Vibration control of an axially moving string system: Wave cancellation method

Applied Mathematics and Computation 175 (2006) 851–863 www.elsevier.com/locate/amc

Vibration control of an axially moving string system: Wave cancellation method Wei Zhang

a,b

, Li-Qun Chen

a,c,*

a

b c

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China College of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350002, China Department of Mechanics, Shanghai University, 99 Shanga Da Road, Shanghai 200436, China

Abstract Active vibration control is investigated for axially moving strings with a tersioner. The tensioner is used as an actuator. The active control law is proposed from the view of the traveling wave propagation. Transverse response of both the uncontrolled and the controlled systems is derived in the frequency domain. The designed controller is numerically tested for the string under impulse and harmonic excitations. Simulation results demonstrate the effectiveness of the controller. Ó 2005 Elsevier Inc. All rights reserved.

1. Introduction Axially moving strings can represent many engineering devices. Despite many advantages of the devices, vibrations associated with the devices have limited their applications. Therefore, it is desirable to introduce suitable control mechanisms to reduce vibrations of axially moving strings in order * Corresponding author. Address: Department of Mechanics, Shanghai University, 99 Shanga Da Road, Shanghai 200436, China. E-mail address: lqchen@staff.shu.edu.cn (L.-Q. Chen).

0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.08.007

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to improve performance of the devices. Many researchers have studied the control of axially moving strings [1]. As serpentine belt-driven systems are widely used in the automobile industry, the vibration analysis has been a challenging subject. A serpentine belt-driven system typically contains a tensioner, while the belt can be modeled as a string providing its bending stiffness is rather small. Ulsoy et al. first considered the coupling between the tensioner and the string [2]. Beikmann et al. developed a prototypical model (two pulleys with a tensioner) to examine this coupling mechanism [3]. Beikmann et al. further demonstrated that the nonlinearity due to finite string stretching may lead to strong coupling between the tensioner and the string [4]. Zhang and Zu derived an explicit closed-form solution for the eigenvalues of a prototypical serpentine belt drive system [5]. Zhang and Zu further studied the auto-parametric resonance and the internal resonance in nonlinear oscillations of the serpentine belt drive system [6,7]. Zhang et al. first considered damping factors in a serpentine belt drive system [8]. However, all the above-mentioned studies have only addressed the vibration analysis of a string with a tensioner, while the tensioner has not been used to suppress the vibration of the string. To address the lack of research in this aspect, the authors investigate reduction of transverse vibration of an axially string by a tensioner, which is an essential part of a serpentine belt drive system. The traveling wave cancellation is applied to reduce the vibration. Such approach has been taken to control transverse vibration of axially string without a dynamic tensioner [9]. This paper investigates active control of an axially moving string via a tensioner. Equations of motion are presented for the system. The equations are transformed into the frequency domain in which the control problem is formulated. The feedback controller is designed by wave cancellation. Numerical results demonstrate the effectiveness of the controller.

2. Equations of motion Fig. 1 shows a moving string with a tensioner. A uniform axially moving string, with the density q, cross-sectional area A, and initial tension P, travels

W(X, T)

V

Χ

Fig. 1. Moving string with a tensioner.

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at the constant axial transport speed V between two boundaries separated by distance L. The dynamic tensioner consists of a tensioner spring with the rotational stiffness K, a rotation arm with length B and mass moment of inertia I, and a small pulley whose geometric size and mass are negligible. h1 and h2 are the alignment angles between the tensioner arm and the adjacent string spans at equilibrium. The contact point of the tensioner pulley divides the string into two spans with the length L1 and L  L1 respectively. The motion of the system can be specified by W(X, T), the transverse displacement of the sting at time T and axial coordinate X, and h(T), the angle displacement of the tensioner arm from equilibrium. For both spans of the string, the linear equation of transverse motion is [3,5]. qðW TT þ 2VW XT þ V 2 W XX Þ  PW XX ¼ F 1 ðX ; T Þ;

0 6 X 6 X a;

T P 0; ð1Þ

2

qðW TT þ 2VW XT þ V W XX Þ  PW XX ¼ F 2 ðX ; T Þ;

X a 6 X 6 L;

T P 0; ð2Þ

where subscripts ‘‘,X’’ and ‘‘,T’’ denote the partial derivative with respect to X and T respectively. F1(X, T) and F2(X, T) are the external forces applied string. The equation of motion for the tensioner arm is  I€ h ¼ ðP  qV 2 ÞðW þ X a ;X cos h2  W X a ;X cos h1 ÞB  Kh  M e ;

