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Control of circular plate vibrations via piezoelectric actuators shunted with a capacitive circuit
Thin-Walled Structures 39 (2001) 83–94 www.elsevier.com/locate/tws
Control of circular plate vibrations via piezoelectric actuators shunted with a capacitive circuit A. Tylikowski
*
Institute of Machine Design Fundamentals, Warsaw University of Technology, Narbutta 84, 02-524 Warsaw, Poland Received 19 September 2000; accepted 23 October 2000
Abstract The purpose of this research is to analyse capacitively shunting distributed piezoelectric elements perfectly glued to the vibrating annular plate excited by harmonic displacement of the inner plate edge. The piezoelement is described by constitutive equations relating stresses and electric displacements with strains and electric field. The equations are coupled with the equations of plate motion by the surface strain terms. At joint circles between annular sections without and with the piezolayers the continuity of plate deflection, slope and curvature is considered. The free stress condition for the piezoelement constrains the strain value at inner and outer piezoelement edges. Results from numerical simulation show an influence of the external shunting capacity on the frequency and space dependent plate response. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Circular plates; Axisymmetric vibrations; Piezoelectric actuators; Capacitive shunting; Distributed approach
1. Introduction Structure vibrations that propagate from engines or other sources may be reduced by passive and active isolation, by active control, or by passive and active vibration absorbers. Of particular interest to this research is the use of passive vibration absor-
* Fax: +48-22-660-8622. E-mail address: [email protected] (A. Tylikowski). 0263-8231/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 0 ) 0 0 0 5 5 - 0
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bers for structure-borne noise/vibration control. Passive vibration absorbers are conceptually simple devices consisting of a mass attached to a structure via a complex spring. The primary function of these devices is to increase the dynamic stiffness and therefore to change the natural absorber frequency. A shunting method has been developed for tuning the natural frequency of a piezoelectric element glued to beam surfaces. The passive vibration absorbers minimise vibration at a specific frequency related with a lightly damped structure. Large response reduction is only possible if the absorber is accurately tuned to the considered frequency. Therefore, an important feature of absorbers is the ability to be accurately tuned to the changing frequency. Tuning a mechanical absorber requires a change in either the mass or stiffness of the device. The electromechanical properties of a piezoceramic forcing element with an external passive electrical shunt circuit are used to alter the natural frequency [1]. An analytical distributed model of a piezoelectric vibration absorber was created to predict changes in natural frequency due to passive electrical shunting. Capacitive shunting alters the natural frequency of the actuators. A passive vibration absorber generally acts to minimise structural vibration at a specific frequency associated with the response of the lightly damped structural mode. This frequency is rarely stationary due to changing velocity. Maximum response reductions however are achived only if the absorber is lightly damped and accurately tuned to the frequency of concern. The vibration of two-dimensional structures excitated by a piezoelectric actuator has been modelled by Dimitriadis et al. [3] for rectangular plates and by Van Niekerk et al. [6] for circular plates. Van Niekerk, Tongue and Packard presented a comprehensive static model for a circular actuator and a coupled circular plate. Their static results were used to predict the dynamic behaviour of the coupled system, particularly to reduce acoustic transmissions. Piezoelectric transducers can be modelled as two-dimensional devices. The approach allows the distributed transducer shape to be included in the control design process for two-dimensional structures as an additional design parameter. The essence of the approach involves replacing the piezoactuators by forces and moments distributed along the piezoelements’ edges (e.g. Kim et al. [5]). The analytical basis for damping structural vibrations with piezoelectric materials and passive electrical circuits has been developed by Hagood and von Flotow [4]. A key feature of the tunable vibration absorber developed by Davis et al. [1,2] is the use of piezoelectric ceramic elements as part of the device stiffness. A discrete model of a piezoceramic vibration absorber was created to predict changes in natural frequency and damping due to passive electrical shunting. A comparison of passive and active damping of thermally induced vibrations of beams with piezolayers was given in [7]. Influence of an actuator shape on stabilisation of plate parametric vibrations has been developed in [8]. The goal of this research is to describe an annular piezoelectric vibration absorber of transverse vibrations of annular plate by using a consistent distributed model based on partial differential equations of system motion. The Kirchhoff annular plate of inner radius R1 and external radius R2 is divided into three sections and the dynamic behaviour of each section is analysed separately (cf. Fig. 1). The analysis is confined to axially symmetric modes. The absorbers are glued to the plate in the second
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
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section a⬍r⬍b. The dynamic extensional strain on the plate surface is calculated by considering the dynamic coupling between the actuator and the plate, and by taking into account a perfect bonding (infinite shear stiffness). The plate motion is described by partial differential equations in the plate transverse displacement. The piezoelement is described by constitutive equations relating stresses and electric displacements with strains and electric field. The equations are coupled with the equations of plate motion by the surface strain term. Along circles of connection between sections the continuity of plate deflection, slope and curvature is considered. The free normal stress condition for the piezoelectric element at the location r=a and r=b constraints the strain values at these two locations. The plate is excited by a motion at the internal edge. The displacement is harmonically varying with a constant frequency. For boundary conditions at clamped and free edges, the joint conditions form a boundary value problem. For single frequency excitation the equations of motion become fourth order ordinary homogeneous differential equations, with solutions in the form of travelling waves in the plate. Results from numerical simulation show an influence of the external shunting capacity on the frequency and space dependent plate response. The effect of changing both resonance frequency and amplitude is observed in going from the open external circuit to the short external circuit.
2. Dynamic equation of axisymmetric plate motion Fig. 1 shows a Kirchhoff thin annular plate with identical piezoceramic elements mounted on opposite sides of the plate. The annular piezolayers perpendicularly polarised to the plate surface are conected with an external passive electric shunt circuit, which can be used to alter the natural frequencies of the continuous structure. The effective bending stiffness of the plate is adjusted electrically using a capacitive shunt-circuit. The piezolayers are perfectly bonded to the plate, which implies strain continuity at the bonding interface. As the axisymmetrical motion is considered the plate is divided into three annular sections as shown in Fig. 1, and the motion of the plate in each section is analysed separately. Consider a finite element of radial length dr in the second section. The radial stresses in the piezolayers are assumed to be uniformly distributed in the direction
Fig. 1.
