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VIBRATION CONTROL OF AN AXIALLY MOVING STRIP BY A NON-LINEAR BOUNDARY CONTROL
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
www.elsevier.com/locate/ifac
VIBRATION CONTROL OF AN AXIALLY MOVING STRIP BY A NON-LINEAR BOUNDARY CONTROL
Ji-Yun Choia, Keum-Shik Hongb*, and Chang-Do Huha a
Department of Mechanical and Intelligent Systems Engineering, Pusan National University, 30 Changjeon-dong, Keumjeong-ku, Pusan, 609-735, Korea. Tel: +82-51-510-1481, Fax: +82-51-514-0685, Email: [email protected] b School of Mechanical Engineering, Pusan National University, 30 Changjeon-dong Kumjeong-ku Pusan 609-735, Korea. Tel: +82-51-510-2454, Fax: +82-51-514-0685 Email: [email protected] * The author to whom all correspondence should be addressed. control
Abstract: In this paper, an active vibration of the translating steel strip in the zinc galvanizing line is investigated. The unwanted vibrations of the strip deteriorate the uniformity of the deposited mass on the surfaces of the steel strip and the consumption of zinc. The translating steel strip is modelled as a moving belt equation by using Hamilton’s principle for the systems with changing mass. The total mechanical energy of the system is considered as a Lyapunov function candidate. A nonlinear boundary control law that assures the exponential stability of the closed loop system is derived. The existence of a closed-loop solution is shown by proving that the closed-loop dynamics is dissipative. Simulation results are provided. Copyright © 2002 IFAC
Keywords: asymptotic stability, axially moving system, boundary control, hyperbolic nonlinear vibrations, zinc galvanizing line. partial differential equation, Lyapunov method,
1. INTRODUCTION The examples of axially moving systems are found in various engineering areas: the steel strip in a zinc galvanizing line, power transmission belts, textile fibers, aerial cable thread lines, magnetic tapes, band saw blades, paper sheets during processing, etc. In such systems, undesirable vibrations of the moving objects occur during the process due to the eccentricity of a pulley, and/or non-stationary speed of a driving motor, and/or aerodynamic excitation from the surrounding environment, and/or nonuniformity of the material. Also, such vibrations result in costly defects on the products. Fig. 1 shows a zinc galvanizing line. The preheated steel strip is passed through a hot zinc tank and then pulled up vertically with zinc coated on it. The control objectives in the galvanizing line are to improve the uniformity of the zinc deposit on the strip surfaces and reduce the zinc consumption. Therefore, an active control of the vibrations, with minimal number of actuators and sensors, is currently the main research issue in these areas.
and Mote (1996), Moon and Wickert (1997), Fung et al. (1999) and Pellicano et al. (1998, 2000). Recently, Wickert (1992) analyzed free nonlinear vibrations of an axially moving elastic tensioned beam over the sub- and super-critical transport speed ranges. Laousy et al. (1996) proposed a stabilizing boundary feedback control law for a rotating body-beam system and showed that the beam vibrations are forced to decay exponentially to zero. Lee and Mote (1996) applied the boundary control technique to control the transverse vibration of an axially moving string and showed the exponential stability of the boundary-controlled string. Moon and Wickert (1997) investigated nonlinear vibrations of a prototypical power transmission belt system, which is excited by pulleys having slight eccentricity, through analytical and experimental methods. Fung et al. (1999) controlled the vibrations of an axially moving string system using a boundary controller derived by including the actuator dynamics in the plant model. Shahruz (2000) also showed that the non-linear string can be stabilized by a linear boundary control.
During the past several decades, axially moving systems have been extensively studied by many researchers including Carrier (1945), Mote (1966), Bapat and Srinivasan (1967), Wickert and Mote (1988, 1990), Morgul (1992), Wickert (1992), Lee
The axially moving systems can be modeled in three different ways: a string equation (Lee and Mote, 1996; Fung et al., 1999; Shahruz, 2000), a beam equation (Wickert, 1992; Pellicano et al., 1998), and a belt equation (Moon and Wickert, 1997; Pellicano
1
et al., 2000) depending on the flexibility of the system considered and the objectives of control. In this paper, a belt equation is used because the length of the steel strip between two supporting points, see Fig. 1, is 17.5 m long and therefore it allows some longitudinal deformation as well as the transversal displacement. The equations of motion of the moving strip, by using the Hamilton’s principle (McIver, 1973) for the systems of changing mass, is first derived. A nonlinear right-boundary control law to suppress the vibrations of the moving strip is derived so that the total vibration energy dissipates at the right boundary. The asymptotic stability of the closed loop system is guaranteed by the Lyapunov second method.
the longitude of motion, v0 be the axial speed of the strip, uˆ ([ ,W ) and wˆ ([ ,W ) be the longitudinal and transversal displacements of the strip, respectively, and L be the length of the strip. Also, let U be the mass per unit area of the strip, A be the cross section area, E be the elastic modulus, and T0 be the tension applied to the strip. Then, with a control force Fc (W ) at the right boundary, the following equations of motion are derived:
§ ©
1 2
· ¹[
U v02uˆ[[ 2v0uˆ[W uˆWW E¨ uˆ[ wˆ [2 ¸
(1)
0,
uˆ([ ,0) uˆ0 ([ ), uˆW ([ ,0) uˆW 0 ([ ), uˆ(0,W ) uˆ( L,W ) 0,
·½ ¹¿[
0, (2)
1 ½ Fc (W ) T0 wˆ [ (L,W ) AEwˆ [ (L,W )®uˆ[ (L,W ) wˆ [2 (L,W )¾ 0, 2 ¯ ¿
(3)
1 2
§ ©
UA v02 wˆ[[ 2v0wˆ[W wˆWW T0wˆ[[ AE®wˆ[ ¨uˆ[ wˆ[2 ¸¾
2. MODELING OF TRANSLATING STEEL STRIP Fig. 1 depicts a continuous hot-dip zinc galvanizing process with an active vibration control. The steel strip, which may vary in width from 800 mm to 1400 mm and in thickness from 1.2 mm to 4.5 mm, is preheated in a continuous annealing furnace and then introduced at a speed about 1 m/sec into a pot of molten zinc at about 450 qC. The steel strip passes under a sink roll and through a pair of stabilizing and correcting rolls to emerge vertically from the pot coated with a layer of zinc. Because there is an increasing demand for greater consistency of the thickness of zinc film, a pair of air knives located about 0.5 m above the surface of the zinc tank, which direct a long thin wedge-shaped jet of high-velocity air toward the strip, are used to control the deposited mass by stripping excess zinc back to the pot. The strip then travels 35 m vertically, while the solidification of the deposited film is enhanced with the help of an air cooler, and 110 m horizontally, cooling as it goes, to a gauge that measures the mass of zinc deposited on the strip surfaces.
¯
wˆ ([ ,0) wˆ 0 ([ ), wˆW ([ ,0) wˆW 0 ([ ), wˆ (0,W ) 0,
and
where (1)-(3) represent the equations of the longitudinal motion, the transversal motion, and the right transversal boundary condition, respectively. [ and W denote w / w[ and w / w W , respectively. 110 m
Tower Roll
17.5 m
Thickness Gauge
Strip Direction of Travel
Pr
Touch Rolls, Hydraulic Actuator
Ps
Ap2 , V p 2
17.5 m
In order to achieve the uniformity of the zinc deposit on the strip surfaces and to reduce the zinc consumption, the strip should pass at equidistance from each of the air knives. But, due to the shifting and vibration of the strip, a discrepancy between the averaged deposited masses on the left and right strip surfaces and the non-uniformity of the deposited mass across the strip occur. The variations in deposited mass will lower the quality of the product.
