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membrane mirrors
Mechanical Systems and Signal Processing 96 (2017) 393–424
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Distributed sensing signal analysis of deformable plate/membrane mirrors Yifan Lu a, Honghao Yue 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 11 January 2017 Received in revised form 11 March 2017 Accepted 19 April 2017
Keywords: Plate/membrane mirror Piezoelectric layer Sensing signal Tension effect
a b s t r a c t Deformable optical mirrors usually play key roles in aerospace and optical structural systems applied to space telescopes, radars, solar collectors, communication antennas, etc. Limited by the payload capacity of current launch vehicles, the deformable mirrors should be lightweight and are generally made of ultra-thin plates or even membranes. These plate/membrane mirrors are susceptible to external excitations and this may lead to surface inaccuracy and jeopardize relevant working performance. In order to investigate the modal vibration characteristics of the mirror, a piezoelectric layer is fully laminated on its non-reflective side to serve as sensors. The piezoelectric layer is segmented into infinitesimal elements so that microscopic distributed sensing signals can be explored. In this paper, the deformable mirror is modeled as a pre-tensioned plate and membrane respectively and sensing signal distributions of the two models are compared. Different pre-tensioning forces are also applied to reveal the tension effects on the mode shape and sensing signals of the mirror. Analytical results in this study could be used as guideline of optimal sensor/actuator placement for deformable space mirrors. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Circular thin plates are commonly used as key components for optical deformable mirror systems in space telescopes [1– 5]. To further reduce the areal density and total mass of the optical system in space, the concept of thin film optics was proposed in recent years and series of membrane deformable mirror prototypes have been developed and tested [6–11]. These high-precision deformable plate/membrane mirrors used in space are usually lightweight and flexible and they often exhibit high flexibility, poor stiffness and low damping properties. Thus, undesirable external induced vibration can become a serious problem during operations, which lead to surface inaccuracy and even jeopardize relevant working performance. Precision sensing and control are therefore necessary to these high-performance optical mirror systems. This study is (1) to investigate spatially distributed microscopic sensing signals of lightweight circular plate/membrane mirrors, and (2) to evaluate the tensioning effects on different signal components of different modes. Sensor and actuator design criteria can be summarized from these results. Due to the distinct direct and converse piezoelectric effects, piezoelectric materials are widely used in sensor and actuator applications where high accuracy and precision are needed [12]. Polyvinylidene fluoride (PVDF) polymeric materials, as an outstanding representative of thin-film piezoelectric materials, are lightweight, dynamically sensitive, and can be easily seg⇑ Corresponding author. E-mail address: [email protected] (H. Yue). http://dx.doi.org/10.1016/j.ymssp.2017.04.028 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
mented and shaped for distributed sensing and control of flexible structures. Accordingly, the distributed piezoelectric layer can serve as distributed sensors and actuators in sensing and control of optical deformable plate/membrane mirrors. Distributed sensing and control of various structures using piezoelectric materials have been investigated over the years [13–17]. Electromechanical sensing properties of flexible rings laminated with distributed piezoelectric sensors were investigated [18]. Static and dynamic control of a simply-supported non-linear circular plate with bimorph fully-covered piezoelectric layers were studied [19]. Modal sensing and actuating signals of thin cylindrical shells and panels were evaluated [20– 23]. Sensing and active control of piezoelectric elastic spherical shells were explored and micro sensing signals were obtained [24–26]. Micro-sensing