Geometry of the plate with actuators.
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perpendicular to the plate due to the small thickness. For the radial actuator motion the dynamics equation is as follows
冉
r
冊
∂sr ∂2 u ⫹sr⫺st ta⫺tr⫽ratar 2 ∂r ∂t
(1)
where sr — radial stress, st — circumferential stress, t — shear stress on the interface surface, u — radial displacement, ta — thickness of the actuator, ra — modified density of the actuator, r — radial coordinate, and t — time. We express the strains in actuators by the radial displacement ∂u u ⑀ r⫽ ⑀t⫽ ∂r r
(2)
Constitutive equations of piezoelectric layers have the following form
冉 冉
冊 冊
u Ea ∂u s r⫽ ⫹n ⫺e3rE3 1−n2a ∂r a r
(3)
Ea u ∂u ⫹n ⫺e3tE3 1−n2a r a ∂r
(4)
du u ⫹e ⫹⑀ E dr 3t r 33 3
(5)
st⫽
D3⫽e3r
where d3r and d3t are transverse piezoelectric constants in the radial and circumferrential direction, respectively. The electric field acting in the direction perpendicular to the plate is denoted by E3, Ea and na are the modulus and Poisson ratio of the actuator, D3 denotes electrical displacement, and ⑀33 is a permittivity coefficient. Eliminating the normal stresses we rewrite the dynamic actuator equation in displacement as follows
再 冋
册
冎
Eata ∂ 1 ∂ ∂2u (ru) r ⫹E (e ⫺e ) ⫺tr⫽r t r 3 3t 3r a a 1−n2a ∂r r ∂r ∂t2
(6)
The equation of the transverse plate motion in the second section w2 is as follows ∂(Tr) ∗ ∂2w2 ⫽rp tpr 2 ∂r ∂t
(7)
where T is shear force, rp the modified plate density calculated according to the rule of mixture r∗p =rp+2ra[(ta)/(tp)], and tp is plate thickness. The balance of moments has the form ∂(Mrr) ⫺Mt⫺Tr⫹ttpr⫽0 ∂r
(8)
where Mr is radial moment, and Mt is circumferential moment. Using Hook’s law we express the moments by the plate transverse displacement in the following form
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
冉 冉
冊
Mr⫽⫺Dp
∂2w2 vp∂w2 ⫹ ∂r2 r ∂r
Mt⫽⫺Dp
1∂w2 ∂2w2 ⫹np 2 r ∂r ∂r
87
(9)
冊
(10)
where Dp=Ept3p/(12(1⫺n2)) is the plate cylindrical stiffness. Eliminating the transverse force we rewrite Eq. (7) in the transverse displacement 1∂(rtpr) ∗ ∂2w2 Dpⵜ4w2⫺ ⫹rp tp 2 ⫽0 r ∂r ∂t
(11)
where ⵜ4 operator has the form ⵜ4⫽
再 冋 冉 冊册冎
1∂ ∂1 ∂ ∂ r r r ∂r ∂r r ∂r ∂r
.
The dynamic extensional strain on the plate surface is calculated by considering the dynamic coupling between the piezolayer and the plate. Taking into account a perfect bonding the plate transverse displacement in the second section is related to the radial actuator displacement by u⫽⫺
tp∂w2 2 ∂r
(12)
Comparing the radial and circumferential strains on the interlayer surface we have ∂u tp∂2w2 ⑀ar⫽ ⫽⑀pr⫽⫺ ∂r 2 ∂r2
(13)
u tp1∂w2 ⑀at⫽ ⫽⑀pt⫽⫺ r 2 r ∂r
(14)
Differentiating Eq. (7) with respect to r, adding to Eq. (11) the terms including t vanish and the following equation is obtained for section 2 (Dp⫹Da)ⵜ4w2⫹r∗p tp
冉 冊
∂2w2 ratat2p 2 ∂2w2 ⫺ ⫽0 a⬍r⬍b ⵜ ∂t2 2 ∂t2
(15)
where the piezoelectric eliminator stiffness is denoted by Da=Eatat2p/2(1⫺n2a). The plate displacement equations for plate sections 1 and 3 have the classical form Dpⵜ4w1⫹rptp
∂2w1 ⫽0 R1⬍r⬍a ∂t2
(16)
Dpⵜ4w3⫹rptp
∂2w3 ⫽0 b⬍r⬍R2 ∂t2
(17)
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Equations (15), (16), and (17) are fourth order linear ordinary homogeneous differential equations. Thus we have to determine 12 constants C1,…,C12 from the boundary conditions. 3. Boundary and joint conditions The boundary conditions at r=R1 and r=R2 of the plate correspond to the clamped and free edge, respectively. Due to the kinematic excitation the inner edge is forced to move harmonically with the amplitude wo and angular velocity w. w1(R1)⫽w0sinwt
冉
Mr(R2)⫽⫺Dp
∂w1 (R )⫽0 ∂r 1
冊|
∂2w3 np∂w3 ⫹ ∂r2 r ∂r
r=R2
(18)
冉
冊|
∂ ∂2w3 1∂w3 ⫽0 T(R2)⫽⫺Dp ⫹ ∂r ∂r2 r ∂r
(19)
r=R2
⫽0 At the joints between sections 1 and 2 and between sections 2 and 3, the continuity in plate deflection, slope and radial moment have to be satisfied w1(a)⫽w2(a)
∂w1 ∂w2 (a)⫽ (a) ∂r ∂r
(20)
w2(b)⫽w3(b)
∂w2 ∂w3 (b)⫽ (b) ∂r ∂r
(21)
Mr1(a)⫽Mr2(a) Mr2(b)⫽Mr3(b)
(22)
The free stress condition for the piezoelectric annular element at r=a and r=b constrains the strains at these circles. Integrating the electric displacement D3 determined by the constitutive Eq. (5) the charge is obtained
冕冉
冊
b
tp Q⫽⫺ 2
e3r
d2w2 1dw2 ⫹e3t 2prdr⫺CpeV 2 dr r dr
a
(23)