Pr
Servo Valve Voltage
P2 , Q 2
Displacement, Velocity
Annealing Furnace
T0
P1 , Q1 A p1 , V p1
Controller
Air Cooler
Air Knife
Stabilizing Roll Correcting Roll
Hot Zinc Tank
Sink Roll
Fig. 1. A translating steel strip in the zinc galvanizing line: control strategy. Y
Fig. 2 shows a schematic of the axially moving steel strip for control system design purpose. Depending on the thickness of the strip and the distance between two support points, the strip can be modelled as one out of three models: a moving beam, a moving string, and a moving belt. In this paper, both the transversal and longitudinal displacements of the strip are initially considered, i.e., the steel strip is modelled as a moving belt.
Boundary Control Force
v0
wˆ ([ , W )
[
Fc (W )
X
uˆ ([ , W )
L
T0
Fig. 2. A schematic of the axially moving steel strip with two intermediate touch rolls. Remark 1: If only linear terms are retained in (1)-(2), they become the travelling rod and tensioned string equations, respectively. Over a technically useful range of parameter values, the speed of the longitudinal waves is significantly faster than the transversal ones (Mote, 1965). On the time scales of
Now, the left boundary at the stabilizing roll in Fig. 2 is assumed fixed. The two touch rolls located in the middle section of the strip will play the right boundary, where the control input force is applied. Let W be the time, [ be the spatial coordinate along
2
the lower transversal modes, the tension variations propagate almost instantaneously as the influence of longitudinal inertial is small (Wickert, 1992).
implementable. However, through the boundary control law design in Section III next, the control force in (12), f c (t ) , will be re-formulated in an
Now, the explicit appearance of uˆ in (2) can be suppressed by approximating the dynamic tension component as a function of time alone: First, the time derivatives of uˆ in (1) are all neglected, i.e., uˆW uˆWW # 0 . Also, since E / U !! v02 , (1) can be
implementable form. Remarks 2. The equilibrium solutions w(x ) of (11) satisfy: 1 2 1 2 ½ 2 , , and wx (1) 0 . ® v 1 vT ³ wx dx¾wxx 0 w(0) 0
1 · § 0 # ¨ uˆ[ wˆ [2 ¸ 2 ¹[ ©
\ (W ) : uˆ[ wˆ [2 ,
(4)
which implies that \ is independent of [. By integrating both sides of (4) with respect to the spatial coordinate from 0 to [ and using uˆ ( 0, W ) 0 , uˆ ([ ,W ) \ (W )[
1 [ 2 wˆ T (T ,W )dT 2 ³0
x
is derived. Therefore, the uˆ -term in (2) and (3) can be approximated in terms of the transversal one as follows: [ L 2 1 [ 2 (7) uˆ ([ ,W ) wˆ (T ,W )dT wˆ (T ,W )dT . 2 ³0
T
T
The substitution of (7) into (2)(3) gives
AE L 2 ˆ[[ 0, (8) wˆT (T,W )dT w 2 ³0 ˆ 0 ([), w ˆW ([,0) w ˆW 0 ([), w ˆ (0,W ) 0, wˆ ([,0) w
UA wˆWW 2v0wˆ[W v02wˆ[[ T0wˆ[[
Fc (W )
(9)
[ , w L
Fc(t) . (10) T0 The substitution of (10) into (8)(9) yields the normalized equations of the transversal motion and the right boundary condition, respectively, as follows:
x
2
³ ¨©
¸ ¹
0
x
2
x
¸ ¹
\v 1 ^w (1, t )w (1, t ) w (0, t )w (0, t )` v ^\ v v 1 v ` ^w (1, t ) w (0, t )` 2 2 T
(11)
t
x
2
t
2
2 x
v3 4 wx (1, t ) wx4 (0, t ) . 8
^
`
x
2 x
(16)
Using wt (0, t ) 0 , (16) further becomes
1 2 1 2 vT wx dx wxx , (11) 2 ³0 w( x,0) w0 ( x), wt ( x,0) wt 0 ( x), w(0, t ) 0,
f c (t )
x
2 T
V (t )
wtt 2vwxt v 2 1 wxx
1 2 1 2 ½ ®1 vT ³ 0 wx dx ¾wx (1, t ), ¯ 2 ¿
¨ ©
2
V (t )
fc(t)
and
0
Using (11), \ (t) u 1 w2 and u xt utt # 0 , (15) yields: x x
and
³
Note that both w-dynamics and u-dynamics are incorporated in (14). The time derivative of V (t ) , V (t ) dV (t ) dt , involves two parts as follows: (15) V (t ) Vt vV x .
T0 1 EA, v v0, t W UAL2 T0 T0 UA
wˆ , v L T
`
2
Note that (8) is a non-linear integro-differential equation. Now, the following new dimensionless variables are introduced. x
x
^
and AE L 2 § · wˆ T (T ,W )dT ¸ wˆ [ ( L,W ). ¨ T0 ³ 0 2 © ¹
2 ³0
x
2
3. BOUNDARY CONTROL LAW The objective is to design a right-boundary control law that guarantees the asymptotic stabilization of the axially moving strip. It is assumed that the dynamics of the strip is well represented by the nonlinear integro-differential equation (11). The Lyapunov second method for proving the stability of the closed-loop system is pursued. A positive definite function, the total mechanical energy V (t ) of the strip, is considered as a Lyapunov functional candidate as follows: 1 1 2 2 V(t) ³ v vux ut wt vwx dx 2 0 2 1§ v2 1§ 1 · 1 · (14) T u w2 dx u w2 dx.
T
2L ³ 0
¿
Section V, v 0.0119 is used. 4. The normalized equation of (4)-(6) using (10) takes the form: 1 1 1 2 (13) u ( x, t ) w 2 ( x, t ) w ( x, t ) dx .
(5)
is derived. Now, observing that the touch rolls at [ = L do not allow any longitudinal displacement, i.e., uˆ ( L,W ) 0 , 1 L 2 (6) \ (W ) wˆ (T ,W )dT t 0, for all W t 0 2L ³ 0
0
Note that the trivial solution w(x) 0 is always a solution. 3. In (11), a critical speed at v 1 exists. That is, the fundamental natural frequency vanishes and divergence instability occurs (Mote, 1965). Thus, in this work it is assumed that v belongs to a subcritical speed range, i.e., v 1 . In simulations in
\ (W ) [ 1 2
where
2
¯
approximated as
*w x2 (0, t )
(11)
v3 w x4 (0, t ) , 8
½ v3 wx (1, t )® \vT2 1 wt (1, t ) *wx (1, t ) wx3 (1, t )¾, 8 ¯ ¿
(12)
(17) where * : v ^\ v 2 v 2 1 v 2 ` ! 0 , because vT2 !! v 2 , T
where the terms wtt , 2vwxt , and w xx represent the local, the Coriolis, and the centrifugal acceleration components, respectively, and f c (t ) is the control
2
v 1 , and
\ t 0 . The first two terms in (17) are always negative. Thus, the following feedback control law, which relates the two terms: wx (1, t ) in
input force to be designed. Note that (12) involves an integral term, which may not be easily 3
(12) and wt (1, t ) , will make (17) negative semidefinite. wt (1, t ) K1 (1 \vT2 ) wx (1, t ) (18) K 2 (1 \ v T2 ) 3 w x3 (1, t )
where L2 and H 0k, l are defined as
To see this, (18) is substituted into (17). Then,
The subscript l in H denotes that the function has a left support. Equation (11) can be written in the state space form as follows: (28) z Az , z (0) / ,
V (t)
(11),( 21)
where K ! 1
*
^
1)
2
!0
v
and K 2 !