characteristics and modal voltages of linear and nonlinear toroidal shells were achieved [27]. Distributed sensing and control of piezoelectric laminated conical shells with different actuator locations were analyzed [28,29]. Distributed modal sensing and actuation of thin-walled rotating paraboloidal shells with different boundary conditions were also studied [30–33]. Neural sensing signals and microscopic actuation of precision parabolic cylindrical shell panels were discussed [34,35]. However, investigation of distributed sensing signals on thin plates and membranes is insufficient. In this study, dynamic governing equations of circular plate and membrane mirrors are presented first. Transverse mode shape functions of the pretensioned plate and membrane mirrors with clamped boundary conditions are derived respectively. Then, the microscopic signal generations of infinitesimal piezoelectric sensor distributed on two different models, i.e., the plate and membrane mirror models, are investigated. Details are described in following sections. 2. Modeling of a flexible thin deformable mirror Based on the Kirchhoff-Love linear thin shell assumptions, the fundamental dynamic equations for a generic doublecurvature thin shell are [12,36]
@ @ @A1 @A2 Q €1 ðN11 A2 Þ þ ðN12 A1 Þ þ N12 N22 þ A1 A2 13 ¼ qhA1 A2 u @ a1 @ a2 @ a2 @ a1 R1
ð1Þ
@ @ @A2 @A1 Q €2 ðN12 A2 Þ þ ðN22 A1 Þ þ N12 N11 þ A1 A2 23 ¼ qhA1 A2 u @ a1 @ a2 @ a1 @ a2 R2
ð2Þ
@ @ N11 N22 €3 ¼ qhA1 A2 u ðQ 13 A2 Þ þ ðQ 23 A1 Þ A1 A2 þ @ a1 @ a2 R1 R2
ð3Þ
Q13 and Q23 are defined as
Q 13 A1 A2 ¼
@ @ @A1 @A2 ðM 11 A2 Þ þ ðM21 A1 Þ þ M 12 M 22 @ a1 @ a2 @ a2 @ a1
ð4Þ
Q 23 A1 A2 ¼
@ @ @A2 @A1 ðM 12 A2 Þ þ ðM22 A1 Þ þ M 21 M 11 @ a1 @ a2 @ a1 @ a2
ð5Þ
Then the vibration equations of a circular thin plate/membrane deformable mirror can be simplified from Eqs. (1)–(5) with specified two Lamé parameters and two radii of curvature. The geometry of the mirror is defined in a polar coordinate system which is shown in Fig. 1, where r and h respectively define the radial and angular directions of the neutral surface and a3 defines the transverse direction of the mirror. The thickness of the mirror is h and the radius is a. The two Lamé parameters of the mirror are A1 = Ar = 1 and A2 = Ah = r; two radii of curvature are R1 = Rr = 1 and R2 = Rh = 1. With the four parameters, one can simplify the double-curvature shell dynamic equations to the dynamic equations of the mirror.
@Nrr 1 @Nrh Nrr Nhh €r þ þ ¼ qhu r @h @r r
ð6Þ
@Nrh 1 @Nhh 2 €h þ þ Nrh ¼ qhu r @h r @r
ð7Þ
3
a
r
h
(Not to scale) Fig. 1. Geometry of a thin deformable mirror.
Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
@Q r3 1 @Q h3 Q r3 1 @ Nhh @u3 @u3 1 @ @u3 @u3 €3 þ ¼ qhu rN rr þ þ þ þ Nrh þ Nrh r @h r @h r @h r @r @r r @r @r @h
395
ð8Þ
Qr3 and Qh3 are defined as
Q r3 ¼
@M rr 1 @Mrh M rr M hh þ þ r @h @r r
Q h3 ¼
@M rh 1 @Mhh 2 þ þ Mrh r @h r @r
ð9Þ ð10Þ
where Nij is the membrane force per unit length; Mij is the bending moment per unit length; Qij is the shear force per unit length; q is the mass density; h is the mirror thickness; ui is the displacements in the i-th direction; and the dot denotes the partial derivation with respect to time. According to the Kirchhoff-Love thin shell assumptions, the transverse shear strains Sr3 and Sh3 and the transverse normal strain S33 are negligible. Radial, angular and in-plane twist strains are respectively defined as
Srr ¼ S0rr þ a3 krr
ð11Þ
Shh ¼ S0hh þ a3 khh
ð12Þ
Srh ¼ S0rh þ a3 krh
ð13Þ
where the membrane strains S0ij are defined as
S0rr ¼
@ur @r
ð14Þ
S0hh ¼
1 @uh ur þ r @h r
ð15Þ
S0rh ¼
@uh uh 1 @ur þ r @h @r r
ð16Þ
The bending strains kij are defined as
krr ¼
@ 2 u3 @r2
ð17Þ
khh ¼
1 @ 2 u3 1 @u3 r 2 @h2 r @r
ð18Þ
krh ¼
2 @u3 2 @ 2 u3 r2 @h r @r@h
ð19Þ
Then, the membrane forces Nij and bending moments Mij of the mirror can be defined as
Nrr ¼ KðS0rr þ lS0hh Þ
ð20Þ
Nhh ¼ KðS0hh þ lS0rr Þ
ð21Þ
Nrh ¼ Nhr ¼
Kð1 lÞ 0 Srh 2
ð22Þ
Mrr ¼ Dðkrr þ lkhh Þ
ð23Þ
Mhh ¼ Dðkhh þ lkrr Þ
ð24Þ
Mrh ¼ Mhr ¼
Dð1 lÞ krh 2
ð25Þ
where K = Yh/(1 l2) is the membrane stiffness; D = Yh3/12(1 l2) is the bending stiffness; Y is Young’s modulus; and l is Poisson’s ratio. Substituting these forces and moments into Eqs. (6)–(10) yields the final dynamic equations of the mirror. Modal vibration analysis of the mirror is presented below.
