vibration control of a fluid-loaded circular plate via pole placement

MECHANICS
vol. 27 No. 1 2008
Lucyna LENIOWSKA*
VIBRATION CONTROL OF A FLUID-LOADED CIRCULAR PLATE
VIA POLE PLACEMENT
SUMMARY
This paper presents the application of active control system to reduce the vibration and the sound radiation of a circular
plate which is fluid-loaded and clamped at a finite baffle. The plate is driven by a uniform harmonic primary force and controlled by a distributed secondary force generated by the piezodisks. The aim of the paper is to design a controller to modify the response of the plant in some desired fashion. The mathematical model of the system includes the influence the
acoustic wave radiated by the plate, Kelvin-Voigt damping in the plate material and viscous fluid damping. For the considered system the transfer function of controller of 4th order, satisfying the polynomial Diofantine equation, is derived. The
influence of the fluid-loading and damping effects on the system response are presented graphically and discussed. The results show that plate modes have been reduced very well by obtained controller for a few assumed values of damping
coefficients.
Keywords: active methods, vibration control, pole placement, PZT actuators, internal and viscous damping
STEROWANIE DRGANIAMI P£YTY KO£OWEJ Z UWZGLÊDNIENIEM ODDZIA£YWANIA OŒRODKA
METOD¥ LOKOWANIA BIEGUNÓW
W pracy przedstawiono zastosowanie aktywnego sterowania do redukcji drgañ i fal akustycznych promieniowanych przez
p³ytê ko³ow¹ utwierdzon¹ na obwodzie. P³yta pobudzana jest do drgañ si³¹ harmoniczn¹ o rozk³adzie równomiernym,
a sterowana za pomoc¹ pary piezodysków PZT. Celem pracy jest zaprojektowanie uk³adu sterowania, który modyfikuje
odpowiedŸ obiektu w po¿¹dany sposób. Model matematyczny rozwa¿anego obiektu uwzglêdnia oddzia³ywanie zwrotne fali
akustycznej, t³umienie wewnêtrzne materia³u oraz t³umienie wiskotyczne. Dla zastosowanego modelu wyznaczono
transmitancjê regulatora czwartego rzêdu, rozwi¹zuj¹c równanie diofantyczne. Wp³yw t³umienia wewnêtrznego, wiskotycznego oraz sprzê¿enia pomiêdzy drgaj¹c¹ struktur¹ i oœrodkiem na odpowiedŸ uk³adu przedstawiono w sposób graficzny i przeanalizowano. Otrzymane wyniki pokazuj¹, ¿e zaprojektowany regulator bardzo dobrze t³umi drgania p³yty dla
kilku za³o¿onych wartoœci wspó³czynników t³umienia.
S³owa kluczowe: metody aktywne, sterowanie drganiami, po³o¿enie biegunów, aktuatory PZT, tarcie wewnêtrzne i wiskotyczne
1. INTRODUCTION
Planar structures as membranes or plates do not usually vibrate in vacuo. They affect and are affected by the surrounding media which were in contact with their surfaces.
The vibration and sound radiation of circular plates have
been studied by many researchers since it is a significant
structural element in many industrial fields. Lord Rayleigh
was the first person who analised the “reaction of the air on a
vibrating circular plate” (Rayleigh 1929), showing that reaction to be equivalent to a virtual mass, and radiation damping
to be added to the plate mass and the mechanical damping.
Modal coupling, radiation efficiency and radiation impedance were employed in the last several years, where the
problem of attenuating structure-borne noise was considered
(Gu 1993; Junger 1986; Rosenhouse 2001). In this case the
cost function is derived from the far-field radiated pressure
(radiation control). Thus, for the closed-loop system re-
sponse, the overall radiation efficiency is reduced while the
overall plate vibration amplitude is not satisfactorily attenuated or sometimes even increased.
