007

Available online 1 March 2007

Abstract

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self- excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main auton- omous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.

Ó 2007 Elsevier B.V. All rights reserved.

PACS: 05.10.a; 45.80.+r

Keywords: Fast parametric excitation; van der Pol equation; Limit cycle; Time delay; Averaging

1. Introduction

Numerous works have been devoted recently to study nontrivial effects of fast harmonic (FH) excitation on mechanical systems using the method of direct partition of motion [1]. This method, based on splitting the dynamic into fast and slow motions, provides an approximation for the small fast dynamic and the main equa- tion governing the averaged slow dynamic. Tcherniak and Thomsen [2] applied this method to study slow dynamic effects of FH excitation for some elastic structures. Thomsen and co-workers [3,4] examined asymptotic properties and analyzed some general effects of systems with strong high-frequency excitation. Recently, Bour- kha and Belhaq [5] examined the effect of FH parametric excitation on self-excited motion in van der Pol pen- dulum. They concluded that the limit cycle can be suppressed by horizontal excitation for a certain value of the excitation frequency and persists in the case of vertical FH excitation. Sah and Belhaq [6] showed that adding time delay in the vertical excitation case can eliminate the limit cycle for certain combinations of parameters.

1007-5704/$ - see front matter Ó2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2007.02.007

*

Corresponding author.

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

Available online at www.sciencedirect.com

Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1706–1713

www.elsevier.com/locate/cnsns

In the present paper, we investigate the effect of horizontal FH excitation and time delay on self-excited motion in van der Pol oscillator. Resonant cases are not considered here. Delayed parametrically excited oscil- lations has been considered in various works. For instance, Insperger and Stepan [7] studied the stability chart for the delayed linear Mathieu equation using the method of exponential multipliers. Morrison and Rand [8]

investigated the stability chart of the delayed nonlinear Mathieu equation and showed that the 2:1 instability region can be eliminated for large delay amplitude, and for appropriately chosen delay periods.

To perform our analysis, we apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator and we average the slow dynamic to obtain a slow flow to be analyzed for equilibria and self-oscillations. The analysis provides analytical approximations of regions in parameter space where self-excited vibrations can be suppressed.

2. Partition of motion and averaging

Vibrations of pendulum with time delay subjected to a horizontal parametric forcing and to a self-excita- tion can be described in non-dimensional form by the following equation

d

x

dt

ða bx

Þ dx

dt þ sin x ¼ aX

cos x cos Xt þ kxðt T Þ ð1Þ

where the parameters a and b are assumed to be small, a is the excitation amplitude, X is the parametric exci- tation frequency, and the parameters k and T are the amplitude of the delay and the delay period, respectively.

Eq. (1) has relevance to regenerative effect in high-speed milling. High-speed milling can induce a rapid para- metric excitation and milling can generate self-oscillations; see for instance [9].

We focus the analysis on small vibrations around the origin by expanding in Taylor’s series up to the third- order terms sin x ’ x dx

and cos x ’ 1 cx

, where the coefficients d ¼ 1=6 and c ¼ 1=2. Eq. (1) becomes

d

x

dt

ða bx

Þ dx

dt þ ð x dx

Þ ¼ aX

ð1 cx

Þ cos Xt þ kx ð t T Þ ð2Þ To implement the method of direct partition of motion [1], we introduce two different time-scales: a fast time T

Xt and a slow time T

t, and we split up x(t) into a slow part z(T

) and a fast part uðT

; T

Þ as follows

xðtÞ ¼ zðT

Þ þ uðT

; T

Þ ð3Þ

and

x ð t T Þ ¼ z ð T

T Þ þ uð T

XT ; T

T Þ ð4Þ

where z describes slow main motions at time-scale of oscillations of the pendulum, and u stands for an over- lay of the fast motions. In Eqs. (3) and (4), indicates that u is small compared to z. Since X is considered as a large parameter we choose X

, for convenience. The fast part u and its derivatives are assumed to have a zero T

