Distributed Parameter Control Design for Vibrating Beams Using Generalized Functions

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DISTRIBUTED PARAMETER CONTROL DESIGN FOR VIBRATING BEAMS USING GENERALIZED FUNCTIONS S. E. Burke and J. E. Hubbard, Jr. C/IIII'/'" St({ r/; f) ml,,'r I.fliltlmtlln, C({III{, rirlgl', ,\1 : 1. ({Ill! J)"/Jfl rt lll l'llt " l ,l/ n lllllliUli FlIgillt'tTillg ..1/lF, ( 'SA

Abstract. The application of polyvinylidene fluoride (PVF 2 ) film to beams for the purpose of active vibration control is investigated for general boundary conditions and non-u ni form spatial distributions, The film offers the possibility of formulating control laws withou t the necessity of modelling the beam in terms of its component vibrational modes. Lyapunov's direct method is used to derive the control laws, The analysis demonstrates that , while for most boundary configurations a spatially uniform control is appropriate, pinned-pin ne d , clamped-sliding , and clamped-clamped beams require non-uniform spatial distributions to be controllable, Candidate distributions are presented for these configurations. As a result of th e investigation a set of guidelines is deduced for the design of non-uniform spatial fil m distributions, Kevwords , Distributed parameter systems, vibration control , Lyapunov methods, dist ribute d parameter actuators, control system synthesis

INTRODUCTION

In this paper an anaiysis of the beam/film system for arbitrary combinations of clamped, pinned , free, and sliding bOl,;ndary conditions is undertaken . Control strategies are then derived using Lyapunov's direct method. It will be shown that the a priori assumption of a non-uniform spatial distribution of PVF 2 can greatly simplify the control formu lation. It is also shown that a certain few beam configurations are not controllable with a spatially uniform control distribution. Candidate spatial control distributions are then presented for two example problems, As a result of this investigation a set of design gu idelines is deduced for the design and application of PVF 2 to the control of beams which preserve the character of controlling all modes,

Structural vibrations in satellites , large space structures , aircraft, and the like are generally lightly damped owing to the low internal damping of the materials used in their construction. Because of weight or other constraints additional passive damping treatments are not often implemented on these structures. Current design practice has investigated the use of discrete actuators to control the vibrations of distributed elastic systems, (See , for example, Gibson , 1981 ; Meirovitch and Bauh, 1984; Balas , 1978; Forward and Swigert, 1981; Haftka, Martinovic, and Schamel, 1985). Since these systems are continuous, and in theory possess an infinite number of degrees of freedom, these control schemes truncate the system model to a finite number of discrete modes (Meirovitch and Bauh, 1984). The number of modes chosen to represent the system as well as the location of the sensors and actuators is difficult to reconcile (Balas, 1978). These control strategies often suffer from contro l/observer spillover and modelling errors (Balas, 1978),

DESIGN AND ANALYSIS The Distributed Parameter Actuator The active element used as a candidate actuator is the piezoelectric polymer polyvinylidene fluoride , or PVF 2 , PVF 2 is a polymer that can be polarized, or made piezoelectrically active, with appropriate processing during manufacture. In its polarized form PVF 2 is essentially a tough , flexible piezoelectric crystal. Polarized PVF 2 is commercially available as a thin polymeric film . The film generally has a layer of nickel or aluminum deposited on each face to conduct a voltage or electric field across its faces,

More recently, an active damper design which circumvents many of the problems associated with modal truncation was developed at MIT (Bailey , 1985). This damper is spatially distributed, and makes use of the piezoelectric polymer polyvinylidene fluoride (PVF 2 ) , The prototype experiment of Bailey and Hubbard (1985) utilized a uniform layer of PVF 2 bonded to one face of a cantilevered beam, They implemented a control strategy using Lyapunov's direct method , and showed that vibrational modes of the beam could be controlled based upon the measurement of the angu lar velocity at the beam's tip, These experiments showed that the first mode's loss factor could be increased by a factor of up to 40, This work has since been expanded upon, and applied to a flexible satellite model (Plump, Hubbard, and Bailey, 1985).

For uniaxially polarized PVF 2, a voltage or fieid applied across its faces results in a longitudinal strain. The strain occurs over the entire plated area of the film, making it a distributed parameter actuator, If the field is varied spatially, the strain will also vary spatially; this gives the added possibility oi varying the control spatially as well as with time.

