Displacement feedback control of beams under moving loads

Journal of Sound and Vibration (1988) 122(3), 457-464

DISPLACEMENT FEEDBACK C O N T R O L

OF BEAMS U N D E R

M O V I N G LOADS J. M. SLOSS AND S. ADALtt

Department of Mathematics, Uni~'ersit), of California, Santa Barbara, California 93106, U.S.A. I. S. SADEK

Department of Mathematical Sciences, University of North Carolina, Wihnington, North Carolina 28403, U.S.A. AND

J. C. BRUCH, JR.

Department of Mechanical and Environmental Enghleering, University of California, Santa Barbara, California 93106, U.S.A. (Received 28 April 1987, and in revised form 7 July 1987) A feedback control problem for a beam under constant and harmonic moving loads is formulated and solved. The objective of the control is to satisfy a barrier condition which places an upper bound on the deflection. The feedback is taken proportional to t h e transverse displacement of the beam. The location and time of the maximum deflection depend on the feedback parameter which is determined from the solution of a minimax problem. Numerical results are given for various problem parameters and the efficiency of the proposed control is investigated. A measure of the amount of force spent in the control process is also determined to assess the effect of displacement constraint on the force expenditure. 1. INTRODUCTION As the minimum weight construction becomes an important design requirement for m a n y dynamically loaded civil engineering structures such as bridges, tall buildings and offshore structures, they are built increasingly more slender and flexible. This, in turn, can lead to excessive vibrations which may need to be actively suppressed to prevent the structure from oscillating beyond acceptable limits. Feedback control can be effectively used for this purpose, as it has been successfully employed in the control o f various structures [1-4]. In the present paper, feedback control is used to suppress the vibration s o f a bridge which is modelled as a simply supported beam under moving loads. Two types of moving loads are treated, namely, concentrated constant and harmonic loads. The control is applied to limit the deflection under the moving load, i.e., the m a x i m u m deflection is bounded from above by a barrier condition:' As the location and time of the m a x i m u m deflection depend on the magnitude of the feedback control parameter which is not known a priori, the problem is solved as a minimax problem. An overview of the field of active structural control has been presented by Zuk [5] in which the future applications of the concept of the so.called " d y n a m i c structures" have also been discussed. Active control of bridges has been studied by Abdel-Rohman and Leipholz [6-8] and by AbdeI-Rohman and Nayfeh [9], the control being exercised by COn leave from the University of Natal, Durban, South Africa. 457 0022-460X/88/090457+08 $03.00/0

9 1988 Academic Press Limited

458

J.M. SLOSS ETAL.

means of stress-couples created by tendons. Control of cable-stayed bridges was investigated in reference [10] in which the control is applied through suspension cables. A recent review of the structural control application in civil and mechanical engineering is given in reference [11]. The recent book by Leiphoiz and Abdel-Rohman [12] gives an up-to-date treatment of the structural control field, with special attention to the control of bridges. In the present article a proportional feedback control is proposed in which the displacement information is used to determine the amount of control at a given time. The feedback parameter is determined so as to satisfy a barrier condition. Numerical results indicate that proportional control can be effectively used in controlling a beam under a moving load. The speed of the moving load is observed to have a definite effect on the efficiency of the control. Relations between the amount of force spent in the control process and the feedback parameter are also shown graphically.

2. FEEDBACK CONTROL PROBLEM The bridge is modelled as a simply supported beam of length L, cross-sectional area A, moment of inertia I and mass density p. With the modulus of elasticity of the beam denoted by E and the deflection function by Y(X, T), the differential equation governing the vibrations of the controlled structure i s

pA Yrr+ EIYxxxx + G( Y; Co) = g(X - VT)Q(T)Po

(1)

in 0 < X < L, where thesubscripts T and X denote differentiation with respect to time and space, respectively. In equation (1), G(Y; Co) = CoY, Co>~O, is the displacement feedback control term [3] and g ( X - V T ) Q ( T ) P o is the forcing function resulting from the action of the concentrated load Po moving across the beam with a uniform velocity V and time dependency of Q(T). ~ denotes the dimensional delta function. It is noted that the feedback control law G ( Y ; Co) = CoY chosen in the present case is formally similar to the equation of a beam on an elastic foundation [13, 14]. Here Co is the feedback parameter to be determined as part of the solution of the control problem. The following dimensionless quantities are defined:

t= VT/L,

x=X/L,

y= Y/L;

v= V(mL/EI) ~/2, C= CoL4/EI;

8 ( x - t) = L ~ ( X - VT),

P = PoL2/EI.