-average, so that h x ð t Þi ¼ z ð T

Þ and h x ð t T Þi ¼ z ð T

T Þ, where hi

R

0

ðÞdT

defines time-averag- ing operator over one period of the fast excitation with the slow time T

fixed. Inserting (3) and (4) into (2) and introducing D

yields

D

z þ D

u þ 2D

D

u þ

D

u aðD

z þ D

u þ D

uÞ þ bðz

D

z þ z

D

u þ z

D

u þ 2zuD

z þ 2zuD

uÞ þ z þ u dðz

þ 3z

uÞ ¼

ðaXÞz cos T

ðaXÞcz

cos T

2ðaXÞczu cos T

ðaXÞcu

cos T

þ kzðT

T Þ þ kuðT

XT ; T

T Þ ð5Þ

Averaging (5) leads to

D

z aD

z þ bz

D

z þ z dz

¼ 2ðaXÞczhucosT

i ðaXÞchu

cosT

i þ kzðT

T Þ ð6Þ Subtracting (6) from (5), an approximate expression for u is obtained by considering only the dominant terms of order

as

D

u ¼ aXð1 cz

ÞcosT

ð7Þ

where it is assumed that aX ¼ Oð

Þ. The stationary solution to the first order for u is written as

u ¼ aXð1 cz

Þcos T

ð8Þ

Retaining the dominant terms of order

in Eq. (6), inserting u from (8) and using that hcos

T

i ¼ 1=2 gives D

z ða bz

ÞD

z þ ð1 ðaXÞ

cÞz þ ððaXcÞ

dÞz

¼ kzðt T Þ ð9Þ The autonomous Eq. (9) governing the slow dynamic of the motion can be examined through analytical predictions.

We apply the averaging method [10,11] by introducing a small parameter l such that a ¼ l~ a, b ¼ l ~ b, c ¼ l~ c, d ¼ l ~ d and k ¼ l ~ k. Then, Eq. (9) reads

€ z lð~ a bz ~

Þ_ z þ x

z þ ðl

ðaXÞ

~ c

l ~ dÞz

¼ l ~ kzðt T Þ ð10Þ where z _ ¼

and x

¼ ð1 ðaXÞ

cÞ. In the case l ¼ 0, Eq. (10) reduces to

€ zðtÞ þ x

zðtÞ ¼ 0 ð11Þ

with the solution

z ð t Þ ¼ R cosðx

t þ /Þ; z _ ð t Þ ¼ Rx

sinðx

t þ /Þ ð12Þ

For l > 0, a solution is sought in the form (12) with R and / are time dependent. Variations of parameters gives the following equations on R(t) and /(t):

RðtÞ ¼ _ l x

sinðx

t þ /ÞF

ðR cosðx

t þ /Þ; x

R sinðx

t þ /Þ; tÞ l

x

sinðx

t þ /ÞF

ðR cosðx

t þ /Þ; x

R sinðx

t þ /Þ; tÞ ð13Þ /ðtÞ ¼ _ l

x

R cosðx

t þ /ÞF

ðR cosðx

t þ /Þ; x

R sinðx

t þ /Þ; tÞ l

x

R cosðx

t þ /ÞF

ðR cosðx

t þ /Þ; x

R sinðx

t þ /Þ; tÞ ð14Þ where

F

ð z; z; _ t Þ ¼ ð ~ a bz ~

Þ z _ þ ~ dz

þ ~ kz ð t T Þ ð15Þ and

F

ðz; z; _ tÞ ¼ ðaXÞ

~ c

z

ð16Þ

with z(t) is given by (12). Using the averaging method [10,11] for small l and replacing the right-hand sides of (13) and (14) by their averages over one 2p-period, since Eq. (10) is autonomous, we obtain:

R _ l x

1 2p

Z

sinðx

t þ /ÞF

dt l

x

1 2p

Z

sinðx

t þ /ÞF

dt ð17Þ

/ _ l x

R

1 2p

Z

cosðx

t þ /ÞF

dt l

x

R

1 2p

Z

cosðx

t þ /ÞF

dt ð18Þ

in which

F

¼ ð ~ a bz ~

Þx

R sinðx

t þ /Þ þ ~ dR

cosðx

t þ /Þ

þ ~ k R ~ cosðx

t x

T þ /Þ ~ ð19Þ and

F

¼ ðaXÞ

~ c

R

cosðx

t þ /Þ

ð20Þ

with R ~ ¼ R ð t T Þ and / ~ ¼ /ð t T Þ. Evaluating the integrals in (17) and (18) yields

R _ ¼ l ~ a 2 R

b ~ 8 R

~ k 2

R ~ x

sinðx

T / ~ þ /Þ

!

ð21Þ / _ ¼ l 3 ~ d

8 R

x

~ k 2

R ~

x

R cosðx

T / ~ þ /Þ

!

þ l

3~ c

8

ðaXÞ

R

x

!