Perhaps the most significant result of these control studies is that the distributed PVF 2 damper led to a control design wherein all modes of the system can potenti ally be contro lled. It has been of interest, then , to extend this research to the control of more general elastic systems,

It should be noted that the present analysis is not limited to the description of only systems that use PVF 2 film as their distributed actuator. The analysis only assumes that the film

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actuator responds to control inputs via longitudinal strain, and is bonded to one face of the beam , hence it can be applied to the study of oth er cand idate materials as we ll. Derivatio n of Beam Govern inq Equatio ns and Control Constraint s

conditions which are at least sufficient for stab ility can be derived (Wang, 1966; Slemrod, 1974). The time derivative of the functional (7) may be cornbined with the governing Eq. (5) to give F' = WI (1,1) '[ Vx (1 ,t) _ w xxx (1,1)

The flexural vibrations of an elastic beam having a layer of PV F 2 bonded to one face are described by the governing equation (Bailey, 1985) [ El Wu - m V(X,t} 1u + Pb~ w n

=0;

0< x< L,

(1)

where the El product is defined in terms of film ('}2 and beam (.), parameters as (2)

- w ux (O .t)

I -

1-

w I (O,t)·[ Vx (O,t)

w x I (1,t) [ V (1 ,t) - w xx (1 ,t)

I

+ w x I (O,t) [ V (O, t) - w xx (O,t) I + fo w xx IV (x,t) dx.

(8)

In light of the boundary cond itions Eq .s (6a) through (6d ). of wh ich there can be ten possible unique combinat ions for a given beam , Eq. (8) can always be written as F' = fow xx fV (x ,t) dx + fcn (w I (;,t) , w x I (S,t) , f(t) , g(t) } (9)

and

hence the control input V(x,t) on ly appears in the spatial integral.

h is thickness, B is the beam's width, and d 3 , is a property of

An example of how Eq . (9) can be used to synthesize and imp le ment a spatia ll y l.!!J.ii.Q..un control can be found in Bailey and Hubbard (1985). They found that, fo r a canti lever beam with a tip mass and a spat ially uniform control distribution, Eq. (9) takes the form

the film . Equation (1) is a linear inhomogeneous partial differential equation , where the inhomogeneous term describes space and time variations of the control distribution mOV(x,t}; mOV(x,t} can be thought of as a distributed moment. Since Eq. (1) describes a Bernoulli-Euler beam , it will be valid for relatively high aspect ratio beams at "Iow" frequenc ies (Graff,1975). For the present , assume that the beam/ film system is homogeneous. Introducing the non-dimensionalizations

F' = V (t)w Xl (1,t) +fcn{w l {1,t},w xl (1,t) ,f(t) , g(t}} ,

where x=1 is the free end. To maximize dF/dt, and hence extract the greatest amount of energy from the system , they designed a controiler where V (t) = -sgn{w xl (1,t}}-V max

x = x / L,

allows Eq. (1) to be written as 0<

x<

1.

(5)

The boundary conditions to be considered here are (~,t) =

(11 )

w=w / L, (4)

w

(10)

w x (~,t) = 0

w (~,t) = 0 , w xx (~,t) = V (~ ,t ) + f(t)

Their experimental resu lts for the beam's first mode are shown in Fig. 1, where the controlled and uncontrolled bearn's peak tip displacement is plotted versus time . The first mode's less factor was increased by more than an order of magnitude for this test, demonstrating the effectiveness of PVF 2 as a distributed actuator. Limitations ef Spatially Uniform Control

(6a) (6b)

w xx (;,t) = V (S,t) + f(t) , w xxx (~,t) = V x (;,t) + g(t}

(6c)

w x (;,t) = 0 , w xxx (~,t) = Vx (;,t) + g(t}

(6d)

The resu lt of Bai ley and Hubbard ean now be extended via the use of generalized functions to more clearly understand the implications of Eq. (9). It will be assumed hereafter that f(t} = g(t) = 0; their inclusion would in no way affect the control laws discussed below. Equation (9) may be written as F' = V (x ,t}·w x I (x,t) ~,o - Vx (x,t)·w I (x,t) ~.o (1 2) + jo w ! (X,t}-Vxx {x,t} dx .