v3 1 d K2. 8(\vT2 1) 4 8
wt (1, t) w (1, t) K 3 2K2 27K2 4K22
3
wt (1, t) K13 wt2 (1, t) , 2K2 27K23 4K22
3
º w ». 2v » wx ¼
(29)
Theorem 2: The transverse dynamics of the axially moving system (11) with boundary control law (18) is dissipative. Proof: The transversal energy of the strip is introduced as follows: 2
E(t) z, z !/ z(t) / 2
vT2 § 1 2 · 1 1 2 (31) 1 1 2 w vw dx ¨ wx dx¸ ³0 wx dx t x ¹ 2 2 ³0 8 © ³0
The substitution of (13) into (31), (31) yields another expression as follows:
(23)
2
1§ v2 1§ 1 1 1 · 1 · 2 E(t) ³ wt vwx dx T ³ ¨ux wx2 ¸ dx ³ ¨ux wx2 ¸dx. 0 0 0 2 2 © 2 ¹ 2 ¹ © (32) Note that (32) is the same form as (14) except the lack of the first term in (14). The time derivative of (32) becomes
wt (1,t) w(1,t) K3 w2(1,t) K3 w2(1,t) 1 3 t 2 3 t 13 t 2 , 2K2 27K2 4K2 2K2 27K2 4K2
E (t )
(24) where the control input is a function of only wt (1, t ) and gains K1 and K 2 .
Et vE x 1
§
³ ^ w vw w vw v ¨©u t
0
x
tt
2 T
xt
u xt w x w xt `dx v ³
All above developments are summarized in the following theorem.
x
1 · wx2 ¸ uxt wxwxt 2 ¹
1 0
^ wt vwx wxt vwxx
1 ·§ 1 · § 1 · ½ § vT2 ¨ u x wx2 ¸¨ u x wx2 ¸ ¨ u x wx2 ¸ ¾dx. (33) 2 ¹© 2 ¹x © 2 ¹x ¿ ©
Theorem 1: Consider the axially moving system (11)-(12). Then, the closed-loop system with the right boundary control law (18) is uniformly asymptotically stable.
The
substitution of (11) into (33), 1 \ (t ) u x w x2 and u xt u tt # 0 , yields:
with
2
E (t )
4. DISSIPATIVE, EXPONENTIAL STABILITY
(11)
\v 1 ^w (1, t)w (1, t) w (0, t)w (0, t)` v \v 1 ^w (1, t ) w (0, t )`. 2 2 T
t
x
2 T
In this section, the dissipativeness and exponential stability of the transversal motion of the axially moving strip with the right boundary control law (18) and control gains (22) is further investigated. In order to analyze this, the state space / is introduced as follows: (25) / : ( w, wt )T w H 01, l , wt L2 ,
^
1
`
where other two roots involving an imaginary number are excluded. Therefore, substituting (23) into (12), the final control input becomes fc(t)
(27)
K2 (1 \vT2 )3 wx3 (1, t) K1(1 \vT2 )wx (1, t) wt (1, t) 0 . (30) In the sequel, because the system is nonlinear, all analyses are performed about the zero equilibrium state.
2 t
`
^
Finally, w x (1, t ) in (18) can be solved in terms of wt (1, t ) as follows: 3
(26)
The domain D ( A) of the nonlinear operator A for the system with the right boundary control law is D( A) : (w, wt )T w H02,l , wt H01,l ,
Therefore, from (20) and (21), K1 and K 2 should satisfy (22) 0 < K1 d 4QK 2 and K 2 t 0.125. It is remarked that (22) is one sufficient condition for assuring that the time derivative of V (t ) along (11) and (18) is negative.
(1 \vT2 )wx (1, t)
`
dx f ,
0 ª «§ vT2 1 2 · w 2 2 «¨¨1 v ³ 0 wx dx ¸¸ 2 2 ¹ wx ¬©
A
(21)
3 1
2
where z ( w, wt ) T / , the operator A : / o / is a nonlinear operator defined as
3
!0 8(\vT2 1) 4 are assumed. The existence of such K1 and K 2 is apparent from the following equations: * v (20) d K1, 2 2 (\vT 1) 2 (\vT2
0
H0k,l : f L2 f c, f cc,, f (k ) L2 , and f (0) 0 .
v3 *w (0, t) wx4 (0, t) K1 (\vT2 1)2 * wx2 (1, t) 8 § v3 · 2 4 (19) ¨¨ K2 (\vT 1) ¸¸wx4 (1, t) d 0, 8 © ¹ 2 x
^f : [0,1] o R ³ f 1
L2 :
2 x
t
x
2 x
(34)
Again, the substitution of (18) into (34) yields: E (t )
(11), (18 )
v 1 d \vT2 (1 \vT2 ) wx2 (1, t ) (1 \vT2 ) 4 wx4 (1, t ) 2 8 v 2 2 . (35) \vT 1 wx (0, t ) d 0 2
`
Now,
4
Thus, from (31), (47), and the semi-group property of the solution, the following inequality is obtained.
d z, z !/ 2 z, Az !/ (11),(18) E (t) d 0 . (36) (11),(18) dt (11),(18)
³
Hence, the closed-loop operator with (18) is dissipative on / . Therefore, a nonlinear C0
f
³0
S (t )
1
1
0
0
³
2 1 v \vT2
³
1 0
xwx wxx dx.
(39)
(41)
0
1
1
2³ xwxwxxdx w (1,t)³ wx2dx. 2 x
0
(43)
0
Finally, (41) is expressed as follows: 1 K(t) E(t) tE (t) wt2 (1, t) wt2 dx (1 v 2 \vT2 )wx2 (1, t)
³
0
1
(1 v 2 \vT2 ) ³ w x2 dx .
dt f ,
(48)
4
(49)
/
d M ce P ct . That is, /
d M c z (0)
/
e P ct .
(50)
z (t )
2 /
d M c 2 e 2 P ct z (0)
2 /
d Me Pt , (51)
2 P c . Therefore,
Fig. 3 shows the 3D plot of the controlled response, which demonstrates the exponentially decaying behavior of the system. The gains used are K1 = 0.0714 and K 2 =10, which satisfy condition (22). The initial condition used is w( x,0) 0.02 sin Sx . In Fig. 4, the transversal displacement at x 0.5 and the control input used are shown. As compared in Fig. 4, the response of the uncontrolled system continuously oscillates with almost same magnitude of the initial condition. This uncontrolled vibration will damp out eventually, but because the damping is so small it will last for a while. On the other hand, with the boundary control, an acceptable level of vibration suppression was achieved within 1.5 sec.