ð22Þ Eqs. (21) and (22) show that R _ and / _ are O(l). We expand in Taylor’s series R ~ and / ~ as

R ~ ¼ Rðt T Þ ¼ RðtÞ T RðtÞ þ _ 1

2 T

RðtÞ þ € ð23Þ

/ ~ ¼ /ðt T Þ ¼ /ðtÞ T /ðtÞ þ _ 1

2 T

/ðtÞ þ € ð24Þ

Then, we replace R ~ and / ~ by R(t) and /(t) in Eqs. (21) and (22) since R _ and / _ and R € and / € are of O(l) and O(l

), respectively [12]. This approximation reduces the infinite dimensional problem into a finite dimensional one by assuming lT is small.

After substituting the above approximation into Eqs. (21) and (22), we obtain the following slow flow R _ ¼ a

2 k 2x

sin x

T

R b

8 R

ð25Þ

/ _ ¼ k 2x

cos x

T þ 3 8x

ðc

ðaXÞ

dÞR

ð26Þ

3. Equilibria and self-excited oscillations

The equilibrium points in Eqs. (25) and (26), corresponding to the periodic motions in the system (9), are obtained by setting R _ ¼ / _ ¼ 0. This gives the two equilibria

R ¼ 0; R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8

b a 2 k

2x

sin x

T

s

ð27Þ The solution R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8 b

ð

0

sin x

T Þ

q corresponding to a periodic motion is real if

0 0.5 1 1.5 2 2.5 3

0 10 20 30 40 50 60 70 80 90 100

T

Ω

λ=0.3 B

A CIII

CII

CI

0 0.5 1 1.5 2 2.5 3

0 10 20 30 40 50 60 70 80 90 100

T

Ω

λ=0.45 B

A CIII

CII

CI

Fig. 1. Comparison between analytical results (solid line) based on conditions CI, CII, and CIII and numerical integration (squares) of the

original system, Eq. (2), for

0:02;

0:5;

1=6. Region A: limit cycle exists. Region B: no limit cycle.

a 2 k

2x

sin x

T P 0 ð28Þ

By setting x

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðaXÞ

c q

, Eq. (28) becomes sin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð aXÞ

c q

T

6 a k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð aXÞ

c q

ð29Þ The above inequality provides the two following conditions, denoted by (CI), corresponding to the birth of the limit cycle

T < 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð aXÞ

c

q arcsin a

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðaXÞ

c

q

ð30Þ

and

T > 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðaXÞ

c

q p arcsin a

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðaXÞ

c

q

ð31Þ

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2

-1.5 -1 -0.5

0

0.5 1 1.5 2

Ω = 40

c d

z(t)

dz(t)/dt

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Ω = 57

z(t)

dz(t)/dt

0 10 20 30 40 50 60 70 80 90 100 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Ω = 40

t

x(t)

0 10 20 30 40 50 60 70 80 90 100 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Ω = 57

t

x(t)

Fig. 2. Phase portraits of slow dynamic z(t), Eq. (9), and time histories of the full motion x(t), Eq. (2), with parameter values as for Fig. 1a

and

2:5. a, c and b, d correspond to regions A and B in Fig. 1a, respectively.

On the other hand, to find the frequency of the limit cycle, we let h ¼ x

t þ / denotes the argument of the cosine in Eq. (12). Then the frequency of limit cycle is

x ¼ dh

dt ¼ x

þ d/

dt ð32Þ

Using Eq. (26) yields x ¼ x

k

2x

cos x

T þ 3 8x

ðc

ðaXÞ

dÞR

ð33Þ

Eq. (33) gives a relationship between the frequency of the limit cycle x, the excitation frequency X, with x

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðaXÞ

c q

, and the delay period T. A condition for the existence of the limit cycle is guaranteed when x is positive, which means that the following conditions, denoted by (CII), obtained from Eq. (33) must be satisfied

T > i x

ln EF i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F

ðG

þ F

E

Þ q

F ðG þ iF Þ 0

@

1

A ð34Þ

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Ω = 60

z(t)

dz(t)/dt

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2

-1.5 -1 -0.5 0 0.5 1 1.5

2 Ω = 75

z(t)

dz(t)/dt

0 10 20 30 40 50 60 70 80 90 100 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Ω = 60

t

x(t)

0 10 20 30 40 50 60 70 80 90 100 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Ω = 75

t

x(t)

Fig. 3. Phase portraits of slow dynamic z(t), Eq. (9), and time histories of the full motion x(t), Eq. (2), with parameter values as for