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3. Free vibration analysis of a thin deformable mirror Note that no external mechanical and electrical excitations are applied to the mirror during the free vibration analysis. The thin deformable mirror is pretensioned and clamped on the boundary. Based on the dynamic equations obtained previously, the vibration behavior of the mirror can be studied. The deformable mirror is modeled as a circular thin plate firstly and the mode shape function for the transverse harmonic oscillation is derived. Then with membrane assumption, the plate mirror model is simplified to a circular membrane model and the mode shape function is also obtained with similar procedure. Its application in sensing signal analysis is investigated later. 3.1. Free vibration of plate mirror model The deformable mirror is modeled as a circular thin plate in this section. It has been demonstrated that for thin shells and plates, the transverse component u3 is usually dominant among the three displacements [12,36]. In previous studies, only the transverse deflection u3 is considered for the explorations of dynamics and piezoelectric sensing/actuation behaviors of different shell structures [21,22,32,34]. Thus only mode shape function in transverse direction is derived here. The transverse dynamic equation of the mirror is
@ 2 u3 1 @u3 1 @ 2 u3 € 3 ¼ Nrr Dr u3 þ qhu þ N þ 2 hh r @r r @h2 @r2
!
4
þ 2Nrh
1 @ 2 u3 1 @u3 r @r@h r 2 @h
!
ð26Þ
For a circular thin plate model with uniform pretension N⁄, we have Nrr = Nhh = N⁄ and Nrh = 0. Thus, the equation of motion for free vibration becomes
€3 ¼ 0 Dr4 u3 N r2 u3 þ qhu
ð27Þ
where the biharmonic operator in polar coordinates is
r ¼r r ¼ 4
2
2
@2 1 @ 1 @2 þ þ @r 2 r @r r2 @h2
!2 ð28Þ
Assume that all points on the mirror vibrate at a natural frequency x, the harmonic oscillatory displacement is
u3 ðr; h; tÞ ¼ U 3 ðr; hÞejxt
ð29Þ
where U3 is the mode shape function in the transverse direction. Substituting Eq. (29) into Eq. (27) yields
Dr4 U 3 N r2 U 3 qhx2 U 3 ¼ 0 Let
k21
ð30Þ
N ¼ 2D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4qhDx2 1þ 1 2 ðN Þ
ð31Þ
N 2D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4qhDx2 1þ þ 1 2 ðN Þ
ð32Þ
k22 ¼
Eq. (30) can then be written as
ðr2 þ k21 Þðr2 k22 ÞU 3 ¼ 0
ð33Þ
For axisymmetric boundary conditions, it is possible to use the separation of the variables r and h to represent U3 in terms of R(r) and H(h)
U 3 ðr; hÞ ¼ RðrÞHðhÞ
ð34Þ
The solutions of H and R can be derived as
H ¼ A cos nðh uÞ
ð35Þ
R ¼ BJ n ðk1 rÞ þ CIn ðk2 rÞ
ð36Þ
where Jn and In are the Bessel function and the modified Bessel function of the first kind of order n, respectively. The boundary conditions, as mentioned above, are, at the boundary radius r = a,
u3 ða; h; tÞ ¼ 0
ð37Þ
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424 Table 1 Values for (k1a)mn and (k2a)mn. N N cr
m
n 0
1
2
k1a
k2a
k1a
k2a
k1a
k2a
0
0 1 2
3.196 6.306 9.440
3.196 6.306 9.440
4.611 7.799 10.958
4.611 7.799 10.958
5.906 9.197 12.402
5.906 9.197 12.402
1
0 1 2
2.954 6.226 9.401
4.840 7.312 10.153
4.478 7.745 10.929
5.895 8.642 11.582
5.822 9.158 12.380
6.971 9.928 12.960
2
0 1 2
2.840 6.165 9.367
6.121 8.210 10.824
4.392 7.700 10.903
6.978 9.418 12.177
5.758 9.123 12.359
7.909 10.613 13.496
Table 2 Values for (ka)mn [36]. m
0 1 2 3
n 0
1
2
3
2.404 3.832 5.135 6.379
5.520 7.016 8.417 9.760
8.654 10.173 11.620 13.017
11.792 13.323 14.796 16.224
Segmented piezoelectric sensors
z r
hs h
(Not to scale)
Plate/membrane mirror
Fig. 2. A plate/membrane mirror laminated with segmented piezoelectric sensors.