(2)

Here m = pAL is the mass of the beam. Substitution of equations (2) into equation (1) gives the non-dimensional form of the equation of motion:

y,,+v-2y~x~+v-2Cy=v-2pQ(t)~(x-t),

0
(3)

Boundary conditions for the simply supported beam are y(0, t ) = ) ~ ( 0 , t) = y ( l , t) = yxx(1, t ) = 0 .

(4)

As the bridge is at rest before the load moves across it, the initial conditions are y(x, 0) = y,(x, 0) = 0.

(5)

Equations (3)-(5) constitute the non-dimensional formulation for a bridge under a concentrated moving load controlled by the feedback term Cy.

CONTROL

OF BEAMS

UNDER

MOVING

LOADS

459

The objective of the control is to determine the feedback coefficient C such that the maximum deflection will not exceed a given value. Thus, the control problem can be stated in the following form: determine C in equation (3) such that max max y(x, t; C)<~d, 1>0

O
(6)

where y(x, t; C) is subject to equations (3)-(5). A solution of the problem that satisfies the barrier condition (6) involves the determination of the maximum point of displacement along the bridge with respect to space and time. For a feasible problem, 0 < d < d ~ should be satisfied, where d, is the maximum displacement of the uncontrolled beam. 3. METHOD OF SOLUTION The solution of the differential equation (3) is assumed to be of the form y(x, t; C) = ~ x/2 sin rrrxq,(t),

(7)

where the qr(t)'s are the principal co-ordinates. It is noted that y(x, t; C) given by equation (7) satisfies the boundary conditions (4). Inserting y(x, t; C) from equation (7) into equation (3), multiplying the resulting expression by x/2sin srrx and integrating with respect to x over the interval [0, 1], one obtains i

~, + to~q, = v ~ v - 2 PQ( t) sin srrt,

(8)

where w z~ - (Tr4s4+ C ) l v z. The solution of equation (8) satisfying the initial Conditions (5) is _

9

q~(t)=(v~P/v2to~)

Io

Q(r) s i n s ~ r r s i n t o s ( t - r ) d r .

(9)

Two types of moving loads will be investigated in the sequel, namely, the constant force for which Q(t) = 1 and the harmonic force for which Q ( t ) = s i n ( t o t + a ) , where to and a are specified constants. The solutions for these cases are as follows. For a moving constant force ( Q ( t ) = 1), the solution for q,(t) follows from equation (9) with Q ( t ) = 1: q,(t) = v'2v-2p(sin sTrt - sTrto~' sin tod)/(to~ - s2~'2).

(10)

For a moving harmonic force ( Q ( t ) = s i n ( t o t + a ) ) , the solution for q~(t) is obtained from equation (9) with Q ( t ) = s i n (tot+a): q,(t) = (P/v~v2)[.R[ I cos w ( t ) - R~.' cos u(t) + (srr/to.,)(R~' cos (w.,i'+ ct) -- R4-' cos (-to, t + a))].

(11)

Here u(t) = (to + srr)t + a, v( t ) = (to - sTr)t + a, R, = to';-( to - sTr) z, R : = t o , - ( t o + 9 R~= (to - t o , ) 2 - s % r 2 and R4= (to + to,)Z-s2~2. Thus, the deflection function y(x, t) given by equation (7) is determined explicitly for any x and t for the moving loads under consideration. The remaining unknowns of the control problem are the location .'Coand time to of the maxirn~m deflection as the load moves across the beam and the amount Of C to be applied to satisfy the barrier condition

(6).

a.MS . LOSS

460

ETAI...

The values of C and (Xo, to) can be determined by solving the min-max problem min [max y(x, t; C

x~ t

C)-dl,

(12)

where O<~x<~ 1 and y(x, t; C) is given by equation (7).

