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Distributed microscopic actuation analysis of paraboloidal membrane shells of different geometric parameters
Mechanical Systems and Signal Processing 103 (2018) 1–22
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Distributed microscopic actuation analysis of paraboloidal membrane shells of different geometric parameters Honghao Yue a, Yifan Lu a,⇑, Zongquan Deng a, Hornsen Tzou b a b
School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang Province 150001, PR China College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China
a r t i c l e
i n f o
Article history: Received 17 August 2017 Received in revised form 20 September 2017 Accepted 1 October 2017
Keywords: Paraboloidal membrane shell Geometric parameter effects Modal control Microscopic actuation Optimal actuator placement
a b s t r a c t Paraboloidal membrane shells of revolution are commonly used as key components for advanced aerospace structures and aviation mechanical systems. Due to their high flexibility and low damping property, active vibration control is of significant importance for these in-orbit membrane structures. To explore the dynamic control behavior of space flexible paraboloidal membrane shells, precision distributed actuation and control effectiveness of free-floating paraboloidal membrane shells with piezoelectric actuators are investigated. Governing equations of the shell structronic system are presented first. Then, distributed control forces and control actions are formulated. A transverse mode shape function of the paraboloidal shell based on the membrane approximation theory and specified boundary condition is assumed in the modal control force analysis. The actuator induced modal control forces on the paraboloidal shell are derived. The expressions of microscopic local modal control forces are obtained by shrinking the actuator area into infinitesimal and the four control components are investigated respectively to predict the spatial microscopic actuation behavior. Geometric parameter (height-radius ratio and shell thickness) effects on the modal actuation behavior are explored when evaluating the micro-control efficiency. Four different cases are discussed and the results reveal the fact that shallow (e.g., antennas/reflectors) and deep (e.g., rocket/missile fairing) paraboloidal shells exhibit totally different modal actuation behaviors due to their curvature differences. Analytical results in this paper can serve as guidelines for optimal actuator placement for vibration control of different paraboloidal structures. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Many advanced aerospace structures and optical systems, such as reflectors, mirrors, antennas, rocket fairings and nozzles, belong to the category of double-curvature paraboloidal shell structures [1]. Due to weight restrictions in launching and space operation, these paraboloidal shells are mostly lightweight and flexible, which tend to exhibit prolonged oscillations and thus influence their precision and accuracy. Accordingly, precision control of structural vibration is of considerable importance not only to their stringent performance requirement, but also to their structural integrity and long-term reliability [2–4]. Accordingly, this study is to investigate the distributed actuation effectiveness of flexible membrane paraboloidal shells and to explore the optimal actuator placement on paraboloidal shell structures. Actuation characteristics of shallow and deep paraboloidal shells are evaluated respectively. ⇑ Corresponding author. E-mail address: [email protected] (Y. Lu). https://doi.org/10.1016/j.ymssp.2017.10.005 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Development of ‘‘smart structures and structronic systems” has demonstrated that various smart materials and control techniques can be applied to control static shapes and/or undesirable vibrations of structures and systems [5,6]. Distributed control of plate and shell structures based on the smart structure technology has been quickly developing in the last twenty years [7–12]. Among various smart materials and structures, the distributed control technique using piezoelectric actuators is widely investigated and applied to engineering systems, ranging from micro-/nano-devices to large aerospace structures. Distributed sensing and control signals of deformable plate membrane mirrors with different pretension forces were investigated by Lu et al. [13,14]. The equations of motion of a free-floating paraboloidal shell laminated with a piezoelectric actuator layer were derived and modal control effectiveness of the actuators was explored by Yue et al. [15]. Distributed control actions of thin cylindrical shells with various piezoelectric [16–18] and electro-optic photostrictive [19] actuators were widely investigated. Distributed modal voltages and their spatial strain characteristics of toroidal shells were investigated by Tzou et al. [20]. Static and dynamic behaviors of spherical and conical shells with piezoelectric actuators were also comprehensively studied [21–24]. Micro-control actions and distributed control effectiveness of segmented actuator patches laminated on hemispheric shells was evaluated by Smithmaitrie et al. [25]. Distributed modal voltage and micro-control characteristics of a simply supported paraboloidal shell were explored by Ding et al. [26,27]. Zhang and Yue et al. experimentally investigated the dynamic vibration control of paraboloidal membrane shells with different control strategies respectively [28–30]. Hu et al. studied the microscopic actuation and optimal actuator locations of parabolic cylindrical shells [31]. In this study, the spatially distributed actuation behavior of piezoelectric actuators on free paraboloidal membrane shells is investigated. The modal control force induced by an arbitrary piezoelectric actuator patch is derived and the microscopic actuation effectiveness is investigated by shrinking the actuator area into infinitesimal. Geometric parameter effects on the microscopic actuation efficiency are taken into account. Paraboloidal membrane shells with two different height-radius ratios, i.e., the shallow and the deep shell, are discussed respectively. Deep/shallow paraboloidal membrane shells with different thickness are also investigated to reveal the influence of shell rigidity on modal control actions. 2. Modal control of paraboloidal membrane shell structronic systems In this section, the governing control equations of a generic paraboloidal structronic shell laminated with a piezoelectric actuator layer are derived. Expression of the modal control force on a paraboloidal shell is formulated by modal expansion method. 2.1. Modeling of piezoelectric laminated paraboloidal membrane shells A structronic system is composed of sensor, actuator, control electronics and elastic structures. Fig. 1 illustrates a generic paraboloidal structronic shell, i.e., an elastic shell laminated with an actuator layer, defined in a tri-orthogonal global coordinate system - XYZ. The neutral surface of the elastic double-curvature shell is defined in a tri-orthogonal curvilinear coordinate system (/, w, a3) where / defines the meridional direction, w the circumferential direction and a3 the transverse direction. d/ denotes an infinitesimal angular change in the meridian direction and dw denotes an infinitesimal angular change in the circumferential direction. Furthermore, ‘‘a” is the radial distance and ‘‘c” is the meridian height at the pole. The Lamé parameters are A1 = b/cos3/ and A2 = (bsin/)/cos/. The meridional radius of curvature is R/ = b/cos3/ and the circumferential radius is Rw = b/cos/, where constant b = a2/2c = 2f, and f is the focal length. The actuator layer, with thickness ha, is perfectly bonded on the shell surface. It is assumed that the actuator is made of bi-axially sensitive piezoelectric material, such that a transverse control voltage can introduce two in-plane normal strains applied to distributed actuation and control of shells. Considering that shallow and deep paraboloidal membrane shells have distinct curvatures and characteristics which are utilized in different aerospace structures and systems, a shallow and a deep paraboloidal shell with a set of distributed actuator patches laminated on its outer surface is shown in Fig. 2(a) and (b), respectively.
Z
Z
Actuator layer 3 2=
Rψsin dψ
ψ
1=
dψ
c
Rψsin Rψ a R
Y
X O2 O1
Y
d *
(Not to scale)
Fig. 1. A generic paraboloidal structronic shell with an actuator layer.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Z
Z
Actuator patches
Δφ
c=1 Δψ
a=2 Y
Y X
φ*=0.7854rad
(a) Z
Z
φ
Actuator patches c=2
Y Y
a=1
Δψ
Y
φ*=1.3258rad
(b) Fig. 2. (a) A shallow paraboloidal shell with distributed actuator patches. (b) A deep paraboloidal shell with distributed actuator patches.