Table 3 Properties and dimensions of the mirror sensing model. Parameters
Symbol
Mirror substrate
PVDF layer
Units
Young’s modules Poisson’s ratio Mass density Thickness Radius Piezoelectric strain constant Dielectric constant
Y, Yp m, mp q, qp h, hs a h31, h32
1.013 106 0.497 1020 0.0015 0.1 / /
2 109 0.29 1800 5 105 / 9.6 103 8.85 1011
N/m2 / kg/m3 m m C/m2 F/m
@u3 ða; h; tÞ ¼ 0 @r
33
ð38Þ
This translates into
RðaÞ ¼ 0
ð39Þ
dR ðaÞ ¼ 0 dr
ð40Þ
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
Substituting Eq. (36) in these conditions gives
"
J n ðk1 aÞ
In ðk2 aÞ
dJn dr
dIn dr
ðk1 aÞ
# B
ðk2 aÞ
C
¼0
ð41Þ
Eq. (41) is satisfied in a meaningful way only if the determinant is 0 and this gives the frequency equation
J n ðk1 aÞ
dIn dJ ðk2 aÞ In ðk2 aÞ n ðk1 aÞ ¼ 0 dr dr
ð42Þ
Combining Eqs. (31) and (32) yields
ðk2 aÞ2 ðk1 aÞ2 ¼
a2 N D
ð43Þ
(Φ )bend,θ
4
0
2
s
s
(Φ )bend,r
5
-5 2
0.1
2
0
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
5
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
0 -5 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
(c)
0.05 θ (rad)
r(m)
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 3. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
Searching for roots k1a and k2a of the simultaneous equations Eqs. (42) and (43), labeled successively m = 0, 1, 2, . . . for each n = 0, 1, 2, . . ., gives the natural frequencies. Values of the roots k1a and k2a are calculated and collected in Table 1, where the critical load Ncr = 14.7D/a2. The natural frequencies are related to these roots by
xmn
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDk21 þ N Þk21 ðDk22 N Þk22 ¼ ¼ qh qh
ð44Þ
5 (Φ )bend,θ
(Φ )bend,r
10
0
s
s
0 -10 2
-5 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 4. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
To find the mode shapes, one can formulate from Eq. (41)
C J ðk1 aÞ ¼ n B In ðk2 aÞ
ð45Þ
Substituting Eqs. (35), (36) and (45) into Eq. (34) gives the mode shape function
J ðk1 aÞ U 3mn ¼ Amn J n ðk1 rÞ n In ðk2 rÞ cos nðh uÞ In ðk2 aÞ
ð46Þ
20 (Φ )bend,θ
(Φ )bend,r
20
10 0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
20
U3
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0 y(m)
0 -0.1 -0.1
x(m)
(e) Mode shape Fig. 5. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
where m, n denote the mode numbers in the radial and angular directions, respectively; Amn is the arbitrary modal amplitude of mode (m, n). 3.2. Free vibration of membrane mirror model Mode shape function of a clamped circular thin plate mirror model has been derived above. However, for some space deformable mirrors made of Kapton [6] or Room Temperature Vulcanizing (RTV) Silicone [7] or Linear Low Density Polyethylene (LLDPE) materials [8], the mirror substrates are such thin and flexible that may be modeled as circular membranes. In classical membrane theories, membranes are considered as pretensioned plates without bending stiffness. For circular membrane model with pretension N⁄, the transverse dynamic equation for free vibration is
10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
50 Φ
0 0
-50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 6. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
€3 ¼ 0 N r2 u3 qhu
ð47Þ
Similarly, substituting Eq. (29) in the equation of motion gives
N r2 U 3 þ qhx2 U 3 ¼ 0
ð48Þ
Defining
k2 ¼
qhx2
ð49Þ
N
10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 7. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
Eq. (48) can then be written as
ðr2 þ k2 ÞU 3 ¼ 0
ð50Þ
Also, the mode shape function U3 is possible to be separated as Eq. (34). The solutions of H and R for each n can be derived
H ¼ A cos nðh uÞ
ð51Þ
R ¼ BJn ðkrÞ þ CY n ðkrÞ
ð52Þ
(Φ )bend,θ
40
0
20 0
s
s
(Φ )bend,r
50
-50 2
-20 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0 y(m)
0 -0.1 -0.1
x(m)
(e) Mode shape Fig. 8. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
where Jn is the Bessel function of the first kind of order n; Yn is the Bessel function of the second kind of order n.Note that
Yð0Þ ¼ 1
ð53Þ
Thus, since it is physically impossible for the membrane to have an infinite deflection at its center, it follows that C = 0. The boundary condition for the membrane model at r = a is
u3 ða; h; tÞ ¼ 0
ð54Þ
which implies that
RðaÞ ¼ 0
ð55Þ
4 (Φ )bend,θ
0 -5
2
s
s
(Φ )bend,r
5
-10 2
2
0.1
0
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0 y(m)
0 -0.1 -0.1
x(m)
(e) Mode shape Fig. 9. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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or that