where Cpe is the capacity of piezolectric elements, and V is the voltage. Taking into account the relation between the voltage and external capacity we have Q⫽CshV
(24)
where Csh is the external circuit capacity. Finally we determine the induced voltage by the harmonic plate displacement in the following way
冉
(Csh⫹Cpe)V⫽⫺tppe3r b
冊
dw2 dw2 (b)⫺a (a) ⫹tpp(e3r⫺e3t)(w2(b)⫺w2(a)) dr dr
(25)
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Eliminating the voltage from Eq. (3) by means of Eq. (25) the radial stresses in piezoelectric elements are as follows sr⫽⫺
冉
冊
冉
dw2 Eatp d2w2 nadw2 tppe23r dw2 (b)⫺a (a) ⫹⫺ b 2 2 ⫹ 2(1−na ) dr r dr (Cpe+Csh)ta dr dr
冊
(26)
tppe3r(e3r−e3t) ⫹ (w (b)⫺w2(a)) (Cpe+Csh)ta 2 Comparing the radial stresses at r=a and r=b to zero we have two boundary conditions depending on the external circuit capacity Csh. When the capacity Csh=0 the open circuit and boundary conditions are as follows
冉
Eatp d2w2 nadw2 ⫹ 2(1−n2a) dr2 r dr
冊
冊|
⫽ r=a
冉
Eatp d2w2 nadw2 ⫺ 2(1−n2a) dr2 r dr
冊|
r=b
冉
tppe23r dw2 (b) ⫽⫺ b Cpeta dr
(27)
dw2 tppe3r(e3r−e3t) ⫺a (a) ⫹ (w2(b)⫺w2(a))⫽0 dr Cpeta Csh→⬁ corresponds to the short circuit, and the boundary conditions have the form
冉
Eatp d2w2 nadw2 ⫹ 2(1−n2a) dr2 r dr
冊|
⫽ r=a
冉
Eatp d2w2 nadw2 ⫺ 2(1−n2a) dr2 r dr
冊|
⫽0
(28)
r=b
Equations (15), (16), (17) with boundary conditions (18), (19), joint conditions (20), (21), (22), free stress conditions (27) or (28) form a boundary value problem.
4. Analytical solution For single frequency excitation w w1(R1,t)⫽wosinwt
(29)
the steady-state solution is analysed. Equations (15), (16), and (17) become fourth order linear ordinary homogeneous differential equations with r dependent coefficients. The spatial term of solutions for w1, w2 and w3 are respectively w1⫽C1J0(r)⫹C2Y0(r)⫹C3I0(r)⫹C4K0(r)
(30)
w2⫽C5J0(ar)⫹C6Y0(ar)⫹C7I0(br)⫹C8K0(br)
(31)
w3⫽C9J0(r)⫹C10Y0(r)⫹C11I0(r)⫹C12K0(r)
(32)
where the zeroth order Bessel functions of the first and the second kind are denoted
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A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
by J0, Y0 and I0, K0, respectively. The wavenumbers , a and b are calculated from the following formulae
4⫽rptpw2/Dp
冋冪冉 冋冪冉
(33)
冊 冊
册 册
a2⫽12
2 4rptp 2 ratat2p ratat2p w2 + w2 w⫹ 2(Dp+Da) Dp+Da 2(Dp+Da)
(34)
b2⫽12
2 ratat2p 4rptp 2 ratat2p w⫺ w2 + w2 2(Dp+Da) Dp+Da 2(Dp+Da)
(35)
The 12 unknown constants in Eqs. (30)–(32) are determined by boundary conditions (18) and (19) at the plate edges and the joint conditions (20)–(22). For a given displacement amplitude wo and excitation frequency w the constants C=[C1,C2,…,C12]T can be calculated from nonhomogeneous linear algebraic equations. GC⫽⌳
(36)
where the matrix G is defined as follows
冤 冥 A 0 B
G⫽ C D 0
0 E F
冤
J0(R1)
A⫽
B⫽
冤
Y0(R1)
I0(R1) K0(R1)
−J1(R1) −Y1(R1) I1(R1) −K1(R1) 0
0
0
0
0
0
0
0
冥
0
0
0
0
0
0
0
0
冥
J0(R2)+ Y0(R2)+ −I0(R2)+ −K0(R2)+ np−1 np−1 np−1 np−1 + J ( R ) + Y ( R ) + I ( R ) + K ( R ) R 2 1 2 R2 1 2 R 2 1 2 R2 1 2
J1(R2)
Y1(R2)
I1(R2)
−K1(R2)
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
C⫽
冤
J0(a)
Y0(a)
I0(a)
K0(a)
−J1(a)
−Y1(a)
I1(a)
−K1(a)
冥
91
(J0(a)+ (Y0(a)+ (−I0(a)+ 2(−K0(a)+ np−1 np−1 np−1 np−1 + J1(a)) + Y1(a)) − I1(a)) + K (a)) a a a a 1 2
0
2
2
0
0
0
−J0(aa)
−Y0(aa)
−I0(ba)
−K0(ba)
J1(aa)
Y1(aa)
−I1(ba)
K1(ba)
2
2
a (J0(aa)+ a (Y0(aa)+ np−1 np−1 + J1(aa)) + Y (aa)) aa aa 1 D⫽ a2(J (ab)+ a2(Y0(ab)+ 0 −J0(aa)+ −Y0(aa)+ 1−np 1−np − J (ab)+ − Y (ab)+ ab 1 ab 1 1−np 1−np + J1(aa)) + Y (aa)) aa 1 aa
2
b (−I0(ba)+ b2(−K0(ba)+ np−1 np−1 + I1(ba)) + K (ba)) ba ba 1
Y0(ab)
I0(bb)
K0(bb)
−J1(ab)
−Y1(ab)
I1(bb)
−K1(bb)
冤
2
a (J0(ab)+ a (Y0(ab)+ b (−I0(bb)+ b2(−K0(bb)+ np−1 np−1 np−1 np−1 + J (ab)) + Y (ab)) + I (bb)) + K (bb)) ab 1 ab 1 bb 1 bb 1 E⫽ a2(−J0(ab)+ a2(−Y0(ab)+ b2(I0(bb)+ b2(K0(bb)+ 1−np 1−np 1−np 1−np + J1(ab))+ + Y1(ab))+ − I1(bb))+ + K (bb))+ ab ab bb bb 1 +ag(J1(aa)+ +ag(Y1(aa)+ −bg(I1(ba)+ +bg(K1(ba)+ −J1(ab)) −Y1(ab)) −I1(bb)) −K1(bb))
F⫽
2
b2(−I0(bb)+ b2(−K0(bb)+ +I0(ba)+ +K0(ba)+ 1−np 1−np + I (bb)+ − K (bb)+ bb 1 bb 1 1−np 1−np − I1(ba)) + K (ba)) ba ba 1
J0(ab) 2
−J0(b)
−Y0(b)
−I0(b)
−K0(b)
J1(b)
Y1(b)
−I1(b)
K1(b)
冥
2(J0(b)+ 2(Y0(b)+ 2(−I0(b)+ 2(−K0(b)+ np−1 np−1 np−1 np−1 + J1(b)) + Y1(b)) − I1(b)) K (b)) b b b b 1
0
0
0
0
0 is the fourth order zero matrix, and the input matrix