4. NUMERICAL RESULTS The effect of feedback control on the beam is studied in this section by comparing the controlled and uncontrolled beams under a unit load P = 1. A measure of the effectiveness of the control is e = lO0(1-d/d,,), where du denotes the maximum deflection of the uncontrolled beam. The curves of e are plotted against C to assess the performance of the control. The measure of the force F spent in reducing the deflection from d~ to d is computed from

F= C 2

foIo

y2(x, t; C) dx dt,

(13)

which is the same measure proposed in reference [3]. Here F given by equation (13) represents the square of the mean square average of the total feedback force Cy in the space-time domain [0, 1] x [0, 1]. Figure 1 shows the curves of l O 0 ( l - d / d , ) vs. C f o r v a r i o u s values of the velocity parameter e for the case of a constant load moving across the beam. Here the relation between C and d i~ obtained by solving the rain-max problem (12) for different values of d. It is observed that the effectiveness of the control monotonically increases with respect to the feedback parameter C, but is not correlated to v. This situation is studied in Figure 2, which shows the curves of d plotted against u for different values of C. These curves are again computed from the solution of problem (12). It is observed that as o increases, the maximum deflection will go through a number of minima and maxima points. The control causes the peak points to be shifted and flattened. Note that the curve for C = 0 is given in reference [13]. The amount of force F, given by equation (13), in achieving a given control for a constant moving load is shown in Figure 3 where F is plotted against d. As expected, the expenditure of force increases with decreasing d. Note that d = d, for F = 0.

!

|

I

oo._60 ~ A

~

' ~

40

0"5

2"0

I

5

2C y V

0

"30 .

I 50

I I(X)

I 150

200

c

Figure I. Efficiencycurves plotted against the feedback parameter C for a constant moving load.

CONTROL !

OF BEAMS UNDER i

I

i

MOVING

I

LOADS

I

i

C--O

.

0-3

o-1

! 0.3

! 04

l 0-5

! o.7

! I-0

I 1-4

I z-o

!

v

Figure 2. Curves of d plotted against u for a constant moving load.

9 i'~

|

I

I

I

I

I

I-0 ,043 5.0~

04

0-2

o

oo I

ooz

003

d Figure 3. The amount o f force spent in the control plotted against d for a constant moving load.

6C

I

2C

~

O

!

!

50

I00 C

!

150

200

F i g u r e 4. Efficiency c u r v e s p l o t t e d a g a i n s t C f o r a h a r m o n i c m o v i n g l o a d f o r co = 1, a n d a = r r / 2 .

461

462

J.M.

ETAL

SLOSS

The effectiveness o f the c o n t r o l in the case o f a h a r m o n i c m o v i n g l o a d is s t u d i e d in F i g u r e 4, w h e r e the curves o f 1 0 0 ( 1 - d i d , ) are p l o t t e d a g a i n s t C for v a r i o u s values o f o with to = 1 a n d ct = rr/2. It is o b s e r v e d that the effectiveness o f the control is s i m i l a r to that o f a b e a m u n d e r a c o n s t a n t load. F i g u r e 5 shows the curves o f d p l o t t e d a g a i n s t v for to = 1 a n d a = ~ ' / 2 , w h e r e the circles on the curves i n d i c a t e the p o i n t s w h e r e r e s o n a n c e occurs. It is o b s e r v e d that the m a x i m u m deflection d o e s not o c c u r at r e s o n a n c e points. This is also the case for a b e a m u n d e r a c o n s t a n t force [13]. T h e curves o f Fvs. d for a h a r m o n i c l o a d are s h o w n in F i g u r e 6 for the s a m e p r o b l e m p a r a m e t e r s as in Figure 4. T h e a p p l i c a t i o n o f f e e d b a c k c o n t r o l c h a n g e s the r e s o n a n c e c h a r a c t e r i s t i c s o f the b e a m . T h e critical velocity o,.r is given b y vcr = (s47r4+

C)t/2/slr

v , = (s47r4+

C)t/2/(to ~: s~r) ( h a r m o n i c force),

( c o n s t a n t force),

(14) (15)

for the s t h m o d e . T h e q~(t) for the r e s o n a n c e case is a g a i n d e t e r m i n e d from e q u a t i o n (9) with o given b y e q u a t i o n (14) o r (15). It is n o t e d that the m a x i m u m deflection d o e s !