2.2. Governing control equations As discussed above, a bi-axially oriented piezoelectric actuator induces two normal control strains when a transverse control voltage is applied across the actuator thickness due to the converse piezoelectric effect. Since the actuator layer is placed away from the neutral surface, these strains not only translate to in-plane control forces, but also induce counteracting control moments. These control forces and moments can be used to control static shape or to counteract dynamic oscillations of the shell. Since the piezoelectric layer is thin as compared with the paraboloidal shell, the elastic and mass properties of the actuators do not affect the elastic shell. The paraboloidal membrane shell system equations can be defined as
cos / @ cos4 / @ cos2 / m cos3 / a €/ ðNm Þ þ ½ðNm ðN Naww Þ þ Q /3 þ q/ ¼ qhu // N // Þ tan / b sin / @w w/ b sin / @/ b sin / ww b
ð1Þ
cos / @ cos4 / @ cos2 / m cos / €w ðNm Naww Þ þ ðNm tan /Þ N þ Q w3 þ qw ¼ qhu b sin / @w ww b sin / @/ /w b sin / w/ b
ð2Þ
cos / @ cos4 / @ cos3 / m cos / m €3 Q w3 þ ðQ /3 tan /Þ ðN// Na// Þ ðNww Naww Þ þ q3 ¼ qhu b sin / @w b sin / @/ b b
ð3Þ
m a a where N m ij and M ij are the mechanical forces and moments; N ij are the actuator induced forces; M ij are the actuator induced moments; qhüi denotes the inertia force effect; q is the mass density; h is the shell thickness; qi is the input force; and the superscripts m and a respectively denote the mechanical (or elastic) and actuator induced components. The transverse shear effects Q/3 and Qw3 are defined as
Q /3 ¼
cos4 / @ m @ a m a 2 sec3 / M w/ þ ½ðMm M Þ tan / ðM M Þ sec / // // ww ww b sin / @w @/
ð4Þ
Q w3 ¼
cos4 / @ @ a m m 2 sec3 / ðM m ðM M Þ þ tan /Þ þ M sec / ww w/ b sin / @w ww @/ w/
ð5Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22 Table 1 Four cases of paraboloidal membrane shells with piezoelectric actuators. Case
Shell structure
1a 1b 2a 2b
Actuator patch
Height c (m)
Radius a (m)
Thickness h (m)
Thickness ha (mm)
1
2
0.04
2
1
0.001 0.002 0.001 0.002
Table 2 Physical properties of PVDF film. Parameters
Value
Unit
Young’s modulus Yp Tensile strength rb Poisson’s ratio mp Mass density q Piezoelectric strain constant d31, d32
2500 200 0.29 1.78 103 21
MPa MPa / kg/m3 pC/N
0.5 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -0.5 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0
(rad)
(b) (Tk )bend ,ψ
(a) (Tk )bend ,φ
500 (T)mem,ψ
500 (T)mem,
0 0
0 -500 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
1000 0 -1000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 3. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 1.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Based on the membrane approximation, the mechanical moments, i.e., the M m ij terms, are neglected in the governing equations. Note that although mechanical moments are neglected, control moments induced by the piezoelectric actuator are still preserved in order to evaluate the overall control actions and microscopic contributing components of actuator patches on the paraboloidal membrane shell. Since this study focuses on the microscopic control actions of actuator patches laminated on the paraboloidal shell, not the shell dynamics, only actuation component expressions are given. According to the thin shell theory, the in-plane shear effects of piezoelectric patch are not considered, namely N aij 0 and M aij 0. Thus, the actuation forces and moments can be expressed as
Na// ¼ d3/ Y p /a ð/; w; tÞ
ð6Þ
Naww ¼ d3w Y p /a ð/; w; tÞ
ð7Þ
Ma// ¼ ra/ d3/ Y p /a ð/; w; tÞ
ð8Þ
Maww ¼ raw d3w Y p /a ð/; w; tÞ
ð9Þ
where d3i is the piezoelectric strain constant; Yp is Young’s modulus of the piezoelectric actuator; ra i is the moment arm, i.e., the distance from shell’s neutral surface to the actuator’s mid plane; and /a(/, w, t) is the control signal.
1 (T)bend,ψ
(T)bend,
1 0 -1 2
*
0 -1 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(rad)
(b) (Tk )bend ,ψ
(a) (Tk )bend ,φ
500 (T)mem,ψ
500 (T)mem,
0 0
0 -500 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0
0 0
(rad)
(d) (Tk )mem,ψ
(c) (Tk )mem,φ
(T)T otal
1000 0 -1000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 4. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
500 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
1000 0 -1000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 5. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 3.
Although the closed-form system equations of paraboloidal shells are defined, the analytical solution procedure of modal analysis is very complicated. Mode shape functions are needed in evaluating spatial modal actuation behaviors of shells. Thus, an assumed transverse mode shape function satisfying the free boundary conditions is selected in Section 3 to evaluate the distributed control effects and micro-control actions of the piezoelectric actuator patches. Note that when evaluating the control actions solely contributed by actuator patches, the mechanical forces and moments are unnecessarily considered in this study. 2.3. Modal control force According to the modal expansion method, the dynamic response of the paraboloidal membrane shell is composed of all participating modes [32]
ui ð/; w; tÞ ¼
1 X
gk ðtÞU ik ð/; wÞ; i ¼ /; w; 3
ð10Þ
k¼1
where gk(t) is the k-th modal participating factor or the k-th modal coordinate and Uik(/, w) is the k-th mode shape function in the i-th direction. By applying the modal orthogonality and substituting the modal expansion into the simplified piezoelectric shell equations, the modal control equation of the paraboloidal shell can be derived as
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
500 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 6. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 4.
g€ k þ 2fk xk g_ k þ x2k gk ¼ F^mk þ F^ck
ð11Þ
^c where fk is the k-th modal damping ratio; xk is the k-th natural frequency; F^m k and F k are the modal excitation contributed by the mechanical force and the piezoelectric actuator, respectively. This study aims to investigate the spatially distributed micro-control actions of piezoelectric patches at various locations of the shell, the mechanical excitation is therefore assumed zero. Expression of the modal control force can be defined as
F^ck ¼
1 qhNk
where N k ¼
Z Z
2
½Lc/ ð/3 ÞU /k þ Lcw ð/3 ÞU wk þ Lc3 ð/3 ÞU 3k w
/
b sin / d/dw cos4 /
ð12Þ
R R P3 2 / ½ i¼1 U 2ik ð/; wÞ bcossin 4 / d/dw and the Love’s control operator is defined as a function of /3. w /
Lc/ ð/3 Þ ¼
a a cos3 / @N // cos6 / @M// 2 b @/ @/ b
ð13Þ
Lcw ð/3 Þ ¼
a a cos / @Nww cos2 / @Mww 2 b sin / @w b sin / @w
ð14Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
0.5 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -0.5 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 7. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 1.
Lc3 ð/3 Þ ¼
2 a cos6 / @ M //
b
2
@/2
þ
2 a cos / @ M ww 2
2 2 2 b sin / @w
a 2 3 cos5 / sin / 2 cos5 / @M// cos4 / a M// þ 2 2 @/ b sin / b
a cos5 / @M ww cos4 / a cos3 / a cos / a N// þ Nww þ 2 M ww þ 2 b b @/ b sin / b
ð15Þ
3. Distributed actuation effectiveness of paraboloidal membrane shells Based on the membrane approximation theory, the mode shape functions of a paraboloidal membrane shell with free boundary condition have been assumed and validated [15].
U /k ¼ Ak cos
ð2k þ 1Þp kþ1 /ðsin /Þ cos kw 2/ kþ1
U wk ¼ Ak cos /ðsin /Þ
sin kw k
U 3k ¼ Ak ðk þ 1Þ cos /ðsin /Þ cos kw
ð16Þ ð17Þ ð18Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
where Ak, the modal amplitude constant of mode k, is chosen so that
Z Z
2
U 23k w
/
b sin / d/dw ¼ 1 cos4 /
ð19Þ
and / is the meridional angle measured from the pole varying from 0 (pole) to /⁄ (boundary rim) for a shell with a closed apex; w defines the circumferential angle which varies from 0 to 2p for a closed paraboloidal shell. Substituting the Love’s control operators and the assumed mode shape functions into Eq. (12), the modal control force can be redefined as
Y p d3i /a ðtÞAk Y p d3i /a ðtÞAk F^ck ¼ Tk ¼ ðT k
q
q
meri
þ Tk
cir
þ Tk
trans Þ
ð20Þ
where Tk denotes the total control action induced by the actuator, and Tk_meri, Tk_cir and Tk_trans are the control actions respectively in the meridional, circumferential and transverse directions. It has been proved that for plates and very shallow shells, the transverse deformation and oscillation usually dominates. This paper mainly focus on the micro-control actions that
1 (T)bend,ψ
(T)bend,
1 0 -1 2
*
0 -1 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 8. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
counteract the transverse oscillation, control forces in meridional and circumferential directions are neglected in derivations for simplicity. Note that this may be inaccurate if one wants to analyze the total dynamic control behavior of deep shells with significant curvature. However, the same procedure can be easily applied to either the meridional or the circumferential vibrations. In the following derivations, Tk_meri and Tk_cir are omitted and Tk_trans is aliased as (Tk)Total. An arbitrary segmented actuator patch ranging from /1 to /2 in the meridional direction and from w1 to w2 in the circumferential direction is assumed laminated on the surface of the shell (see Fig. 2). Since the piezoelectric actuator patch is ultrathin and lightweight as compared with the elastic shell, material and stiffness of the actuator layers are neglected in the shell system equations. The electrode resistance is neglected and the potential is constant over the effective electrode surface of a piezoelectric patch. Thus, the actuator control signal applied to a segmented actuator patch is uniformly distributed which can be defined by the patch boundaries as
/a ð/; w; tÞ ¼ /a ðtÞ½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ
ð21Þ
where us() is the unit step function: us ðx xi Þ ¼ 1 when x > xi , and us ðx xi Þ ¼ 0 when x < xi . /a(t) is the imposed control voltage. Substituting Eqs. (15), (18) in Eq. (12) and taking into account Eqs. (6)–(9), (19) and (21), the modal control force induced by the actuator patch can be obtained as
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
100 (T)mem,
0 0
0 -100 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 9. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 3.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ra cos6 / @ ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ 2 @/ q hN k w / b 2 3 cos5 / sin / 2 cos5 / ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ 2 b sin / cos4 / ½u ð/ / Þ u ð/ / Þ½u ðw w Þ u ðw w Þ þ s s s s 1 2 1 2 2 b ra cos2 / @ þ ½us ð/ /1 Þ us ð/ /2 Þ 2 ½dðw w1 Þ dðw w2 Þ 2 hNk @w b sin / cos5 / ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ 2 b sin / cos4 / ½u ð/ / Þ u ð/ / Þ½u ðw w Þ u ðw w Þ s s s s 1 2 1 2 2 b 1 cos3 / ½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ hNk b 1 cos / ½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ hNk b 2 b sin / k ðk þ 1Þ cos /ðsin /Þ cos kw d/dw cos4 /
Y p d3i /a ðtÞAk F^ck ¼
Z Z
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
0 0
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
100 (T)mem,
ð22Þ
0 -100 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 10. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 4.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
where d() is the Dirac delta function and dðx xi Þ ¼ 1 when x ¼ xi , and dðx xi Þ ¼ 0 when x – xi . It is assumed that the actuator is made of hexagonal piezoelectric materials, i.e., d3/ = d3w = d3i; both the actuator and the shell have uniform thickness, i.e., r a/ ¼ r aw ¼ r a h=2 One can find that the total modal control force consists of four contributing components resulting from both membrane forces and bending moments.