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A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
Table 1 Material parameters used in the calculations Material
Plate-steel
Actuator–PZTG-1195
Density (kg/m3) Modulus (N/m2) Thickness (m) Piezoelectric const. (m/V) Inner radius (m) Outer radius (m)
7800 21.6×1010 0.002 — 0.02 0.15
7275 63×109 0.0002 1.9×10−10 0.03 0.07
⌳⫽w0[1,0,0,0,0,0,0,0,0,0,0,0]T
5. Results and conclusions Numerical calculations based on the formulas presented in the previous sections are performed for a wide range of angular frequency and wo=0.001 m and e3r=e3t. To include the damping effect in the plate material the Voigt–Kelvin model with a complex Young’s modulus Ep and with a retardation time l=5×10−6 s is used in the numerical calculations. Due to a harmonic oscillation the Young’s modulus of the plate Ep is replaced by Ep(1+ilw), where i=√−1. The parameters of the plate and piezoelectric elements used in the calculations are listed in Table 1. Fig. 2 shows the response of the plate outer edge (displacement) to the harmonic displacement of the inner plate edge for the extreme values of shunting capacity
Fig. 2.
Plate transverse response 20 lg兩w兩 at the outer edge.
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
93
Fig. 3. Details of plate transverse response 20 lg兩w兩 near the first resonance region.
Csh=0, Csh → ⬁ corresponding to the open and short circuit, respectively. Fig. 3 demonstrates details of the response in the vicinity of w=7000 1/s. Both the open circuit and the short circuit plots are shown. A reduction of approximately 15 dB may be seen at a frequency of w=6800 1/s while going from a large value of shunting capacity to the open circuit. Approaching the open circuit the resonance frequency is approximately 8% larger than the short circuit resonance frequency (i.e. when Csh is very large compared to Cpe. Fig. 4 illustrates the effect of varying the shunt capacity on the spatial shape of forced vibration at w=7000 1/s. The open circuit and the short circuit correspond to Csh=0.01 Cpe and Csh=1000Cpe, respectively. The influence of a light material damping is negligible out of the resonance (cf. Fig. 5). A dynamic distributed model has been developed which is able to predict the behaviour of the piezoelectric vibration absorber. Electrically shunting piezoelectric annular plates with a capacitive electrical impedance have changed boundary con-
Fig. 4.
Spatial plate transverse response w in the first resonance region.
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A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
Fig. 5.
Influence of material damping on frequency characteristics.
ditions at the edges of the piezolayers, thus changing the dynamic frequency and spatial characteristics of the plate with an annular piezoelectric absorber.
Acknowledgements This work was supported by the State Committee for Scientific Research under KBN Grant No 7T07A04414.
References [1] Davis CL, Lesieutre GA. An actively-tuned solid state piezoelectric vibration absorber. Proc SPIE Conf Smart Struct 1998;3327:169–82. [2] Davis CL, Lesieutre GA, Dosch J. A tunable electrically shunted piezoceramic vibration absorber. Proc SPIE Conf Smart Struct 1997;3045:51–9. [3] Dimitriadis E, Fuller CR, Rogers CA. Piezoelectric actuators for distributed vibration excitation of thin plates. ASME J Appl Mech 1991;113:100–7. [4] Hagood NW, Von Flotov A. Damping of structural vibrations with piezoelectric materials and passive electric networks. J Sound Vibr 1991;146:243–68. [5] Kim SJ, Sonti VR, Jones JD. Equivalent forces and wavenumbers spectra of shaped piezoelectric actuators. In: Proceedings of the Second Conference on Recent Advances in Active Control of Sound and Vibration. Lancaster/Basel: Technomic Publications, 1993. p. 216–27. [6] Van Niekerk JL, Tongue BH, Packard AK. Active control of a circular plate to reduce transient noise transmission. J Sound Vibr 1995;183:643–62. [7] Tylikowski A, Hetnarski RB. Passive and active damping of thermally induced vibrations. In: Proc Third Internat Congr Thermal Stresses. Krako´w: Cracow Institute of Technology, 1999. p. 573–6. [8] Tylikowski A, Hetnarski RB. Stabilisation of mechanical systems with distributed piezoelectric sensors and actuators. Proc SPIE Annual Symp Math Contr Smart Struct 1998;3323:346–56.