Fig. 1(a) and

0:5. a, c and b, d correspond to regions A and B in Fig. 1a, respectively.

and

T < i x

ln

EF þ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F

ðG

þ F

E

Þ q

F ðG þ iF Þ 0

@

1

A ð35Þ

where E ¼ x

þ

ðcð1 x

Þ dÞ, F ¼

ðcðx

1Þ þ dÞ, G ¼

and i ¼ ffiffiffiffiffiffiffi p 1

. A third condition (CIII) obtained from the expression of x

reads

X < 1

a p ffiffiffi c ð36Þ

This condition (CIII) as well as the conditions (CI), Eqs. (30) and (31), and (CII), Eqs. (34) and (35), are plot- ted in Fig. 1. These three combined conditions delimit region of suppression of limit cycle in the parameter plane (T, X). Results obtained by numerical integration (squares) done in Matlab by using the integrating function dde23 [13] are also reported. As it can be seen in Fig. 1, the region (B) where self-excited oscillations are absent increases by increasing the delay amplitude k. For large values of delay amplitude k, self-oscillations can be suppressed for moderate values of X in the vicinity of T ¼

. In contrast, the case without delay requires large values of X to suppress limit cycle. Indeed, Fig. 1b shows that for T ¼ 0, the limit cycle disappears for X ¼ 70:7 and for T ¼

the limit cycle vanishes for X ¼ 32:8.

Phase portraits of slow dynamic (Eq. (9), Fig. 2a and b) and time histories of the corresponding full motion (Eq. (2), Fig. 2c and d) are shown in Fig. 2 for k ¼ 0:3, T ¼ 2:5 and for different values of X. This illustrates the elimination of the limit cycle as we move from region A to region B, in Fig. 1. Similar phase portraits and time histories are shown in Fig. 3 for k ¼ 0:3 and T ¼ 0:5. Phase portraits and time histories in Figs. 2 and 3 are obtained by numerical integration [13].

4. Conclusion

In this work, we have investigated the interaction effect of horizontal FH parametric excitation and time delay on self-oscillation in van der Pol pendulum. We have used the method of direct partition of motion to obtain the main equation governing the slow dynamic and applied the averaging technique on the slow dynamic to derive the slow flow. This slow flow is analyzed to locate regions where cycle limit can be elimi- nated. Results from a numerical study were reported and compared to analytical approximations showing good qualitative agreement. It was shown that adding delay in the horizontal FH excitation case increases sig- nificantly the region in the (T, X) plane where undesirable self-excited vibration can be eliminated. In contrast to the case without delay that requires large values of X to eliminate self-excited vibration, the elimination here can be achieved for moderate values of X.

References

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

[2] Tcherniak D, Thomsen JJ. Slow effects of fast harmonic excitation for elastic strucures. Nonlinear Dyn 1998;17:227–46.

[3] Fidlin A, Thomsen JJ. Non-trivial effect of strong high-frequency excitation on a nonlinear controlled system. In: Proceeding of the XXI international congress of theoretical and applied mechanics. 15–21 August 2004, Warsaw.

[4] Thomsen JJ. Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothing. J Sound Vib 2002;253(4):807–31.

[5] Bourkha R, Belhaq M. Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos, Soliton & Fractals, in press, doi:10.1016/j.chaos.2006.03.099.

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

Chaos, Soliton & Fractals, in press, doi:10.1016/j.chaos.2006.10.040.

[7] Insperger T, Stepan G. Stability chart for the delayed Mathieu equation. Proc Royal Soc, Math, Phys Eng Sci 2002;458:1989–98.

[8] Morrison TM, Rand RH. 2:1 Resonance in the delayed nonlinear Mathieu equation. Nolinear Dyn, in press, doi:10.1007/s11071-006- 9162-5.

[9] Stepan G, Szalai R, Mann BP, Bayly PV, Insperger T, Gradisek J, et al. Nonlinear dynamics of high-speed milling, analyses,

numerics and experiments. J Vib Acoust 2005;127:197–203.

[10] Rand RH. Lecture notes on nonlinear vibrations (version 52). Available from: <http://www.tam.cornell.edu/randdocs/nlvibe52.pdf>

(2005).

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

[12] Wirkus S, Rand RH. Dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn 2002;30:205–21.

[13] Shampine LF, Thompson S. Solving delay differential equations with dde23. <http://www.radford.edu/~thompson/webddes/

tutorial.pdf> (2000).