J n ðkaÞ ¼ 0
ð56Þ
For a giving n, Eq. (56) has an infinite number of roots (ka)mn identified by m = 0, 1, 2, . . . in ascending order. With similar procedure of solving the plate model above, the natural frequencies of the membrane are given by
xmn
ðkaÞmn ¼ a
sffiffiffiffiffiffi N qh
ð57Þ
The mode shape function is derived as
U 3mn ¼ Amn J n ðkmn rÞ cos nðh uÞ
ð58Þ
5 (Φ )bend,θ
(Φ )bend,r
10
0
s
s
0 -10 2
-5 2
0.1
0.1 0.05
0.05 θ (rad)
θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 10. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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A few values for (ka)mn are listed in Table 2. Eq. (56) indicates that for the membrane model, the pretension N⁄ has no influence on the values of (ka)mn. In other words, the pretension N⁄ does not affect the natural frequencies and mode shape function of the membrane mirror model. Application of the mode shape functions on distributed sensing signal analysis of the plate/membrane mirror fully covered with infinitesimal piezoelectric sensors are discussed in the next section. 4. Distributed sensing signals of a thin deformable mirror A piezoelectric layer is fully laminated on the non-reflective side of the mirror in order to evaluate the distributed sensing signals. To demonstrate spatially distributed sensing signals, the sensor layer is segmented into a series of infinitesimal ele-
20 (Φ )bend,θ
(Φ )bend,r
20
10 0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
20
U3
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 11. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
(b) ( s)bend,
1
0
U3
s
50 Φ
r(m)
0 0
-50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 12. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
ments to study the microscopic spatial distribution of modal sensing signals responding to modal oscillations of the plate/ membrane mirror (see Fig. 2). The thickness of the piezoelectric sensing layer hs is thin as compared with the plate/membrane mirror thickness h, usually less than 1/10 and in this study hs/h = 1/30. Thus, it is assumed that: (1) mass density and elastic stiffness of the sensing layer are neglected; (2) the neutral surface of the mirror is not influenced by the sensing layer; and (3) sensors are perfectly bonded on the mirror so that they exhibit the elastic strain consistent to the mirror substrate [26–34]. It is worth noting that as the distributed sensor, only the direct piezoelectric effect is considered. So one can define an open-circuit voltage /s in the transverse direction as
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1 U3
0
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 13. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system. s
/s ¼
h Se
Z Z h
r
ðh31 Srr þ h32 Shh Þrdrdh
ð59Þ
where hs is the distributed sensor thickness; Se is the effective sensor electrode area; h3i is the piezoelectric constant indicating a signal generation in the transverse direction due to the strain in the ith direction; Substituting Eqs. (14), (15) and Eqs. (17), (18) into Eqs. (11), (12) gives the radial and angular strains of the plate/membrane mirror. Then substituting Eqs. (11) and (12) into the electric signal equation Eq. (59) yields the signal generation of a sensor patch due to the transverse deflection: s
h / ¼ e S s
!# ! Z Z " @ 2 u3 1 @ 2 u3 1 @u3 s s rdrdh h31 r 2 þ h32 r 2 r @h2 r @r @r h r
ð60Þ
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40 (Φ )bend,θ
(Φ )bend,r
50
20 0
s
s
0 -50 2
-20 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0 -1 2
0.1
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 14. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
where rs denotes the distance between the neutral surface of the mirror and the mid-surface of the piezoelectric sensor. For a sensor layer with constant thickness, the bending arm rs=(h + hs)/2 h/2, when h hs. Furthermore, when reducing the sensor patch area to an infinitesimal element, the localized microscopic sensing signal at an arbitrary point (r, h) on the mirror can be described as
" s
s
/ ðr; hÞ ¼ h h31 r
s
@ 2 u3 2 @r
! þ h32 r
s
1 @ 2 u3 1 @u3 2 r @h2 r @r
!# ð61Þ
This distributed microscopic sensing signal can be divided into two contributing components: (1) the bending strain contribution in the radial direction (/s)bend,r; (2) the bending strain contribution in the angular direction (/s)bend,h.