³
0
t C 2
Therefore, the parameters in normalized equation (11) are v 0.0119 , vT 35.857 , and x [0,1] .
Also, the last two terms of (41) satisfy the following equalities. 1 1 (42) 2 xwxtwtdx wt2(1,t) wt2dx,
³
:
Numerical simulations by using a finite difference scheme are performed. Typical parameter values of the steel strip are: 2 2 E 2 u 1011 N/m , A 1.4 u 0.0045 m , T0 9,800 kN, 2 (52) U 7,850 kg/m , v0 1.67 m/s, and L 17.5 m.
2
0
2
K (t ) 2
5. NUMERICAL SIMULATIONS
where C ! 2 is a constant. Hence, the following holds: (40) 0 d t C E (t ) d K (t ) d (t C ) E (t ) , for t ! C sufficiently large. With the use of (11) and (13), the differentiation of (41) with respect to time yields: 1 K(t) E(t) tE (t) 2 xwxtwt dx (11)
/
f
2 2 where M z (0) / M c and P the theorem is proved.
0
d ³ wx2 dx ³ wt vwx 2 dx d 2 E (t ) d CE (t ) ,
4
S (t ) z (0) dt ³
/
E (t )
2 ³ xw x wt vwx dx d ³ xw x dx ³ wt vwx dx 2
:
From (31),
³0
0
0
f
E 2 (t ) dt ³ E 2 (t ) dt
S (t ) z(0) dt f .
z (t )
where E(t) is defined in (31). The last term of (38) satisfies the following inequalities: 0
:
: 0
Then, by the semigroup theorem (Pazy, 1983), there exist constants P c ! 0 and M c ! 0 such that
Proof: To prove that the system decays exponentially to zero, a positive definite function by following the approach in (Lee and Mote, 1996) is introduced. 1 (38) K (t ) tE (t ) x^2wx wt vwx `dx, t t 0 ,
1
³
where z (0) D ( A) . (48) implies that
Theorem 3: The axially moving system (11) under boundary control law (18) with the control gains in (22) is exponentially stable. That is, there exist constants P ! 0 and M ! 0 such that (37) E (t ) d Me Pt , t t 0 .
1
E 2 (t )dt d³
semigroup S (t ) of contraction on / is generated (Kǀmura, 1967; Pazy, 1983; Chen, 1993), where S (t ) is a bounded operator on / for t t 0 .
1
f 0
(44)
0
The substitution of (21) into (44) yields: K (t ) (11),( 21) E (t ) K12 (1 \vT2 ) 2 wx2 (1, t ) 2K1K2 (1\vT2 )4 wx4 (1, t) K22 (1\vT2 )6 wx6 (1,t) (1 v 2 \vT2 )wx2 (1, t) tE (t ) 1
1
0
0
³ wt2 dx (1 v 2 \vT2 )³ wx2 dx ,
(45)
where E (t ) d 0 . Thus, noting that E(t) and wx (1, t ) are bounded, (45) is negative for a sufficiently large time : . That is, when t ! : , (45) satisfies the following inequality. (46) K (t ) d 0 . From (40) and (46), the following holds: K (t ) , t ! : . (47) E (t ) d
Fig. 3. 3D plot of the controlled response with control gains of K1 0.0714 , K 2 10 .
t C
5
String by Linear Boundary Feedback. Automatica, 35(1), 177-181. Kǀmura, Y., (1967). Nonlinear Semi-Groups in Hilbert Space. Journal of the Mathematical Society of Japan, 19, 493-507. Kreyszig, E., (1989). Introductory Functional Analysis with Applications. John Wiley & Sons, 73-77. Lee, S. Y., and Mote, C. D. (1996). Vibration Control of an Axially Moving String by Boundary Control. ASME Journal of Dynamics Systems, Measurement, and Control, 118(1), 6674. Laousy, H., Xu, C. Z., & Sallet, G. (1996) Boundary Feedback Stabilization of a Rotating BodyBeam System. IEEE Transactions on automatic control, 41(2), 241-244. McIver, D. B. (1973). Hamilton’s Principle forSystems of Changing Mass. Journal of Engineering Mathematics, 7(3), 249-261. Moon, J., and Wickert, J. A. (1997). Non-linear Vibration of Power Transmission Belts. Journal of Sound and Vibration, 200(4), 419-431. Morgul, O. (1992). Dynamics Boundary Control of a Euler-Bernoulli Beam. IEEE Transactions On Automatic Control, 37(5), 639-642. Mote, C. D. (1965). A Study of Band Saw Vibration. Journal of the Franklin Institute, 279, 430-444. Pazy, A., (1983). Semigruops of Linear Operators and Applications to Partial Differential Equations, New York. Pellicano, F., Vestroni, F., and Fregolent, A. (2000). Experimental and Theoretical Analysis of a Power Transmission Belt. ASME Conference on Vibration and Control of Continuous Systems, 107, 71-78. Pellicano, F., and Zirilli, F. (1998). Boundary Layers and Non-Linear Vibrations in an Axially Moving Beam. International Journal of NonLinear Mechanics, 33(4), 691-694. Shahruz, S. M. (1998). Boundary Control of the Axially Moving Kirchhoff String. Automatica, 34(10), 1273-1277. Shahruz, S. M. (2000). Boundary control of a nonlinear axially moving string. International Journal of Robust Nonlinear Control, 10, 17-25. Wickert, J. A. (1992). Non-Linear Vibration of a Travelling Tensioned Beam. International Journal of Non-Linear Mechanics, 27(3), 503506. Wickert, J. A., and Mote, C. D. (1988). Current Research on the Vibration and Stability of Axially Moving Materials. Shock and Vibration Digest, 20(5), 3-13. Wickert, J. A., and Mote, C. D. (1990). Classical Vibration Analysis of Axially Moving Continua. Journal of Applied Mechanics, 57, 738-744.
