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Active control of flexural vibrations
Journal of Sound and Vibration (1987) 114(2), 253-270
ACTIVE C O N T R O L OF FLEXURAL VIBRATIONS B. R. MACE Department of ltfechanical Engineering, The University of Auckland, Private Bag, Auckland, New Zealand (Received 12 March 1986, and in revised form 3 June 1986) The active control of disturbances propagating along a waveguide is considered with particular reference to the control of ftexural vibrations in thin beams. The control system consists of a number of point sensors and point exciters distributed in some manner along the waveguide. The sensor measurements are used as inputs to a controller which forms the required excitations such that incoming disturbances are completely cancelled. A method by which this ideal controller can be synthesized is presented. The stability of the system is considered. In general the system reflects the downstream cancelled disturbances while upstream disturbances are both reflected and transmitted. Active control of flexural waves in beams is then considered and two possible active control systems are examined. The first consists of measurements of displacement and rotation at a point and the application of a point force and point moment while the second comprises displacement measurements at two points and the application of two point forces. In both cases the ideal active controller is derived and some aspects of the systems' performances examined. 1. INTRODUCTION Active control of an unwanted disturbance consists of cancelling the disturbance by the deliberate addition of a second disturbance, equal in magnitude but opposite in sign. It has in the past found applications in the control of noise and vibrations. As far as vibrations are concerned single and multiple degree of freedom systems can be controlled by applying forces whose magnitudes and phases are determined by a controller. The inputs to the controller are displacements or velocities measured at various points in the system. The controller may be designed by using an eigenvalue assignment method [1-4] so that the natural frequencies and damping factors of the controlled system can be prescribed. Other control techniques such as optimal control [5-7], adaptive control [8] and the use of observers [9-11] may alternatively be used in the design of a suitable controller. Continuous systems on the other hand have an infinite number of degrees of freedom. They have in general been treated analytically by way of modal decomposition with the number of modes retained in the analytical model being truncated [12-20]. Essentially they are then treated as multiple degree of freedom systems with control being applied to a number of those modes retained. However the effects of the uncontrolled and unmodelled modes deteriorate the performance of the system [13-16]. Firstly observation spillover occurs in that the responses in these modes contribute to the measured displacements and velocities and hence to the control forces. Secondly these forces, intended to control specific modes, will however excite other modes unless the spatial distribution o f the control forces is orthogonal to the mode shapes of the uncontrolled modes. This is control spillover. Improved estimates of the response in each mode can be found by use of the so-called modal filter [19, 20] in which the responses at a number of points are weighted according 253 0022-460x/87/080253+ 18 S03.00/0 O 1987 Academic Press Inc. (London) Limited
254
n.R. MACE
to the mode shape. A limitation of modal control methods is that a great many modes may need to be considered especially if it is desired to control the vibrations of large structures or over a wide frequency range. As an alternative to the modal approach the response of the system can be described in terms of point and transfer mobilities. The authors of references [21-23] have adopted such an approach, with the vibrations of a continuous system being damped by applying forces proportional to velocity. An alternative approach to active vibration control of continuous systems is presented in this paper. Disturbances propagate through a continuous structure as waves. If some point on the structure (which acts as a waveguide) can be fixed then passage of waves past this point will not occur. The general scheme of the active controller is to measure incoming disturbances and from these measurements determine the excitations to be applied to the structure such that their effects identically cancel the incoming disturbances. If those regions where disturbances originate are surrounded by such controllers then they may be isolated from the remainder o f the structure. Similarly if a point on each transmission path leading to a disturbance-free region is fixed then that region will remain disturbance free. The characteristics of the incoming waves which are to be cancelled depend only on the properties of the structure at the point where the controller is attached. The structure need not therefore be uniform and only knowledge of its local properties is necessary. In the next section the characteristics of wave motion in thin beams are reviewed. Then the requirements o f a system for the active control of disturbances in a waveguide are presented. Although derived with flexural vibrations in mind, the formulation is general so that the results can easily be extended to include the effects of other modes of transmission such as longitudinal or torsional vibrations. Since the control scheme is similar to the approach adopted in noise control applications, where the wave-like nature of the problem is more evident, the relevance to some schemes of noise control in ducts is then briefly examined. Finally two possible systems for the active control of flexural vibrations are examined in more detail. The first o f these, in which measurements of displacement and rotation at a single point are used and a single point force and point moment are applied, has the merit of theoretical neatness while the second, in which only displacements and forces are used, would have a more practical application. The analysis is carried out in the frequency domain so that all variables are assumed to vary as exp (itot) and this dependence will in general be suppressed.
2. WAVE MOTION IN BEAMS Consider a thin beam lying along the x-axis. If it is assumed that the effects of shear deformation and rotary inertia can be neglected then the displacement w(x, t) satisfies the equation E1 04W/OX4"q- In O2w/Ol2----p(x, t),
(1)
where E1 and m are the flexural rigidity and mass per unit length of the beam and p(x, t) is the applied force per unit length (a list of nomenclature is given in the Appendix). This assumption imposes a frequency limit above which this simple model ceases to be valid. The corresponding shear force V and bending moment M as defined in Figure 1 are then
V = - E I 03w/ax 3,
~! = - E I 02w/Ox 2.
(2)
ACTIVE
CONTROL
OF
FLEXURAL
255
VIBRATIONS
W
T M(I
v __
_ x
Figure I. Definitionof positive shear force and bending moment. If one now assumes a time dependence of the form exp (itot), then in the absence of external forces (p =0) the solution of the homogeneous equation (1) can be written as the sum o f four flexural wave components.
w(x)=a+e-ik"~+a-e
+ -raNe
-ran
i/cx - -
-k.x --
ek.~:
(3)
where the amplitudes a may be complex and the time dependence exp (itot) has been suppressed. The wavenumber k is given by
k = 4n,f~-n-n~z~2/E1
(4)
and is real and positive, unless damping is present when it will have a negative (and usually small) imaginary part. The system is dispersive since k is not proportional to to. The four wave components in equation (3) represent respectively positive-going (a +) + and negative-going ( a - ) propagating waves and the two near field components aN and aTv which can be regarded as positive- and negative-going attenuating waves which decay exponentially. Their amplitudes decrease rapidly with distance, by a factor of over 500 in one wavelength, and can therefore be ignored at sufficiently large distances. However although the overall structure may be large compared to a wavelength the individual components of an active control system and adjacent boundaries and discontinuities in the structure may not be (especially if a compact system is desired) and so the full effects of the near fields must be included. In general therefore the displacement at any point consists of four components. It is convenient to follow the notation of reference [24] and group the wave amplitudes into 2 x 1 vectors of positive-going waves a + and negative-going waves a-: i.e., a+ =
[o;] a
,
a- =
(o} a~,
.
(5)
The beam displacement, slope, bending moment and shear force are given in terms of the wave amplitudes by iV
{(1/k)Ow/Ox}
r 1 1-1 + rl 1-1 =L-, -,J" +L, ,J='
[alEIk~]=II - 1 1 § I 1 - 1 1 _ { VlEIk3J L-, ,J: +L, -,J"
(6)
It is immediately apparent that in the absence o f applied forces the wave amplitudes along a uniform length o f beam are related by a+(xo+ x) =fa+(xo),
a-(xo) =f_a-(xo+x),
(7)
where the diagonal field transfer matrix or propagation matrix f is given by =
e_kx
9
(8)
256
B.R. MACE
At sufficiently large distances the near fields can be ignored and the propagation matrix reduces to
If waves are incident upon some discontinuity or boundary then the amplitudes of the reflected and transmitted waves are given by the product of reflection and transmission matrices and the vector o f incident wave amplitudes. These matrices can be determined by considering continuity and equilibrium at the discontinuity [24]. The beam may be excited by applied forces and moments which have the effect of injecting waves into the beam. If attention is restricted to point forces P and moments M , then at the point of application there are discontinuities in the shear force and bending moment in the beam with resulting discontinuities in the waves a and b on either side of the excitation point (see Figure 2). Thus 1~1 =
EI(O2w_/OX2-O2w+/Ox2),
P = El(O3w+/Ox3-O3w_/~x3),
(10)
from which it follows that b+-a * =qp+q^t,
a--b-
= qp-q~f,
(11)
{:1 _ 4 EM l k 2.
(12)
where the vectors of injected wave amplitudes are qP
4 E l k 3,
=
qM =
Usually the propagating wave components are the most important in determining the power flow into the beam. However, if adjacent discontinuities are close enough the near field terms may play an important role.
a
~
~- b~
____2"#...__ o-~
I ..4---.---- b-
Figure 2. Point excitations.
3. ACTIVE CONTROL IN A WAVEGUIDE In general the disturbances in the waveguide will be a set of n,,. waves travelling in both the positive and negative directions, the number of different wave types depending on the nature of the waveguide. The propagation matrix _f will be diagonal and its j t h element will be exp (-ikjx), /,) being the wavenumber appropriate to the jth wave type. The general scheme for active control is outlined in Figure 3. A set of waves a + propagates along the waveguide. A number of measurements m are taken by sensors distributed in some manner over the region from a to b and control excitations Q are applied by actuators in another region from c to d. These regions may overlap or be
ACTIVE
CONTROL
OF FLEXURAL
f
257
VIBRATIONS
Controller , ~ Q=Om
Measurementsm
ExcitationQ
t
1
L SensorsJ
t Exc,ers3
+ 0
,-
I
o
I
b
I
c
d
--d
Figure 3. Active control scheme. distinct. The controller has a transfer function _(3 and forms the required excitations Q = Gm such that they cancel the incident waves a + whose effects no longer propagate downstream. In practice the controller transfer function would only be an approximation to the ideal and there would therefore be a residual downstream propagation. Unless special measures are taken the excitations will also generate backward travelling waves a - which are effectively reflections of a +. Furthermore an upstream disturbance d- may not only propagate through the control region but unless special steps are taken will also contribute to m and hence give rise to reflected waves d +. Finally it should be noted that feedback will usually occur in that the waves created by the excitations may themselves contribute to the measurements. The individual components of the system will now be examined in more detail. 3.1.