Control of circular plate vibrations via piezoelectric actuators shunted with a capacitive circuit A. Tylikowski
*
Institute of Machine Design Fundamentals, Warsaw University of Technology, Narbutta 84, 02-524 Warsaw, Poland Received 19 September 2000; accepted 23 October 2000
Abstract The purpose of this research is to analyse capacitively shunting distributed piezoelectric elements perfectly glued to the vibrating annular plate excited by harmonic displacement of the inner plate edge. The piezoelement is described by constitutive equations relating stresses and electric displacements with strains and electric field. The equations are coupled with the equations of plate motion by the surface strain terms. At joint circles between annular sections without and with the piezolayers the continuity of plate deflection, slope and curvature is considered. The free stress condition for the piezoelement constrains the strain value at inner and outer piezoelement edges. Results from numerical simulation show an influence of the external shunting capacity on the frequency and space dependent plate response. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Circular plates; Axisymmetric vibrations; Piezoelectric actuators; Capacitive shunting; Distributed approach
1. Introduction Structure vibrations that propagate from engines or other sources may be reduced by passive and active isolation, by active control, or by passive and active vibration absorbers. Of particular interest to this research is the use of passive vibration absor-
* Fax: +48-22-660-8622. E-mail address: [email protected] (A. Tylikowski). 0263-8231/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 0 ) 0 0 0 5 5 - 0
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bers for structure-borne noise/vibration control. Passive vibration absorbers are conceptually simple devices consisting of a mass attached to a structure via a complex spring. The primary function of these devices is to increase the dynamic stiffness and therefore to change the natural absorber frequency. A shunting method has been developed for tuning the natural frequency of a piezoelectric element glued to beam surfaces. The passive vibration absorbers minimise vibration at a specific frequency related with a lightly damped structure. Large response reduction is only possible if the absorber is accurately tuned to the considered frequency. Therefore, an important feature of absorbers is the ability to be accurately tuned to the changing frequency. Tuning a mechanical absorber requires a change in either the mass or stiffness of the device. The electromechanical properties of a piezoceramic forcing element with an external passive electrical shunt circuit are used to alter the natural frequency [1]. An analytical distributed model of a piezoelectric vibration absorber was created to predict changes in natural frequency due to passive electrical shunting. Capacitive shunting alters the natural frequency of the actuators. A passive vibration absorber generally acts to minimise structural vibration at a specific frequency associated with the response of the lightly damped structural mode. This frequency is rarely stationary due to changing velocity. Maximum response reductions however are achived only if the absorber is lightly damped and accurately tuned to the frequency of concern. The vibration of two-dimensional structures excitated by a piezoelectric actuator has been modelled by Dimitriadis et al. [3] for rectangular plates and by Van Niekerk et al. [6] for circular plates. Van Niekerk, Tongue and Packard presented a comprehensive static model for a circular actuator and a coupled circular plate. Their static results were used to predict the dynamic behaviour of the coupled system, particularly to reduce acoustic transmissions. Piezoelectric transducers can be modelled as two-dimensional devices. The approach allows the distributed transducer shape to be included in the control design process for two-dimensional structures as an additional design parameter. The essence of the approach involves replacing the piezoactuators by forces and moments distributed along the piezoelements’ edges (e.g. Kim et al. [5]). The analytical basis for damping structural vibrations with piezoelectric materials and passive electrical circuits has been developed by Hagood and von Flotow [4]. A key feature of the tunable vibration absorber developed by Davis et al. [1,2] is the use of piezoelectric ceramic elements as part of the device stiffness. A discrete model of a piezoceramic vibration absorber was created to predict changes in natural frequency and damping due to passive electrical shunting. A comparison of passive and active damping of thermally induced vibrations of beams with piezolayers was given in [7]. Influence of an actuator shape on stabilisation of plate parametric vibrations has been developed in [8]. The goal of this research is to describe an annular piezoelectric vibration absorber of transverse vibrations of annular plate by using a consistent distributed model based on partial differential equations of system motion. The Kirchhoff annular plate of inner radius R1 and external radius R2 is divided into three sections and the dynamic behaviour of each section is analysed separately (cf. Fig. 1). The analysis is confined to axially symmetric modes. The absorbers are glued to the plate in the second
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section a⬍r⬍b. The dynamic extensional strain on the plate surface is calculated by considering the dynamic coupling between the actuator and the plate, and by taking into account a perfect bonding (infinite shear stiffness). The plate motion is described by partial differential equations in the plate transverse displacement. The piezoelement is described by constitutive equations relating stresses and electric displacements with strains and electric field. The equations are coupled with the equations of plate motion by the surface strain term. Along circles of connection between sections the continuity of plate deflection, slope and curvature is considered. The free normal stress condition for the piezoelectric element at the location r=a and r=b constraints the strain values at these two locations. The plate is excited by a motion at the internal edge. The displacement is harmonically varying with a constant frequency. For boundary conditions at clamped and free edges, the joint conditions form a boundary value problem. For single frequency excitation the equations of motion become fourth order ordinary homogeneous differential equations, with solutions in the form of travelling waves in the plate. Results from numerical simulation show an influence of the external shunting capacity on the frequency and space dependent plate response. The effect of changing both resonance frequency and amplitude is observed in going from the open external circuit to the short external circuit.
2. Dynamic equation of axisymmetric plate motion Fig. 1 shows a Kirchhoff thin annular plate with identical piezoceramic elements mounted on opposite sides of the plate. The annular piezolayers perpendicularly polarised to the plate surface are conected with an external passive electric shunt circuit, which can be used to alter the natural frequencies of the continuous structure. The effective bending stiffness of the plate is adjusted electrically using a capacitive shunt-circuit. The piezolayers are perfectly bonded to the plate, which implies strain continuity at the bonding interface. As the axisymmetrical motion is considered the plate is divided into three annular sections as shown in Fig. 1, and the motion of the plate in each section is analysed separately. Consider a finite element of radial length dr in the second section. The radial stresses in the piezolayers are assumed to be uniformly distributed in the direction
Fig. 1.