where; is the bou;,dary point x=O or x=1. These describe clamped , pinned, free, and sliding boundary conditions, respectively . The function f(t) may describe , for example, a torsional spring , damper, or rotary inertia, while g (t) can represent a linear spring, dashpot, or translational mass ; their particular form will not affect the ensuing control strategies. The second, or direct method of Lyapunov is used as the technique for derivi ng the sys tem's control constraint equations (Kalman and Bartram , 1960) . Choosing the Lyapunov functional (7)

for the beam is particularly advantageous, for it represents the total system energy. Vibration damping can then be implemented based on total energy considerations. This approach has the advantage over conventional methods of enabling one to deal directly with the distributed, continuous system model without resorting to approximations. As a result,

Beams with clamped ends, where Eq. {6 a} holds, will not display bour.dary control in terms of linear and or angular velocities, as Wx and w must vanish at the boundaries. We define here a spatially uniform controller in terms of step functions h(x-c) as V (x,t) = V max [ h (x) - h (x - 1) I p(t}

(13)

Vxx = Vmax [O'(x} - O'(x - 1) 1p{t}

(14)

for which

Th is distribution will on ly produce boundary control terms in Eq. (10) in terms of the linear and angular velocities at x=O and x=1. Thus, a clamped-clamped beam is not controlla ble using a spat ially uniform contro l. It can be shown using Eq. (10), however, that a beam clamped at one end with a pinned boundary condition at the other is. controllable with such a

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odd modes have even symmetry , one is led to choose a spatially non-uniform control distribution of the form

spatial distribution. In this case (10) takes the form (1 5)

A(x) = (1 - x)h(x) + (x - 1)·h(x - 1) where x=1 is the pin ned end. On the other hand, a clam ped-s liding beam configuration won't be controllable with the uniform distribution Eq. (11); this spatial control distribution only imposes point moments at the boundaries, and the sliding end, like the root , must have Wx = O. As a further example, consider a pinned-pinned beam with the uniform control distribution Eq. (11). Equation (10) becomes (16)

F' = 2 V maJ( p(t) [ w x t (1 ,t) - w x t (O,t) ]

The beam is controllable for motions such that the the beam's slopes at the ends are unequal. Physically, this corresponds to odd-order modes of the pinned-pinned beam. However, even-order modes have slopes at the boundaries that are equal, hence F' = 0 for these modes. Since the time derivative of the Lyapunov functionai corresponds here to (instantaneous) power flow, this implies no energy can be extracted from or added to the system for these vibrational modes. Parenthetically , one also notes that, for even-order modes and the stated uniform contro l distributio n, the integrand in Eq. (9) is the product of an odd and an even (about mid span) function: the integral must vanish over the beam's length for the pinned-pinned beam . Similarly, a free-free beam with uniform spatial control yields the same result as Eq. (16), in that even modes are not controllable using angular ve locity measurements at the boundaries. Thus, in spite of the utility and simplicity of the uniform spatial control for most boundary condition combinations, there exist configurations for which it is inappropriate. It is of use, then , to consider non-uniform spatial control distributions to determine whether problematic configurations can be controlled.

(20)

Th is distribution is shown in Fig. 3a. Th is control can be implemented in practice, for example, by varying the plating on the PVFc layer over the beam's length, or by varying the film's thickness (subject to the constraint, in the present analysis, that such a variation does not severely affect the homogeneity assumption).

Perlorming the integration in Eq. (19) gives

P = AV max p(t) '!l'(t) [ 1jI'(x) A(X) \ ,0 + fo 'V(x) A"(x) dx ]

(21)

Using the properties of generalized functions (Kaplan, 1962; Lighthill, 1958), one finds (22)

A"(x) = O'(x) - 8(x) + 8(x - 1) . Equation (21) can then be written as P = -2A V max p(t) '!l'(t)'1jI'(0) = -2V max p(t)·w x t (O,t) .