SENSORS
AND
MEASUREMENTS
The sensor region contains n, sensors located at positions xi as indicated in Figure 4. The sensors are here assumed to give point measurements of any field variable such as displacement or slope or any of their time derivatives. If the wave amplitudes at the j t h sensor are aj ~ and b~ then the output sj of the sensor will be
sj=
[ajJaf+[fljJb;, j = 1 , 2 . . . n,, (13) IajJ and [fljJ depend on the type of sensor being used. For
where the 1 x n,. row vectors example [ajJ = [fljJ = I1 1 9 9 9 lJ K for sensors that measure displacement and IajJ = -[fljJ = [-ik~,-ik2,...,-ikn, IK for those that measure slope. Here K is the gain of the sensor and may contain factors of ico if the sensors measure time derivatives. The S!
S2
I'I !
53
J..... i
0
Figure 4. Sensor region.
Sn s
"'"
4
b-
258
B.R. r,IACE
vector of sensor outputs s can be formed from equation (13) by noting that a~ =_f(xj)a § and b7 = f ( x , , ' - x ~ ) b - . Thus s = _aa§ +_/3b - ,
(14)
where _a a n d / 3 are ns • nw sensor matrices w h o s e j t h rows are [ajJ_f(xj) and [/3,J_f(x,,, -x~) respectively. A set o f n,, measurements m are to be used as inputs to the controller. While in m a n y cases the sensor outputs can be used directly as the required measurements this need not be the case. In general the measurements will be some c o m b i n a t i o n o f t h e sensor outputs: m = A_ts= _Aa++ _Bb-.
(15)
Here _A and _B are n,,, • n~,. measurement matrices whose elements d e p e n d on the type o f sensor used, their spatial distribution and the m a n n e r in which their outputs are combined to form the measurements. If the sensor outputs are to be used directly then A_I will be the unit matrix. Note that the orders o f n , , n~ and n,,, need not be equal, but some restrictions apply. Firstly n, ~> n,.. if the system is to be observable and for full cancellation to be possible and n, ~<2n,. if the sensor outputs are to be linearly independent (not strictly necessary). Similarly nm ~> n,, for observability since if n,,, < n,,. there will be unobservable c o m p o n e n t s in a § which will then act as disturbance inputs to the system. Also if n,,, > nw then the measurements, which are o f couse estimates o f the wave amplitudes in a § will not be independent. Since n,,, independent measurements can be made n,,, = n,,. will be assumed henceforth. Advantages do however o c c u r if ns > n , since then A/ can be chosen such that the sensitivity o f the measurements to the disturbances b- is reduced. For example if n~ = 2n,.. then A_J can be chosen such that _M_B= B = 0 and hence the upstream disturbances b- will no longer have any effect on the system. The sensor part o f the system is shown in a block diagram in Figure 5. 3.2.
ACTUATORS
AND
EXCITATIONS
The exciter region contains np actuators located at positions x~ as shown in Figure 6. They are assumed to provide point inputs p, in any form such as forces or moments. The a+
~--m
b-
Figure 5. Block diagram of sensors.
Pl P2 C-~. i
T i-~
P3
P~r ~d
....................T x,,p 9
c
Figure 6. Exciter region.
rI d
§
ACTIVE CONTROL OF FLEXURAL VIBRATIONS
259
controller provides n~ excitations Q which will often be such point excitations but in general (as in the case of sensor outputs and measurements) a single excitation may give rise to a number of point inputs. The j t h point input pj will produce waves
d] = 8p,,
c~- = 3"pj,
(I 6)
in the waveguide where the n~. x 1 column vectors 8 and 3' depend on the type of exciter being used. For a force-like exciter 8 = 3' and 8 = -3" for a moment-like exciter. The total output from the exciters will thus be d+ = _zip,
c- = _Fp,
(17)
where the ith colunms o f the n,. x np matrices _A and F are {A}i
=f(x,,,-x~)8,
{I'}~ =_f(x~)3",
(18)
and give the contribution of the ith exciter to the total output. However the point inputs p will be some combination of the excitations Q: i.e., p = _PQ where _P will of course be the unit matrix if point excitations are to be used directly. Consequently d § =
_DQ,
_D = _zl_P,
c-
=
CQ,
_C =/_'_P,
(19)
where the elements of the n , x n, excitation matrices _C and D depend on the type of actuators used, their spatial distribution and the manner in which the excitations are combined to give the actuator inputs. The number o f excitations ne ~> n~. for the system to be controllable and full cancellation to be possible although only n,, independent excitations can be formed from the measurements. Consequently n, = n,. will be assumed henceforth. Advantages occur if np> n~, since _P can then be chosen such that the magnitudes of the unwanted negative going waves c- can be reduced. For example if n, = 2n,,. then _19 can be chosen such that _/'_P= _C = 0_ and hence no upstream components c- will be created. A block diagram of the exciters is shown in Figure 7.
I o
---J
11
p ~_d + Figure 7. Block diagram of exciters.
3.3. MEASUREMENT/EXCITATION INTERACTION
It is apparent that the exciter outputs may be detected by the sensors. This feedback can, amongst other things, cause the overall system to become unstable. If the sensor and exciter regions (see Figure 3) do not overlap then this interaction occurs due to the negative going waves c- produced by the exciters. These then contribute an amount m = _HQ,
/__/= _B_f,b_C,
(20)
to the measurement m. Clearly the feedback _H can be eliminated if either _B or _C is zero: i.e., if either the excitations do not produce or the measurements do not respond to negative going waves.
260
a.R. M^CE
I f t h e r e is overlap then the feedback matrix H is slightly more complex. The contribution to the kth sensor output sk from the j t h point input pj is sk = H*~jp~ where
m,:{
ro,
x,
[BkJ_f(xj-xD'~p,,
xk <~ xjJ"
(21)
Thus m = _HQ where H_ = M H * P .
(22)
Once again it is possible, providing there are sufficient sensors and exciters, to make /4 = 0 and hence eliminate feedback. 3.4. THE CONTROLLER Figure 8 shows a block d i a g r a m of the active control system. T h e controller _G (which m a y be an a n a l o g u e or digital device) forms the excitations Q from the m e a s u r e m e n t s m such that Q = _Gin.
(23)
The controller _(3 and the feedback element H together have a transfer function T = [1 - Q_H ]-t _G,
(24)
where _/is the n, x n, unit matrix. The u p s t r e a m waves d- act as an input disturbance to the system and a second set o f upstream waves a - may be credited as a result o f the applied excitation.
~ , d§
0 §
Figure 8. Block diagram of active control system. If the disturbances d- are ignored then the d o w n s t r e a m waves d + consist of two p a n s . The first, _faaa +, is due to the input waves, _fad being the p r o p a g a t i o n matrix from a to d, and the second part o f d § will be due to the excitations. Thus d + =f,,da ++ D T A a + = [fan + O [ ! - Q_H]-I _G_A]a§ In the ideal case, when full cancellation occurs, d §
(25)
0 and hence
T = -_D-tf_,,a_A -'.
(26)
Hence the controller required to achieve full cancellation is _(3 = [_H + T - ' ] - ' = [_H - _A_ff~a~_D] - '
(27)
and the control excitations are O = - _D-'_f,,a a +.
(28)
ACTIVE CONTROL OF FLEXURAL VIBRATIONS
261
Thus given that the individual components in the system are known equation (27) gives a method by which an ideal controller _G can be synthesized such that complete control can be achieved. The controller _G effectively has two elements, _G' and _H' as indicated in Figure 9 where _H'-~ _H is intended to subtract the feedback in the physical system and _G'=--_D-I_f,,a_A-I provides the required transfer function T. These two elements have individually much .simpler frequency responses than _G alone In practice the controller will not be ideal and the residual downstream propagation is given by equation (25). The stability o f the system can be determined from the Nyquist plots of the eigenvalues of (-O_H) which must all satisfy the Nyquist criterion if the system is to be stable. If the system is unstable or if the relative stability is insufficient then it will be necessary to include filters in the controller to remove the undesired frequency components. For this reason and because of difficulties in accurately synthesizing _G the controller transfer function will differ from the ideal given by equation (27). The residual transmission through tile system will then be d + = [f_,,a + _D[I._ - _G_H]-' _G_A]a+.
(29)
From this the actual attenuation achieved can be found. Although the control forces cancel the downstream transmission of the incoming waves a § they create additional reflected waves
a- = _paa*,
(30)
where the reflection matrix at point a is
e~ =f_,~C_T_A.
(3 i)
If the ideal controller (equation (27)) is used then
P_a =
-f,,_C_D-' f_,,a.
(32)
These reflections are o f course zero if the excitations are arranged such that C = 0. The reflections may create standing waves or resonances upstream of the controller. Furthermore since the control system is active in that it may add energy to the waveguide it is possible (if_p= has eigenvalues whose magnitudes are greater than one) for forced unstable oscillations to occur in the section of the waveguide upstream of the controller. In a similar manner an upstream disturbance d- will act as an input to the system and cause a spurious reflection d + = _Pad-,
(33)
where the reflection matrix at point d is
e a = _DT_B_fab.
(34)
e a = -f_,,a-A - t _B_fab.