Y p d3i /a ðtÞAk ^ Y p d3i /a ðtÞAk ^ F^ck ¼ ½ðT k ÞTotal ¼ ½ðT k Þbend;/ þ ðT^ k Þbend;w þ ðT^ k Þmem;/ þ ðT^ k Þmem;w
q
ð23Þ
q
where ðT^ k ÞTotal denotes the total modal control force; ðT^ k Þbend;/ and ðT^ k Þbend;w denote the two components induced by the bendand ðT^ k Þ denote the two components induced by the membrane forces in / ing moments in / and w directions; ðT^ k Þ mem;/
mem;w
and w directions, respectively. Accordingly, these four components can be further expressed as
ra ðk þ 1Þ h
Z Z @ kþ1 ½dð/ /1 Þ dð/ /2 Þþ ðsin /Þ cos3 /½us ðw w1 Þ us ðw w2 Þ @/ w /
ðsin /Þ cos2 /ð3 sin / 2Þ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þþ kþ1 ðsin /Þ cos /½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ cos kwd/dw k
2
20 (T)bend,ψ
(T)bend,
50 0 -50 2
0 -20 2
*
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
0 0
(rad)
(b) (Tk )bend ,ψ
5000 (T)mem,ψ
5000 (T)mem,
0 -5000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ 4
x 10 1 (T)T otal
ðT^ k Þbend;/ ¼
0 -1 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 11. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 1.
ð24Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ðT^ k Þbend;w ¼
ra ðk þ 1Þ h
Z Z @ k1 ½dðw w1 Þ dðw w2 Þþ ðsin /Þ ðcos /Þ1 ½us ð/ /1 Þ us ð/ /2 Þ @w w /
ðsin /Þ cos2 /½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ
ð25Þ
k
ðsin /Þ ðT^ k Þmem;/ ¼ ðT^ k Þmem;w ¼
bðk þ 1Þ h bðk þ 1Þ h
Z Z w
kþ1
cos /½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ cos kwd/dw
½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þðsin /Þ
kþ1
cos kwd/dw
ð26Þ
/
Z Z w
½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ /
kþ1
ðsin /Þ cos2 /
cos kwd/dw
ð27Þ
When the actuator patch moves along the /-direction with the same spatial interval, namely D/ remains a constant, the actuator area also varies correspondingly due to the curvature difference. In order to explore the modal control capability per actuator area, modal control force induced by an actuator patch is normalized by dividing it by the corresponding actuator area as
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(b) (Tk )bend ,ψ
5000 (T)mem,ψ
2000 (T)mem,
0 0
(rad)
0 -2000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 12. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ðF^ck Þeff ¼
F^ck Sa
ð28Þ
where Sa denotes the effective area of the actuator patch
Z
Sa ¼ S
½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 ÞdS
ð29Þ
Eq. (28) gives the expression of a so-called average control effect. Then the actuator area Sa is shrunken into infinitesimal so that modal control force at an arbitrary position can be precisely investigated. The expression of the microscopic modal control force can be defined as
Y p d3i /a ðtÞAk ~ Y p d3i /a ðtÞAk ~ F~ck ¼ ½ðT k ÞTotal ¼ ½ðT k Þbend;/ þ ðT~ k Þbend;w þ ðT~ k Þmem;/ þ ðT~ k Þmem;w
q
ð30Þ
q
and its four components are derived as follows
r a ðk þ 1Þ 2
b h
k2
sin
2
/ cos5 /½ðk þ 5k þ 4Þ cos4 / ð6k þ 7Þ cos2 / þ 3 cos kw
ð31Þ
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
0 0
(rad)
(b) (Tk )bend ,ψ
5000 (T)mem,ψ
2000 (T)mem,
0 -2000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
5000 (T)T otal
ðT~ k Þbend;/ ¼
0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 13. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 3.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ðT~ k Þbend;w ¼
r a ðk þ 1Þ 2
b h
k2
sin
2
/ cos3 /½ðk þ 1Þ cos4 / cos2 / k cos kw
ð32Þ
ðT~ k Þmem;/ ¼
kþ1 k cos4 / sin / cos kw bh
ð33Þ
ðT~ k Þmem;w ¼
kþ1 k cos2 / sin / cos kw bh
ð34Þ
The four individual actuation components will then be evaluated to determine their respective contributions in different modes of different paraboloidal shell structures. 4. Evaluation of microscopic actuation effectiveness Spatial distribution of microscopic actuation effectiveness of free paraboloidal membrane shells are investigated in this section. Furthermore, in order to evaluate the geometric parameter effects on microscopic control behavior, two cases, i.e., a shallow and a deep paraboloidal membrane shell, are studied respectively. For each case, two different shell thickness, i.e., h = 0.001 m and h = 0.002 m, are considered to reveal the influence of shell rigidity on modal control actions. Table 1 shows the
200 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -200 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(rad)
(b) (Tk )bend ,ψ
(a) (Tk )bend ,φ
5000 (T)mem,ψ
2000 (T)mem,
0 0
0 -2000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 14. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 4.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
geometric parameters of four different cases of paraboloidal membrane shells with distributed piezoelectric actuators. Case 1/Case 2 stands for the shallow/deep shell while suffix a/b denotes the thickness h = 0.001 m/0.002 m. For each case, the actuator patch has a constant thickness of ha = 0.04 mm. Table 2 presents the physical properties of the thin film polyvinylidene fluoride (PVDF) which serves as the actuator layer. Analyses of the microscopic actuation distributions would provide a criterion for optimal actuator placement to achieve the maximum control efficiency. The total microscopic actuation effectiveness and its contributing components of four cases are investigated and spatial distributions are illustrated. Case 1a: shallow shell, h = 0.001 m In this case, a shallow paraboloidal membrane with thickness h = 0.001 m is investigated. The total microscopic control force and its four components of modes k = 1–4 are plotted in Figs. 3–6. In each figure, the subgraph (a) denotes the component resulting from the meridional bending moment ðT~ k Þ ; (b) denotes the component resulting from the circumferbend;/
ential bending moment ðT~ k Þbend;w ; (c) denotes the component resulting from the meridional membrane force ðT~ k Þmem;/ ; (d) denotes the component resulting from the circumferential membrane force ðT~ k Þmem;w ; and (e) denotes the summation of all contributing components, namely, the total modal control force ðT~ k Þ . Total
Case 1b: shallow shell, h = 0.002 m In this case, a shallow paraboloidal membrane with thickness h = 0.002 m is studied. Distribution of the microscopic modal actuation forces are presented in Figs. 7–10. The subgraph arrangement is the same as before.