This paper investigates the application of an active control system to reduce the vibration of a circular plate which is
fluid-loaded and clamped at a finite baffle. The formal solution of the fluid-plate coupled equation is presented for a
plate driven by a uniform harmonic primary force and controlled by a distributed secondary force generated by
piezodisks. For the considered fluid-plate system, the state
space realization is given. Three parameters which characterise fluid density, plate material internal damping and viscous fluid damping are included in the considered model. It
is well- known that the dynamic behaviour of the linear systems depends strongly on the location of the models’ roots
(zeros and poles). To examine the stability of the system and
the influence of fluid-loading as well as structural internal
damping and viscous damping, the roots of the system with
* Institute of Technology, Rzeszow University of Technology, [email protected]
18
MECHANICS Vol. 27 No. 1 2008
assumed parameters were plotted on a complex plane. This
enables the use of a method for determining the transfer
function of a controller which places the poles of the
closed-system at some desired locations to suppress a plate
vibration. The effects of fluid-loading and viscous damping
on the system response are observed and presented graphically. Finally, the feedback control law is developed for the
minimization of the plate vibration.
2. FLEXURAL VIBRATIONS
OF A CIRCULAR PLATE
The structure under study is a vibrating circular plate of radius a, having a constant thickness h (Fig. 1), surrounded by
a lossless medium with static density r0. It is assumed that
the plate, clamped in a flat, rigid and finite baffle of radius b,
(b>r>a, z=0) is made of a homogeneous isotropic material
with density r, and has Kelvin-Voigt internal damping.
It is assumed that the plate is excited on one side by a uniform periodic force with constant amplitude F0 generated by
a loudspeaker
f w ( r, t ) = F0 e – iwt
for
0£r £a
(2.2)
and it radiates into free space filled with fluid of density r 0 .
The system model is formulated when taking into account
the coupling effect between the structure and the acoustic
medium, so the third component of the right hand side of
equation (2.2), f p ( r, t ), represents the acoustic fluid-loading
acting on the plate as an additional force. The value of this
force exerted by the fluid on the plate surface can be calculated as follows (Junger 1986):
f p ( r, t ) = – p( r, z , t )
z=0
(2.3)
where p( r, z = 0, t ) is the acoustic pressure at the point on the
surface of the plate. The acoustic waves propagate through
the fluid must satisfy the wave equation (Malecki 1964):
1 ¶ 2 p( r, z , t )
Ñ 2 p( r, z , t ) =
(2.4)
c2
¶t 2
where Ñ 2 is the two-dimensional Laplace operator, and c is
sound velocity in the fluid. At the fluid-structure interface,
the pressure must satisfy the boundary condition (Junger
1986):
¶p( r, z , t )
¶n
Fig. 1. A circular plate in a rigid baffle of radius b
In the case being considered, the applied loading and end
restraints of the circular plate are independent of the angle j
(axially symmetrical vibrations), thus we can write the governing differential equation of the forced motion of the plate
as follows (Leniowska 2005):
BÑ 4 w( r, t ) + R
+ rh
¶
2
¶t 2
¶
¶
[Ñ 4 w( r, t )] + g w( r, t ) +
¶t
¶t
(2.1)
w( r, t ) = f w ( r, t ) + f s ( r, t ) + f p ( r, t )
where:
B = Eh 3 / 12(1– v 2 ) – bending stiffness of the plate,
E, n and R – Young’s modulus, Poisson’s
ratio and Kelvin-Voigt damping coefficient for the plate,
r – density for the combined structure,
g – viscous fluid damping coefficient.
The displacement w(r,t) and its derivative ¶w( r, t ) / ¶r
satisfy the boundary condition for a clamped plate: they both
equal zero at the edge of the plate.
z = 0 =– r0
¶2
¶t 2
&& ( r, t ) (2.5)
w( r, t ) = – r 0 w
with n denoting the normal to the structure. In the case in
which the acoustic pressure radiates from the plate vibrating
harmonically, the wave equation reduces to the Helmholtz
equation
(Ñ 2 + k 0 ) p( r, z ) = 0
(2.6)
where:
p(r, z) – pressure amplitude,
k0 = w/c – acoustic wave number at frequency w.