(35)
In the ideal case
I'n
H'
["
Figure 9. Equivalent form of controller _G.
262
n.R. MACE
These reflections can be made zero if the sensors are arranged such that B = 0 but if they exist then they may produce effects similar to those caused by the reflected waves a-. The total transmission through the system will be the sum of the direct component'_fd~dand those waves additionally created by the excitations, giving a - = Ydad-,
(36)
where the transmission matrix _ra~ is
"rda = [fcaC- T_Flfdb+ fda] = [--P~a,~Pd + fda]
(37, 38)
in the ideal case. 4. ACTIVE CONTROL OF DUCT NOISE The control of duct noise by active methods has received much attention in the past. In this section some of the approaches adopted will be briefly described in the context of the general method of active control in a waveguide presented in the previous section. If it can be assumed that the higher order duct modes are negligible then only a single wave a § or a- can propagate in each direction in the duct. Thus n~,=ne = n,, = 1. Furthermore the air is non-dispersive so that the wavenumber k = to/c is directly proportional to frequency. The propagation matrix f ( x ) then reduces to e x p ( - i k x ) = exp ( - i t o x / c ) and represents a time delay of x / c seconds. A number of studies have been made of the problem of reducing the effect of feedback which acts to destabilize the system and make the required controller frequency response non-uniform. With no feedback loop only a time delay in the controller is necessary. Swinbanks [25] analyzed the case of two loudspeakers which are phased such that the net output is a downstream travelling wave. Hence C = _0and there are two point actuators (np = 2). Such an array responds uniformly over a fairly narrow bandwidth and so the frequency range of operation is restricted unless a more sophisticated controller is used. He also considered how the bandwidth could be improved by using three loudspeakers (np = 3 ) and how the outputs of three microphones (ns--3) could be phased so that they detect only the downstream propagating noise (i.e., B = _0). In the Jessel and Mangiante arrangement [26] a three output monopole plus dipole combination was discussed (np = 3). The Chelsea dipole [27] (np = 2, n5 = 1) consists of a microphone located centrally between two loudspeakers. In this system the sensor and exciter regions overlap and the loudspeaker outputs are phased to give zero net output at the microphone. Thus feedback is eliminated by combining loudspeaker outputs in a particular way such that _H = _0. The performance of a general distribution of acoustic sources was analyzed in reference [28]. In the monopole system, comprising one microphone and one loudspeaker (n, = 1, np = 1), no attempt is made to break the feedback path, resulting in the need for a slightly more sophisticated controller which compensates for this feedback [29]. In addition if the positions of the microphone and loudspeaker are made to coincide then _f~a = I. For A = D = I the required controller characteristics become sign inversion and infinite gain. This is the so-called tightly coupled monopole [30]. The maximum gain realizable is limited by spurious measurement noise and instability gives an upper frequency limit. Digital control methods can give much improved performance since the construction of digital filters with quite complex frequency responses is straightforward. Thus in most recent practical implementations digital techniques are preferred. Two examples of reduction of feedback effects have been provided by Ross [31] (n5 =3, np= l) and La Fontaine and Shepherd [32] (n~ = 2, np = 2). The filters in these cases were constructed by using least squares system identification methods. Adaptive control methods [33, 34], which have been applied to monopole systems, would seem to hold great promise.
ACTIVE C O N T R O L OF FLEX U R A L VIBRATIONS
263
5. ACTIVE CONTROL OF FLEXURAL VIBRATION Flexural vibration in a thin beam comprises two wave components as described in section 2 and thus n,, = 2. While in theory the sensors may measure either displacement or rotation or their respective velocities or accelerations, in practice rotational measurements are not easy to obtain. Similarly actuators applying forces are easier to realize than those applying moments. In this section two possible control schemes are discussed. It should perhaps be noted that in any control scheme the final point of application of a control force will have no displacement or rotation in order that no energy can pass this point. It will therefore appear fully clamped as far as the incident waves are concerned. 5.1. SINGLE POINT MEASUREMENT AND EXCITATION
Consider the situation where two sensors are located at a point. One measures displacement, the other rotation. A control point force P and point moment M are applied at a second point a distance l away as shown in Figure 10. The sensor outputs s and point inputs p are then
s=[ow/Oxl,
P= M "
(39)
It is convenient to define the measurements m and excitations Q to be
w m={(l/k)aw/Ox} ,
~ P/4EIk3~ Q=LM/4EIk2 J.
(40)
From equations (15), (12) and (19) the measurement matrices _Aand _Band the excitation matrices _C and _D are _A=[1 i
_11],
_/3=[I
:],
_C=[_-i1
-:],
_D=[-i 1
_11]
(41)
while
,. ,b , _
_
0]
_
e -~'
,
(42)
where ,u = kl, k being the wavenumber, will have a small negative imaginary part if damping is present. The transfer function of the ideal controller (equation (27)) is 1 ~ sinh # +sin ~ -G=4(cos# cosh/.t-1) L-(cosh#-cos/x)
cosh/.t-cos ~] sin/.t-sinhp. J"
(43)
Note that for real p. _G is real. This is to be expected because incoming waves a + are reflected since the excitation point is effectively clamped. Indeed it is straightforward to show that the reflection matrix (equation (32)) is given by e,, = _f-ref,_
(44)
awlax i~,. a,b
M
L c,d
Figure 10. Single point active control system.
LI.R.MACE
264 where
-i
-(l+i)]
-re = - ( l - i )
(45)
i
is the reflection matrix for a clamped end [24]. The reflections therefore produce standing waves and the applied forces do no work since they are in phase with the displacements. The magnitudes o f the elements of _G with/1 real are shown in Figure 11 where they are plotted against dimensionless frequency/'2 = # ~ = oJ,,/-mi-4/El. _G exhibits resonances at those values of/.t for which cos/.t cosh ~ - 1= 0 (i.e., /z =0, 4-730, 7 . 8 5 3 . . . when .(2 =0, 22-37, 6 1 . 6 7 . . . ). These correspond to the natural frequencies of a clampedclamped beam of length L Since the beam is effectively clamped at the excitation point then at these frequencies a component of the vibration cannot be detected by the sensors.
1
i
i
I
I
i
U :II :II -II
"2;~05!..
0
f
t I I 20 40 Dimensionless frequency .1~
I
60
Figure !1. Single point controller transfer function _G~vs. dimensionless frequency n ; pc real.
- - -, It,I,
IG~,I; . . . . , I c , I.
. Ic,,I;
In practice the controller would tend to ring and it would be desirable to filter these frequency components out o f t h e measured signals. In any event small differences between the actual and ideal controllers may lead to poor performance and even amplification at these frequencies. The feedback matrix is
/-/= [1 i -i] e-"9 + [_-i1 i]e-~
(46)
and varies smoothly with frequency. The stability of the controller can be determined from the Nyquist plots of the eigenvalues o f ( - G _H). These are given in Figure 12 where a small amount of damping has been assumed (p. = ,u0(l -i0.01)). It can be seen that the system is stable. One eigenvalue, corresponding loosely to the near field, decays rapidly towards the origin while the other, corresponding to the propagating component, clearly shows the resonances inherent in the controller _G. In addition to cancelling the incident waves a + the active controller will both reflect and transmit disturbances d-. For the scheme under consideration the reflection matrix
265
ACTIVE CONTROL OF FLEXURAL VIBRATIONS i
I
I/
'
//
'
~-'----
"
....
l //[(..//
~
/
E
"i""%'.."'%. ~ /
/
/
/
"%,,~ I
I
I
I
I
O
-2
Real part Figure 12. Nyquist plot of the two eigenvalues of the single point controller,/~ =/ao(l -i0-01).
ea (equations
(33) and (34)) is given by _Pa = f-ref,
(47)
where &, the reflection matrix for a clamped end, is given by equation (44). As far as the disturbances are concerned the beam therefore appears to be clamped at the sensor location. The transmission matrix (equation (36)) is given by
za~ =[[! -rcf_rc_fl.
(48)
r,, is shown in Figure 13. There are frequency regions where the direct component _fd,, d- and the component induced by the applied forces interfere constructively and destructively, the maximum transmitted amplitude being twice that of the incident. Due to the exp (-/z) terms the other components of _r decay with increasing frequency and are of less significance. 5.2. T w o POINT M E A S U R E M E N T A N D EXCITATION
As a second example consider the case where the displacements are measured at two points and vibrations are to be controlled by two point forces. For simplicity the separation of both measurement and excitation points is I and the second measurement point B coincides with the first excitation point C as shown in Figure 14. The measurements and excitations are taken to be m=
I Pc/4Elk'Sl Q=[Po/4EIk3J
tIwAi'wJo
(49)
and the measurement and excitation matrices, which are now frequency dependent, become e -i"
e-"
'
B=
1
'
-
e-"J'
ke-"
I
(50)
266
B. R. MACE I
I
I
I
I
2
I
0
T
f
I
P
20 40 O,mensionlessfrequer~y,f/
T
60
Figure 13. Transmission coefficient rjt for single point controller vs. dimensionless frequency//; # real. wz
wB , Pc
Po
c ~__.~I_.._a,.c~~
!
O
b, c
d
Figure 14. Two point active control system.