20 (T)bend,ψ
(T)bend,
50 0 -50 2
0 -20 2
*
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
2000 (T)mem,
0 0
0 -2000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 15. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 1.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Case 2a: deep shell, h = 0.001 m In this case, a deep paraboloidal membrane with thickness h = 0.001 m is investigated. Distribution of the microscopic modal actuation forces are shown in Figs. 11–14. Also, the subgraph arrangement remains the same as in Case 1. Case 2b: deep shell, h = 0.002 m In this case, a deep paraboloidal membrane with thickness h = 0.002 m is explored. Figs. 15–18 illustrate the microscopic modal control force distribution of the shell. All the subgraphs are arranged the same as before. Distributed microscopic control effects generated by infinitesimal actuator elements on paraboloidal membrane shells of different geometric parameters can be observed from Figs. 3–18. One should notice that the modal control force or any of its component exhibits either positive or negative value at various positions of the paraboloidal shell. Correspondingly, at different locations of the shell, the piezoelectric actuator induces either positive or negative control force. The positive control force can be used to suppress the vibration of the structure while the negative one is on the contrary. However, since the sign of the induced control force can be easily determined by the sign of the input control voltage, negative control force can then be easily transformed to positive one with an appropriate control strategy. So in this paper, we focus only on the absolute value of the modal control force. For each mode, the peak and valley of the modal control force indicate the maximum control efficiency induced by a certain input control signal. Therefore, in each figure, the location where the maximum control force is obtained is the so-called optimal actuator location for the corresponding mode and component. Furthermore, in
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
1000 (T)mem,
0 0
0 -1000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 16. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
order to make it more intuitive for readers to understand the actuation effects of each component in each case, magnitudes of the microscopic control forces are summarized in Tables 2–6. From Figs. 3–18 one can find that the membrane component dominates the total control force, while in contrast, the bending component is relatively small. This phenomenon is more significant in the shallow shell case. For each case, distributions of the two membrane modal control components are quite similar. However, the two bending component distributions are markedly different. Also, the membrane and bending control components exhibit marked difference in their distributions. Referring to Figs. 3–18, the thickness effects on the modal control force can be observed. For a shallow/deep shell of different thickness, spatial distributions of total modal control force and the four components are the same. Magnitudes of the membrane components vary due to the thickness difference, however, magnitudes of the bending components remain unchanged. At the pole of the paraboloidal shell, both membrane control components are zero, while the two bending control components are opposite each other. Therefore, total modal control forces are always zero at the pole, which means that piezoelectric actuators are noneffective at this location. It can also be concluded that for the kth mode, either shallow or deep shell, there are 2k positions where the maximum control force can be achieved, i.e., there are 2k so-called optimal actuator locations on the paraboloidal membrane shell. Although we have artificially divided the modal control force into four components, in practical application, we mainly care about the total control effect. The spatial distributions of total control force of shallow and deep shell are significantly different. The most effective control area on the shallow shell is at the edge for all modes (k = 1–4); while for the deep shell, the optimal control location moves from
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
1000 (T)mem,
0 0
0 -1000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 17. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 3.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
200 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -200 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
1000 (T)mem,
0 0
(rad)
0 -1000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0
(rad)
( )
(e) Tk
Total
Fig. 18. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 4.
Table 3 Maximum actuation magnitudes of ðT~ k Þbend;/ . ðT~ k Þbend;/
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
0.37 0.75 0.46 0.36
0.37 0.75 0.46 0.36
23.4 48.0 28.5 23.2
23.4 48.0 28.5 23.2
Deep (a = 1 m, c = 2 m)
/⁄/2 to the boundary with the increasing of the mode k from 1 to 4. Additionally, it is obvious that the control effect of the deep shell is much larger than that of the shallow shell. Quantitative analyses can be obtained referring to Tables 3–7. By comparing Tables 3, 4 with Tables 5, 6 we can find that the membrane control effects are much larger than the bending control effects. In Case 2, i.e., the deep shell, the bending component is more effective than that of the shallow shell. Additionally, it can be inferred that proportion of the bending component will gradually increase when the shell thickness grows. Note that the research object of this paper is paraboloidal membrane shell, whose thickness is relatively thin compared with the physical dimensions. Membrane approximation is adopted in the process of modeling. Hence, it is reasonable that the membrane components dominate the total control effect. When the thickness of the shell becomes larger, other approximate
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Table 4 Maximum actuation magnitudes of ðT~ k Þbend;w . ðT~ k Þbend;w
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
0.21 0.79 1.24 1.73
0.21 0.79 1.24 1.73
13.3 50.6 79.5 110.5
13.3 50.6 79.5 110.5
Deep (a = 1 m, c = 2 m)
Table 5 Maximum actuation magnitudes of ðT~ k Þmem;/ . ðT~ k Þmem;/
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
286.1 222.2 183.2 156.3
143.1 111.1 91.6 78.1
2289.7 1775.6 1463.4 1249.0
1144.9 887.8 731.7 624.5
Deep (a = 1 m, c = 2 m)
Table 6 Maximum actuation magnitudes of ðT~ k Þmem;w . ðT~ k Þmem;w
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
384.9 375.0 353.6 312.5
192.4 187.5 176.8 156.3
3077.3 2998.8 2973.3 2962.3
1538.7 1499.4 1486.6 1481.2
Deep (a = 1 m, c = 2 m)
Table 7 Maximum actuation magnitudes of ðT~ k ÞTotal . ðT~ k ÞTotal
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
657.0 578.1 531.6 470.6
328.7 289. 5 266.4 236.2
5282.1 4666.9 4334.1 4134.4
2656.1 2358.9 2203.9 2117.3
Deep (a = 1 m, c = 2 m)
solution approaches (e.g., bending approximation) may be used which is not in the scope of this article. Tables 3–6 indicate that for both paraboloidal shells, when the thickness increases from 0.001 m to 0.002 m, the bending control effects remain constant, while the membrane control effects become half. Table 7 shows that when the thickness is doubled, the total control effect becomes about half. For each case, with the increasing of mode k, both membrane control components decrease; the bending component in circumferential direction increases; the bending component in meridional direction varies irregularly; and the total control effect, determined by the membrane components, decrease. For a specific mode, control effects of all components of the deep shell are much larger than those of the shallow shell. The total modal control force of deep shell is approximately 8 times that of the shallow shell. For thickness h = 0.001 m, when k grows from 1 to 4, the ratio also augments slightly as follows: 8.04, 8.07, 8.16 and 8.79. 5. Conclusion In order to control the vibration of flexible paraboloidal shell structronic systems, precision distributed microscopic actuation effectiveness of free paraboloidal membrane shells with piezoelectric actuator elements was investigated in this study. Microscopic distributed actuators laminated on deep and shallow paraboloidal shells were designed first. Modal control actions and their contributing components on paraboloidal shells of different geometric parameters were analyzed and their effective locations were identified. Parametric studies suggest that (1) The distributed membrane control effects dominate in the total control force for a given mode. The circumferential membrane micro-control effect is the most prominent component; the meridional membrane component is placed second; while the two bending control components are relatively small in both shallow and deep shells.
H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
21
(2) The total control effect and its four contributing components of the deep paraboloidal shell are larger than those of the shallow shell, due to the curvature effect. (3) The most effective control area on the shallow shell is at the edge for all modes; the optimal control location on the deep shell moves from /⁄/2 to the boundary when k grows. (4) For the kth mode, there are 2k positions on the paraboloidal shell where the control effect reaches a maximum. (5) The total control effect decreases when the natural mode k increases. (6) The total control effect increases when the shell thickness decreases, due to its reduced flexural rigidity. In summary, this study presents a rather comprehensive investigation of distributed actuation of paraboloidal membrane shells. Analyses suggest that distributed actuation characteristics depends on a number of key design parameters, including actuator locations, natural modes, shell curvatures and thickness, and control signals, etc. Analytical results in this study can serve as guidelines of optimal actuator placement for different paraboloidal membrane shell structures.
Acknowledgments This research is supported by National Natural Science Foundation of China (Grant No. 51175103) and Self-Planned Task of State Key Laboratory of Robotics and System (HIT), China (Grant No. SKLRS201301B).
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[28] Y. Zhang, X. Zhang, S.J. Elliott, Numerical and experimental investigations of actuating characteristics of laminated polyvinylidene fluoride actuator used on paraboloidal shells, J. Sound Vib. 331 (2012) 510–521, https://doi.org/10.1016/j.jsv.2011.09.024. [29] H. Yue, Y. Lu, Z. Deng, H. Tzou, Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. 82 (2017) 279–295, https://doi.org/10.1016/j.ymssp.2016.05.023. [30] H. Yue, Y. Lu, Z. Deng, H. Tzou, Modal sensing and control of paraboloidal shell structronic system, Mech. Syst. Signal Process. 100 (2018) 647–661, https://doi.org/10.1016/j.ymssp.2017.08.005. [31] S.D. Hu, H. Li, H.S. Tzou, Precision microscopic actuations of parabolic cylindrical shell reflectors, J. Vib. Acoust. 137 (2015) 11008, https://doi.org/ 10.1115/1.4028341. [32] M. Amabili, Nonlinear vibrations and stability of shells and plates, Cambridge University Press, New York, 2008.