The goal in the control problem is to determine a control
force f s ( r, t ) which, when applied to the plate (realized via
a voltage u(t) for the assumed actuators), leads to a reduced
level of vibration. The second component in the Eq. (2.1)
represents such a wanted control force, f s = ( r, t ), which will
cancel the plate vibrations. The location of actuators (a pair
of piezodisks) is assumed to be in the middle of the plate.
3. STATE-SPACE SYSTEM MODEL
To approximate the plate dynamics, a Fourier-Bessel
expansion of the plate displacement is used to discretize the
infinite dimensional system (2.1). The plate displacement
can be approximated by
N
w N ( r, t ) = å sm ( t )w m ( r )
(3.1)
m
19
Lucyna LENIOWSKA
VIBRATION CONTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMENT
where N is considered to be a finite number suitably large for
the accurate modelling the system dynamics and w m ( r )is the
(0, m) plate mode described as follows (Malecki, 1964)
é æ
r ö J (g ) æ
r öù
w m ( r ) = u 0 m êJ 0 ç g m ÷ – 0 m I 0 ç g m ÷ú (3.2)
a ø I 0 (g m ) è
a øû
ë è
J 0 ( x ), I 0 ( x )designate the cylinder functions, g m = k m a is the
m-th root of the frequency equation and sm ( t ) is the corresponding modal amplitude in time t. The eigenfunctions satisfy the ortogonality condition and can be normalized as follows:
ò ò wm (r )dS = pa
2
2
(3.3)
S
(n´n) state matrix, B is the (n´m) control input matrix, V is
(1´n) disturbance matrix, described as follows:
0
é
A=ê
–1 2
ë– ( I + E) W
1
ù
–(m 2 + m 1 W )(I + E) úû
0
0
é
ù
ù
é
B =ê
, V =ê
ú
ú
–1
–1
ë( I + E ) K w û
ë( I + E) K s û
2
–1
(3.12)
In above expression I denotes identity matrix, KS and KW
are the coefficient vectors, E represents fluid-plate interaction matrix, W = diag [ w1 , w2 ,.. , w N ]. It is assumed, that the
response of the considered plate to the applied force distribution is measured by a set of linearly independent point sensors, situated at locations r on the plate. The output equation
in matrix form is
In a similar way let us expand the right side of the plate
equation of motion (2.1) into series:
y( t ) = Cx( t )
(3.13)
where
N
f wN ( r, t ) = å z m ( t )w m ( r )
(3.4)
m
N
f sN ( r, t ) = å u m ( t )w m ( r )
(3.5)
m
N
f p ( r, t ) = p N ( r, z = 0, t ) = å p m ( t )w m ( r )
(3.6)
m
Inserting above expansions into the equation (2.1), multiplying both sides by the orthogonal eigenfunction w n ( r ), and
integrating over the surface of the structure, the governing
equation of motion can be re-expressed as:
N
å[ s&&m (t ) + (m 1w2m + m 2 )s&m (t ) + wm2 sm (t ) = zm (t ) + um (t ) + pm (t )]
z m (t )ü
ï
u m ( t ) ý = ò ò f j ( r, t )w m ( r )dS , j = w, s, p; m =1, 2, .. , N (3.8)
p m ( t )ïþ S
stand for the generalised modal forces
(3.9)
The modal model presented above can now be expressed
in the state-space format. The state vector is defined as follows:
(3.10)
Equation (3.7) can be expressed as (Leniowska 2005)
x& ( t ) = Ax( t ) + Bu ( t ) + Vz ( t )
(3.11)
where the dot denotes differentiation with respect to time, x
is the (n´1) state vector, u is (m´1) control vector, and A is
20
ù
ú
ú(3.14)
ú
0
L
ú
L w N ( r1 ) ú
ú
:
ú
L w N ( rNv )û
0
Nc and Nv denote the number of displacement and velocity
sensors respectively, w i ( r j ) is a value of i-th eigenfunction
at j-th measurement point.