For full cancellation to occur the controller should have the transfer function 1 [10 - ( s i n h 2 ~ - s i n 2 ~ ) -G - 2(sinh/.L _ sin/_t ) (s'(-~nh ~ ' p . ~
] J,
(51)
where/.t = kl again. Once more _G, shown in Figure 15, is real and the incident waves are reflected. Since Q = _Gm the second line of equation (51) indicates that the second force Po depends only on the second measured displacement wt~. This is because the point D is effectively clamped. Since Po can only produce cancelling waves proportional to [-i - l J T (equation (12)) then the amplitudes ofthose waves incident upon D must have amplitudes in the proportion of [-i - l J T. Hence the function of Pc is to shape the incoming waves in such a fashion that, after they have propagated over a distance l, the amplitudes of the two components have this desired ratio. Indeed it is straightforward to show that the wave amplitudes c *+ as defined in Figure 14 are 1 [-e-(i'-m' } ( c + - i c ~ ) c *+ - ! _ e_(l_i)~,
(52)
and thus d *§ =_fc*+ - 1 _ e_(,_i) .
{-:}
(c § - ic~).
(53)
ACTIVE
CONTROL
OF
FLEXURAL
267
VIBRATIONS
lo)
0
I 2
0
I 4
l 6
8
10
I 8
10
Dimen,slonless frequency ,12
I
I
I
(b)
-5
tO
-15
-20
I
0
2
T
4
I
6
Dimensionless frequency 12
Figure 15. Two point controller transfer function _Gvs. dimensionless frequency .O;/,t real. (a) GII = G22; (b) Gt2.
Note that these amplitudes depend only on ( c + - i c ~ ) . The displacement measured at B gives (c § - ic~,) and that measured at A is simply used in conjunction with this to determine the excitation required to correctly shape the incoming wave amplitudes. As in the previous example this system is stable. Upstream disturbances d- are reflected and transmitted, the transmission matrix _z exhibiting frequency ranges of constructive and destructive interference. 6. CONCLUDING REMARKS In the preceding sections active control in a waveguide was discussed. The special case of control of flexural vibrations was considered and two possible control schemes outlined. Control is possible if a suitable two-input two-output controller _G can be designed. In this approach to active vibration control the overall size of the structure being controlled
268
a.R. MACE
is irrelevant. It may thus be very large and hence a very large number of modes may be affected. The control system itself may however be compact. In a practical implementation a controller would be designed to operate over a certain frequency range and a rational approximation to the irrational transfer functions required can be made. It would also be desirable to select the size of the system so that controller resonances lie outside the frequency range of interest. It may also be necessary to consider the sensor and exciter dynamics. The examples considered consisted o f the minimum number of sensors and exciters necessary to fully cancel the vibrations. It is to be expected that more complex systems, containing additional sensors a n d / o r exciters, would enable improved practical performance to be obtained, for example by removing sensor/exciter interaction.
ACKNOWLEDGMENTS The author gratefully acknowledges the financial assistance provided by the University of Auckland Research Fund and the Research Committee of the U.G.C.
REFERENCES 1. B. PORTER and A. BRADStlAW 1972 Journal of Mechanical Engineering Science 14, 307-311. Synthesis of active controllers for vibratory systems. 2. B. PORTER and A. BRADSHAW 1974 Journal of Mechanical Enghzeering Science 16, 95-100. Synthesis of active dynamic controllers for vibratory systems. 3. B. PORTER and T. R. CROSSLEY 1972 Modal Control. London: Taylor & Francis. 4. C. R. MARTIN and T. T. SOONG 1976 Proceedings of the American Society of Civil Engineers, Journal of the Engineering Mechanics Division 613-623. Modal control of multistory structures. 5. H. KWAKERNAAKand R. SIVAN 1972 Linear Optimal Control Systems. New York: WileyInterscience. 6. M. ABDEL-ROHMAN and H. H. E. LEIPHOLZ 1979 Proceedings of the American Society of Civil Engineers, Journal of the Engineering Afechanics Division 105, 1007-1023. A general approach to active structural control. 7. M. ABDEL-ROHMAN and H. H. E. LEIPitOLZ 1980 Proceedings of the American Society of Civil Engineers, Journal of the Structural Division 106, 663-677. Automatic active control of structures. 8. K. J. ASTROM and B. WI'I'q'ENMARK19 Computer Controlled Systems. Englewood Cliffs, NJ: Prentice Hall. 9. O. G. LLJENBERGER 1971 Institute of Electrical and Electronic Engineers Transactions on Automatic Control 16, 596-602. An introduction to observers. 10. J. S. BURDESS and A. V. METCALFE 1983 Journal of Sound and Vibration 91,447-459. Active control of forced harmonic vibration in finite degree of freedom structures with negligible natural damping. 11. J. S. BURDESS and A. V. METCALFE 1985 Transactions of the American Society of Mechanical Engineers, Journal of Vibration, Acoustics, Stress and Reliability in Design 107, 33-37. The active control of forced vibration produced by arbitrary disturbances. 12. J. ROORDA 1975 Proceedings of the American Society of Civil Engineers, Journal of the Structural Division 3, 505-521. Tendon control in tall structures. 13. M. J. BALAS 1978 Journal of Optimisation Theory and Applications 25, 415-435. Active control of flexible systems. 14. M. J. BALAS 1978 Institute of Electrical and Electronic Engineers Transactions on Automatic Control 23, 673-679. Feedback control of flexible systems. 15. L. MEIROVITCH and tt. BARUH 1981 Jounlal of Optimisation Theory and Applications 35, 31-44. Effect of damping on observation spillover instability. 16. L. MEIROVITCtl and H. BARUtl 1983 Journal of Optimisation Theory and Applications 39, 269-291. On the problem of observation spillover in self-adjoint distributed parameter systems. 17. E. LUZZATO and M. JEAN 1983 Journal of Sound and Vibration 86, 455-473. Mechanical analysis of active vibration damping in continuous structures.
ACTIVE CONTROL OF FLEXURAL VIBRATIONS
269
18. E. LUZZATO 1983 Journal of Sound and Vibration 91, 161-183. Active protection of domains of a vibrating structure by using optimal control theory: a model determination. 19. L. MEIROVlTCIt and H. BARUII 1982 Journal of Guidance 5, 60-66. Control of self-adjoint distributed parameter systems. 20. L. MEIROVITCll and L. M. SILVERBERG 1984 Journal of Sound and Vibration 97, 489-498. Active vibration suppression of a cantilever wing. 21. T. tt. ROCKWELL and J. M. LAW'triER 1964 Journal of the Acoustical Society of America 36, 1507-1515. Theoretical and experimental results on active vibration dampers. 22. L. A. WALKER and P. P. YANESKE 1976 Journal of Sound and Vibration 46, 157-176. Characteristics of an active feedback system for the control of plate vibrations. 23. L. A. WALKER and P. P. YANESKE 1976 Journal of Sound and Vibration 46, 177-193. The damping of plate vibrations by means of multiple active control systems. 24. B. R. MACE 1984 Journal of Sound and Vibration 97, 237-246. Wave reflection and transmission in beams. 25. M. A. SWINBANKS 1973 Journal of Sound and Vibration 27, 411-436. The active control of sound propagation in long ducts. 26. M. J. M. JESSEL and G. A. MANGIANTE 1972 Journal of Sound and Vibration 23, 383-390. Active sound absorbers in an air duct. 27. Kh. EGHTESADI and H. G. LEVENTHALL 1981 Journal of Sound and Vibration 75, 127-134. Active attenuation of noise: the Chelsea dipole. 28. Kh. EGIITESADI and H. G. LEVENTtlALL 1983 Journal of Sound and Vibration 91, 11-19. A study of n-souce active attenuator arrays for noise in ducts. 29. Kh. EGilTESADI and H. G. LEVENTIIALL 1982 Journal of the Acoustical Society of America 71,608-611. Active attenuation of noise--the monopole system. 30. Kh. EGHTESAD1, W. K. W. I-lONG and H. G. LEVENTI-tALL 1983 Noise Control Engineering Journal 20, 16-20. The tight-coupled monopole active attenuator in a duct. 31. C. F. ROSS 1982 Journal of Sound and Vibration 80, 373-380. An algorithm for designing a broadband active sound control system. 32. R. F. LA FONTAINE and I. C. SHEPHERD 1983 Journal of Sound and Vibration 91,351-362. An experimental study of a broadband active attenuator for cancellation of random noise in ducts. 33. C. F. R o s s 1982 Journal of Sound and Vibration 80, 381-388. An adaptive digital filter for broadband active sound control. 34. A. ROURE 1985 Journal of Sound and Vibration 101,429-441. Self-adaptive broadband active sound control system. APPENDIX: NOMENCLATURE a, b, c, d abcd
A,
_C,_D
_f G H
k m
AI A_I nr nm np ns nw
P
P P
q.~l, qP Q
boundaries of active control region, Figure 3 vectors of wave amplitudes at corresponding points measurement matrices, equation (15) excitation matrices, equation (19) wave propagation matrix, equation (8) controller transfer function, Q = Gm measurement--excitation feedback matrix, equation (20) wavenumber vector of measurements applied point moment, Figure 2 bending moment in beam, Figure 1 sensor output combination matrix, m = AJs number of control excitations applied number of measurements taken number of point actuators number of point sensors number of wave motions in waveguide vector of actuator point inputs applied point force, Figure 2 control excitation combination matrix, p = P Q vectors of waves injected in beam by a point moment and a point force, equation (12) vector of control excitations
270 $ T
v IV
!,_a /z
e
!