T P 0;

ð3Þ

where Me is the the external moment applied by the tensioner arm. Introduce the following dimensionless parameters sffiffiffiffiffiffiffiffi rffiffiffi X W P q L FL ; x¼ ; w¼ ; t¼T ; c ¼ V ; l¼ ; f ¼ L L P L P qL2 I B L L ; b ¼ ; k ¼ K ; me ¼ M e . 3 L P P qL Then Eqs. (1)–(3) can be respectively transformed into ie ¼

wtt ðx; tÞ þ 2cwxt ðx; tÞ  ð1  c2 Þwxx ðx; tÞ ¼ f1 ðx; tÞ; 0 6 x 6 xa ; t P 0; 2

wtt ðx; tÞ þ 2cwxt ðx; tÞ  ð1  c Þwxx ðx; tÞ ¼ f2 ðx; tÞ; xa 6 x 6 1; t P 0; € ¼ ð1  c2 Þðwx ðxþ ;tÞ cosh2  wx ðx ; tÞ cosh1 Þb  khðtÞ  me ðtÞ; t P 0. ie hðtÞ a a

ð4Þ ð5Þ ð6Þ

The boundary conditions for Eqs. (4) and (5) are wð0; tÞ ¼ 0;

þ wðl 1 ; tÞ ¼ wðl1 ; tÞ ¼ bhðtÞ sin u;

wð1; tÞ ¼ 0.

ð7Þ

The initial conditions for Eqs. (4)–(6) are wðx; 0Þ ¼ u0 ðxÞ; wt ðx; 0Þ ¼ v0 ðxÞ; hð0Þ ¼ a; ht ð0Þ ¼ b.

ð8Þ ð9Þ

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3. Transfer function for uncontrolled string The transfer function of a distributed system contains the information required to predict the system response under given initial and external disturbances [10]. The transfer function of a distributed parameter system is the Laplace transform of its Greens function. Therefore, the control of a distributed system can be formulated in terms of transfer functions, and the analysis of control laws can be carried out in the s domain. The Laplace transform of Eqs. (4)–(9) with respect to t gives o o2  ðx; sÞ  ð1  c2 Þ 2 w  ðx; sÞ ¼ f 1e ðx;sÞ; w ox ox o f 1e ¼ f 1 ðx; sÞ þ su0 ðxÞ þ v0 ðxÞ þ 2c u0 ðxÞ; ox o o2  ðx; sÞ  ð1  c2 Þ 2 w  ðx;sÞ þ 2cs w  ðx; sÞ ¼ f 2e ðx;sÞ; s2 w ox ox o f 2e ¼ f 2 ðx; sÞ þ su0 ðxÞ þ v0 ðxÞ þ 2c u0 ðxÞ; ox   o o 2 2 þ   ðxa ; sÞ cosh1 b  k  wðxa ;sÞ cosh2  w hðsÞ þ s2 a þ b; ie s hðsÞ ¼ ð1  c Þ ox ox  ðsÞ ¼ bhðsÞ;  ðxþ ; sÞ ¼ vðsÞ sin u; v  ð0; sÞ ¼ 0; w  ð1; sÞ ¼ 0; w  ðx ; sÞ ¼ w w  ðx;sÞ þ 2cs s2 w

a

a

ð10aÞ ð10bÞ ð10cÞ ð10dÞ ð10eÞ ð10fÞ

 ð; sÞ, f 1e ð; sÞ, f 2e ð; sÞ,  where w hðsÞ and  vðsÞ are the Laplace transforms of w( Æ ,t), f1( Æ ,t), f2( Æ ,t), h (t) and v(t) respectively, and s is the complex Laplace transform variable. The response solution to Eqs. (10a–d) is Z xa  1 ðx; sÞ ¼ G1 ðx; n; sÞf 1e ðn; sÞ dn þ h1 ðx; sÞ wðxa ; sÞ; ð11aÞ w 0 Z xa  1;x ðx; sÞ ¼ w Gf 1 ðx; n; sÞf 1e ðn; sÞ dn þ hf 1 ðx; sÞ wðxa ; sÞ; ð11bÞ 0 Z 1  2 ðx; sÞ ¼ G2 ðx; n; sÞf 2e ðn; sÞ dn þ h2 ðx; sÞ wðxa ; sÞ; ð11cÞ w xa