Geometry of the plate with actuators.
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perpendicular to the plate due to the small thickness. For the radial actuator motion the dynamics equation is as follows
冉
r
冊
∂sr ∂2 u ⫹sr⫺st ta⫺tr⫽ratar 2 ∂r ∂t
(1)
where sr — radial stress, st — circumferential stress, t — shear stress on the interface surface, u — radial displacement, ta — thickness of the actuator, ra — modified density of the actuator, r — radial coordinate, and t — time. We express the strains in actuators by the radial displacement ∂u u ⑀ r⫽ ⑀t⫽ ∂r r
(2)
Constitutive equations of piezoelectric layers have the following form
冉 冉
冊 冊
u Ea ∂u s r⫽ ⫹n ⫺e3rE3 1−n2a ∂r a r
(3)
Ea u ∂u ⫹n ⫺e3tE3 1−n2a r a ∂r
(4)
du u ⫹e ⫹⑀ E dr 3t r 33 3
(5)
st⫽
D3⫽e3r
where d3r and d3t are transverse piezoelectric constants in the radial and circumferrential direction, respectively. The electric field acting in the direction perpendicular to the plate is denoted by E3, Ea and na are the modulus and Poisson ratio of the actuator, D3 denotes electrical displacement, and ⑀33 is a permittivity coefficient. Eliminating the normal stresses we rewrite the dynamic actuator equation in displacement as follows
再 冋
册
冎
Eata ∂ 1 ∂ ∂2u (ru) r ⫹E (e ⫺e ) ⫺tr⫽r t r 3 3t 3r a a 1−n2a ∂r r ∂r ∂t2
(6)
The equation of the transverse plate motion in the second section w2 is as follows ∂(Tr) ∗ ∂2w2 ⫽rp tpr 2 ∂r ∂t
(7)
where T is shear force, rp the modified plate density calculated according to the rule of mixture r∗p =rp+2ra[(ta)/(tp)], and tp is plate thickness. The balance of moments has the form ∂(Mrr) ⫺Mt⫺Tr⫹ttpr⫽0 ∂r
(8)
where Mr is radial moment, and Mt is circumferential moment. Using Hook’s law we express the moments by the plate transverse displacement in the following form
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
冉 冉
冊
Mr⫽⫺Dp
∂2w2 vp∂w2 ⫹ ∂r2 r ∂r
Mt⫽⫺Dp
1∂w2 ∂2w2 ⫹np 2 r ∂r ∂r
87
(9)
冊
(10)
where Dp=Ept3p/(12(1⫺n2)) is the plate cylindrical stiffness. Eliminating the transverse force we rewrite Eq. (7) in the transverse displacement 1∂(rtpr) ∗ ∂2w2 Dpⵜ4w2⫺ ⫹rp tp 2 ⫽0 r ∂r ∂t
(11)
where ⵜ4 operator has the form ⵜ4⫽
再 冋 冉 冊册冎
1∂ ∂1 ∂ ∂ r r r ∂r ∂r r ∂r ∂r
.
The dynamic extensional strain on the plate surface is calculated by considering the dynamic coupling between the piezolayer and the plate. Taking into account a perfect bonding the plate transverse displacement in the second section is related to the radial actuator displacement by u⫽⫺
tp∂w2 2 ∂r
(12)
Comparing the radial and circumferential strains on the interlayer surface we have ∂u tp∂2w2 ⑀ar⫽ ⫽⑀pr⫽⫺ ∂r 2 ∂r2
(13)
u tp1∂w2 ⑀at⫽ ⫽⑀pt⫽⫺ r 2 r ∂r
(14)
Differentiating Eq. (7) with respect to r, adding to Eq. (11) the terms including t vanish and the following equation is obtained for section 2 (Dp⫹Da)ⵜ4w2⫹r∗p tp
冉 冊
∂2w2 ratat2p 2 ∂2w2 ⫺ ⫽0 a⬍r⬍b ⵜ ∂t2 2 ∂t2
(15)
where the piezoelectric eliminator stiffness is denoted by Da=Eatat2p/2(1⫺n2a). The plate displacement equations for plate sections 1 and 3 have the classical form Dpⵜ4w1⫹rptp
∂2w1 ⫽0 R1⬍r⬍a ∂t2
(16)
Dpⵜ4w3⫹rptp
∂2w3 ⫽0 b⬍r⬍R2 ∂t2
(17)
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Equations (15), (16), and (17) are fourth order linear ordinary homogeneous differential equations. Thus we have to determine 12 constants C1,…,C12 from the boundary conditions. 3. Boundary and joint conditions The boundary conditions at r=R1 and r=R2 of the plate correspond to the clamped and free edge, respectively. Due to the kinematic excitation the inner edge is forced to move harmonically with the amplitude wo and angular velocity w. w1(R1)⫽w0sinwt
冉
Mr(R2)⫽⫺Dp
∂w1 (R )⫽0 ∂r 1
冊|
∂2w3 np∂w3 ⫹ ∂r2 r ∂r
r=R2
(18)
冉
冊|
∂ ∂2w3 1∂w3 ⫽0 T(R2)⫽⫺Dp ⫹ ∂r ∂r2 r ∂r
(19)
r=R2
⫽0 At the joints between sections 1 and 2 and between sections 2 and 3, the continuity in plate deflection, slope and radial moment have to be satisfied w1(a)⫽w2(a)
∂w1 ∂w2 (a)⫽ (a) ∂r ∂r
(20)
w2(b)⫽w3(b)
∂w2 ∂w3 (b)⫽ (b) ∂r ∂r
(21)
Mr1(a)⫽Mr2(a) Mr2(b)⫽Mr3(b)
(22)
The free stress condition for the piezoelectric annular element at r=a and r=b constrains the strains at these circles. Integrating the electric displacement D3 determined by the constitutive Eq. (5) the charge is obtained
冕冉
冊
b
tp Q⫽⫺ 2
e3r
d2w2 1dw2 ⫹e3t 2prdr⫺CpeV 2 dr r dr
a
(23)