0..I
D
13. R. M A C E
vector of sensor outputs controller plus feedback transfer function, equation (24) shear force in beam, Figure 1 beam displacement sensor matrices, equations (13) and (14) actuator matrices, equations (17) and (18) dimensionless wavenumber matrix of reflection coefficients matrix of transmission coefficients frequency dimensionless frequency, .O = ,u 2 = t o nf~d4/EI ,
Subscripts a,b,c,d pertaining to points a, b, c, d i,j pertaining to element i,j near field wave component N Superscripts + -
propagating in the positive direction propagating in the negative direction
ACTIVE C O N T R O L OF FLEXURAL VIBRATIONS B. R. MACE Department of ltfechanical Engineering, The University of Auckland, Private Bag, Auckland, New Zealand (Received 12 March 1986, and in revised form 3 June 1986) The active control of disturbances propagating along a waveguide is considered with particular reference to the control of ftexural vibrations in thin beams. The control system consists of a number of point sensors and point exciters distributed in some manner along the waveguide. The sensor measurements are used as inputs to a controller which forms the required excitations such that incoming disturbances are completely cancelled. A method by which this ideal controller can be synthesized is presented. The stability of the system is considered. In general the system reflects the downstream cancelled disturbances while upstream disturbances are both reflected and transmitted. Active control of flexural waves in beams is then considered and two possible active control systems are examined. The first consists of measurements of displacement and rotation at a point and the application of a point force and point moment while the second comprises displacement measurements at two points and the application of two point forces. In both cases the ideal active controller is derived and some aspects of the systems' performances examined. 1. INTRODUCTION Active control of an unwanted disturbance consists of cancelling the disturbance by the deliberate addition of a second disturbance, equal in magnitude but opposite in sign. It has in the past found applications in the control of noise and vibrations. As far as vibrations are concerned single and multiple degree of freedom systems can be controlled by applying forces whose magnitudes and phases are determined by a controller. The inputs to the controller are displacements or velocities measured at various points in the system. The controller may be designed by using an eigenvalue assignment method [1-4] so that the natural frequencies and damping factors of the controlled system can be prescribed. Other control techniques such as optimal control [5-7], adaptive control [8] and the use of observers [9-11] may alternatively be used in the design of a suitable controller. Continuous systems on the other hand have an infinite number of degrees of freedom. They have in general been treated analytically by way of modal decomposition with the number of modes retained in the analytical model being truncated [12-20]. Essentially they are then treated as multiple degree of freedom systems with control being applied to a number of those modes retained. However the effects of the uncontrolled and unmodelled modes deteriorate the performance of the system [13-16]. Firstly observation spillover occurs in that the responses in these modes contribute to the measured displacements and velocities and hence to the control forces. Secondly these forces, intended to control specific modes, will however excite other modes unless the spatial distribution o f the control forces is orthogonal to the mode shapes of the uncontrolled modes. This is control spillover. Improved estimates of the response in each mode can be found by use of the so-called modal filter [19, 20] in which the responses at a number of points are weighted according 253 0022-460x/87/080253+ 18 S03.00/0 O 1987 Academic Press Inc. (London) Limited
254
n.R. MACE
to the mode shape. A limitation of modal control methods is that a great many modes may need to be considered especially if it is desired to control the vibrations of large structures or over a wide frequency range. As an alternative to the modal approach the response of the system can be described in terms of point and transfer mobilities. The authors of references [21-23] have adopted such an approach, with the vibrations of a continuous system being damped by applying forces proportional to velocity. An alternative approach to active vibration control of continuous systems is presented in this paper. Disturbances propagate through a continuous structure as waves. If some point on the structure (which acts as a waveguide) can be fixed then passage of waves past this point will not occur. The general scheme of the active controller is to measure incoming disturbances and from these measurements determine the excitations to be applied to the structure such that their effects identically cancel the incoming disturbances. If those regions where disturbances originate are surrounded by such controllers then they may be isolated from the remainder o f the structure. Similarly if a point on each transmission path leading to a disturbance-free region is fixed then that region will remain disturbance free. The characteristics of the incoming waves which are to be cancelled depend only on the properties of the structure at the point where the controller is attached. The structure need not therefore be uniform and only knowledge of its local properties is necessary. In the next section the characteristics of wave motion in thin beams are reviewed. Then the requirements o f a system for the active control of disturbances in a waveguide are presented. Although derived with flexural vibrations in mind, the formulation is general so that the results can easily be extended to include the effects of other modes of transmission such as longitudinal or torsional vibrations. Since the control scheme is similar to the approach adopted in noise control applications, where the wave-like nature of the problem is more evident, the relevance to some schemes of noise control in ducts is then briefly examined. Finally two possible systems for the active control of flexural vibrations are examined in more detail. The first o f these, in which measurements of displacement and rotation at a single point are used and a single point force and point moment are applied, has the merit of theoretical neatness while the second, in which only displacements and forces are used, would have a more practical application. The analysis is carried out in the frequency domain so that all variables are assumed to vary as exp (itot) and this dependence will in general be suppressed.
2. WAVE MOTION IN BEAMS Consider a thin beam lying along the x-axis. If it is assumed that the effects of shear deformation and rotary inertia can be neglected then the displacement w(x, t) satisfies the equation E1 04W/OX4"q- In O2w/Ol2----p(x, t),
(1)
where E1 and m are the flexural rigidity and mass per unit length of the beam and p(x, t) is the applied force per unit length (a list of nomenclature is given in the Appendix). This assumption imposes a frequency limit above which this simple model ceases to be valid. The corresponding shear force V and bending moment M as defined in Figure 1 are then
V = - E I 03w/ax 3,
~! = - E I 02w/Ox 2.
(2)
ACTIVE
CONTROL
OF
FLEXURAL
255
VIBRATIONS
W
T M(I
v __
_ x
Figure I. Definitionof positive shear force and bending moment. If one now assumes a time dependence of the form exp (itot), then in the absence of external forces (p =0) the solution of the homogeneous equation (1) can be written as the sum o f four flexural wave components.
w(x)=a+e-ik"~+a-e
+ -raNe
-ran
i/cx - -
-k.x --
ek.~:
(3)
where the amplitudes a may be complex and the time dependence exp (itot) has been suppressed. The wavenumber k is given by
k = 4n,f~-n-n~z~2/E1
(4)
and is real and positive, unless damping is present when it will have a negative (and usually small) imaginary part. The system is dispersive since k is not proportional to to. The four wave components in equation (3) represent respectively positive-going (a +) + and negative-going ( a - ) propagating waves and the two near field components aN and aTv which can be regarded as positive- and negative-going attenuating waves which decay exponentially. Their amplitudes decrease rapidly with distance, by a factor of over 500 in one wavelength, and can therefore be ignored at sufficiently large distances. However although the overall structure may be large compared to a wavelength the individual components of an active control system and adjacent boundaries and discontinuities in the structure may not be (especially if a compact system is desired) and so the full effects of the near fields must be included. In general therefore the displacement at any point consists of four components. It is convenient to follow the notation of reference [24] and group the wave amplitudes into 2 x 1 vectors of positive-going waves a + and negative-going waves a-: i.e., a+ =
[o;] a
,
a- =
(o} a~,
.
(5)
The beam displacement, slope, bending moment and shear force are given in terms of the wave amplitudes by iV
{(1/k)Ow/Ox}
r 1 1-1 + rl 1-1 =L-, -,J" +L, ,J='
[alEIk~]=II - 1 1 § I 1 - 1 1 _ { VlEIk3J L-, ,J: +L, -,J"
(6)
It is immediately apparent that in the absence o f applied forces the wave amplitudes along a uniform length o f beam are related by a+(xo+ x) =fa+(xo),
a-(xo) =f_a-(xo+x),
(7)
where the diagonal field transfer matrix or propagation matrix f is given by =
e_kx
9
(8)
256
B.R. MACE
At sufficiently large distances the near fields can be ignored and the propagation matrix reduces to
If waves are incident upon some discontinuity or boundary then the amplitudes of the reflected and transmitted waves are given by the product of reflection and transmission matrices and the vector o f incident wave amplitudes. These matrices can be determined by considering continuity and equilibrium at the discontinuity [24]. The beam may be excited by applied forces and moments which have the effect of injecting waves into the beam. If attention is restricted to point forces P and moments M , then at the point of application there are discontinuities in the shear force and bending moment in the beam with resulting discontinuities in the waves a and b on either side of the excitation point (see Figure 2). Thus 1~1 =
EI(O2w_/OX2-O2w+/Ox2),
P = El(O3w+/Ox3-O3w_/~x3),
(10)
from which it follows that b+-a * =qp+q^t,
a--b-
= qp-q~f,
(11)
{:1 _ 4 EM l k 2.
(12)
where the vectors of injected wave amplitudes are qP
4 E l k 3,
=
qM =
Usually the propagating wave components are the most important in determining the power flow into the beam. However, if adjacent discontinuities are close enough the near field terms may play an important role.
a
~
~- b~
____2"#...__ o-~
I ..4---.---- b-
Figure 2. Point excitations.
3. ACTIVE CONTROL IN A WAVEGUIDE In general the disturbances in the waveguide will be a set of n,,. waves travelling in both the positive and negative directions, the number of different wave types depending on the nature of the waveguide. The propagation matrix _f will be diagonal and its j t h element will be exp (-ikjx), /,) being the wavenumber appropriate to the jth wave type. The general scheme for active control is outlined in Figure 3. A set of waves a + propagates along the waveguide. A number of measurements m are taken by sensors distributed in some manner over the region from a to b and control excitations Q are applied by actuators in another region from c to d. These regions may overlap or be
ACTIVE
CONTROL
OF FLEXURAL
f
257
VIBRATIONS
Controller , ~ Q=Om
Measurementsm
ExcitationQ
t
1
L SensorsJ
t Exc,ers3
+ 0
,-
I
o
I
b
I
c
d
--d
Figure 3. Active control scheme. distinct. The controller has a transfer function _(3 and forms the required excitations Q = Gm such that they cancel the incident waves a + whose effects no longer propagate downstream. In practice the controller transfer function would only be an approximation to the ideal and there would therefore be a residual downstream propagation. Unless special measures are taken the excitations will also generate backward travelling waves a - which are effectively reflections of a +. Furthermore an upstream disturbance d- may not only propagate through the control region but unless special steps are taken will also contribute to m and hence give rise to reflected waves d +. Finally it should be noted that feedback will usually occur in that the waves created by the excitations may themselves contribute to the measurements. The individual components of the system will now be examined in more detail. 3.1.