 2;x ðx; sÞ ¼ w

Z

1

Gf 2 ðx; n; sÞf 2e ðn; sÞ dn þ hf 2 ðx; sÞ wðxa ; sÞ;

ð11dÞ

xa

where the transfer functions, W01(x, n, s), Gf1(x, n, s), W02(x, n, s) and Gf2(x, n, s), are given respectively by 8 k2 ðxnÞ e  ek2 xk1 n  ek2 ðxa nÞk1 ðxa xÞ þ ek2 xa k1 ðxa þnxÞ > > ; > > 2sð1  eðk2 k1 Þxa Þ > > > < 0 6 n 6 x; ð12aÞ G1 ðx; n; sÞ ¼ > ek1 ðxnÞ  ek2 xk1 n  ek2 ðxa nÞk1 ðxa xÞ þ ek2 ðxa nþxÞk1 xa > > ; > > > 2sð1  eðk2 k1 Þxa Þ > : x 6 n 6 xa ;

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8 k ð1x Þþk ðx 1þxnÞ e2 a 1 a þ ek2 ðxnÞ  ek2 ð1nÞk1 ð1xÞ  ek1 ðxa nÞk2 ðxa xÞ > > > ; > > 2sð1  eðk2 k1 Þð1xa Þ Þ > > > > > < xa 6 n G2 ðx; n; sÞ ¼ 6 x; ð12bÞ > > k k2 ð1xa þxnÞk1 ð1xa Þ k2 ð1nÞk1 ð1xÞ k1 ðxa nÞk2 ðxa xÞ > > e ðx  nÞ þ e  e  e 1 > > ; > > 2sð1  eðk2 k1 Þð1xa Þ Þ > > : x 6 n 6 1; 8 k2 ðxnÞ k  ek2 xk1 n Þ þ k1 ðek2 xa k1 ðxa þnxÞ  ek2 ðxa nÞk1 ðxa xÞ Þ 2 ðe > > ; > > 2sð1  eðk2 k1 Þxa Þ > > > < 0 6 n 6 x; Gf 1 ðx; n; sÞ ¼ ð12cÞ > k1 ðek1 ðxnÞ  ek2 ðxa nÞk1 ðxa xÞ Þ þ k2 ðek2 ðxa nþxÞk1 xa  ek2 xk1 n Þ > > ; > > > 2sð1  eðk2 k1 Þxa Þ > : x 6 n 6 xa ; 8 k ðek2 ðxnÞ  ek1 ðxa nÞk2 ðxa xÞ Þ þ k1 ðek2 ð1xa Þþk1 ðxa 1þxnÞ  ek2 ð1nÞk1 ð1xÞ Þ 2 > > ; > > 2sð1  eðk2 k1 Þð1xa Þ Þ > > > < x 6 n 6 x; a Gf 2 ðx; n; sÞ ¼ k1 ðxnÞ > k ðe  ek2 ð1nÞk1 ð1xÞ Þ þ k2 ðek2 ð1xa þxnÞk1 ð1xa Þ  ek1 ðxa nÞk2 ðxa xÞ Þ 1 > > ; > > > 2sð1  eðk2 k1 Þð1xa Þ Þ > : xa 6 n 6 1;

ð12dÞ

where k1 = 1/(1  c) and k1 = 1/(1 + c). The influence functions, h1(x, s), h2(x, s), hf1(x, s), and hf2(x, s), due to inhomogeneous boundary conditions are h1 ðx; sÞ ¼

ek1 ðxa xÞ  ek2 xk1 xa ek2 ðxa xÞ  ek2 ð1xa Þk1 ð1xÞ ; h2 ðx; sÞ ¼ ; ðk k Þx a 2 1 1e 1  eðk2 k1 Þð1xa Þ

ð12eÞ k1 ðxa xÞ

hf 1 ðx; sÞ ¼

k1 e

k2 xk1 xa

 k2 e 1  eðk2 k1 Þxa

; hf 2 ðx; sÞ ¼

k2 e

k2 ðxa xÞ

k2 ð1xa Þk1 ð1xÞ

 k1 e 1  eðk2 k1 Þð1xa Þ

.