4. SYSTEM DYNAMICS AND DAMPING EFFECTS
where
é s ( t )ù
x( t ) = ê
ú
ë s& ( t )û
L
:
(3.7)
m=1
m 1 = R / B and m 2 = g / rh
0 ù
éC
C =ê a
ú=
0
C
vû
ë
0
é w1 ( r1 ) L w N ( r1 )
ê :
:
ê
êw ( r ) L w N ( rNc )
0
= ê 1 Nc
w1 ( r1 )
ê 0
ê :
:
ê
0
w
(
1 rNv )
ë
To examine the possibility of instability of the system and the
influence of fluid loading as well as structural internal damping and viscous damping, it is convenient to plot the roots of
the system for assumed parameters on a complex plane. For
the analysis in this paper, water, propane, air and aluminium
were chosen to respectively represent the acoustic fluid material and material of the plate, to ensure strong, moderate
and light coupling between the structural and acoustic response. In the Figure 2 the location of poles and zeros of the
eight-order system is presented in air, for two values of parameter m1 (and constant m2). It can be seen that the values of
internal damping coefficient m1 have considerable influence
on systems’ root locations.
The following rules may be formulated when the parameter m1 increases (Fig. 2):
– the complex roots of the system migrate to the circle
with the radius of 1/(2m1),
– the number of complex roots diminishes – they become
real roots,
– the real roots move along the real axis: half of them migrate towards the middle of the circle and the second
half to the – ¥.
MECHANICS Vol. 27 No. 1 2008
Fig. 2. Distribution of poles and zeros of the considered system in air for m 2 = 0.5; a) m 1 = 0.00002; b) m 1 = 0.00004; o – zeros; § – poles
On the basis of the pole locations we can make a conclusion about the system dynamics which may be observed on
the Bode diagram (Fig. 3). Figure 3 presents the results obtained in air for three values of internal damping coefficient
m1 and constant value of fluid damping parameter m2, when
the acoustic and structural wave numbers are equal: kv0= kp.
It can be seen that for m1 = 0.0001122, the resonance frequency of the first and second mode of the considered plate
are dominated. However, for lower values of m1 the third resonance frequency close to 500 Hz appears and the phase
characteristic becomes more complicated.
Fig. 3. Bode diagram of the fluid-plate system in air:
1) m 1 = 0.0001122; 2) m 1 = 0.000022; 3) m 1 = 0.000044
It is worth to note that the modification of parameter m2,
assuming the linear bandwidth 0.1–100 sN/m3, does not
change root locations noticeably and can be observed on the
Bode diagram in the vicinity of the resonance peaks. The
third parameter included in the model derived, the
r
fluid-loading term e 0 = 0 (Crighton 1989), is helpful for
rhk 0
examining the influence of the fluid surrounding the considered plate. Taking some values of the fluid density r0 (m1 and
m2 are constant now) it may be seen (Fig. 4) how the localization of system roots is changed for different values of e0.
For the increasing density of the surrounding fluid medium, the roots of the considered system move to point (0, 0)
on the complex plane. The Bode diagram (Fig. 4) reveals an
additional feature which is important for the correct design
of controller transfer function, namely a phase shift, especially for low frequency radiated acoustic waves (Fig. 4a).
It can be seen that when the fluid density alters (increases), the resonance frequencies and the appropriate
phase characteristics diminish. As a result of the fluid coupling the response of the plate in fluid can be significantly
different from responses in vacuo. It can also be observed,
that the effect of the fluid-loading on the considered plate is
dependent on the frequency of the vibration. In the case of
lower frequencies, the shift of the resonance peak is greater
(Fig. 5a) and when the operating frequency increases it diminishes. To best illustrate this effect, Figure 6 shows the
Bode diagrams of the system in water (where the strong coupling can be theoretically assumed) for the ‘lower’ (Fig. 6a)
and ‘higher’ frequencies of acoustic waves (Fig. 6b).