!

i

.

i

I

|

I

i

t.4

2"0

2.6

o.~ c=o

0-2

oq

0-3

0'4

0"5

0-7

FO V

Figure 5. Curves of d plotted against v for a harmonic moving load for co= 1, a = 7r/2. Circles indicate critical velocities. I

06

I

i

I

I

04 it.

0"2

v-0.25 0

0

05

OOI

1.0 0,02

2-0 0"03

d

Figure 6. The amouqt of force spent in the control plotted against d for a harmonic moving load with t, = l and a = ,'r/2.

CONTROL OF BEAMS UNDER MOVING LOADS

463

not increase indefinitely when the load moves at the critical speed since the beam is under the action of the load for 0 ~ t ~ 1. In fact, Figure 5 indicates that the maximum deflection at v = vcr is less than that for some v < vcr.

5. CONCLUDING REMARKS The problem of keeping a beam subject to a moving load above a given barrier has been solved analytically. The beam simulates a bridge under dynamic loadings and the active control is provided by means of displacement feedback. The magnitude of the feedback parameter is computed from the barrier condition by determining the location and time of the maximum deflection as the load moves across the beam. This computation leads to a minimax solution procedure in which the feedback parameter is obtained such that the maximum deflection at any time does not exceed the distance to the barrier. Two cases of the moving loads are treated, those of a constant load and of a harmonic load. Solutions for both cases are obtained by eigenfunction expansion in which the feedback term is taken into account. Numerical results provide a quantitative assessment of the effectiveness o f the proposed feedback mechanism as well as an indication of the amount o f force spent in the control process. It is observed that the maximum displacement and the expenditure of control force depend considerably on the velocity of the moving load, although increasing the velocity may, in general, lead to a decrease or an increase in these quantities. The critical velocity becomes a function of the control parameter, higher values o f which lead to higher critical velocities.

ACKNOWLEDGMENT The work o f the second author was supported by a grant from the South African Foundation for Research Development.

REFERENCES 1. H. HORIKAWA, E. H. DOWELL and F. C. MOON 1978 International Journal of Solids and Structures, 14, 821-839. Active feedback control of a beam subjected to a nonconservative force. 2. L. MEIROVITCH and L. M. SILVERBERG 1984 Journal of Sound and Vibration 97, 489-498. Active vibration suppression of a cantilever wing. 3. H. H. E. LEIPHOLZ and M. ABDEL-ROHMAN 1986 lngenieur-Archiv 56, 55-70. Active control of elastic plates. 4. P. E. O'DONOGHUE and S. N. ATLUR! 1986 Computers and Structures, 23, 199-209. Control of dynamic response of a continuum model of a large space structure. 5. W. ZUK 1980 Solid Mechanics Archives 5, 75-90. The past and future of active structural control systems. 6. M. ABDEL-ROHMANand H. H. E. LEIPHOLZ 1978 American Society of Civil Engineers Journal of Structural Division 104 (ST8), 1251-1266. Active control of flexible structures. 7. M. ABDEL-ROHMAN, V. QUINTANA and H. H. E. LEIPHOLZ 1980 American Society of Civil Engineers Journal of Engineering Mechanics 106 (EM 1), 57-73. Optimal control of civil engineering structures. 8. M. ABDEL-ROHMANand H. H. E. LEIPHOLZ 1980 American Society of Civil Engineers Journal of Structural Division 106 (ST3), 663-677. Automatic active control of structures. 9. M. ABDEL-ROHMAN and A. H. NAYFEH 1987 American Society of Civil Engineers Journal of Engineering Mechanics, 113 (EM3), 335-348. Active control of nonlinear oscillations in bridges. 10. J. N. YANG and F. GIANNOPOULOS 1979 American Society of Civil Engineers Journal of Engineering Mechanics 105 (EM4), 677-694. Control and stability of cable-stayed bridge. 11. A. M. REINHORN and G. D. MANOLIS 1985 The Shock and Vibration Digest, 17(10), 7-16. Current state of knowledge on structural control.

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12. H. H. E. LE1PHOLZ and M. ABDEL-ROHMAN 1986 Control of Structures. Dordrecht. Martinus NijhofI. 13. G. B. WARBURTON 1964 The Dynamical Behavior of Structures. Oxford, Pergamon Press, second edition 1976. 14. L. FRYBA 1972 Vibration of Solids and Structures under Moving Loads. Groningen: Noordhoff International.