where Cpe is the capacity of piezolectric elements, and V is the voltage. Taking into account the relation between the voltage and external capacity we have Q⫽CshV
(24)
where Csh is the external circuit capacity. Finally we determine the induced voltage by the harmonic plate displacement in the following way
冉
(Csh⫹Cpe)V⫽⫺tppe3r b
冊
dw2 dw2 (b)⫺a (a) ⫹tpp(e3r⫺e3t)(w2(b)⫺w2(a)) dr dr
(25)
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89
Eliminating the voltage from Eq. (3) by means of Eq. (25) the radial stresses in piezoelectric elements are as follows sr⫽⫺
冉
冊
冉
dw2 Eatp d2w2 nadw2 tppe23r dw2 (b)⫺a (a) ⫹⫺ b 2 2 ⫹ 2(1−na ) dr r dr (Cpe+Csh)ta dr dr
冊
(26)
tppe3r(e3r−e3t) ⫹ (w (b)⫺w2(a)) (Cpe+Csh)ta 2 Comparing the radial stresses at r=a and r=b to zero we have two boundary conditions depending on the external circuit capacity Csh. When the capacity Csh=0 the open circuit and boundary conditions are as follows
冉
Eatp d2w2 nadw2 ⫹ 2(1−n2a) dr2 r dr
冊
冊|
⫽ r=a
冉
Eatp d2w2 nadw2 ⫺ 2(1−n2a) dr2 r dr
冊|
r=b
冉
tppe23r dw2 (b) ⫽⫺ b Cpeta dr
(27)
dw2 tppe3r(e3r−e3t) ⫺a (a) ⫹ (w2(b)⫺w2(a))⫽0 dr Cpeta Csh→⬁ corresponds to the short circuit, and the boundary conditions have the form
冉
Eatp d2w2 nadw2 ⫹ 2(1−n2a) dr2 r dr
冊|
⫽ r=a
冉
Eatp d2w2 nadw2 ⫺ 2(1−n2a) dr2 r dr
冊|
⫽0
(28)
r=b
Equations (15), (16), (17) with boundary conditions (18), (19), joint conditions (20), (21), (22), free stress conditions (27) or (28) form a boundary value problem.
4. Analytical solution For single frequency excitation w w1(R1,t)⫽wosinwt
(29)
the steady-state solution is analysed. Equations (15), (16), and (17) become fourth order linear ordinary homogeneous differential equations with r dependent coefficients. The spatial term of solutions for w1, w2 and w3 are respectively w1⫽C1J0(r)⫹C2Y0(r)⫹C3I0(r)⫹C4K0(r)
(30)
w2⫽C5J0(ar)⫹C6Y0(ar)⫹C7I0(br)⫹C8K0(br)
(31)
w3⫽C9J0(r)⫹C10Y0(r)⫹C11I0(r)⫹C12K0(r)
(32)
where the zeroth order Bessel functions of the first and the second kind are denoted
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by J0, Y0 and I0, K0, respectively. The wavenumbers , a and b are calculated from the following formulae
4⫽rptpw2/Dp
冋冪冉 冋冪冉
(33)
冊 冊
册 册
a2⫽12
2 4rptp 2 ratat2p ratat2p w2 + w2 w⫹ 2(Dp+Da) Dp+Da 2(Dp+Da)
(34)
b2⫽12
2 ratat2p 4rptp 2 ratat2p w⫺ w2 + w2 2(Dp+Da) Dp+Da 2(Dp+Da)
(35)
The 12 unknown constants in Eqs. (30)–(32) are determined by boundary conditions (18) and (19) at the plate edges and the joint conditions (20)–(22). For a given displacement amplitude wo and excitation frequency w the constants C=[C1,C2,…,C12]T can be calculated from nonhomogeneous linear algebraic equations. GC⫽⌳
(36)
where the matrix G is defined as follows
冤 冥 A 0 B
G⫽ C D 0
0 E F
冤
J0(R1)
A⫽
B⫽
冤
Y0(R1)
I0(R1) K0(R1)
−J1(R1) −Y1(R1) I1(R1) −K1(R1) 0
0
0
0
0
0
0
0
冥
0
0
0
0
0
0
0
0
冥
J0(R2)+ Y0(R2)+ −I0(R2)+ −K0(R2)+ np−1 np−1 np−1 np−1 + J ( R ) + Y ( R ) + I ( R ) + K ( R ) R 2 1 2 R2 1 2 R 2 1 2 R2 1 2
J1(R2)
Y1(R2)
I1(R2)
−K1(R2)
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
C⫽
冤
J0(a)
Y0(a)
I0(a)
K0(a)
−J1(a)
−Y1(a)
I1(a)
−K1(a)
冥
91
(J0(a)+ (Y0(a)+ (−I0(a)+ 2(−K0(a)+ np−1 np−1 np−1 np−1 + J1(a)) + Y1(a)) − I1(a)) + K (a)) a a a a 1 2
0
2
2
0
0
0
−J0(aa)
−Y0(aa)
−I0(ba)
−K0(ba)
J1(aa)
Y1(aa)
−I1(ba)
K1(ba)
2
2
a (J0(aa)+ a (Y0(aa)+ np−1 np−1 + J1(aa)) + Y (aa)) aa aa 1 D⫽ a2(J (ab)+ a2(Y0(ab)+ 0 −J0(aa)+ −Y0(aa)+ 1−np 1−np − J (ab)+ − Y (ab)+ ab 1 ab 1 1−np 1−np + J1(aa)) + Y (aa)) aa 1 aa
2
b (−I0(ba)+ b2(−K0(ba)+ np−1 np−1 + I1(ba)) + K (ba)) ba ba 1
Y0(ab)
I0(bb)
K0(bb)
−J1(ab)
−Y1(ab)
I1(bb)
−K1(bb)
冤
2
a (J0(ab)+ a (Y0(ab)+ b (−I0(bb)+ b2(−K0(bb)+ np−1 np−1 np−1 np−1 + J (ab)) + Y (ab)) + I (bb)) + K (bb)) ab 1 ab 1 bb 1 bb 1 E⫽ a2(−J0(ab)+ a2(−Y0(ab)+ b2(I0(bb)+ b2(K0(bb)+ 1−np 1−np 1−np 1−np + J1(ab))+ + Y1(ab))+ − I1(bb))+ + K (bb))+ ab ab bb bb 1 +ag(J1(aa)+ +ag(Y1(aa)+ −bg(I1(ba)+ +bg(K1(ba)+ −J1(ab)) −Y1(ab)) −I1(bb)) −K1(bb))
F⫽
2
b2(−I0(bb)+ b2(−K0(bb)+ +I0(ba)+ +K0(ba)+ 1−np 1−np + I (bb)+ − K (bb)+ bb 1 bb 1 1−np 1−np − I1(ba)) + K (ba)) ba ba 1
J0(ab) 2
−J0(b)
−Y0(b)
−I0(b)
−K0(b)
J1(b)
Y1(b)
−I1(b)
K1(b)
冥
2(J0(b)+ 2(Y0(b)+ 2(−I0(b)+ 2(−K0(b)+ np−1 np−1 np−1 np−1 + J1(b)) + Y1(b)) − I1(b)) K (b)) b b b b 1
0
0
0
0
0 is the fourth order zero matrix, and the input matrix