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4 (Φ )bend,θ
0 -5
s
s
(Φ )bend,r
5
-10 2
2 0 -2 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 15. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
/s ðr; hÞ ¼ ð/s Þbend;r þ ð/s Þbend;h
ð62Þ
Substituting Eq. (29) into Eq. (61) and (62) gives the expressions of the two components s
ð/s Þbend;r ¼ h h31 r s ð1Þ s
ð/s Þbend;h ¼ h h32 r s ð1Þ
@ 2 U 3mn @r2 1 @ 2 U 3mn 1 @U 3mn þ r2 @h2 r @r
ð63Þ ! ð64Þ
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5 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-5 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0 s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 16. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
For the two different mirror models discussed in Section 3, i.e., the plate mirror model and the membrane mirror model, the mode shape functions U3mn are different and thus lead to the different sensing signal /s. Substituting mode shape functions of the two mirror model, Eqs. (46) and (58), into expressions of the two sensing signal components respectively gives (1) For plate mirror model s
ð/ Þbend;r
" # @ 2 J n ðwrÞ J n ðwaÞ @ 2 In ð/rÞ cos nh ¼ h h31 r ð1ÞAmn @r 2 In ð/aÞ @r 2 s
s
ð65Þ
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20 (Φ )bend,θ
(Φ )bend,r
20
10 0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1 U3
20
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 17. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
1 J n ðwaÞ 1 @J n ðwrÞ Jn ðwaÞ @In ð/rÞ 2 J ðwrÞ ð/rÞ ðn Þ þ cos nh I n r2 n In ð/aÞ r @r In ð/aÞ @r
s
ð/s Þbend;h ¼ h h31 r s ð1ÞAmn
ð66Þ
(2) For membrane mirror model
@ 2 J n ðkrÞ cos nh @r 2 1 1 @J n ðkrÞ s cos nh ¼ h h31 r s ð1ÞAmn 2 J n ðkrÞðn2 Þ þ r r @r s
ð/s Þbend;r ¼ h h31 r s ð1ÞAmn
ð67Þ
ð/s Þbend;h
ð68Þ
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10 (Φ )bend,θ
(Φ )bend,r
50
0
s
s
0 -50 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
50 Φ
0 0
-50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 18. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
In order to simplify the expression, the coordinate axes are coincide with the symmetry axes of the mode shape and thus u = 0. Since only the transverse displacement is considered and u3 has no contribution to the membrane strains S0ij , there is no signal component related to the membrane strain. The two bending signal components will be evaluated and compared to determine their respective contributions in future design applications. In the following section, specific material properties and mirror dimensions will be given and spatial modal sensing signals of the two models will be explored as well as the tension effect. 5. Evaluation of modal sensing signals of different models In this section, spatial distributions of modal sensing signals of two mirror models are investigated. The mirror is clamped on the boundary with pretension N⁄ and is discussed in two conditions: (1) the plate mirror model; and (2) the membrane
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 19. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
mirror model. Analyzing results of the sensing signal distributions would provide a design criterion to identify the optimal sensor locations to maximize the monitoring and feedback control effects. It has been derived above that the sensing signal consists of two contributing components: (1) the signal component induced by the bending strain in the radial direction (/s)bend,r; and (2) the signal component induced by the bending strain in the angular direction (/s)bend,h.The total sensing signal of the mirror and its signal components are evaluated and plotted together with corresponding mode shapes. Note that the mirror sensing signal distribution greatly depends on the transverse mode shape function U3mn. For the membrane mirror model, as discussed previously, the pretension N⁄ does not affect the U3mn and therefore has no influence on the sensing signal distributions. However, for the plate mirror model, U3mn is influenced by the eigenvalues k1a and k2a according to Eq. (46), which is determined by the pretension N⁄ in a given boundary condition (see Table 1). Thus, different pretension forces (N⁄= 0, Ncr, 2Ncr) are applied to the mirror to evaluate the tension effects on modal sensing signals. Material
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(Φ )bend,θ
40
0
20 0
s
s
(Φ )bend,r
50
-50 2
-20 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 20. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
properties and some dimensions of the mirror and the piezoelectric sensors are summarized in Table 3. The material parameters of piezoelectric polyvinylidene fluoride (PVDF) are used for piezoelectric sensors [12] and properties of the AFIT thin deformable mirror are used for mirror substrate [2]. Without loss of generality, the arbitrary modal amplitude Amn is assumed to be unity. 5.1. The plate mirror model Sensing signals of the plate mirror model are investigated in this section. Spatial distributions of mirror’s modal sensing signals and their signal components under different pretension forces are evaluated and their maximal signal magnitudes are identified respectively.