An examination of the system dynamics shows that the
wavenumber spectrum of the sound pressure can be divided
into two domains. Below the null frequency defined as
fn =
r 0 c0
2prh
,
the surrounding fluid mass-loads the plate and the effective
mass of the plate increases. In this case the fluid-loading acts
mainly as a mass-loading and the resonances peaks are
moved towards lower frequencies.
21
Lucyna LENIOWSKA
VIBRATION CONTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMENT
a)
b)
Fig. 4. Distribution of poles of the considered system for
m 1 = 0.00002, m 2 = 0.5 and three kinds of the fluid density: (*)
r0=1000; (o) r0=500; (+) ro=1.2
a)
Fig. 6. Bode diagram of the fluid-plate system in water, for
m 1 = 0.00002, m 2 = 01
. and three value of the frequency of the acoustic pressure: a) (—)k0 = 0.42 (100 Hz); (––)k0 = 1.26 (300 Hz);
(––)k0 = 2.09 (500 Hz); b) (––)k0 = 25.13 (6000 Hz); (––)k0 = 41.88
(10000 Hz); (—)k0 = 62.83 (15000 Hz)
b)
On the other hand, for frequencies of acoustic response f > fn,
the fluid acts as a dampener – the effect of the fluid-loading is
small and it has dissipating character.
5. FEEDBACK CONTROL
The aim of the project is to design a control system to modify
the response of the plant in some desired fashion. The closed-loop setup is sketched in the Figure 7.
Fig. 5. Bode diagram of the fluid-plate system for m 1 = 0.00002,
. and three kinds of the fluid density: (—) r0=1000; (—)
m 2 = 01
r0=500; (—) r0= 1.2; a) f =100 Hz; b) f = 875 Hz
22
Fig. 7. Closed-loop setup
MECHANICS Vol. 27 No. 1 2008
For the assumed measurement point (SISO system), the
considered structure has a following transfer function
m
m–1
+ ... + b0
b( s ) bm s + bm – 1 s
G( s ) =
=
–
1
m
m
a( s ) a m s + a m – 1 s
+ .. . + a 0
(5.1)
The closed-loop transfer function is given by
G z (s ) =
G ( s )R ( s )
b( s ) p( s )
=
1+ R ( s )G ( s ) a( s ) p( s ) + b( s )q( s )
(5.2)
We seek a controller
R (s ) =
n
n–1
+ ... + q 0
q( s ) q n s + q n – 1 s
=
p( s ) p n s n + p n – 1 s n – 1 + .. . + p 0
éa0
êa
ê 1
êM
êa
ê m
ê0
êM
ê
êë 0
0
a0
a1
M
0
M
b0
b1
0
b0
L
O
O 0
O a0
O a1
M
b1
M
O
O
O
am
M O M
0 L am
bm
M
0
bm O
M O
0 L
0 ù é p0 ù é d 0 ù
ú
ê
M ú ê p1 ú ê d 1 ú
úê ú
0 úê M ú ê M ú
ú
ê
b 0 ú ê p m ú = ê M ú(5.5)
úê ú
b1 ú ê q 0 ú ê M ú
ú
ê
M úú êê M úú ê M ú
b m úû êë q m úû ëêd m + n úû
or graphically by using MATLAB Root Locus Tool (in this
case the Diofantine equation is solved as well).
Since we assumed two complex poles and two complex
zeros corresponding to the desired controller dynamics
(Fig. 9).
(5.3)
of order n satisfying the polynomial Diophantine equation
(Astrom 1990):
a( s ) p( s ) + b( s )q( s ) = d ( s ) =
= d m + n s m + n + d m + n – 1 s m + n – 1 + ... + d 0
(5.4)
where d(s) denotes a designed polynomial of m+n order
which moved the system roots to some predefined location.
The order of the controller should be equal to the number of
considered modes in the system model. Since we are basically
interested in damping the dominating vibrating modes and a loworder controller is generally preferred for physical implementation reasons, we take into account first four modes only.
The open-loop zeros and poles locations of the plant are
depicted in the Figure 8.