With Boundary Control
Without Control
Displacement w(0.5,t) [m]
0.02
0.01
0.00
-0.01
-0.02 0.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
Time [sec]
0.04 0.03
Control input [N]
0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 0.0
0.5
1.0
1.5
Time [sec]
Fig. 4. Simulation results of the axially moving strip with control gains of K1 0.0714 , K 2 10 . Top: transversal displacement at x=0.5, w(0.5,t) Bottom: control input 6. CONCLUSIONS In this paper, a boundary control law for the axially moving belt was proposed. The boundary control law obtained, which is different from that derived by using a string equation, was implemented in the form of a negative feedback control of the transversal velocity of the belt at the right end. The controller dissipates the total energy of an axially moving belt system. The simulation results also show that transverse vibrations decay exponentially. ACKNOWLEDGMENT This work was supported by the Brain Korea 21 Program of the Ministry of Education and Human Resources, Korea. REFERENCES Bapat, V. A., and Srinivasan, P. (1967). Nonlinear transverse oscillations in traveling strings by the method of harmonic balance. Journal of Applied Mechanics, 34, 775-777. Carrier, G. F. (1945). On the Nonlinear Vibration Problem of the Elastic String. Quarterly of Applied Mathematics, 3, 157-165. Chen, G., and Zhou, J., (1993). Vibration and Damping in Distributed Systems, CRC Press, Floria. Fung, R. F., Wu, J. W., and Wu, S. L. (1999). Exponential Stability of an Axially Moving
6
www.elsevier.com/locate/ifac
VIBRATION CONTROL OF AN AXIALLY MOVING STRIP BY A NON-LINEAR BOUNDARY CONTROL
Ji-Yun Choia, Keum-Shik Hongb*, and Chang-Do Huha a
Department of Mechanical and Intelligent Systems Engineering, Pusan National University, 30 Changjeon-dong, Keumjeong-ku, Pusan, 609-735, Korea. Tel: +82-51-510-1481, Fax: +82-51-514-0685, Email: [email protected] b School of Mechanical Engineering, Pusan National University, 30 Changjeon-dong Kumjeong-ku Pusan 609-735, Korea. Tel: +82-51-510-2454, Fax: +82-51-514-0685 Email: [email protected] * The author to whom all correspondence should be addressed. control
Abstract: In this paper, an active vibration of the translating steel strip in the zinc galvanizing line is investigated. The unwanted vibrations of the strip deteriorate the uniformity of the deposited mass on the surfaces of the steel strip and the consumption of zinc. The translating steel strip is modelled as a moving belt equation by using Hamilton’s principle for the systems with changing mass. The total mechanical energy of the system is considered as a Lyapunov function candidate. A nonlinear boundary control law that assures the exponential stability of the closed loop system is derived. The existence of a closed-loop solution is shown by proving that the closed-loop dynamics is dissipative. Simulation results are provided. Copyright © 2002 IFAC
Keywords: asymptotic stability, axially moving system, boundary control, hyperbolic nonlinear vibrations, zinc galvanizing line. partial differential equation, Lyapunov method,
1. INTRODUCTION The examples of axially moving systems are found in various engineering areas: the steel strip in a zinc galvanizing line, power transmission belts, textile fibers, aerial cable thread lines, magnetic tapes, band saw blades, paper sheets during processing, etc. In such systems, undesirable vibrations of the moving objects occur during the process due to the eccentricity of a pulley, and/or non-stationary speed of a driving motor, and/or aerodynamic excitation from the surrounding environment, and/or nonuniformity of the material. Also, such vibrations result in costly defects on the products. Fig. 1 shows a zinc galvanizing line. The preheated steel strip is passed through a hot zinc tank and then pulled up vertically with zinc coated on it. The control objectives in the galvanizing line are to improve the uniformity of the zinc deposit on the strip surfaces and reduce the zinc consumption. Therefore, an active control of the vibrations, with minimal number of actuators and sensors, is currently the main research issue in these areas.
and Mote (1996), Moon and Wickert (1997), Fung et al. (1999) and Pellicano et al. (1998, 2000). Recently, Wickert (1992) analyzed free nonlinear vibrations of an axially moving elastic tensioned beam over the sub- and super-critical transport speed ranges. Laousy et al. (1996) proposed a stabilizing boundary feedback control law for a rotating body-beam system and showed that the beam vibrations are forced to decay exponentially to zero. Lee and Mote (1996) applied the boundary control technique to control the transverse vibration of an axially moving string and showed the exponential stability of the boundary-controlled string. Moon and Wickert (1997) investigated nonlinear vibrations of a prototypical power transmission belt system, which is excited by pulleys having slight eccentricity, through analytical and experimental methods. Fung et al. (1999) controlled the vibrations of an axially moving string system using a boundary controller derived by including the actuator dynamics in the plant model. Shahruz (2000) also showed that the non-linear string can be stabilized by a linear boundary control.
During the past several decades, axially moving systems have been extensively studied by many researchers including Carrier (1945), Mote (1966), Bapat and Srinivasan (1967), Wickert and Mote (1988, 1990), Morgul (1992), Wickert (1992), Lee
The axially moving systems can be modeled in three different ways: a string equation (Lee and Mote, 1996; Fung et al., 1999; Shahruz, 2000), a beam equation (Wickert, 1992; Pellicano et al., 1998), and a belt equation (Moon and Wickert, 1997; Pellicano
1
et al., 2000) depending on the flexibility of the system considered and the objectives of control. In this paper, a belt equation is used because the length of the steel strip between two supporting points, see Fig. 1, is 17.5 m long and therefore it allows some longitudinal deformation as well as the transversal displacement. The equations of motion of the moving strip, by using the Hamilton’s principle (McIver, 1973) for the systems of changing mass, is first derived. A nonlinear right-boundary control law to suppress the vibrations of the moving strip is derived so that the total vibration energy dissipates at the right boundary. The asymptotic stability of the closed loop system is guaranteed by the Lyapunov second method.
the longitude of motion, v0 be the axial speed of the strip, uˆ ([ ,W ) and wˆ ([ ,W ) be the longitudinal and transversal displacements of the strip, respectively, and L be the length of the strip. Also, let U be the mass per unit area of the strip, A be the cross section area, E be the elastic modulus, and T0 be the tension applied to the strip. Then, with a control force Fc (W ) at the right boundary, the following equations of motion are derived:
§ ©
1 2
· ¹[
U v02uˆ[[ 2v0uˆ[W uˆWW E¨ uˆ[ wˆ [2 ¸
(1)
0,
uˆ([ ,0) uˆ0 ([ ), uˆW ([ ,0) uˆW 0 ([ ), uˆ(0,W ) uˆ( L,W ) 0,
·½ ¹¿[
0, (2)
1 ½ Fc (W ) T0 wˆ [ (L,W ) AEwˆ [ (L,W )®uˆ[ (L,W ) wˆ [2 (L,W )¾ 0, 2 ¯ ¿
(3)
1 2
§ ©
UA v02 wˆ[[ 2v0wˆ[W wˆWW T0wˆ[[ AE®wˆ[ ¨uˆ[ wˆ[2 ¸¾
2. MODELING OF TRANSLATING STEEL STRIP Fig. 1 depicts a continuous hot-dip zinc galvanizing process with an active vibration control. The steel strip, which may vary in width from 800 mm to 1400 mm and in thickness from 1.2 mm to 4.5 mm, is preheated in a continuous annealing furnace and then introduced at a speed about 1 m/sec into a pot of molten zinc at about 450 qC. The steel strip passes under a sink roll and through a pair of stabilizing and correcting rolls to emerge vertically from the pot coated with a layer of zinc. Because there is an increasing demand for greater consistency of the thickness of zinc film, a pair of air knives located about 0.5 m above the surface of the zinc tank, which direct a long thin wedge-shaped jet of high-velocity air toward the strip, are used to control the deposited mass by stripping excess zinc back to the pot. The strip then travels 35 m vertically, while the solidification of the deposited film is enhanced with the help of an air cooler, and 110 m horizontally, cooling as it goes, to a gauge that measures the mass of zinc deposited on the strip surfaces.
¯
wˆ ([ ,0) wˆ 0 ([ ), wˆW ([ ,0) wˆW 0 ([ ), wˆ (0,W ) 0,
and
where (1)-(3) represent the equations of the longitudinal motion, the transversal motion, and the right transversal boundary condition, respectively. [ and W denote w / w[ and w / w W , respectively. 110 m
Tower Roll
17.5 m
Thickness Gauge
Strip Direction of Travel
Pr
Touch Rolls, Hydraulic Actuator
Ps
Ap2 , V p 2
17.5 m
In order to achieve the uniformity of the zinc deposit on the strip surfaces and to reduce the zinc consumption, the strip should pass at equidistance from each of the air knives. But, due to the shifting and vibration of the strip, a discrepancy between the averaged deposited masses on the left and right strip surfaces and the non-uniformity of the deposited mass across the strip occur. The variations in deposited mass will lower the quality of the product.