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Distributed microscopic actuation analysis of paraboloidal membrane shells of different geometric parameters Honghao Yue a, Yifan Lu a,⇑, Zongquan Deng a, Hornsen Tzou b a b
School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang Province 150001, PR China College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China
a r t i c l e
i n f o
Article history: Received 17 August 2017 Received in revised form 20 September 2017 Accepted 1 October 2017
Keywords: Paraboloidal membrane shell Geometric parameter effects Modal control Microscopic actuation Optimal actuator placement
a b s t r a c t Paraboloidal membrane shells of revolution are commonly used as key components for advanced aerospace structures and aviation mechanical systems. Due to their high flexibility and low damping property, active vibration control is of significant importance for these in-orbit membrane structures. To explore the dynamic control behavior of space flexible paraboloidal membrane shells, precision distributed actuation and control effectiveness of free-floating paraboloidal membrane shells with piezoelectric actuators are investigated. Governing equations of the shell structronic system are presented first. Then, distributed control forces and control actions are formulated. A transverse mode shape function of the paraboloidal shell based on the membrane approximation theory and specified boundary condition is assumed in the modal control force analysis. The actuator induced modal control forces on the paraboloidal shell are derived. The expressions of microscopic local modal control forces are obtained by shrinking the actuator area into infinitesimal and the four control components are investigated respectively to predict the spatial microscopic actuation behavior. Geometric parameter (height-radius ratio and shell thickness) effects on the modal actuation behavior are explored when evaluating the micro-control efficiency. Four different cases are discussed and the results reveal the fact that shallow (e.g., antennas/reflectors) and deep (e.g., rocket/missile fairing) paraboloidal shells exhibit totally different modal actuation behaviors due to their curvature differences. Analytical results in this paper can serve as guidelines for optimal actuator placement for vibration control of different paraboloidal structures. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Many advanced aerospace structures and optical systems, such as reflectors, mirrors, antennas, rocket fairings and nozzles, belong to the category of double-curvature paraboloidal shell structures [1]. Due to weight restrictions in launching and space operation, these paraboloidal shells are mostly lightweight and flexible, which tend to exhibit prolonged oscillations and thus influence their precision and accuracy. Accordingly, precision control of structural vibration is of considerable importance not only to their stringent performance requirement, but also to their structural integrity and long-term reliability [2–4]. Accordingly, this study is to investigate the distributed actuation effectiveness of flexible membrane paraboloidal shells and to explore the optimal actuator placement on paraboloidal shell structures. Actuation characteristics of shallow and deep paraboloidal shells are evaluated respectively. ⇑ Corresponding author. E-mail address: [email protected] (Y. Lu). https://doi.org/10.1016/j.ymssp.2017.10.005 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Development of ‘‘smart structures and structronic systems” has demonstrated that various smart materials and control techniques can be applied to control static shapes and/or undesirable vibrations of structures and systems [5,6]. Distributed control of plate and shell structures based on the smart structure technology has been quickly developing in the last twenty years [7–12]. Among various smart materials and structures, the distributed control technique using piezoelectric actuators is widely investigated and applied to engineering systems, ranging from micro-/nano-devices to large aerospace structures. Distributed sensing and control signals of deformable plate membrane mirrors with different pretension forces were investigated by Lu et al. [13,14]. The equations of motion of a free-floating paraboloidal shell laminated with a piezoelectric actuator layer were derived and modal control effectiveness of the actuators was explored by Yue et al. [15]. Distributed control actions of thin cylindrical shells with various piezoelectric [16–18] and electro-optic photostrictive [19] actuators were widely investigated. Distributed modal voltages and their spatial strain characteristics of toroidal shells were investigated by Tzou et al. [20]. Static and dynamic behaviors of spherical and conical shells with piezoelectric actuators were also comprehensively studied [21–24]. Micro-control actions and distributed control effectiveness of segmented actuator patches laminated on hemispheric shells was evaluated by Smithmaitrie et al. [25]. Distributed modal voltage and micro-control characteristics of a simply supported paraboloidal shell were explored by Ding et al. [26,27]. Zhang and Yue et al. experimentally investigated the dynamic vibration control of paraboloidal membrane shells with different control strategies respectively [28–30]. Hu et al. studied the microscopic actuation and optimal actuator locations of parabolic cylindrical shells [31]. In this study, the spatially distributed actuation behavior of piezoelectric actuators on free paraboloidal membrane shells is investigated. The modal control force induced by an arbitrary piezoelectric actuator patch is derived and the microscopic actuation effectiveness is investigated by shrinking the actuator area into infinitesimal. Geometric parameter effects on the microscopic actuation efficiency are taken into account. Paraboloidal membrane shells with two different height-radius ratios, i.e., the shallow and the deep shell, are discussed respectively. Deep/shallow paraboloidal membrane shells with different thickness are also investigated to reveal the influence of shell rigidity on modal control actions. 2. Modal control of paraboloidal membrane shell structronic systems In this section, the governing control equations of a generic paraboloidal structronic shell laminated with a piezoelectric actuator layer are derived. Expression of the modal control force on a paraboloidal shell is formulated by modal expansion method. 2.1. Modeling of piezoelectric laminated paraboloidal membrane shells A structronic system is composed of sensor, actuator, control electronics and elastic structures. Fig. 1 illustrates a generic paraboloidal structronic shell, i.e., an elastic shell laminated with an actuator layer, defined in a tri-orthogonal global coordinate system - XYZ. The neutral surface of the elastic double-curvature shell is defined in a tri-orthogonal curvilinear coordinate system (/, w, a3) where / defines the meridional direction, w the circumferential direction and a3 the transverse direction. d/ denotes an infinitesimal angular change in the meridian direction and dw denotes an infinitesimal angular change in the circumferential direction. Furthermore, ‘‘a” is the radial distance and ‘‘c” is the meridian height at the pole. The Lamé parameters are A1 = b/cos3/ and A2 = (bsin/)/cos/. The meridional radius of curvature is R/ = b/cos3/ and the circumferential radius is Rw = b/cos/, where constant b = a2/2c = 2f, and f is the focal length. The actuator layer, with thickness ha, is perfectly bonded on the shell surface. It is assumed that the actuator is made of bi-axially sensitive piezoelectric material, such that a transverse control voltage can introduce two in-plane normal strains applied to distributed actuation and control of shells. Considering that shallow and deep paraboloidal membrane shells have distinct curvatures and characteristics which are utilized in different aerospace structures and systems, a shallow and a deep paraboloidal shell with a set of distributed actuator patches laminated on its outer surface is shown in Fig. 2(a) and (b), respectively.
Z
Z
Actuator layer 3 2=
Rψsin dψ
ψ
1=
dψ
c
Rψsin Rψ a R
Y
X O2 O1
Y
d *
(Not to scale)
Fig. 1. A generic paraboloidal structronic shell with an actuator layer.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Z
Z
Actuator patches
Δφ
c=1 Δψ
a=2 Y
Y X
φ*=0.7854rad
(a) Z
Z
φ
Actuator patches c=2
Y Y
a=1
Δψ
Y
φ*=1.3258rad
(b) Fig. 2. (a) A shallow paraboloidal shell with distributed actuator patches. (b) A deep paraboloidal shell with distributed actuator patches.