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Table 4 Maximum signal magnitudes of (/s)bend,r of the plate mirror model. Sensing signal
ð/s Þbend;r
Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude
Trend
N* = 0
N* = 1 Ncr
N* = 2 Ncr
3.617 14.950 33.414 8.266 18.828 37.003 11.688 23.279 36.689
3.182 14.555 33.140 9.252 19.251 36.797 12.635 24.134 37.480
2.987 15.293 32.902 10.217 20.124 36.611 13.536 24.942 38.230
N/A
Table 5 Maximum signal magnitudes of (/s)bend,h of the plate mirror model. Sensing signal
ð/s Þbend;h
Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude
Trend
N* = 0
N* = 1 Ncr
N* = 2 Ncr
3.617 14.950 33.414 2.765 8.219 16.165 6.574 15.857 28.840
3.182 14.555 33.140 2.629 8.103 16.075 6.371 15.724 28.737
2.987 14.262 32.902 2.546 8.008 15.995 6.224 15.605 28.640
Table 6 Maximum signal magnitudes of /s of the plate mirror model. Sensing signal
/s
Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude
Trend
N* = 0
N* = 1 Ncr
N* = 2 Ncr
7.234 29.900 66.828 8.945 26.509 52.291 12.398 30.850 56.085
6.363 29.109 66.280 9.251 26.121 52.029 12.636 30.566 55.894
5.974 28.524 65.803 10.218 25.828 51.794 13.537 30.312 55.712
5.1.1. Case 1: pretension N⁄=0 In this case, the pretension force applied on the boundary of the mirror is set to zero. Figs. 3–6 show the distribution of mirror modal sensing signals with modes (m, n) from (0,0) to (1,1) and Figs. 7 and 8 illustrate that generated in higher natural modes, i.e., (0,2) and (2,0). In each figure, the top-left (a) denotes the signal component resulting from the radial bending stain (/s)bend,r; the top-right (b) denotes the signal component resulting from the angular bending strain(/s)bend,h; the bottom-left (c) denotes the summation of both contributing components, i.e., total signal /s; and the bottom-right (d) denotes the mode shape plotted in the polar coordinate system. 5.1.2. Case 2: pretension N⁄=Ncr Pretension force N⁄ = Ncr (Ncr = 14.7D/a2) is applied to the mirror in this case. Modal sensing signal distributions are presented in Figs. 9–14 below. Also, figure (a), (b), (c) and (d) denote (/s)bend,r, (/s)bend,h, total signal /s and the mode shape, respectively. 5.1.3. Case 3: pretension N⁄=2Ncr In this case, pretension force N⁄ increases to 2Ncr (Ncr = 14.7D/a2) and is applied to the mirror. Modal sensing signal distributions are presented in Figs. 15–20 below. Also, figure (a), (b), (c) and (d) denote (/s)bend,r, (/s)bend,h, total signal /s and the mode shape correspondingly.
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3 (Φ )bend,θ
2 0
s
s
(Φ )bend,r
4
-2 2
2
0.1
2 1 0
0.1 0.05
0.05 θ (rad)
0 0
θ (rad)
r(m)
5
1
0
0
-5 2
0.1
-1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 21. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
5.1.4. Modal sensing signal analysis From Figs. 3–20 one can find that the sensing signal distribution changes slightly when the pretension N⁄ increases. In order to evaluate the sensing signals of the mirror under different pretension forces of different modes, modal signal magnitudes of various components of each mode (m, n) are summarized in Tables 4–6. Due to the clamped condition, the mode shape of the mirror reduces to zero at the boundary r = a. Also, the signal component (/s)bend,h = 0 of all modes at the bound ary r = a. This can be demonstrated by the definition of (/s)bend,h, referring to Eq. (64), that @U@r3mn r¼a ¼ 0 achieves according to 2 U 3mn the clamped boundary condition. With Eq. (46) one can derive that @ @h ¼ 0, which leads to ð/s Þbend;h ¼ 0. From Tables 2 r¼a
r¼a
4–6 it can be observed that with the increasing of mode m, all sensing signal components are also enhanced. However, this is
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
(Φ )bend,θ
2
0
0
s
s
(Φ )bend,r
5
-5 2
-2 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
10
1
0
0
-10 2
0.1
-1 2
0.1
0.05 θ (rad)
(c)
0.05 θ (rad)
r(m)
0 0
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 22. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
not always applicable with mode n. When n = 0, the maximal signal magnitude of (/s)bend,r and (/s)bend,h are the same. While with the growing of mode n = 2, (/s)bend,r component dominates obviously. However when n = 3, the gap of the two components narrows. Furthermore, when the pretension force N⁄ augments, only the (/s)bend,h component decreases in all modes. For most modes, the total sensing signal /s decreases, while mode (0,1) and (0,2) are the opposite. Variation tendency of each sensing signal component with time is also concluded in Tables 4–6.
5.2. The membrane mirror model For a membrane mirror model with arbitrary pretension N⁄, Figs. 21–26 present the spatial distribution of modal sensing signals and the corresponding mode shapes. In these figures, plots of component signals, total signal and mode
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
(Φ )bend,θ
20
10
10 0
s
0
s
(Φ )bend,r
20
-10 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
20
U3
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 23. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
shapes are arranged the same as before. Also, the maximum sensing signal magnitude of each mode is presented in Table 7. It can also be observed from Figs. 21–26 that owing to the clamped boundary condition, the mode shapes of the mirror maintain zero as well as the total sensing signals at r = a. This can be derived from Eq. (48) that when r = a, it is obvious that U3mn = 0 and therefore r2U3mn = 0. Referring to the definition of total sensing signal in Eq. (62), we have /s = 0, which is different from the plate mirror model where (/s)bend,h = 0. Additionally, it is easy to find that for all modes, the trends of /s and the mode shape consistent with each other qualitatively. Similarly with the plate mirror model, it can be concluded from Table 7 that all sensing signal components increase with the growing of mode m while mode n does not have the same effect. Moreover, the (/s)bend,r component starts to dominate when n = 2 and increases little when n = 3, which is the same with
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
50
1
0
0
-50 2
0.1
-1 2
0.1
0.05 θ (rad)
0 0
(c)
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
0.05 θ (rad)
r(m)
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 24. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
plate mirror model. Comparing Table 7 with Tables 4–6 one can find that the sensing signal magnitudes of the membrane mirror model are lower than that of the plate mirror model for each component and all modes. Moreover, from the mathematical procedure of solving U3mn of the two models one can find that for the membrane model, the bending stiffness D is neglected as a simplification, which leads to a weaker membrane boundary condition, i.e., Eq. (54), as compared with the boundary conditions, Eqs. (37) and (38), of the plate model. As a consequence, pretension N⁄ will not affect the transverse mode shape and sensing signal of the mirror. Actually, this is to some extent physically unreasonable since the bending stiffness should never be zero even though it can be quite small for an ultra-thin and soft space membrane mirror.