(23)

The product '!l'(t)'1jI'(0) is, of course, the angular velocity at the boundary x=O. In order to extract the most power from this system (for the given spatial control) the time portion of the control p(t) should be (24)

p(t) = sgn [w x t (0, t ) ]

yields an admissable solution form , since the governing equation (5) is linear. Following Komkov (19 72 ), write

The physical significance of the control described by Eq. (20) is most easily seen by examining the inhomogeneous governing Eq. (5). The forcing term in Eq. (5) due to the chosen distribution will involve point forces and a point moment, as shown in Eq. (22), and illustrated in Fig. 3b. Since translational motion is prohibited by the pinned boundary conditions, terms involving the delta functions will vanish in the time derivative of the Lyapunov functional , as seen in Eq. (12). Thus, the control appears to the system as a boundary moment controller ~ ~. Forcing the spatial distribution to zero "smoothly" (e.g. without a discontinuity in amplitude) at x=1 does not give rise to a point moment there, precluding the result for the uniform control distribution, Eq. (16). Discontinuities in amplitude give rise to point moments, while discontinuities in slope give rise to point forces. This is analagous to the interpretation of static loading distributions on beams in elementary structural mechanics texts. One might alternatively choose a somewhat more general distribution of the form

V (x,t) = VmaJ( A(x)·p(t) ,

A(x) = (1 - bx)·h(x) + (bx - 1)·h(x - 1)

APPLICATION OF SPATIALLY NON-UNIFORM CONTROL Case l ' Pinned-Pinned Beam The pinned-pinned beam will be discussed in detail as a su itable model problem for non-uniform spatial vibration control. Writing the beam's displacement w(x,t) as w (x,t) = A 1jI(x)''!l(t)

(17)

(18)

where it is required that IA (x) 1< 1, p(t) 1< 1, which imposes saturation limits on the available control. Then, Eq. (9) becomes P = AV maJ( p(t) '!l'(t) fo 1jI"(x) A(X) dx .

(19)

The integral in Eq . (19) could be maximized with a constant amplitude spatial control distribution that "switched" in space at zeroes of the beam curvature . Such a spatial control distribution and its graphical interpretation are sketched in Fig.s 2a and 2b. This "spatial bang-bang" controller, however, would require an inordinate amount of associated instrumentation to construct, and would necessitate segmenting the film over the beam's span at the zeroes of the beam's curvature for some number of modes n, thus imposing a bandwidth limit on the effectivenes of the controller (the scheme would require an infinite number of segments to control all modes) . However, noting that the pinned-pinned beam's even modes exhibit odd symmetry about x=t/2 ,and

(25)

°

< b < 2. This permits the recovery of the uniform where control distribution result, Eq. (16), by choosing b = 0. However, setting b = 2 gives

P = -2Vmaxp(t)[wxt(0,t)+wxt(1,t)].

(26)

In this case one can now control even-order modes, but not odd modes. This occurs because , for b = 2, the distribution of Eq. (25) has odd symmetry about x = '/2: A(x) changes sign there. Both even and odd modes are weighted equally when the modal weighting factor b equals 1 in Eq. (25). By choosing < b < 1, one can a modal weighting factor for the distribution weight the odd modes more than the even modes, with the converse true for 1 < b < 2. This might be exploited in the design of systems requiri ng the control of only certain modes. Additionally, one can easily show that this distribution is appropriate for the control of a beam with clamped-sliding boundary conditions.

°

S . E. Hurkcallcll E. HlIi>i>arcl .il.

Ilil Case 2' The Clamped-Clamoed Beam

distributions, and for other systems as well using identical arguments.

One might naturally ask whether the "ramped" spatial distribution will work for the clamped-clamped beam, which was shown to be unaffected by the uniform control. Since the boundary conditions for this system require vanishing slope and disp lacement at x:o and x:1, the time derivative of the functional (12) can be written as (27) The control distribution of Eq. (20), as previously described, will produce point forces and a point moment at the boundaries. Equation (27) can then be written as

which of course must vanish. Thus, in order to extract a non-vanishing quantity at the boundaries from Eq. (27) , the second derivative of the control distribution A(X) must have a ~:o

In order to Estimate the effectiveness of the control described in Eq. (17) for the pinned-pinned beam, an analysis of the its energy extraction compared to any inherent passive structural damping is now presented. This beam configuration was chosen as a model owing to its analytical simplicity . The r th mode of the pinned-pinned beam is assumed to have the form w (x,t) : A sin (nrrx) cos (n~ t ) .