2.2. Governing control equations As discussed above, a bi-axially oriented piezoelectric actuator induces two normal control strains when a transverse control voltage is applied across the actuator thickness due to the converse piezoelectric effect. Since the actuator layer is placed away from the neutral surface, these strains not only translate to in-plane control forces, but also induce counteracting control moments. These control forces and moments can be used to control static shape or to counteract dynamic oscillations of the shell. Since the piezoelectric layer is thin as compared with the paraboloidal shell, the elastic and mass properties of the actuators do not affect the elastic shell. The paraboloidal membrane shell system equations can be defined as
cos / @ cos4 / @ cos2 / m cos3 / a €/ ðNm Þ þ ½ðNm ðN Naww Þ þ Q /3 þ q/ ¼ qhu // N // Þ tan / b sin / @w w/ b sin / @/ b sin / ww b
ð1Þ
cos / @ cos4 / @ cos2 / m cos / €w ðNm Naww Þ þ ðNm tan /Þ N þ Q w3 þ qw ¼ qhu b sin / @w ww b sin / @/ /w b sin / w/ b
ð2Þ
cos / @ cos4 / @ cos3 / m cos / m €3 Q w3 þ ðQ /3 tan /Þ ðN// Na// Þ ðNww Naww Þ þ q3 ¼ qhu b sin / @w b sin / @/ b b
ð3Þ
m a a where N m ij and M ij are the mechanical forces and moments; N ij are the actuator induced forces; M ij are the actuator induced moments; qhüi denotes the inertia force effect; q is the mass density; h is the shell thickness; qi is the input force; and the superscripts m and a respectively denote the mechanical (or elastic) and actuator induced components. The transverse shear effects Q/3 and Qw3 are defined as
Q /3 ¼
cos4 / @ m @ a m a 2 sec3 / M w/ þ ½ðMm M Þ tan / ðM M Þ sec / // // ww ww b sin / @w @/
ð4Þ
Q w3 ¼
cos4 / @ @ a m m 2 sec3 / ðM m ðM M Þ þ tan /Þ þ M sec / ww w/ b sin / @w ww @/ w/
ð5Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22 Table 1 Four cases of paraboloidal membrane shells with piezoelectric actuators. Case
Shell structure
1a 1b 2a 2b
Actuator patch
Height c (m)
Radius a (m)
Thickness h (m)
Thickness ha (mm)
1
2
0.04
2
1
0.001 0.002 0.001 0.002
Table 2 Physical properties of PVDF film. Parameters
Value
Unit
Young’s modulus Yp Tensile strength rb Poisson’s ratio mp Mass density q Piezoelectric strain constant d31, d32
2500 200 0.29 1.78 103 21
MPa MPa / kg/m3 pC/N
0.5 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -0.5 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0
(rad)
(b) (Tk )bend ,ψ
(a) (Tk )bend ,φ
500 (T)mem,ψ
500 (T)mem,
0 0
0 -500 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
1000 0 -1000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 3. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 1.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Based on the membrane approximation, the mechanical moments, i.e., the M m ij terms, are neglected in the governing equations. Note that although mechanical moments are neglected, control moments induced by the piezoelectric actuator are still preserved in order to evaluate the overall control actions and microscopic contributing components of actuator patches on the paraboloidal membrane shell. Since this study focuses on the microscopic control actions of actuator patches laminated on the paraboloidal shell, not the shell dynamics, only actuation component expressions are given. According to the thin shell theory, the in-plane shear effects of piezoelectric patch are not considered, namely N aij 0 and M aij 0. Thus, the actuation forces and moments can be expressed as
Na// ¼ d3/ Y p /a ð/; w; tÞ
ð6Þ
Naww ¼ d3w Y p /a ð/; w; tÞ
ð7Þ
Ma// ¼ ra/ d3/ Y p /a ð/; w; tÞ
ð8Þ
Maww ¼ raw d3w Y p /a ð/; w; tÞ
ð9Þ
where d3i is the piezoelectric strain constant; Yp is Young’s modulus of the piezoelectric actuator; ra i is the moment arm, i.e., the distance from shell’s neutral surface to the actuator’s mid plane; and /a(/, w, t) is the control signal.
1 (T)bend,ψ
(T)bend,
1 0 -1 2
*
0 -1 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(rad)
(b) (Tk )bend ,ψ
(a) (Tk )bend ,φ
500 (T)mem,ψ
500 (T)mem,
0 0
0 -500 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0
0 0
(rad)
(d) (Tk )mem,ψ
(c) (Tk )mem,φ
(T)T otal
1000 0 -1000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 4. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
500 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
1000 0 -1000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 5. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 3.
Although the closed-form system equations of paraboloidal shells are defined, the analytical solution procedure of modal analysis is very complicated. Mode shape functions are needed in evaluating spatial modal actuation behaviors of shells. Thus, an assumed transverse mode shape function satisfying the free boundary conditions is selected in Section 3 to evaluate the distributed control effects and micro-control actions of the piezoelectric actuator patches. Note that when evaluating the control actions solely contributed by actuator patches, the mechanical forces and moments are unnecessarily considered in this study. 2.3. Modal control force According to the modal expansion method, the dynamic response of the paraboloidal membrane shell is composed of all participating modes [32]
ui ð/; w; tÞ ¼
1 X
gk ðtÞU ik ð/; wÞ; i ¼ /; w; 3
ð10Þ
k¼1
where gk(t) is the k-th modal participating factor or the k-th modal coordinate and Uik(/, w) is the k-th mode shape function in the i-th direction. By applying the modal orthogonality and substituting the modal expansion into the simplified piezoelectric shell equations, the modal control equation of the paraboloidal shell can be derived as
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
500 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -500 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 6. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.001 m, k = 4.
g€ k þ 2fk xk g_ k þ x2k gk ¼ F^mk þ F^ck
ð11Þ
^c where fk is the k-th modal damping ratio; xk is the k-th natural frequency; F^m k and F k are the modal excitation contributed by the mechanical force and the piezoelectric actuator, respectively. This study aims to investigate the spatially distributed micro-control actions of piezoelectric patches at various locations of the shell, the mechanical excitation is therefore assumed zero. Expression of the modal control force can be defined as
F^ck ¼
1 qhNk
where N k ¼
Z Z
2
½Lc/ ð/3 ÞU /k þ Lcw ð/3 ÞU wk þ Lc3 ð/3 ÞU 3k w
/
b sin / d/dw cos4 /
ð12Þ
R R P3 2 / ½ i¼1 U 2ik ð/; wÞ bcossin 4 / d/dw and the Love’s control operator is defined as a function of /3. w /
Lc/ ð/3 Þ ¼
a a cos3 / @N // cos6 / @M// 2 b @/ @/ b
ð13Þ
Lcw ð/3 Þ ¼
a a cos / @Nww cos2 / @Mww 2 b sin / @w b sin / @w
ð14Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
0.5 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -0.5 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 7. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 1.
Lc3 ð/3 Þ ¼
2 a cos6 / @ M //
b
2
@/2
þ
2 a cos / @ M ww 2
2 2 2 b sin / @w
a 2 3 cos5 / sin / 2 cos5 / @M// cos4 / a M// þ 2 2 @/ b sin / b
a cos5 / @M ww cos4 / a cos3 / a cos / a N// þ Nww þ 2 M ww þ 2 b b @/ b sin / b
ð15Þ
3. Distributed actuation effectiveness of paraboloidal membrane shells Based on the membrane approximation theory, the mode shape functions of a paraboloidal membrane shell with free boundary condition have been assumed and validated [15].
U /k ¼ Ak cos
ð2k þ 1Þp kþ1 /ðsin /Þ cos kw 2/ kþ1
U wk ¼ Ak cos /ðsin /Þ
sin kw k
U 3k ¼ Ak ðk þ 1Þ cos /ðsin /Þ cos kw
ð16Þ ð17Þ ð18Þ
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
where Ak, the modal amplitude constant of mode k, is chosen so that
Z Z
2
U 23k w
/
b sin / d/dw ¼ 1 cos4 /
ð19Þ
and / is the meridional angle measured from the pole varying from 0 (pole) to /⁄ (boundary rim) for a shell with a closed apex; w defines the circumferential angle which varies from 0 to 2p for a closed paraboloidal shell. Substituting the Love’s control operators and the assumed mode shape functions into Eq. (12), the modal control force can be redefined as
Y p d3i /a ðtÞAk Y p d3i /a ðtÞAk F^ck ¼ Tk ¼ ðT k
q
q
meri
þ Tk
cir
þ Tk
trans Þ
ð20Þ
where Tk denotes the total control action induced by the actuator, and Tk_meri, Tk_cir and Tk_trans are the control actions respectively in the meridional, circumferential and transverse directions. It has been proved that for plates and very shallow shells, the transverse deformation and oscillation usually dominates. This paper mainly focus on the micro-control actions that
1 (T)bend,ψ
(T)bend,
1 0 -1 2
*
0 -1 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
200 (T)mem,
0 0
0 -200 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 8. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
counteract the transverse oscillation, control forces in meridional and circumferential directions are neglected in derivations for simplicity. Note that this may be inaccurate if one wants to analyze the total dynamic control behavior of deep shells with significant curvature. However, the same procedure can be easily applied to either the meridional or the circumferential vibrations. In the following derivations, Tk_meri and Tk_cir are omitted and Tk_trans is aliased as (Tk)Total. An arbitrary segmented actuator patch ranging from /1 to /2 in the meridional direction and from w1 to w2 in the circumferential direction is assumed laminated on the surface of the shell (see Fig. 2). Since the piezoelectric actuator patch is ultrathin and lightweight as compared with the elastic shell, material and stiffness of the actuator layers are neglected in the shell system equations. The electrode resistance is neglected and the potential is constant over the effective electrode surface of a piezoelectric patch. Thus, the actuator control signal applied to a segmented actuator patch is uniformly distributed which can be defined by the patch boundaries as
/a ð/; w; tÞ ¼ /a ðtÞ½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ
ð21Þ
where us() is the unit step function: us ðx xi Þ ¼ 1 when x > xi , and us ðx xi Þ ¼ 0 when x < xi . /a(t) is the imposed control voltage. Substituting Eqs. (15), (18) in Eq. (12) and taking into account Eqs. (6)–(9), (19) and (21), the modal control force induced by the actuator patch can be obtained as
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
100 (T)mem,
0 0
0 -100 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 9. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 3.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ra cos6 / @ ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ 2 @/ q hN k w / b 2 3 cos5 / sin / 2 cos5 / ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ 2 b sin / cos4 / ½u ð/ / Þ u ð/ / Þ½u ðw w Þ u ðw w Þ þ s s s s 1 2 1 2 2 b ra cos2 / @ þ ½us ð/ /1 Þ us ð/ /2 Þ 2 ½dðw w1 Þ dðw w2 Þ 2 hNk @w b sin / cos5 / ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ 2 b sin / cos4 / ½u ð/ / Þ u ð/ / Þ½u ðw w Þ u ðw w Þ s s s s 1 2 1 2 2 b 1 cos3 / ½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ hNk b 1 cos / ½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ þ hNk b 2 b sin / k ðk þ 1Þ cos /ðsin /Þ cos kw d/dw cos4 /
Y p d3i /a ðtÞAk F^ck ¼
Z Z
2 (T)bend,ψ
(T)bend,
0.5 0 -0.5 2
*
0 -2 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (a) (Tk )bend ,φ
0 0
(rad)
(b) (Tk )bend ,ψ
200 (T)mem,ψ
100 (T)mem,
ð22Þ
0 -100 2
*
0 -200 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
500 0 -500 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 10. Spatial distributions of microscopic actuation effectiveness of shallow shell, h = 0.002 m, k = 4.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
where d() is the Dirac delta function and dðx xi Þ ¼ 1 when x ¼ xi , and dðx xi Þ ¼ 0 when x – xi . It is assumed that the actuator is made of hexagonal piezoelectric materials, i.e., d3/ = d3w = d3i; both the actuator and the shell have uniform thickness, i.e., r a/ ¼ r aw ¼ r a h=2 One can find that the total modal control force consists of four contributing components resulting from both membrane forces and bending moments.