SENSORS
AND
MEASUREMENTS
The sensor region contains n, sensors located at positions xi as indicated in Figure 4. The sensors are here assumed to give point measurements of any field variable such as displacement or slope or any of their time derivatives. If the wave amplitudes at the j t h sensor are aj ~ and b~ then the output sj of the sensor will be
sj=
[ajJaf+[fljJb;, j = 1 , 2 . . . n,, (13) IajJ and [fljJ depend on the type of sensor being used. For
where the 1 x n,. row vectors example [ajJ = [fljJ = I1 1 9 9 9 lJ K for sensors that measure displacement and IajJ = -[fljJ = [-ik~,-ik2,...,-ikn, IK for those that measure slope. Here K is the gain of the sensor and may contain factors of ico if the sensors measure time derivatives. The S!
S2
I'I !
53
J..... i
0
Figure 4. Sensor region.
Sn s
"'"
4
b-
258
B.R. r,IACE
vector of sensor outputs s can be formed from equation (13) by noting that a~ =_f(xj)a § and b7 = f ( x , , ' - x ~ ) b - . Thus s = _aa§ +_/3b - ,
(14)
where _a a n d / 3 are ns • nw sensor matrices w h o s e j t h rows are [ajJ_f(xj) and [/3,J_f(x,,, -x~) respectively. A set o f n,, measurements m are to be used as inputs to the controller. While in m a n y cases the sensor outputs can be used directly as the required measurements this need not be the case. In general the measurements will be some c o m b i n a t i o n o f t h e sensor outputs: m = A_ts= _Aa++ _Bb-.
(15)
Here _A and _B are n,,, • n~,. measurement matrices whose elements d e p e n d on the type o f sensor used, their spatial distribution and the m a n n e r in which their outputs are combined to form the measurements. If the sensor outputs are to be used directly then A_I will be the unit matrix. Note that the orders o f n , , n~ and n,,, need not be equal, but some restrictions apply. Firstly n, ~> n,.. if the system is to be observable and for full cancellation to be possible and n, ~<2n,. if the sensor outputs are to be linearly independent (not strictly necessary). Similarly nm ~> n,, for observability since if n,,, < n,,. there will be unobservable c o m p o n e n t s in a § which will then act as disturbance inputs to the system. Also if n,,, > nw then the measurements, which are o f couse estimates o f the wave amplitudes in a § will not be independent. Since n,,, independent measurements can be made n,,, = n,,. will be assumed henceforth. Advantages do however o c c u r if ns > n , since then A/ can be chosen such that the sensitivity o f the measurements to the disturbances b- is reduced. For example if n~ = 2n,.. then A_J can be chosen such that _M_B= B = 0 and hence the upstream disturbances b- will no longer have any effect on the system. The sensor part o f the system is shown in a block diagram in Figure 5. 3.2.
ACTUATORS
AND
EXCITATIONS
The exciter region contains np actuators located at positions x~ as shown in Figure 6. They are assumed to provide point inputs p, in any form such as forces or moments. The a+
~--m
b-
Figure 5. Block diagram of sensors.
Pl P2 C-~. i
T i-~
P3
P~r ~d
....................T x,,p 9
c
Figure 6. Exciter region.
rI d
§
ACTIVE CONTROL OF FLEXURAL VIBRATIONS
259
controller provides n~ excitations Q which will often be such point excitations but in general (as in the case of sensor outputs and measurements) a single excitation may give rise to a number of point inputs. The j t h point input pj will produce waves
d] = 8p,,
c~- = 3"pj,
(I 6)
in the waveguide where the n~. x 1 column vectors 8 and 3' depend on the type of exciter being used. For a force-like exciter 8 = 3' and 8 = -3" for a moment-like exciter. The total output from the exciters will thus be d+ = _zip,
c- = _Fp,
(17)
where the ith colunms o f the n,. x np matrices _A and F are {A}i
=f(x,,,-x~)8,
{I'}~ =_f(x~)3",
(18)
and give the contribution of the ith exciter to the total output. However the point inputs p will be some combination of the excitations Q: i.e., p = _PQ where _P will of course be the unit matrix if point excitations are to be used directly. Consequently d § =
_DQ,
_D = _zl_P,
c-
=
CQ,
_C =/_'_P,
(19)
where the elements of the n , x n, excitation matrices _C and D depend on the type of actuators used, their spatial distribution and the manner in which the excitations are combined to give the actuator inputs. The number o f excitations ne ~> n~. for the system to be controllable and full cancellation to be possible although only n,, independent excitations can be formed from the measurements. Consequently n, = n,. will be assumed henceforth. Advantages occur if np> n~, since _P can then be chosen such that the magnitudes of the unwanted negative going waves c- can be reduced. For example if n, = 2n,,. then _19 can be chosen such that _/'_P= _C = 0_ and hence no upstream components c- will be created. A block diagram of the exciters is shown in Figure 7.
I o
---J
11
p ~_d + Figure 7. Block diagram of exciters.
3.3. MEASUREMENT/EXCITATION INTERACTION
It is apparent that the exciter outputs may be detected by the sensors. This feedback can, amongst other things, cause the overall system to become unstable. If the sensor and exciter regions (see Figure 3) do not overlap then this interaction occurs due to the negative going waves c- produced by the exciters. These then contribute an amount m = _HQ,
/__/= _B_f,b_C,
(20)
to the measurement m. Clearly the feedback _H can be eliminated if either _B or _C is zero: i.e., if either the excitations do not produce or the measurements do not respond to negative going waves.
260
a.R. M^CE
I f t h e r e is overlap then the feedback matrix H is slightly more complex. The contribution to the kth sensor output sk from the j t h point input pj is sk = H*~jp~ where
m,:{
ro,
x,
[BkJ_f(xj-xD'~p,,
xk <~ xjJ"
(21)
Thus m = _HQ where H_ = M H * P .
(22)
Once again it is possible, providing there are sufficient sensors and exciters, to make /4 = 0 and hence eliminate feedback. 3.4. THE CONTROLLER Figure 8 shows a block d i a g r a m of the active control system. T h e controller _G (which m a y be an a n a l o g u e or digital device) forms the excitations Q from the m e a s u r e m e n t s m such that Q = _Gin.
(23)
The controller _(3 and the feedback element H together have a transfer function T = [1 - Q_H ]-t _G,
(24)
where _/is the n, x n, unit matrix. The u p s t r e a m waves d- act as an input disturbance to the system and a second set o f upstream waves a - may be credited as a result o f the applied excitation.
~ , d§
0 §
Figure 8. Block diagram of active control system. If the disturbances d- are ignored then the d o w n s t r e a m waves d + consist of two p a n s . The first, _faaa +, is due to the input waves, _fad being the p r o p a g a t i o n matrix from a to d, and the second part o f d § will be due to the excitations. Thus d + =f,,da ++ D T A a + = [fan + O [ ! - Q_H]-I _G_A]a§ In the ideal case, when full cancellation occurs, d §
(25)
0 and hence
T = -_D-tf_,,a_A -'.
(26)
Hence the controller required to achieve full cancellation is _(3 = [_H + T - ' ] - ' = [_H - _A_ff~a~_D] - '
(27)
and the control excitations are O = - _D-'_f,,a a +.
(28)
ACTIVE CONTROL OF FLEXURAL VIBRATIONS
261
Thus given that the individual components in the system are known equation (27) gives a method by which an ideal controller _G can be synthesized such that complete control can be achieved. The controller _G effectively has two elements, _G' and _H' as indicated in Figure 9 where _H'-~ _H is intended to subtract the feedback in the physical system and _G'=--_D-I_f,,a_A-I provides the required transfer function T. These two elements have individually much .simpler frequency responses than _G alone In practice the controller will not be ideal and the residual downstream propagation is given by equation (25). The stability o f the system can be determined from the Nyquist plots of the eigenvalues of (-O_H) which must all satisfy the Nyquist criterion if the system is to be stable. If the system is unstable or if the relative stability is insufficient then it will be necessary to include filters in the controller to remove the undesired frequency components. For this reason and because of difficulties in accurately synthesizing _G the controller transfer function will differ from the ideal given by equation (27). The residual transmission through tile system will then be d + = [f_,,a + _D[I._ - _G_H]-' _G_A]a+.
(29)
From this the actual attenuation achieved can be found. Although the control forces cancel the downstream transmission of the incoming waves a § they create additional reflected waves
a- = _paa*,
(30)
where the reflection matrix at point a is
e~ =f_,~C_T_A.
(3 i)
If the ideal controller (equation (27)) is used then
P_a =
-f,,_C_D-' f_,,a.
(32)
These reflections are o f course zero if the excitations are arranged such that C = 0. The reflections may create standing waves or resonances upstream of the controller. Furthermore since the control system is active in that it may add energy to the waveguide it is possible (if_p= has eigenvalues whose magnitudes are greater than one) for forced unstable oscillations to occur in the section of the waveguide upstream of the controller. In a similar manner an upstream disturbance d- will act as an input to the system and cause a spurious reflection d + = _Pad-,
(33)
where the reflection matrix at point d is
e a = _DT_B_fab.
(34)
e a = -f_,,a-A - t _B_fab.