Fig. 9. Poles and zeros of closed-loop fluid-plate system in air,
o – zeros; x – poles; g – active point of root locus
After a series of attempts we obtained fourth-order controller
with the following values (Tab. 1).
Table 1
Poles and zeros of designed controller
Gain
Poles
Zeros
–209
±392 i
–96
±250 i
–279
±1500 i
–205
800 i
200
Fig. 8. Poles and zeros of open-loop fluid-plate system in air:
m 1 = 0.00004; o – zeros; n – poles
In theory, if the system is controllable, the poles and zeros
can be placed anywhere to improve close system performance. It can be done in two ways: analytically by solving
the linear system of equation with (m+n)´(m+n) non-singular Sylvester matrix:
On the Bode magnitude plot of the Figure 10, we can notice the significant reduction in the resonance peaks of the
dominating vibrating modes of the closed-loop system in air.
It can be seen that the uncontrolled plate response vibrates significantly while the controller causes that plate
dominant modes have been reduced very well for two values
of internal damping coefficient. However, there are some
disadvantages of pole placement control. First of them is
steady-state error which appears usually in closed-loop
system (Fig. 11).
23
Lucyna LENIOWSKA
VIBRATION CONTROL OF A FLUID-LOADED CIRCULAR PLATE VIA POLE PLACEMENT
a)
This problem can be improved by an appropriate reference signal correction or by applying cascade structure of
control system. The next disadvantage of polynomial pole
placement control is numerical troubles for higher order of
Diofantine equation.
It is well-known that linear algebra problems involving
polynomials are generally ill-conditioned. A frequency scaling or alternative polynomial bases (Chebyshev, Bernstein,
orthogonal polynomials) can probably help to improve numerical conditioning.
6. FINAL REMARKS
b)
Fig. 10. Compared Bode magnitude plots of the open-loop and closed-loop system in air for and two values of internal damping:
a) m1 = 0.0001122; b) m2 = 0.000022; — open-loop; - - - closed-loop
The problem of active suppression of the plate vibration of
the fluid-loaded plate has been solved by means of using pole
placement method. The mathematical model of the considered system includes the influence the acoustic wave radiated by the plate interacting on its surface due to the coupling
mechanism. In addition, Kelvin-Voigt damping in the plate
material and viscous fluid damping has been taken into account. It has been observed that the fluid-loading has two
main effects on the vibrating plate – in dependence on the
operating frequency it causes either damping or a mass-loading. The obtained controller was verified by testing the
plate displacement response obtained for chirp disturbance.
Comparing the structural and acoustic responses for the open
and close-loop systems, it can be seen that the obtained controller was damped the output signal very well. The main disadvantage of this method: a steady-state error can be improved by an appropriate correction of reference signal or by
applying a cascade structure of control system.
REFERENCES
Astrom K.L., Wittenmark B. 1990: Computer Controlled Systems. Theory
and Design, 2nd ed., Prentice Hall, Inc., ISBN:0-13-168600-3.
Crighton D.G., 1989: Fluid loading – the interaction between sound and
vibration. Journal of Sound and Vibration, 133, Issue 1, 1–27.
Gu Y., Fuller C.R. 1993: Active control of sound radiation from a fluid-loaded rectangular uniform plate. The Journal of the Acoustical Society of America, vol. 93, Issue 1, 337–345.
Junger M.C., Feit D. 1986: Sound, Structures and Their Interaction. Massachusetts Institute of Technology Press, Boston, ISBN: 0262100347.
Leniowska L. 2005: Modelling of piezoceramic actuators of circular plate
to reduce noise and vibration, Molecular and Quantum Acoustics,
vol. 26, 191–204.
Malecki I. 1964: Theory of waves and acoustic systems (in Polish). PWN,
Warszawa.
Fig. 11. The step response of the closed-loop system for m 2 = 01
.
and m 1 = 0.0001122
Rayleigh J.W. 1929: Theory of Sound. MacMillan, London.
Rosenhouse G. 2001: Active Noise Control. WIT Press, London.