Y p d3i /a ðtÞAk ^ Y p d3i /a ðtÞAk ^ F^ck ¼ ½ðT k ÞTotal ¼ ½ðT k Þbend;/ þ ðT^ k Þbend;w þ ðT^ k Þmem;/ þ ðT^ k Þmem;w
q
ð23Þ
q
where ðT^ k ÞTotal denotes the total modal control force; ðT^ k Þbend;/ and ðT^ k Þbend;w denote the two components induced by the bendand ðT^ k Þ denote the two components induced by the membrane forces in / ing moments in / and w directions; ðT^ k Þ mem;/
mem;w
and w directions, respectively. Accordingly, these four components can be further expressed as
ra ðk þ 1Þ h
Z Z @ kþ1 ½dð/ /1 Þ dð/ /2 Þþ ðsin /Þ cos3 /½us ðw w1 Þ us ðw w2 Þ @/ w /
ðsin /Þ cos2 /ð3 sin / 2Þ½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þþ kþ1 ðsin /Þ cos /½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ cos kwd/dw k
2
20 (T)bend,ψ
(T)bend,
50 0 -50 2
0 -20 2
*
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
0 0
(rad)
(b) (Tk )bend ,ψ
5000 (T)mem,ψ
5000 (T)mem,
0 -5000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ 4
x 10 1 (T)T otal
ðT^ k Þbend;/ ¼
0 -1 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 11. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 1.
ð24Þ
13
H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ðT^ k Þbend;w ¼
ra ðk þ 1Þ h
Z Z @ k1 ½dðw w1 Þ dðw w2 Þþ ðsin /Þ ðcos /Þ1 ½us ð/ /1 Þ us ð/ /2 Þ @w w /
ðsin /Þ cos2 /½dð/ /1 Þ dð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ
ð25Þ
k
ðsin /Þ ðT^ k Þmem;/ ¼ ðT^ k Þmem;w ¼
bðk þ 1Þ h bðk þ 1Þ h
Z Z w
kþ1
cos /½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ cos kwd/dw
½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þðsin /Þ
kþ1
cos kwd/dw
ð26Þ
/
Z Z w
½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 Þ /
kþ1
ðsin /Þ cos2 /
cos kwd/dw
ð27Þ
When the actuator patch moves along the /-direction with the same spatial interval, namely D/ remains a constant, the actuator area also varies correspondingly due to the curvature difference. In order to explore the modal control capability per actuator area, modal control force induced by an actuator patch is normalized by dividing it by the corresponding actuator area as
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(b) (Tk )bend ,ψ
5000 (T)mem,ψ
2000 (T)mem,
0 0
(rad)
0 -2000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 12. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ðF^ck Þeff ¼
F^ck Sa
ð28Þ
where Sa denotes the effective area of the actuator patch
Z
Sa ¼ S
½us ð/ /1 Þ us ð/ /2 Þ½us ðw w1 Þ us ðw w2 ÞdS
ð29Þ
Eq. (28) gives the expression of a so-called average control effect. Then the actuator area Sa is shrunken into infinitesimal so that modal control force at an arbitrary position can be precisely investigated. The expression of the microscopic modal control force can be defined as
Y p d3i /a ðtÞAk ~ Y p d3i /a ðtÞAk ~ F~ck ¼ ½ðT k ÞTotal ¼ ½ðT k Þbend;/ þ ðT~ k Þbend;w þ ðT~ k Þmem;/ þ ðT~ k Þmem;w
q
ð30Þ
q
and its four components are derived as follows
r a ðk þ 1Þ 2
b h
k2
sin
2
/ cos5 /½ðk þ 5k þ 4Þ cos4 / ð6k þ 7Þ cos2 / þ 3 cos kw
ð31Þ
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
0 0
(rad)
(b) (Tk )bend ,ψ
5000 (T)mem,ψ
2000 (T)mem,
0 -2000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
5000 (T)T otal
ðT~ k Þbend;/ ¼
0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 13. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 3.
15
H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
ðT~ k Þbend;w ¼
r a ðk þ 1Þ 2
b h
k2
sin
2
/ cos3 /½ðk þ 1Þ cos4 / cos2 / k cos kw
ð32Þ
ðT~ k Þmem;/ ¼
kþ1 k cos4 / sin / cos kw bh
ð33Þ
ðT~ k Þmem;w ¼
kþ1 k cos2 / sin / cos kw bh
ð34Þ
The four individual actuation components will then be evaluated to determine their respective contributions in different modes of different paraboloidal shell structures. 4. Evaluation of microscopic actuation effectiveness Spatial distribution of microscopic actuation effectiveness of free paraboloidal membrane shells are investigated in this section. Furthermore, in order to evaluate the geometric parameter effects on microscopic control behavior, two cases, i.e., a shallow and a deep paraboloidal membrane shell, are studied respectively. For each case, two different shell thickness, i.e., h = 0.001 m and h = 0.002 m, are considered to reveal the influence of shell rigidity on modal control actions. Table 1 shows the
200 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -200 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(rad)
(b) (Tk )bend ,ψ
(a) (Tk )bend ,φ
5000 (T)mem,ψ
2000 (T)mem,
0 0
0 -2000 2
*
0 -5000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 14. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.001 m, k = 4.
16
H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
geometric parameters of four different cases of paraboloidal membrane shells with distributed piezoelectric actuators. Case 1/Case 2 stands for the shallow/deep shell while suffix a/b denotes the thickness h = 0.001 m/0.002 m. For each case, the actuator patch has a constant thickness of ha = 0.04 mm. Table 2 presents the physical properties of the thin film polyvinylidene fluoride (PVDF) which serves as the actuator layer. Analyses of the microscopic actuation distributions would provide a criterion for optimal actuator placement to achieve the maximum control efficiency. The total microscopic actuation effectiveness and its contributing components of four cases are investigated and spatial distributions are illustrated. Case 1a: shallow shell, h = 0.001 m In this case, a shallow paraboloidal membrane with thickness h = 0.001 m is investigated. The total microscopic control force and its four components of modes k = 1–4 are plotted in Figs. 3–6. In each figure, the subgraph (a) denotes the component resulting from the meridional bending moment ðT~ k Þ ; (b) denotes the component resulting from the circumferbend;/
ential bending moment ðT~ k Þbend;w ; (c) denotes the component resulting from the meridional membrane force ðT~ k Þmem;/ ; (d) denotes the component resulting from the circumferential membrane force ðT~ k Þmem;w ; and (e) denotes the summation of all contributing components, namely, the total modal control force ðT~ k Þ . Total
Case 1b: shallow shell, h = 0.002 m In this case, a shallow paraboloidal membrane with thickness h = 0.002 m is studied. Distribution of the microscopic modal actuation forces are presented in Figs. 7–10. The subgraph arrangement is the same as before.