(35)
In the ideal case
I'n
H'
["
Figure 9. Equivalent form of controller _G.
262
n.R. MACE
These reflections can be made zero if the sensors are arranged such that B = 0 but if they exist then they may produce effects similar to those caused by the reflected waves a-. The total transmission through the system will be the sum of the direct component'_fd~dand those waves additionally created by the excitations, giving a - = Ydad-,
(36)
where the transmission matrix _ra~ is
"rda = [fcaC- T_Flfdb+ fda] = [--P~a,~Pd + fda]
(37, 38)
in the ideal case. 4. ACTIVE CONTROL OF DUCT NOISE The control of duct noise by active methods has received much attention in the past. In this section some of the approaches adopted will be briefly described in the context of the general method of active control in a waveguide presented in the previous section. If it can be assumed that the higher order duct modes are negligible then only a single wave a § or a- can propagate in each direction in the duct. Thus n~,=ne = n,, = 1. Furthermore the air is non-dispersive so that the wavenumber k = to/c is directly proportional to frequency. The propagation matrix f ( x ) then reduces to e x p ( - i k x ) = exp ( - i t o x / c ) and represents a time delay of x / c seconds. A number of studies have been made of the problem of reducing the effect of feedback which acts to destabilize the system and make the required controller frequency response non-uniform. With no feedback loop only a time delay in the controller is necessary. Swinbanks [25] analyzed the case of two loudspeakers which are phased such that the net output is a downstream travelling wave. Hence C = _0and there are two point actuators (np = 2). Such an array responds uniformly over a fairly narrow bandwidth and so the frequency range of operation is restricted unless a more sophisticated controller is used. He also considered how the bandwidth could be improved by using three loudspeakers (np = 3 ) and how the outputs of three microphones (ns--3) could be phased so that they detect only the downstream propagating noise (i.e., B = _0). In the Jessel and Mangiante arrangement [26] a three output monopole plus dipole combination was discussed (np = 3). The Chelsea dipole [27] (np = 2, n5 = 1) consists of a microphone located centrally between two loudspeakers. In this system the sensor and exciter regions overlap and the loudspeaker outputs are phased to give zero net output at the microphone. Thus feedback is eliminated by combining loudspeaker outputs in a particular way such that _H = _0. The performance of a general distribution of acoustic sources was analyzed in reference [28]. In the monopole system, comprising one microphone and one loudspeaker (n, = 1, np = 1), no attempt is made to break the feedback path, resulting in the need for a slightly more sophisticated controller which compensates for this feedback [29]. In addition if the positions of the microphone and loudspeaker are made to coincide then _f~a = I. For A = D = I the required controller characteristics become sign inversion and infinite gain. This is the so-called tightly coupled monopole [30]. The maximum gain realizable is limited by spurious measurement noise and instability gives an upper frequency limit. Digital control methods can give much improved performance since the construction of digital filters with quite complex frequency responses is straightforward. Thus in most recent practical implementations digital techniques are preferred. Two examples of reduction of feedback effects have been provided by Ross [31] (n5 =3, np= l) and La Fontaine and Shepherd [32] (n~ = 2, np = 2). The filters in these cases were constructed by using least squares system identification methods. Adaptive control methods [33, 34], which have been applied to monopole systems, would seem to hold great promise.
ACTIVE C O N T R O L OF FLEX U R A L VIBRATIONS
263
5. ACTIVE CONTROL OF FLEXURAL VIBRATION Flexural vibration in a thin beam comprises two wave components as described in section 2 and thus n,, = 2. While in theory the sensors may measure either displacement or rotation or their respective velocities or accelerations, in practice rotational measurements are not easy to obtain. Similarly actuators applying forces are easier to realize than those applying moments. In this section two possible control schemes are discussed. It should perhaps be noted that in any control scheme the final point of application of a control force will have no displacement or rotation in order that no energy can pass this point. It will therefore appear fully clamped as far as the incident waves are concerned. 5.1. SINGLE POINT MEASUREMENT AND EXCITATION
Consider the situation where two sensors are located at a point. One measures displacement, the other rotation. A control point force P and point moment M are applied at a second point a distance l away as shown in Figure 10. The sensor outputs s and point inputs p are then
s=[ow/Oxl,
P= M "
(39)
It is convenient to define the measurements m and excitations Q to be
w m={(l/k)aw/Ox} ,
~ P/4EIk3~ Q=LM/4EIk2 J.
(40)
From equations (15), (12) and (19) the measurement matrices _Aand _Band the excitation matrices _C and _D are _A=[1 i
_11],
_/3=[I
:],
_C=[_-i1
-:],
_D=[-i 1
_11]
(41)
while
,. ,b , _
_
0]
_
e -~'
,
(42)
where ,u = kl, k being the wavenumber, will have a small negative imaginary part if damping is present. The transfer function of the ideal controller (equation (27)) is 1 ~ sinh # +sin ~ -G=4(cos# cosh/.t-1) L-(cosh#-cos/x)
cosh/.t-cos ~] sin/.t-sinhp. J"
(43)
Note that for real p. _G is real. This is to be expected because incoming waves a + are reflected since the excitation point is effectively clamped. Indeed it is straightforward to show that the reflection matrix (equation (32)) is given by e,, = _f-ref,_
(44)
awlax i~,. a,b
M
L c,d
Figure 10. Single point active control system.
LI.R.MACE
264 where
-i
-(l+i)]
-re = - ( l - i )
(45)
i
is the reflection matrix for a clamped end [24]. The reflections therefore produce standing waves and the applied forces do no work since they are in phase with the displacements. The magnitudes o f the elements of _G with/1 real are shown in Figure 11 where they are plotted against dimensionless frequency/'2 = # ~ = oJ,,/-mi-4/El. _G exhibits resonances at those values of/.t for which cos/.t cosh ~ - 1= 0 (i.e., /z =0, 4-730, 7 . 8 5 3 . . . when .(2 =0, 22-37, 6 1 . 6 7 . . . ). These correspond to the natural frequencies of a clampedclamped beam of length L Since the beam is effectively clamped at the excitation point then at these frequencies a component of the vibration cannot be detected by the sensors.
1
i
i
I
I
i
U :II :II -II
"2;~05!..
0
f
t I I 20 40 Dimensionless frequency .1~
I
60
Figure !1. Single point controller transfer function _G~vs. dimensionless frequency n ; pc real.
- - -, It,I,
IG~,I; . . . . , I c , I.
. Ic,,I;
In practice the controller would tend to ring and it would be desirable to filter these frequency components out o f t h e measured signals. In any event small differences between the actual and ideal controllers may lead to poor performance and even amplification at these frequencies. The feedback matrix is
/-/= [1 i -i] e-"9 + [_-i1 i]e-~
(46)
and varies smoothly with frequency. The stability of the controller can be determined from the Nyquist plots of the eigenvalues o f ( - G _H). These are given in Figure 12 where a small amount of damping has been assumed (p. = ,u0(l -i0.01)). It can be seen that the system is stable. One eigenvalue, corresponding loosely to the near field, decays rapidly towards the origin while the other, corresponding to the propagating component, clearly shows the resonances inherent in the controller _G. In addition to cancelling the incident waves a + the active controller will both reflect and transmit disturbances d-. For the scheme under consideration the reflection matrix
265
ACTIVE CONTROL OF FLEXURAL VIBRATIONS i
I
I/
'
//
'
~-'----
"
....
l //[(..//
~
/
E
"i""%'.."'%. ~ /
/
/
/
"%,,~ I
I
I
I
I
O
-2
Real part Figure 12. Nyquist plot of the two eigenvalues of the single point controller,/~ =/ao(l -i0-01).
ea (equations
(33) and (34)) is given by _Pa = f-ref,
(47)
where &, the reflection matrix for a clamped end, is given by equation (44). As far as the disturbances are concerned the beam therefore appears to be clamped at the sensor location. The transmission matrix (equation (36)) is given by
za~ =[[! -rcf_rc_fl.
(48)
r,, is shown in Figure 13. There are frequency regions where the direct component _fd,, d- and the component induced by the applied forces interfere constructively and destructively, the maximum transmitted amplitude being twice that of the incident. Due to the exp (-/z) terms the other components of _r decay with increasing frequency and are of less significance. 5.2. T w o POINT M E A S U R E M E N T A N D EXCITATION
As a second example consider the case where the displacements are measured at two points and vibrations are to be controlled by two point forces. For simplicity the separation of both measurement and excitation points is I and the second measurement point B coincides with the first excitation point C as shown in Figure 14. The measurements and excitations are taken to be m=
I Pc/4Elk'Sl Q=[Po/4EIk3J
tIwAi'wJo
(49)
and the measurement and excitation matrices, which are now frequency dependent, become e -i"
e-"
'
B=
1
'
-
e-"J'
ke-"
I
(50)
266
B. R. MACE I
I
I
I
I
2
I
0
T
f
I
P
20 40 O,mensionlessfrequer~y,f/
T
60
Figure 13. Transmission coefficient rjt for single point controller vs. dimensionless frequency//; # real. wz
wB , Pc
Po
c ~__.~I_.._a,.c~~
!
O
b, c
d
Figure 14. Two point active control system.