(34)

(28)

p : - W x t (0 , t ) - W t (0, t ) + W t (1, t) ,

term (or terms) proportional to o"(x - ~), where

SPATIALLY NON-UNIFORM CONTROL AND ENERGY EXTRACTION

or 1;:1 . This

would require the spatial control distribution A(X) to contain delta functions (e.g. point actuation), which is not physically real izable w ith the PVF 2 film. As a result , alternative distributions must be considered. The physics of tne problem , combined with insights gained from the previous analyses, provides an appropriate solution. A clamped-clamped beam will have modes with either a vanishing displacement or a vanishing slope at the center, but never both . A distribution A(X) with a discontinuous amplitude at x: ' /2 (giving a point moment, and hence an angular velocity term) and a discontinuous slope there as well (giving a point force , resulting in a linear velocity term) of the form (29) will then give the result (30) The distribution in Eq. (29) and its effective loading are sketched in Fig.s 4a and 4b. As seen, this distribution places a point moment control (for even-order modes) and a point load (for odd-order modes) at mid span. To extract the most energy from the system, choose

The energy extraction can be augmented by making the amplitude and slope discontinuities more "severe" by choosing

This describes a single mode having some initial dispiacement and zero initial velocity. It is further assumed for the purpose of illustration that the modes remain uncoupled . (This is not rigourously correct.) The frequency n2rr2 is chosen so as to satisfy the beam's dispersion re lation. The beam's initial strain energy is

The energy the control , Eq. (18) , can extract over a half-period of oscillation (using the form of A(x) found in Eq. (20) and p(t) in Eq. (24)) is found by integrating Eq. (23) over a half period ; (36)

Eecnt,ol, TI2 : - 2n rr A Vmax .

If the passive internal damping is modeled as a viscous loss proportional to the beam's transverse velocity , over a half period of oscillation it will extract energy of the amount E.-iscous,TI2 : -"

n~ A2 1 4

,

(37)

where" is the associated modal loss factor. Thus, over a half period of oscillation, the controller will extract more energy from the beam than the viscous damper if 8 Vmax 1 "nrrA > 1 ,

(38)

Thus, the controller is most effective at low frequencies (where structural damping is generally insufficient to damp modes) and low amplitudes. This result has been observed experimentally (Plump, Hubbard, and Bailey, 1985). The two damping mechanisms, then, complement each other. Essentially, one can regard the result (38) as a bandwidth limitation on the controller's effectiveness versus the passive damping.

DESIGN GUIDELINES This distribution is depicted in Fig. 8. Substituting Eq. (32) into Eq. (27) gives

This control introduces the added complexity of requiring the spatial control distribution A (x) to change sign in space: damping capability may have to be traded for ease of implementation. These basic deSign considerations can readily be used to synthesize spatial control distributions that will weight the linear velocity measurement more than the angular velocity , and hence control odd-order modes more than even-order modes , by removing the amplitude discontinuity in A (x). The converse holds true for the even-order modes. The important result, however, is the realization that a control scheme for ihe clamped-clamped beam can be constructed using spatially discontinuous control

The results of the previous sections can be formalized into a set of design guidelines: 1) The PVF / film will only yield controls using the Lyapunov method in terms of linear and angular velocity feedback. This may affect the choice of sensors and sensor placement for a given beam configuration. 2) Uniform control distributions are effective for many beam configurations. This control might be choosen for the sake of simplicity , unless other considerations preclude it. 3) The PVF z film will be most effective at "Iow" frequencies and "Iow" amplitudes; "Iow" must of course be quantified in terms of the specific system configuration.

lli:j

Di,trihutl'd Parallll'tl'r COlltrol Dl'sigll 4) Discontinuities in amplitude will produce point moments , and will result in controls in terms of angular velocity at the point of discontinuity. The amplitude of these terms in the control law is proportional to the magnitude of the amplitude discontinuity. 5) Discontinuities in slope will produce point forces, and will result in controls in terms of linear velocity at the point of discontinuity. The amplitude of these terms in the control law is proportional to the magnitude of the slope discontinuity. 6) The location of the discontinuities can be varied so as to implement a control which will affect only certain modes. CONCLUSIONS For a number of boundary condition combinations a beam can be controlled effectively with a spatially uniform layer of PVF 2 · This technique has been demonstrated experimentally for cantilevered beams. However, certain configurations were shown herein to not be amenable to a spatially uniform controller. Also, it may be desirable to formulate a control solely in terms of certain states. As a result, non-uniform spatial control distributions were investigated, and potential implementations were studied for pinned-pinned and clamped-clamped beams. Through these analyses insights were gained into the design of spatially discontinuous controllers , and a set of design guidelines were formulated. The advantage of the control distribution discussed for the pinned-pinned and clamped-clamped beams is that they describe ~ spatial configurations of control input that can be implemented on real systems. Further, they provide insight about controllability and observability that appeal to one's intuition, rather than to mathematical obfuscation. The requisite states that need to be measured for implementing these controls are realizable. Additionally, at no time was it necessary to model the system in terms of its component modes, hence the controls won't suffer due to model truncation. Thus, the approach provides insight into a design methodology for applying film to (nearly) arbitrary beam configurations , for all modes. Using this, one can design systems to selectively control certain modes for specialized applications. At present, experiments to study the application of spatially non-uniform controllers are being designed.