Pr
Servo Valve Voltage
P2 , Q 2
Displacement, Velocity
Annealing Furnace
T0
P1 , Q1 A p1 , V p1
Controller
Air Cooler
Air Knife
Stabilizing Roll Correcting Roll
Hot Zinc Tank
Sink Roll
Fig. 1. A translating steel strip in the zinc galvanizing line: control strategy. Y
Fig. 2 shows a schematic of the axially moving steel strip for control system design purpose. Depending on the thickness of the strip and the distance between two support points, the strip can be modelled as one out of three models: a moving beam, a moving string, and a moving belt. In this paper, both the transversal and longitudinal displacements of the strip are initially considered, i.e., the steel strip is modelled as a moving belt.
Boundary Control Force
v0
wˆ ([ , W )
[
Fc (W )
X
uˆ ([ , W )
L
T0
Fig. 2. A schematic of the axially moving steel strip with two intermediate touch rolls. Remark 1: If only linear terms are retained in (1)-(2), they become the travelling rod and tensioned string equations, respectively. Over a technically useful range of parameter values, the speed of the longitudinal waves is significantly faster than the transversal ones (Mote, 1965). On the time scales of
Now, the left boundary at the stabilizing roll in Fig. 2 is assumed fixed. The two touch rolls located in the middle section of the strip will play the right boundary, where the control input force is applied. Let W be the time, [ be the spatial coordinate along
2
the lower transversal modes, the tension variations propagate almost instantaneously as the influence of longitudinal inertial is small (Wickert, 1992).
implementable. However, through the boundary control law design in Section III next, the control force in (12), f c (t ) , will be re-formulated in an
Now, the explicit appearance of uˆ in (2) can be suppressed by approximating the dynamic tension component as a function of time alone: First, the time derivatives of uˆ in (1) are all neglected, i.e., uˆW uˆWW # 0 . Also, since E / U !! v02 , (1) can be
implementable form. Remarks 2. The equilibrium solutions w(x ) of (11) satisfy: 1 2 1 2 ½ 2 , , and wx (1) 0 . ® v 1 vT ³ wx dx¾wxx 0 w(0) 0
1 · § 0 # ¨ uˆ[ wˆ [2 ¸ 2 ¹[ ©
\ (W ) : uˆ[ wˆ [2 ,
(4)
which implies that \ is independent of [. By integrating both sides of (4) with respect to the spatial coordinate from 0 to [ and using uˆ ( 0, W ) 0 , uˆ ([ ,W ) \ (W )[
1 [ 2 wˆ T (T ,W )dT 2 ³0
x
is derived. Therefore, the uˆ -term in (2) and (3) can be approximated in terms of the transversal one as follows: [ L 2 1 [ 2 (7) uˆ ([ ,W ) wˆ (T ,W )dT wˆ (T ,W )dT . 2 ³0
T
T
The substitution of (7) into (2)(3) gives
AE L 2 ˆ[[ 0, (8) wˆT (T,W )dT w 2 ³0 ˆ 0 ([), w ˆW ([,0) w ˆW 0 ([), w ˆ (0,W ) 0, wˆ ([,0) w
UA wˆWW 2v0wˆ[W v02wˆ[[ T0wˆ[[
Fc (W )
(9)
[ , w L
Fc(t) . (10) T0 The substitution of (10) into (8)(9) yields the normalized equations of the transversal motion and the right boundary condition, respectively, as follows:
x
2
³ ¨©
¸ ¹
0
x
2
x
¸ ¹
\v 1 ^w (1, t )w (1, t ) w (0, t )w (0, t )` v ^\ v v 1 v ` ^w (1, t ) w (0, t )` 2 2 T
(11)
t
x
2
t
2
2 x
v3 4 wx (1, t ) wx4 (0, t ) . 8
^
`
x
2 x
(16)
Using wt (0, t ) 0 , (16) further becomes
1 2 1 2 vT wx dx wxx , (11) 2 ³0 w( x,0) w0 ( x), wt ( x,0) wt 0 ( x), w(0, t ) 0,
f c (t )
x
2 T
V (t )
wtt 2vwxt v 2 1 wxx
1 2 1 2 ½ ®1 vT ³ 0 wx dx ¾wx (1, t ), ¯ 2 ¿
¨ ©
2
V (t )
fc(t)
and
0
Using (11), \ (t) u 1 w2 and u xt utt # 0 , (15) yields: x x
and
³
Note that both w-dynamics and u-dynamics are incorporated in (14). The time derivative of V (t ) , V (t ) dV (t ) dt , involves two parts as follows: (15) V (t ) Vt vV x .
T0 1 EA, v v0, t W UAL2 T0 T0 UA
wˆ , v L T
`
2
Note that (8) is a non-linear integro-differential equation. Now, the following new dimensionless variables are introduced. x
x
^
and AE L 2 § · wˆ T (T ,W )dT ¸ wˆ [ ( L,W ). ¨ T0 ³ 0 2 © ¹
2 ³0
x
2
3. BOUNDARY CONTROL LAW The objective is to design a right-boundary control law that guarantees the asymptotic stabilization of the axially moving strip. It is assumed that the dynamics of the strip is well represented by the nonlinear integro-differential equation (11). The Lyapunov second method for proving the stability of the closed-loop system is pursued. A positive definite function, the total mechanical energy V (t ) of the strip, is considered as a Lyapunov functional candidate as follows: 1 1 2 2 V(t) ³ v vux ut wt vwx dx 2 0 2 1§ v2 1§ 1 · 1 · (14) T u w2 dx u w2 dx.
T
2L ³ 0
¿
Section V, v 0.0119 is used. 4. The normalized equation of (4)-(6) using (10) takes the form: 1 1 1 2 (13) u ( x, t ) w 2 ( x, t ) w ( x, t ) dx .
(5)
is derived. Now, observing that the touch rolls at [ = L do not allow any longitudinal displacement, i.e., uˆ ( L,W ) 0 , 1 L 2 (6) \ (W ) wˆ (T ,W )dT t 0, for all W t 0 2L ³ 0
0
Note that the trivial solution w(x) 0 is always a solution. 3. In (11), a critical speed at v 1 exists. That is, the fundamental natural frequency vanishes and divergence instability occurs (Mote, 1965). Thus, in this work it is assumed that v belongs to a subcritical speed range, i.e., v 1 . In simulations in
\ (W ) [ 1 2
where
2
¯
approximated as
*w x2 (0, t )
(11)
v3 w x4 (0, t ) , 8
½ v3 wx (1, t )® \vT2 1 wt (1, t ) *wx (1, t ) wx3 (1, t )¾, 8 ¯ ¿
(12)
(17) where * : v ^\ v 2 v 2 1 v 2 ` ! 0 , because vT2 !! v 2 , T
where the terms wtt , 2vwxt , and w xx represent the local, the Coriolis, and the centrifugal acceleration components, respectively, and f c (t ) is the control
2
v 1 , and
\ t 0 . The first two terms in (17) are always negative. Thus, the following feedback control law, which relates the two terms: wx (1, t ) in
input force to be designed. Note that (12) involves an integral term, which may not be easily 3
(12) and wt (1, t ) , will make (17) negative semidefinite. wt (1, t ) K1 (1 \vT2 ) wx (1, t ) (18) K 2 (1 \ v T2 ) 3 w x3 (1, t )
where L2 and H 0k, l are defined as
To see this, (18) is substituted into (17). Then,
The subscript l in H denotes that the function has a left support. Equation (11) can be written in the state space form as follows: (28) z Az , z (0) / ,
V (t)
(11),( 21)
where K ! 1
*
^
1)
2
!0
v
and K 2 !