For full cancellation to occur the controller should have the transfer function 1 [10 - ( s i n h 2 ~ - s i n 2 ~ ) -G - 2(sinh/.L _ sin/_t ) (s'(-~nh ~ ' p . ~
] J,
(51)
where/.t = kl again. Once more _G, shown in Figure 15, is real and the incident waves are reflected. Since Q = _Gm the second line of equation (51) indicates that the second force Po depends only on the second measured displacement wt~. This is because the point D is effectively clamped. Since Po can only produce cancelling waves proportional to [-i - l J T (equation (12)) then the amplitudes ofthose waves incident upon D must have amplitudes in the proportion of [-i - l J T. Hence the function of Pc is to shape the incoming waves in such a fashion that, after they have propagated over a distance l, the amplitudes of the two components have this desired ratio. Indeed it is straightforward to show that the wave amplitudes c *+ as defined in Figure 14 are 1 [-e-(i'-m' } ( c + - i c ~ ) c *+ - ! _ e_(l_i)~,
(52)
and thus d *§ =_fc*+ - 1 _ e_(,_i) .
{-:}
(c § - ic~).
(53)
ACTIVE
CONTROL
OF
FLEXURAL
267
VIBRATIONS
lo)
0
I 2
0
I 4
l 6
8
10
I 8
10
Dimen,slonless frequency ,12
I
I
I
(b)
-5
tO
-15
-20
I
0
2
T
4
I
6
Dimensionless frequency 12
Figure 15. Two point controller transfer function _Gvs. dimensionless frequency .O;/,t real. (a) GII = G22; (b) Gt2.
Note that these amplitudes depend only on ( c + - i c ~ ) . The displacement measured at B gives (c § - ic~,) and that measured at A is simply used in conjunction with this to determine the excitation required to correctly shape the incoming wave amplitudes. As in the previous example this system is stable. Upstream disturbances d- are reflected and transmitted, the transmission matrix _z exhibiting frequency ranges of constructive and destructive interference. 6. CONCLUDING REMARKS In the preceding sections active control in a waveguide was discussed. The special case of control of flexural vibrations was considered and two possible control schemes outlined. Control is possible if a suitable two-input two-output controller _G can be designed. In this approach to active vibration control the overall size of the structure being controlled
268
a.R. MACE
is irrelevant. It may thus be very large and hence a very large number of modes may be affected. The control system itself may however be compact. In a practical implementation a controller would be designed to operate over a certain frequency range and a rational approximation to the irrational transfer functions required can be made. It would also be desirable to select the size of the system so that controller resonances lie outside the frequency range of interest. It may also be necessary to consider the sensor and exciter dynamics. The examples considered consisted o f the minimum number of sensors and exciters necessary to fully cancel the vibrations. It is to be expected that more complex systems, containing additional sensors a n d / o r exciters, would enable improved practical performance to be obtained, for example by removing sensor/exciter interaction.
ACKNOWLEDGMENTS The author gratefully acknowledges the financial assistance provided by the University of Auckland Research Fund and the Research Committee of the U.G.C.
REFERENCES 1. B. PORTER and A. BRADStlAW 1972 Journal of Mechanical Engineering Science 14, 307-311. Synthesis of active controllers for vibratory systems. 2. B. PORTER and A. BRADSHAW 1974 Journal of Mechanical Enghzeering Science 16, 95-100. Synthesis of active dynamic controllers for vibratory systems. 3. B. PORTER and T. R. CROSSLEY 1972 Modal Control. London: Taylor & Francis. 4. C. R. MARTIN and T. T. SOONG 1976 Proceedings of the American Society of Civil Engineers, Journal of the Engineering Mechanics Division 613-623. Modal control of multistory structures. 5. H. KWAKERNAAKand R. SIVAN 1972 Linear Optimal Control Systems. New York: WileyInterscience. 6. M. ABDEL-ROHMAN and H. H. E. LEIPHOLZ 1979 Proceedings of the American Society of Civil Engineers, Journal of the Engineering Afechanics Division 105, 1007-1023. A general approach to active structural control. 7. M. ABDEL-ROHMAN and H. H. E. LEIPitOLZ 1980 Proceedings of the American Society of Civil Engineers, Journal of the Structural Division 106, 663-677. Automatic active control of structures. 8. K. J. ASTROM and B. WI'I'q'ENMARK19 Computer Controlled Systems. Englewood Cliffs, NJ: Prentice Hall. 9. O. G. LLJENBERGER 1971 Institute of Electrical and Electronic Engineers Transactions on Automatic Control 16, 596-602. An introduction to observers. 10. J. S. BURDESS and A. V. METCALFE 1983 Journal of Sound and Vibration 91,447-459. Active control of forced harmonic vibration in finite degree of freedom structures with negligible natural damping. 11. J. S. BURDESS and A. V. METCALFE 1985 Transactions of the American Society of Mechanical Engineers, Journal of Vibration, Acoustics, Stress and Reliability in Design 107, 33-37. The active control of forced vibration produced by arbitrary disturbances. 12. J. ROORDA 1975 Proceedings of the American Society of Civil Engineers, Journal of the Structural Division 3, 505-521. Tendon control in tall structures. 13. M. J. BALAS 1978 Journal of Optimisation Theory and Applications 25, 415-435. Active control of flexible systems. 14. M. J. BALAS 1978 Institute of Electrical and Electronic Engineers Transactions on Automatic Control 23, 673-679. Feedback control of flexible systems. 15. L. MEIROVITCH and tt. BARUH 1981 Jounlal of Optimisation Theory and Applications 35, 31-44. Effect of damping on observation spillover instability. 16. L. MEIROVITCtl and H. BARUtl 1983 Journal of Optimisation Theory and Applications 39, 269-291. On the problem of observation spillover in self-adjoint distributed parameter systems. 17. E. LUZZATO and M. JEAN 1983 Journal of Sound and Vibration 86, 455-473. Mechanical analysis of active vibration damping in continuous structures.
ACTIVE CONTROL OF FLEXURAL VIBRATIONS
269
18. E. LUZZATO 1983 Journal of Sound and Vibration 91, 161-183. Active protection of domains of a vibrating structure by using optimal control theory: a model determination. 19. L. MEIROVlTCIt and H. BARUII 1982 Journal of Guidance 5, 60-66. Control of self-adjoint distributed parameter systems. 20. L. MEIROVITCll and L. M. SILVERBERG 1984 Journal of Sound and Vibration 97, 489-498. Active vibration suppression of a cantilever wing. 21. T. tt. ROCKWELL and J. M. LAW'triER 1964 Journal of the Acoustical Society of America 36, 1507-1515. Theoretical and experimental results on active vibration dampers. 22. L. A. WALKER and P. P. YANESKE 1976 Journal of Sound and Vibration 46, 157-176. Characteristics of an active feedback system for the control of plate vibrations. 23. L. A. WALKER and P. P. YANESKE 1976 Journal of Sound and Vibration 46, 177-193. The damping of plate vibrations by means of multiple active control systems. 24. B. R. MACE 1984 Journal of Sound and Vibration 97, 237-246. Wave reflection and transmission in beams. 25. M. A. SWINBANKS 1973 Journal of Sound and Vibration 27, 411-436. The active control of sound propagation in long ducts. 26. M. J. M. JESSEL and G. A. MANGIANTE 1972 Journal of Sound and Vibration 23, 383-390. Active sound absorbers in an air duct. 27. Kh. EGHTESADI and H. G. LEVENTHALL 1981 Journal of Sound and Vibration 75, 127-134. Active attenuation of noise: the Chelsea dipole. 28. Kh. EGIITESADI and H. G. LEVENTtlALL 1983 Journal of Sound and Vibration 91, 11-19. A study of n-souce active attenuator arrays for noise in ducts. 29. Kh. EGilTESADI and H. G. LEVENTIIALL 1982 Journal of the Acoustical Society of America 71,608-611. Active attenuation of noise--the monopole system. 30. Kh. EGHTESAD1, W. K. W. I-lONG and H. G. LEVENTI-tALL 1983 Noise Control Engineering Journal 20, 16-20. The tight-coupled monopole active attenuator in a duct. 31. C. F. ROSS 1982 Journal of Sound and Vibration 80, 373-380. An algorithm for designing a broadband active sound control system. 32. R. F. LA FONTAINE and I. C. SHEPHERD 1983 Journal of Sound and Vibration 91,351-362. An experimental study of a broadband active attenuator for cancellation of random noise in ducts. 33. C. F. R o s s 1982 Journal of Sound and Vibration 80, 381-388. An adaptive digital filter for broadband active sound control. 34. A. ROURE 1985 Journal of Sound and Vibration 101,429-441. Self-adaptive broadband active sound control system. APPENDIX: NOMENCLATURE a, b, c, d abcd
A,
_C,_D
_f G H
k m
AI A_I nr nm np ns nw
P
P P
q.~l, qP Q
boundaries of active control region, Figure 3 vectors of wave amplitudes at corresponding points measurement matrices, equation (15) excitation matrices, equation (19) wave propagation matrix, equation (8) controller transfer function, Q = Gm measurement--excitation feedback matrix, equation (20) wavenumber vector of measurements applied point moment, Figure 2 bending moment in beam, Figure 1 sensor output combination matrix, m = AJs number of control excitations applied number of measurements taken number of point actuators number of point sensors number of wave motions in waveguide vector of actuator point inputs applied point force, Figure 2 control excitation combination matrix, p = P Q vectors of waves injected in beam by a point moment and a point force, equation (12) vector of control excitations
270 $ T
v IV
!,_a /z
e
!
0..I
D
13. R. M A C E
vector of sensor outputs controller plus feedback transfer function, equation (24) shear force in beam, Figure 1 beam displacement sensor matrices, equations (13) and (14) actuator matrices, equations (17) and (18) dimensionless wavenumber matrix of reflection coefficients matrix of transmission coefficients frequency dimensionless frequency, .O = ,u 2 = t o nf~d4/EI ,
Subscripts a,b,c,d pertaining to points a, b, c, d i,j pertaining to element i,j near field wave component N Superscripts + -
propagating in the positive direction propagating in the negative direction