Gibson, J.S. (1981). An Analysis of Optimal Model Regulation: Convergence and Stability. SIAM Journal of Control and Optjmazatjon • .la(5), 686-707. Graft, K.F. (1975). Wave Motion in Elastic Solids. Ohio State University Press, 142, 180-194. Haftka, R.T., Martinovic, Z.N., Hallauer,W.L., and Schamel, G. (1985). Sensitivity of Optimized Control Systems to Minor Structural Modifications·, Proceedings AIAA/ASMEI ASCE/AHS 26th Structures Structural Dynamics and Materials Conference. Kalmann, R.E., and Bartram , J.E . (1960). Control Systems Analysis and Design Via the 'Second' Method of Lyapunov. Journal of Basic Engineering Transactions of the ASME . 371-400. Kaplan, W. (1962) . Operational Methods for linear Systems Addison-Wesley, Reading MA, 44-63. Komkov, V. (1972). Optimal Control Theory for Damping pf Vibrations of Elastic Systems. Springer-Verlag, Berlin. lighthill, M.J. (1958). An Introductjon to Fourier Analysis and Generalized Functions . Cambridge University Press, Cambridge, England. Meirovitch, L., and Bauh, H. (1984). Nonlinear Control of an Experimental Beam by IMSC. AIAA preprint #83-0855. Plump, J., Hubbard, J.E., and Bailey, T. (1985). Nonlinear Control of a Distributed System: Simulation and Experirr.ental Results. presented at the 1985 ASME Winter Annual Meeting. Slemrod, M. (1974). An application of Maximal Dissipative Sets in Control Theory. Journal of Mathematical AnalysiS and Applicatipns ~(2), 369-387, Wang, P.K.C. (1966). Stability Analysis of Elastic and Aeroelastic Systems via Liapunov's Direct Method. Journal of the Frankljn Ins'jtute 2.8.1(1),51-72.

ACKNOWLEDGEMENT This work was supported by the Charles Stark Draper Labratory under contract #19480.

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REFERENCES Bailey, T. (1984 ). Distributed Parameter Active Vibration Ccnt~ol of a Cantilevered Beam Using a Distributed Parameter Actuator. MS/BS Thesis, Department of Mechanical Engineering, MIT.

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CONSTANT AMPLITUDE CONTROL: V m • x

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Bailey, T., and Hubbard, J.E. (1985) . Distributed Piezoelectric Polymer Active Vibration Control of a Cantilever Beam. Journal of Guidance a(5), 605-611.

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20

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40

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Balas, M.J. (1987). Active Control of Flexible Systems. ~ of Optimization Thecrv and App lications. 2..5.(3), 415-436. Balas , M.J. (1978). Feedback Control of Flexible Systems. IEEE Transact ions on Automatic Control ~(4), 673-679. Forward , R.L., and Swigert, C.J. (1981). Electronic Damping of Orthogonal Bending Modes in a Cylindrical Mast. AIAA papers 81-4017/4018, Journal of Spacecraft and Rockets

Fig. 1: First mode test results for cantilever: logarithmic plot of peak tip displacement decay

s. E. Burke dlld J L ]]lIbl w ·d .1r.

]tifi

1

r--

r--

r--

d 2 !\(X) d. 2

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x 0

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a(.- 1)

-

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Fig . 2a: Pericdic spatially discontinuous control distribution

....

Fig. 3b: EHective loading distribution for "ramped" control

i\(X)

°OL---------------Jl~--------------JL------ x

"2

Fig. 2b: Effective loading fo r distribution of Fig. 2a

Fig. 4a: Spatial control distribution for the clamped beam

a(x - 1)

i\( X )

Fig. 3a: Linearly-varying ("ramped") spatial distribution

o~---------------r---+------------L---+----'

Fig. 4b : Effective loading distribution for the clamped beam