v3 1 d K2. 8(\vT2 1) 4 8
wt (1, t) w (1, t) K 3 2K2 27K2 4K22
3
wt (1, t) K13 wt2 (1, t) , 2K2 27K23 4K22
3
º w ». 2v » wx ¼
(29)
Theorem 2: The transverse dynamics of the axially moving system (11) with boundary control law (18) is dissipative. Proof: The transversal energy of the strip is introduced as follows: 2
E(t) z, z !/ z(t) / 2
vT2 § 1 2 · 1 1 2 (31) 1 1 2 w vw dx ¨ wx dx¸ ³0 wx dx t x ¹ 2 2 ³0 8 © ³0
The substitution of (13) into (31), (31) yields another expression as follows:
(23)
2
1§ v2 1§ 1 1 1 · 1 · 2 E(t) ³ wt vwx dx T ³ ¨ux wx2 ¸ dx ³ ¨ux wx2 ¸dx. 0 0 0 2 2 © 2 ¹ 2 ¹ © (32) Note that (32) is the same form as (14) except the lack of the first term in (14). The time derivative of (32) becomes
wt (1,t) w(1,t) K3 w2(1,t) K3 w2(1,t) 1 3 t 2 3 t 13 t 2 , 2K2 27K2 4K2 2K2 27K2 4K2
E (t )
(24) where the control input is a function of only wt (1, t ) and gains K1 and K 2 .
Et vE x 1
§
³ ^ w vw w vw v ¨©u t
0
x
tt
2 T
xt
u xt w x w xt `dx v ³
All above developments are summarized in the following theorem.
x
1 · wx2 ¸ uxt wxwxt 2 ¹
1 0
^ wt vwx wxt vwxx
1 ·§ 1 · § 1 · ½ § vT2 ¨ u x wx2 ¸¨ u x wx2 ¸ ¨ u x wx2 ¸ ¾dx. (33) 2 ¹© 2 ¹x © 2 ¹x ¿ ©
Theorem 1: Consider the axially moving system (11)-(12). Then, the closed-loop system with the right boundary control law (18) is uniformly asymptotically stable.
The
substitution of (11) into (33), 1 \ (t ) u x w x2 and u xt u tt # 0 , yields:
with
2
E (t )
4. DISSIPATIVE, EXPONENTIAL STABILITY
(11)
\v 1 ^w (1, t)w (1, t) w (0, t)w (0, t)` v \v 1 ^w (1, t ) w (0, t )`. 2 2 T
t
x
2 T
In this section, the dissipativeness and exponential stability of the transversal motion of the axially moving strip with the right boundary control law (18) and control gains (22) is further investigated. In order to analyze this, the state space / is introduced as follows: (25) / : ( w, wt )T w H 01, l , wt L2 ,
^
1
`
where other two roots involving an imaginary number are excluded. Therefore, substituting (23) into (12), the final control input becomes fc(t)
(27)
K2 (1 \vT2 )3 wx3 (1, t) K1(1 \vT2 )wx (1, t) wt (1, t) 0 . (30) In the sequel, because the system is nonlinear, all analyses are performed about the zero equilibrium state.
2 t
`
^
Finally, w x (1, t ) in (18) can be solved in terms of wt (1, t ) as follows: 3
(26)
The domain D ( A) of the nonlinear operator A for the system with the right boundary control law is D( A) : (w, wt )T w H02,l , wt H01,l ,
Therefore, from (20) and (21), K1 and K 2 should satisfy (22) 0 < K1 d 4QK 2 and K 2 t 0.125. It is remarked that (22) is one sufficient condition for assuring that the time derivative of V (t ) along (11) and (18) is negative.
(1 \vT2 )wx (1, t)
`
dx f ,
0 ª «§ vT2 1 2 · w 2 2 «¨¨1 v ³ 0 wx dx ¸¸ 2 2 ¹ wx ¬©
A
(21)
3 1
2
where z ( w, wt ) T / , the operator A : / o / is a nonlinear operator defined as
3
!0 8(\vT2 1) 4 are assumed. The existence of such K1 and K 2 is apparent from the following equations: * v (20) d K1, 2 2 (\vT 1) 2 (\vT2
0
H0k,l : f L2 f c, f cc,, f (k ) L2 , and f (0) 0 .
v3 *w (0, t) wx4 (0, t) K1 (\vT2 1)2 * wx2 (1, t) 8 § v3 · 2 4 (19) ¨¨ K2 (\vT 1) ¸¸wx4 (1, t) d 0, 8 © ¹ 2 x
^f : [0,1] o R ³ f 1
L2 :
2 x
t
x
2 x
(34)
Again, the substitution of (18) into (34) yields: E (t )
(11), (18 )
v 1 d \vT2 (1 \vT2 ) wx2 (1, t ) (1 \vT2 ) 4 wx4 (1, t ) 2 8 v 2 2 . (35) \vT 1 wx (0, t ) d 0 2
`
Now,
4
Thus, from (31), (47), and the semi-group property of the solution, the following inequality is obtained.
d z, z !/ 2 z, Az !/ (11),(18) E (t) d 0 . (36) (11),(18) dt (11),(18)
³
Hence, the closed-loop operator with (18) is dissipative on / . Therefore, a nonlinear C0
f
³0
S (t )
1
1
0
0
³
2 1 v \vT2
³
1 0
xwx wxx dx.
(39)
(41)
0
1
1
2³ xwxwxxdx w (1,t)³ wx2dx. 2 x
0
(43)
0
Finally, (41) is expressed as follows: 1 K(t) E(t) tE (t) wt2 (1, t) wt2 dx (1 v 2 \vT2 )wx2 (1, t)
³
0
1
(1 v 2 \vT2 ) ³ w x2 dx .
dt f ,
(48)
4
(49)
/
d M ce P ct . That is, /
d M c z (0)
/
e P ct .
(50)
z (t )
2 /
d M c 2 e 2 P ct z (0)
2 /
d Me Pt , (51)
2 P c . Therefore,
Fig. 3 shows the 3D plot of the controlled response, which demonstrates the exponentially decaying behavior of the system. The gains used are K1 = 0.0714 and K 2 =10, which satisfy condition (22). The initial condition used is w( x,0) 0.02 sin Sx . In Fig. 4, the transversal displacement at x 0.5 and the control input used are shown. As compared in Fig. 4, the response of the uncontrolled system continuously oscillates with almost same magnitude of the initial condition. This uncontrolled vibration will damp out eventually, but because the damping is so small it will last for a while. On the other hand, with the boundary control, an acceptable level of vibration suppression was achieved within 1.5 sec.