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Table 1 Material parameters used in the calculations Material
Plate-steel
Actuator–PZTG-1195
Density (kg/m3) Modulus (N/m2) Thickness (m) Piezoelectric const. (m/V) Inner radius (m) Outer radius (m)
7800 21.6×1010 0.002 — 0.02 0.15
7275 63×109 0.0002 1.9×10−10 0.03 0.07
⌳⫽w0[1,0,0,0,0,0,0,0,0,0,0,0]T
5. Results and conclusions Numerical calculations based on the formulas presented in the previous sections are performed for a wide range of angular frequency and wo=0.001 m and e3r=e3t. To include the damping effect in the plate material the Voigt–Kelvin model with a complex Young’s modulus Ep and with a retardation time l=5×10−6 s is used in the numerical calculations. Due to a harmonic oscillation the Young’s modulus of the plate Ep is replaced by Ep(1+ilw), where i=√−1. The parameters of the plate and piezoelectric elements used in the calculations are listed in Table 1. Fig. 2 shows the response of the plate outer edge (displacement) to the harmonic displacement of the inner plate edge for the extreme values of shunting capacity
Fig. 2.
Plate transverse response 20 lg兩w兩 at the outer edge.
A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
93
Fig. 3. Details of plate transverse response 20 lg兩w兩 near the first resonance region.
Csh=0, Csh → ⬁ corresponding to the open and short circuit, respectively. Fig. 3 demonstrates details of the response in the vicinity of w=7000 1/s. Both the open circuit and the short circuit plots are shown. A reduction of approximately 15 dB may be seen at a frequency of w=6800 1/s while going from a large value of shunting capacity to the open circuit. Approaching the open circuit the resonance frequency is approximately 8% larger than the short circuit resonance frequency (i.e. when Csh is very large compared to Cpe. Fig. 4 illustrates the effect of varying the shunt capacity on the spatial shape of forced vibration at w=7000 1/s. The open circuit and the short circuit correspond to Csh=0.01 Cpe and Csh=1000Cpe, respectively. The influence of a light material damping is negligible out of the resonance (cf. Fig. 5). A dynamic distributed model has been developed which is able to predict the behaviour of the piezoelectric vibration absorber. Electrically shunting piezoelectric annular plates with a capacitive electrical impedance have changed boundary con-
Fig. 4.
Spatial plate transverse response w in the first resonance region.
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A. Tylikowski / Thin-Walled Structures 39 (2001) 83–94
Fig. 5.
Influence of material damping on frequency characteristics.
ditions at the edges of the piezolayers, thus changing the dynamic frequency and spatial characteristics of the plate with an annular piezoelectric absorber.
Acknowledgements This work was supported by the State Committee for Scientific Research under KBN Grant No 7T07A04414.
References [1] Davis CL, Lesieutre GA. An actively-tuned solid state piezoelectric vibration absorber. Proc SPIE Conf Smart Struct 1998;3327:169–82. [2] Davis CL, Lesieutre GA, Dosch J. A tunable electrically shunted piezoceramic vibration absorber. Proc SPIE Conf Smart Struct 1997;3045:51–9. [3] Dimitriadis E, Fuller CR, Rogers CA. Piezoelectric actuators for distributed vibration excitation of thin plates. ASME J Appl Mech 1991;113:100–7. [4] Hagood NW, Von Flotov A. Damping of structural vibrations with piezoelectric materials and passive electric networks. J Sound Vibr 1991;146:243–68. [5] Kim SJ, Sonti VR, Jones JD. Equivalent forces and wavenumbers spectra of shaped piezoelectric actuators. In: Proceedings of the Second Conference on Recent Advances in Active Control of Sound and Vibration. Lancaster/Basel: Technomic Publications, 1993. p. 216–27. [6] Van Niekerk JL, Tongue BH, Packard AK. Active control of a circular plate to reduce transient noise transmission. J Sound Vibr 1995;183:643–62. [7] Tylikowski A, Hetnarski RB. Passive and active damping of thermally induced vibrations. In: Proc Third Internat Congr Thermal Stresses. Krako´w: Cracow Institute of Technology, 1999. p. 573–6. [8] Tylikowski A, Hetnarski RB. Stabilisation of mechanical systems with distributed piezoelectric sensors and actuators. Proc SPIE Annual Symp Math Contr Smart Struct 1998;3323:346–56.