20 (T)bend,ψ
(T)bend,
50 0 -50 2
0 -20 2
*
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
2000 (T)mem,
0 0
0 -2000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 15. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 1.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Case 2a: deep shell, h = 0.001 m In this case, a deep paraboloidal membrane with thickness h = 0.001 m is investigated. Distribution of the microscopic modal actuation forces are shown in Figs. 11–14. Also, the subgraph arrangement remains the same as in Case 1. Case 2b: deep shell, h = 0.002 m In this case, a deep paraboloidal membrane with thickness h = 0.002 m is explored. Figs. 15–18 illustrate the microscopic modal control force distribution of the shell. All the subgraphs are arranged the same as before. Distributed microscopic control effects generated by infinitesimal actuator elements on paraboloidal membrane shells of different geometric parameters can be observed from Figs. 3–18. One should notice that the modal control force or any of its component exhibits either positive or negative value at various positions of the paraboloidal shell. Correspondingly, at different locations of the shell, the piezoelectric actuator induces either positive or negative control force. The positive control force can be used to suppress the vibration of the structure while the negative one is on the contrary. However, since the sign of the induced control force can be easily determined by the sign of the input control voltage, negative control force can then be easily transformed to positive one with an appropriate control strategy. So in this paper, we focus only on the absolute value of the modal control force. For each mode, the peak and valley of the modal control force indicate the maximum control efficiency induced by a certain input control signal. Therefore, in each figure, the location where the maximum control force is obtained is the so-called optimal actuator location for the corresponding mode and component. Furthermore, in
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
1000 (T)mem,
0 0
0 -1000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 16. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 2.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
order to make it more intuitive for readers to understand the actuation effects of each component in each case, magnitudes of the microscopic control forces are summarized in Tables 2–6. From Figs. 3–18 one can find that the membrane component dominates the total control force, while in contrast, the bending component is relatively small. This phenomenon is more significant in the shallow shell case. For each case, distributions of the two membrane modal control components are quite similar. However, the two bending component distributions are markedly different. Also, the membrane and bending control components exhibit marked difference in their distributions. Referring to Figs. 3–18, the thickness effects on the modal control force can be observed. For a shallow/deep shell of different thickness, spatial distributions of total modal control force and the four components are the same. Magnitudes of the membrane components vary due to the thickness difference, however, magnitudes of the bending components remain unchanged. At the pole of the paraboloidal shell, both membrane control components are zero, while the two bending control components are opposite each other. Therefore, total modal control forces are always zero at the pole, which means that piezoelectric actuators are noneffective at this location. It can also be concluded that for the kth mode, either shallow or deep shell, there are 2k positions where the maximum control force can be achieved, i.e., there are 2k so-called optimal actuator locations on the paraboloidal membrane shell. Although we have artificially divided the modal control force into four components, in practical application, we mainly care about the total control effect. The spatial distributions of total control force of shallow and deep shell are significantly different. The most effective control area on the shallow shell is at the edge for all modes (k = 1–4); while for the deep shell, the optimal control location moves from
100 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -100 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(rad)
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
1000 (T)mem,
0 0
0 -1000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0 (e)
(rad)
(T )
k Total
Fig. 17. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 3.
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
200 (T)bend,ψ
(T)bend,
50 0 -50 2
*
0 -200 2
*
*/2 ψ (rad)
0 0
*/2 ψ (rad)
(rad)
(a) (Tk )bend ,φ
(b) (Tk )bend ,ψ
2000 (T)mem,ψ
1000 (T)mem,
0 0
(rad)
0 -1000 2
*
0 -2000 2
*
*/2 ψ (rad)
*/2 ψ (rad)
(rad)
0 0 (c) (Tk )mem,φ
0 0
(rad)
(d) (Tk )mem,ψ
(T)T otal
5000 0 -5000 2
* */2 ψ (rad)
0 0
(rad)
( )
(e) Tk
Total
Fig. 18. Spatial distributions of microscopic actuation effectiveness of deep shell, h = 0.002 m, k = 4.
Table 3 Maximum actuation magnitudes of ðT~ k Þbend;/ . ðT~ k Þbend;/
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
0.37 0.75 0.46 0.36
0.37 0.75 0.46 0.36
23.4 48.0 28.5 23.2
23.4 48.0 28.5 23.2
Deep (a = 1 m, c = 2 m)
/⁄/2 to the boundary with the increasing of the mode k from 1 to 4. Additionally, it is obvious that the control effect of the deep shell is much larger than that of the shallow shell. Quantitative analyses can be obtained referring to Tables 3–7. By comparing Tables 3, 4 with Tables 5, 6 we can find that the membrane control effects are much larger than the bending control effects. In Case 2, i.e., the deep shell, the bending component is more effective than that of the shallow shell. Additionally, it can be inferred that proportion of the bending component will gradually increase when the shell thickness grows. Note that the research object of this paper is paraboloidal membrane shell, whose thickness is relatively thin compared with the physical dimensions. Membrane approximation is adopted in the process of modeling. Hence, it is reasonable that the membrane components dominate the total control effect. When the thickness of the shell becomes larger, other approximate
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H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
Table 4 Maximum actuation magnitudes of ðT~ k Þbend;w . ðT~ k Þbend;w
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
0.21 0.79 1.24 1.73
0.21 0.79 1.24 1.73
13.3 50.6 79.5 110.5
13.3 50.6 79.5 110.5
Deep (a = 1 m, c = 2 m)
Table 5 Maximum actuation magnitudes of ðT~ k Þmem;/ . ðT~ k Þmem;/
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
286.1 222.2 183.2 156.3
143.1 111.1 91.6 78.1
2289.7 1775.6 1463.4 1249.0
1144.9 887.8 731.7 624.5
Deep (a = 1 m, c = 2 m)
Table 6 Maximum actuation magnitudes of ðT~ k Þmem;w . ðT~ k Þmem;w
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
384.9 375.0 353.6 312.5
192.4 187.5 176.8 156.3
3077.3 2998.8 2973.3 2962.3
1538.7 1499.4 1486.6 1481.2
Deep (a = 1 m, c = 2 m)
Table 7 Maximum actuation magnitudes of ðT~ k ÞTotal . ðT~ k ÞTotal
Shallow (a = 2 m, c = 1 m)
Mode
h = 0.001 m
h = 0.002 m
h = 0.001 m
h = 0.002 m
k=1 k=2 k=3 k=4
657.0 578.1 531.6 470.6
328.7 289. 5 266.4 236.2
5282.1 4666.9 4334.1 4134.4
2656.1 2358.9 2203.9 2117.3
Deep (a = 1 m, c = 2 m)
solution approaches (e.g., bending approximation) may be used which is not in the scope of this article. Tables 3–6 indicate that for both paraboloidal shells, when the thickness increases from 0.001 m to 0.002 m, the bending control effects remain constant, while the membrane control effects become half. Table 7 shows that when the thickness is doubled, the total control effect becomes about half. For each case, with the increasing of mode k, both membrane control components decrease; the bending component in circumferential direction increases; the bending component in meridional direction varies irregularly; and the total control effect, determined by the membrane components, decrease. For a specific mode, control effects of all components of the deep shell are much larger than those of the shallow shell. The total modal control force of deep shell is approximately 8 times that of the shallow shell. For thickness h = 0.001 m, when k grows from 1 to 4, the ratio also augments slightly as follows: 8.04, 8.07, 8.16 and 8.79. 5. Conclusion In order to control the vibration of flexible paraboloidal shell structronic systems, precision distributed microscopic actuation effectiveness of free paraboloidal membrane shells with piezoelectric actuator elements was investigated in this study. Microscopic distributed actuators laminated on deep and shallow paraboloidal shells were designed first. Modal control actions and their contributing components on paraboloidal shells of different geometric parameters were analyzed and their effective locations were identified. Parametric studies suggest that (1) The distributed membrane control effects dominate in the total control force for a given mode. The circumferential membrane micro-control effect is the most prominent component; the meridional membrane component is placed second; while the two bending control components are relatively small in both shallow and deep shells.
H. Yue et al. / Mechanical Systems and Signal Processing 103 (2018) 1–22
21
(2) The total control effect and its four contributing components of the deep paraboloidal shell are larger than those of the shallow shell, due to the curvature effect. (3) The most effective control area on the shallow shell is at the edge for all modes; the optimal control location on the deep shell moves from /⁄/2 to the boundary when k grows. (4) For the kth mode, there are 2k positions on the paraboloidal shell where the control effect reaches a maximum. (5) The total control effect decreases when the natural mode k increases. (6) The total control effect increases when the shell thickness decreases, due to its reduced flexural rigidity. In summary, this study presents a rather comprehensive investigation of distributed actuation of paraboloidal membrane shells. Analyses suggest that distributed actuation characteristics depends on a number of key design parameters, including actuator locations, natural modes, shell curvatures and thickness, and control signals, etc. Analytical results in this study can serve as guidelines of optimal actuator placement for different paraboloidal membrane shell structures.
Acknowledgments This research is supported by National Natural Science Foundation of China (Grant No. 51175103) and Self-Planned Task of State Key Laboratory of Robotics and System (HIT), China (Grant No. SKLRS201301B).
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