ð12fÞ  ðxa ; sÞ, substitution of Eqs. (9), (10f), (11b), (11d), In order to get the solution w (12c) and(12d) into (10e) gives C 0 ðsÞ  ðxa ; sÞ ¼ ; ð13aÞ w D0 ðsÞ where Z 1 C 0 ðsÞ ¼ ð1  c2 Þb2 sin uðcos h2 Gf 2 ðxa ; n; sÞfe2 ðn; sÞdn xa Z xa Gf 1 ðxa ; n; sÞfe1 ðn; sÞdnÞ þ ðsa þ bÞb sin u; ð13bÞ  cos h1 0

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D0 ðsÞ ¼ s2 ie þ k  ð1  c2 Þb2 sin uðhf 2 ðxa ; sÞ cos h2  hf 1 ðxa ; sÞ cos h1 Þ; ð13cÞ k1  k2 eðk2 k1 Þxa k2  k1 eðk2 k1 Þð1xa Þ ; h ðx ; sÞ ¼ . ð13dÞ f 2 a 1  eðk2 k1 Þxa 1  eðk2 k1 Þð1xa Þ Substitution of Eqs. (13a–d) into Eqs. (11a) and (11b) leads to the prediction of open-loop system response Z xa  1 ðx; sÞ ¼ w W 01 ðx; n; sÞfe1 ðn; sÞdn þ H 1 ðx; sÞ; ð14aÞ hf 1 ðxa ; sÞ ¼

0

 2 ðx; sÞ ¼ w

Z

1

W 02 ðx; n; sÞfe2 ðn; sÞdn þ H 2 ðx; sÞ;

ð14bÞ

xa

where the transfer functions of the open-loop system W01(x, n, s) and W02(x, n, s) are given by W 01 ðx; n; sÞ ¼ G1 ðx; n; sÞ 

h1 ð1  c2 Þb2 sin u cos h1 Gf 1 ðxa ; n; sÞ ; DðsÞ

h2 ð1  c2 Þb2 sin u cos h2 Gf 2 ðxa ; n; sÞ . DðsÞ Correspondingly, H1(x, s) and H2(x, s) are given by W 02 ðx; n; sÞ ¼ G2 ðx; n; sÞ þ

H 1 ðx; sÞ ¼

h1 ðx;sÞð1  c2 Þb2 sin ucosh2

xa

2

h2 ðx; sÞð1  c Þb sin ucosh1

ð14dÞ

Gf 2 ðxa ; n; sÞfe2 ðn; sÞdn þ h1 ðx; sÞðsa þ bÞb sin u DðsÞ

2

H 2 ðx; sÞ ¼ 

R1

ð14cÞ

R xa 0

;

ð14eÞ Gf 1 ðxa ; n;sÞf1e ðn; sÞdn þ h2 ðx;sÞðsa þ bÞb sinu ; DðsÞ

ð14fÞ where

8 k2 ðek2 ðxa nÞ  ek2 xa k1 n Þ þ k1 ðek2 xa k1 n  ek2 ðxa nÞ Þ > > > ; > > 2sð1  eðk2 k1 Þxa Þ > > < 0 6 n 6 x; ð14gÞ Gf 1 ðxa ; n; sÞ ¼ k1 ðxa nÞ > k  ek2 ðxa nÞ Þ þ k2 ðek2 nk1 xa  ek2 xa k1 n Þ 1 ðe > > ; > > 2sð1  eðk2 k1 Þxa Þ > > : x 6 n 6 xa ; 8 k > 2 ðek2 ðxa nÞ  ek1 ðxa nÞ Þ þ k1 ðek2 ð1xa Þþk1 ð2xa 1nÞ  ek2 ð1nÞk1 ð1xa Þ Þ > > ; > > 2sð1  eðk2 k1 Þð1xa Þ Þ > > < xa 6 n 6 x; Gf 2 ðxa ; n; sÞ ¼ k1 ðxa nÞ > k  ek2 ð1nÞk1 ð1xa Þ Þ þ k2 ðek2 ð1nÞk1 ð1xa Þ  ek1 ðxa nÞ Þ 1 ðe > > ; > > 2sð1  eðk2 k1 Þð1xa Þ Þ > > : xa 6 n 6 1. ð14hÞ

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4. Transfer function for controlled string In order to derive the control law, the solutions of the uncontrolled axially moving have been briefly described in previous section. Now consider the design of the control law for the system. It is often desirable to restricting the response of some part of a system resulting from external disturbance. Therefore, in this paper, controller designs for the left part of the axially moving string at xa are considered. The control force applied at the tensioner is expressed in the transfer function form  e ðsÞ ¼ RðsÞhðsÞ; m