³
0
t C 2
Therefore, the parameters in normalized equation (11) are v 0.0119 , vT 35.857 , and x [0,1] .
Also, the last two terms of (41) satisfy the following equalities. 1 1 (42) 2 xwxtwtdx wt2(1,t) wt2dx,
³
:
Numerical simulations by using a finite difference scheme are performed. Typical parameter values of the steel strip are: 2 2 E 2 u 1011 N/m , A 1.4 u 0.0045 m , T0 9,800 kN, 2 (52) U 7,850 kg/m , v0 1.67 m/s, and L 17.5 m.
2
0
2
K (t ) 2
5. NUMERICAL SIMULATIONS
where C ! 2 is a constant. Hence, the following holds: (40) 0 d t C E (t ) d K (t ) d (t C ) E (t ) , for t ! C sufficiently large. With the use of (11) and (13), the differentiation of (41) with respect to time yields: 1 K(t) E(t) tE (t) 2 xwxtwt dx (11)
/
f
2 2 where M z (0) / M c and P the theorem is proved.
0
d ³ wx2 dx ³ wt vwx 2 dx d 2 E (t ) d CE (t ) ,
4
S (t ) z (0) dt ³
/
E (t )
2 ³ xw x wt vwx dx d ³ xw x dx ³ wt vwx dx 2
:
From (31),
³0
0
0
f
E 2 (t ) dt ³ E 2 (t ) dt
S (t ) z(0) dt f .
z (t )
where E(t) is defined in (31). The last term of (38) satisfies the following inequalities: 0
:
: 0
Then, by the semigroup theorem (Pazy, 1983), there exist constants P c ! 0 and M c ! 0 such that
Proof: To prove that the system decays exponentially to zero, a positive definite function by following the approach in (Lee and Mote, 1996) is introduced. 1 (38) K (t ) tE (t ) x^2wx wt vwx `dx, t t 0 ,
1
³
where z (0) D ( A) . (48) implies that
Theorem 3: The axially moving system (11) under boundary control law (18) with the control gains in (22) is exponentially stable. That is, there exist constants P ! 0 and M ! 0 such that (37) E (t ) d Me Pt , t t 0 .
1
E 2 (t )dt d³
semigroup S (t ) of contraction on / is generated (Kǀmura, 1967; Pazy, 1983; Chen, 1993), where S (t ) is a bounded operator on / for t t 0 .
1
f 0
(44)
0
The substitution of (21) into (44) yields: K (t ) (11),( 21) E (t ) K12 (1 \vT2 ) 2 wx2 (1, t ) 2K1K2 (1\vT2 )4 wx4 (1, t) K22 (1\vT2 )6 wx6 (1,t) (1 v 2 \vT2 )wx2 (1, t) tE (t ) 1
1
0
0
³ wt2 dx (1 v 2 \vT2 )³ wx2 dx ,
(45)
where E (t ) d 0 . Thus, noting that E(t) and wx (1, t ) are bounded, (45) is negative for a sufficiently large time : . That is, when t ! : , (45) satisfies the following inequality. (46) K (t ) d 0 . From (40) and (46), the following holds: K (t ) , t ! : . (47) E (t ) d
Fig. 3. 3D plot of the controlled response with control gains of K1 0.0714 , K 2 10 .
t C
5
String by Linear Boundary Feedback. Automatica, 35(1), 177-181. Kǀmura, Y., (1967). Nonlinear Semi-Groups in Hilbert Space. Journal of the Mathematical Society of Japan, 19, 493-507. Kreyszig, E., (1989). Introductory Functional Analysis with Applications. John Wiley & Sons, 73-77. Lee, S. Y., and Mote, C. D. (1996). Vibration Control of an Axially Moving String by Boundary Control. ASME Journal of Dynamics Systems, Measurement, and Control, 118(1), 6674. Laousy, H., Xu, C. Z., & Sallet, G. (1996) Boundary Feedback Stabilization of a Rotating BodyBeam System. IEEE Transactions on automatic control, 41(2), 241-244. McIver, D. B. (1973). Hamilton’s Principle forSystems of Changing Mass. Journal of Engineering Mathematics, 7(3), 249-261. Moon, J., and Wickert, J. A. (1997). Non-linear Vibration of Power Transmission Belts. Journal of Sound and Vibration, 200(4), 419-431. Morgul, O. (1992). Dynamics Boundary Control of a Euler-Bernoulli Beam. IEEE Transactions On Automatic Control, 37(5), 639-642. Mote, C. D. (1965). A Study of Band Saw Vibration. Journal of the Franklin Institute, 279, 430-444. Pazy, A., (1983). Semigruops of Linear Operators and Applications to Partial Differential Equations, New York. Pellicano, F., Vestroni, F., and Fregolent, A. (2000). Experimental and Theoretical Analysis of a Power Transmission Belt. ASME Conference on Vibration and Control of Continuous Systems, 107, 71-78. Pellicano, F., and Zirilli, F. (1998). Boundary Layers and Non-Linear Vibrations in an Axially Moving Beam. International Journal of NonLinear Mechanics, 33(4), 691-694. Shahruz, S. M. (1998). Boundary Control of the Axially Moving Kirchhoff String. Automatica, 34(10), 1273-1277. Shahruz, S. M. (2000). Boundary control of a nonlinear axially moving string. International Journal of Robust Nonlinear Control, 10, 17-25. Wickert, J. A. (1992). Non-Linear Vibration of a Travelling Tensioned Beam. International Journal of Non-Linear Mechanics, 27(3), 503506. Wickert, J. A., and Mote, C. D. (1988). Current Research on the Vibration and Stability of Axially Moving Materials. Shock and Vibration Digest, 20(5), 3-13. Wickert, J. A., and Mote, C. D. (1990). Classical Vibration Analysis of Axially Moving Continua. Journal of Applied Mechanics, 57, 738-744.
With Boundary Control
Without Control
Displacement w(0.5,t) [m]
0.02
0.01
0.00
-0.01
-0.02 0.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
Time [sec]
0.04 0.03
Control input [N]
0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 0.0
0.5
1.0
1.5
Time [sec]
Fig. 4. Simulation results of the axially moving strip with control gains of K1 0.0714 , K 2 10 . Top: transversal displacement at x=0.5, w(0.5,t) Bottom: control input 6. CONCLUSIONS In this paper, a boundary control law for the axially moving belt was proposed. The boundary control law obtained, which is different from that derived by using a string equation, was implemented in the form of a negative feedback control of the transversal velocity of the belt at the right end. The controller dissipates the total energy of an axially moving belt system. The simulation results also show that transverse vibrations decay exponentially. ACKNOWLEDGMENT This work was supported by the Brain Korea 21 Program of the Ministry of Education and Human Resources, Korea. REFERENCES Bapat, V. A., and Srinivasan, P. (1967). Nonlinear transverse oscillations in traveling strings by the method of harmonic balance. Journal of Applied Mechanics, 34, 775-777. Carrier, G. F. (1945). On the Nonlinear Vibration Problem of the Elastic String. Quarterly of Applied Mathematics, 3, 157-165. Chen, G., and Zhou, J., (1993). Vibration and Damping in Distributed Systems, CRC Press, Floria. Fung, R. F., Wu, J. W., and Wu, S. L. (1999). Exponential Stability of an Axially Moving
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