ð15Þ

where the control law R(s) is the sum of the transfer function of the actuator at the tensioner and the sensor at xa. The sensor, the control law, and the actuator are a feedback controller making a closed-loop system with the axially moving string system. Here for simplicity, zero initial conditions are assumed. Let the force fe2(n, s) be zero. The Laplace transform of Eq. (6) with respect to t gives   o o     e ðsÞ; wðxþ w ðx hðsÞ ¼ ð1  c2 Þ i e s2  ; sÞ cos h  ; sÞ cos h 2 1 b  k hðsÞ þ m a a ox ox ð16Þ  e ðsÞ is the Laplace transform of me(t). Substitution of Eqs. (10f), (11b) where m  ðxa ; sÞ and (11d) into (16) yields the solution w  ðxa ; sÞ ¼ w

CðsÞ ; DðsÞ

ð17aÞ

CðsÞ ¼ ð1  c2 Þb2 sin u cos h1

Z

xa

Gf 1 ðxa ; n; sÞfe1 ðn; sÞdn;

ð17bÞ

0

DðsÞ ¼ RðsÞ þ s2 ie þ k  ð1  c2 Þb2 sin uðhf 2 ðxa ; sÞ cos h2  hf 1 ðxa ; sÞ cos h1 Þ. ð17cÞ Consider a concentrated load applied at x = xk f ðx; sÞ ¼ dðx  xk Þf ðsÞ; 0 6 xk 6 xa ; 1

k

ð18Þ

where xk is the location of the concentrated load and the Laplace transform of the load is f k ðsÞ. Application of Eqs. (17) and (18) to (11a) and (11c) gives the closed-loop response of the string  1 ðx; sÞ ¼ W c1 ðx; xk ; sÞf k ðsÞ ¼ ðcr1 ðsÞek2 x þ cl1 ðsÞek1 x Þf k ðsÞ; w

0 6 x 6 xa ; ð19aÞ

 2 ðx; sÞ ¼ W c2 ðx; xk ; sÞf k ðsÞ ¼ ðcr2 ðsÞek2 x þ cl2 ðsÞek1 x Þf k ðsÞ; w

xa 6 x 6 1; ð19bÞ

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W. Zhang, L.-Q. Chen / Appl. Math. Comput. 175 (2006) 851–863

where cr1 ðsÞ ¼

ek2 xk  ek1 xk þ q1 ðsÞ; 2sð1  eðk2 k1 Þxa Þ

ð19cÞ

xk 6 x 6 xa ;

ek2 xa k1 ðxa þxk Þ  ek2 ðxa xk Þk1 xa  q1 ðsÞ; xk 6 x 6 xa ; 2sð1  eðk2 k1 Þxa Þ ek2 ðxa xk Þk1 xa  ek1 xk cr1 ðsÞ ¼ þ q2 ðsÞ; 0 6 x 6 xk ; 2sð1  eðk2 k1 Þxa Þ

cl1 ðsÞ ¼

cl1 ðsÞ ¼

ek1 xk  ek2 ðxa xk Þk1 xa  q2 ðsÞ; 2sð1  eðk2 k1 Þxa Þ

ð19dÞ ð19eÞ ð19fÞ

0 6 x 6 xk ;

ð1  c2 Þb2 sin u cos h1 ek1 xa

q1 ðsÞ ¼

2sð1  eðk2 k1 Þxa Þ2 

k2 ðek2 ðxa xk Þ  ek2 xa Þ þ k1 ðek2 xa k1 xk  ek2 ðxa xk Þ Þ ; DðsÞ

ð19gÞ

ð1  c2 Þb2 sin u cos h1 ek1 xa

q1 ðsÞ ¼

2sð1  eðk2 k1 Þxa Þ 

k2 ðek2 ð2xa xk Þk1 xa  ek2 xa k1 xk Þ þ k1 ðek1 ðxa xk Þ  ek2 ðxa xk Þ Þ ; DðsÞ

ð19hÞ

ek2 xa ð1  c2 Þb2 sin u cos h1 Gf 1 ðxa ; xk ; sÞ ; 2sð1  eðk2 k1 Þð1xa Þ ÞDðsÞ

ð19iÞ

cr2 ðsÞ ¼  cl2 ðsÞ ¼

2

ek2 ð1xa Þk1 ð1  c2 Þb2 sin u cos h1 Gf 1 ðxa ; xk ; sÞ ; 2sð1  eðk2 k1 Þð1xa Þ ÞDðsÞ

xa 6 x 6 1;

xa 6 x 6 1; ð19jÞ

where Wc1(x, xk, s) and Wc2(x, xk, s) are the closed-loop transfer functions of the two parts of the string respectively, and are the sum of the right propagating wave and the left propagating wave transfer functions, respectively. Inverse Laplace transform of cr ðsÞek2 x f k ðsÞ and cl ðsÞek1 x f k ðsÞ gives the right and the left disturbance propagating waves, respectively. This is verified by letting fk(t) = eixt. The frequency response is wðx; tÞ ¼ cr ðixÞek2 xþixt þ cl ðixÞek1 xþixt .

ð20Þ

Obviously, the time-varying functions ek2 xþixt and ek1 xþixt represent the right and the left disturbance propagating waves, respectively, and the wave veloci1 ties, respectively, are k1 2 ¼ 1 þ c and k1 ¼ 1  c, where cr(ix) and cl(ix) are the coefficients of the right and left propagating waves, respectively. The control law can be derived by canceling the reflected wave of the left part of the string at xa. Let the left propagating wave coefficient in [xk, xa] be zero cl1 ðsÞ ¼ 0

xk 6 x 6 xa .

ð21Þ

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Solving Eq. (21) leads to the feedback control law for the system RðsÞ ¼ ðs2 ie þ k  ð1  c2 Þb2 sin uðhf 2 ðxa ; sÞ cos h2  hf 1 ðxa ; sÞ  cos h1 Þ þ

ðk1  k2 Þb2 sin u cos h1 . 1  eðk2 k1 Þxa

ð22Þ

Substitution of Eq. (22) into Eqs. (19) gives the closed-loop response of the controlled string  1 ðx; sÞ ¼ w

ek2 ðxxk Þ  ek2 xk1 xk þ ek2 ðxþxa Þk1 ðxa þxk Þ  ek2 ðxa xk þxÞk1 xa  f k ðsÞ; xk 6 x 6 xa ; 2sð1  eðk2 k1 Þxa Þ

ð23aÞ

ð1  cÞðek1 ðxxk Þ  ek2 xk1 xk þ ek2 ðxþxa xk Þk1 xa  ek2 ðxa xk Þk1 ðxa xÞ Þ  f k ðsÞ 4sð1  eðk2 k1 Þxa Þ ð1  cÞðek2 ð2xa xk Þk1 ð2xa xÞ  ek2 xa k1 ðxa þxk xÞ þ ek2 ðxþxa Þk1 ðxa þxk Þ  ek2 ð2xa xk þxÞ2k1 xa Þ þ 4sð1  eðk2 k1 Þxa Þ   f k ðsÞ; 0 6 x 6 xk ; ð23bÞ

 1 ðx; sÞ ¼ w

h2 ðx; sÞ ðk1  2k2 Þek2 xa k1 xk þ ðk2  2k1 Þek2 ðxa xk Þ þ k1 ek1 ðxa xk Þ þ k2 ek2 ð2xa xk Þk1 xa ; 2s k1  k2 xa 6 x 6 1. ð23cÞ

w2 ðx; sÞ ¼

Hence the closed-loop transfer functions are W c1 ðx;n;sÞ ¼

ek2 ðxfÞ  ek2 xk1 n þ ek2 ðxþxa Þk1 ðxa þnÞ  ek2 ðxa nþxÞk1 xa ; 2sð1  eðk2 k1 Þxa Þ

ð24aÞ

n 6 x 6 xa ; nÞþek2 ðxþxa nÞk1 xa ek2 ðxa nÞk1 ðxa xÞ

W c1 ðx;n;sÞ ¼ þ

ð1  cÞðek1 ðxfÞ  ek2 ðxk1 4sð1  eðk2 k1 Þxa Þ

Þ

ð1  cÞðek2 ð2xa nÞk1 ð2xa xÞ  ek2 xa k1 ðxa þnxÞ þ ek2 ðxþxa Þk1 ðxa þnÞ  ek2 ð2xa nþxÞ2k1 xa Þ ; 4sð1  eðk2 k1 Þxa Þ

ð24bÞ

0 6 x 6 n; W c2 ðx;n;sÞ ¼ xa 6 n 6 1.

h2 ðx;sÞ ðk1  2k2 Þe 2s

k2 xa k1 n

k2 ðxa nÞ

k1 ðxa nÞ

þ ðk2  2k1 Þe þ k1 e k1  k2

þ k2 e

k2 ð2xa nÞk1 xa

;

ð24cÞ

Eq. (22) shows that the feedback controller R(s) consists of velocity sensors, proportional gains and time delays. This controller is easy to implement for the system. R(s) is independent of xk and f k ðsÞ, and it means that, as long as the excitation is located between 0 and xa, the designed controller is applicable to the control of the string transverse vibration under various kinds of loading conditions.

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5. Numerical simulations Numerical simulations will be performed to demonstrate the effectiveness of the controller designed via wave cancellation. In this paper, the method of the numerical inversion of the Laplace transform (Durbin 1974) is employed to approximate the simulation results. For N equidistant time points, tj = jDt = T/N, j = 1, 2, . . . , N  1, where T is the upper time. For Eq. (19a), the formulation for the numerical inversion of the Laplace transforms gives !!   NX SUM 1 2p 2p  1 ðx; tj Þ ¼  1 ðx; sk Þ cos kj Reð w1 ðx; aÞÞ þ Re w w þ i sin kj ; 2 N N m¼0 ð25Þ where sk = a + i2pk/T, m = 0, 1, 2, . . . NSUM. Durbin verified that aT = 5–10 give good results for NSUM ranging from 50 to 5000 [11]. In this paper, select a = 0.25, T = 20, and NSUM = 2000. The parameters of the axial moving string system used in the simulations are [5]

Fig. 2. The time response of the string under the impulse excitation: (a), (b) for uncontrolled string and (c), (d) for controlled string.

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Fig. 3. The time response of the angle of the tensioner arm under the impulse excitation: (a) uncontrolled string and (b) controlled string.

Fig. 4. The time response of the string under the harmonic excitation: (a), (b) for uncontrolled string and (c), (d) for controlled string.

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Fig. 5. The time response of the angle of the tensioner arm under the harmonic excitation: (a) uncontrolled string and (b) controlled string.

I ¼ 0:001165 kg m2 ; B ¼ 0:097 m;

P ¼ 60 N;

q ¼ 0:1029 kg=m;

K ¼ 54:37 Nm=rad; L ¼ 0:6 m;

L ¼ 0:6 m;

V ¼ 0:2 m=s;

xa ¼ 0:5.

Unless otherwise specified, let the initial condition and f 2e ðx; sÞ be zero. Firstly, the response of the system under the initial disturbance condition is considered. A impulse excitation f1 ðx; tÞ ¼ 0:01dðx  0:2ÞdðtÞ ð26Þ is applied at x = 0.2. The response is calculated at x = 0.1 and x = 0.4, respectively. The numerical results of the uncontrolled and controlled string to the excitation are shown in Figs. 2 and 3. It is seen that, with the designed controller, the transverse displacements of the system and the angle displacement of the tensioner arm go to zero quickly. This verifies the effectiveness of the control law given in this paper under the initial disturbance condition. Next, consider the response of the string to the harmonic excitation f1 ðx; tÞ ¼ 0:01 sin xt

ð27Þ

applied at x = 0.2, where x = 47.1. The response is also calculated at x = 0.1 and x = 0.4 respectively. Results for the disturbance are show in Figs. 4 and 5. It is seen that, with the controller, the vibration of the string is attenuated, the amplitudes of the controlled string is far smaller than the amplitudes of the uncontrolled string, and the angle displacement of the tensioner arm is also reduced. Hence, the feedback controller is also feasible to harmonic excitations. 6. Conclusions Active vibration control of the axially moving string based on the principle of the wave propagation is proposed. The control problem is formulated in the

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863

s domain and the responses of both the uncontrolled and controlled systems are given by their transfer functions. The controller designed by wave cancellation consists of velocity sensors, proportional gains and time delays. It is shown that, with the controller, the transverse vibration of the axially moving string is reduced. The numerical results demonstrate the effectiveness of the controller.

Acknowledgements The research is supported by the National Natural Science Foundation of China (Project No. 10472060), Natural Science Foundation of Shanghai Municipality (Project No. 04ZR14058) and Shanghai Leading Academic Discipline Project (Project No. Y0103).

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