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5 (Φ )bend,θ
(Φ )bend,r
10
0
s
s
0 -10 2
-5 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
10
1
0
0
-10 2
0.1
-1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 25. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
6. Conclusions In this paper, a mathematical model of a deformable plate/membrane mirror is established. Dynamic characteristic of plate and membrane models are studied respectively and transverse mode shape functions are derived and used for distributed signal analysis. Total micro-sensing signal and its two components of different mirror models are presented. Furthermore, effects of different pretension forces on mirror modal sensing behavior are investigated. Based on the detailed analysis, the following conclusions can be drawn:
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(Φ )bend,θ
40
0
20 0
s
s
(Φ )bend,r
50
-50 2
-20 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 26. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
(1) For thin deformable plate/membrane mirrors, the distributed microscopic sensing signal could be divided into two bending components, i.e., the (/s)bend,r and the (/s)bend,h. However, the membrane strains have no contribution to the sensing signals induced by transverse vibration. Note that only linear free vibration is considered in the study. When it comes to large amplitude nonlinear vibrations, the problem would be more complicated. (2) For the membrane mirror model, pretension N⁄ affects neither the mode shape nor the sensing signal distributions. While for the plate mirror model, pretension force will influence both. With the increasing of N⁄, sensing signal component (/s)bend,h decreases in all modes. Also, for most modes, the total sensing signal /s reduces. (3) For each microscopic sensing signal component and total signal /s, the signal magnitudes of all modes of the membrane mirror model are lower than that of the plate mirror model.
Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
423
Table 7 Maximum sensing signal magnitudes of membrane mirror model. Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude ð/s Þbend;r
ð/s Þbend;h
/s
2.169 11.426 28.084 4.540 15.215 31.970 5.578 15.290 29.005
2.169 11.426 28.084 1.982 6.643 13.971 4.946 13.284 25.317
4.338 22.853 56.169 6.407 21.424 45.113 9.623 25.772 48.984
(4) For both plate and membrane mirror models, with the increasing of mode m, all sensing signal components also increase. However, when mode n increases, the sensing signal magnitudes show an irregular variation. (5) Due to the clamped boundary condition of the mirror, all mode shapes of two models reduce to zero at the boundary. Additionally, for plate mirror model, the signal component (/s)bend,h = 0 when r = a. However, for membrane mirror model, the total signal /s = 0 at the boundary. (6) The plate mirror model is physically more reasonable compared with the membrane mirror model since it take into account the bending stiffness and the pretension effect. Distributed sensing signals are location and modal dependent. Based on the mathematical model and signal analysis in this paper, one can get some design criteria for optimal sensor placement on lightweight deformable mirrors. These data can also be used to develop dynamic vibration control of large-scale space flexible structures. Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant No. 51175103) and Self-Planned Task of State Key Laboratory of Robotics and System (HIT) (Grant No. SKLRS201301B). Prof. Tzou would like to thank the ‘‘Ph. D. Supervisor Program” and the ‘‘111 Project” (B07018) at the Harbin Institute of Technology sponsored by the Chinese Ministry of Education. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Distributed sensing signal analysis of deformable plate/membrane mirrors Yifan Lu a, Honghao Yue 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 11 January 2017 Received in revised form 11 March 2017 Accepted 19 April 2017
Keywords: Plate/membrane mirror Piezoelectric layer Sensing signal Tension effect
a b s t r a c t Deformable optical mirrors usually play key roles in aerospace and optical structural systems applied to space telescopes, radars, solar collectors, communication antennas, etc. Limited by the payload capacity of current launch vehicles, the deformable mirrors should be lightweight and are generally made of ultra-thin plates or even membranes. These plate/membrane mirrors are susceptible to external excitations and this may lead to surface inaccuracy and jeopardize relevant working performance. In order to investigate the modal vibration characteristics of the mirror, a piezoelectric layer is fully laminated on its non-reflective side to serve as sensors. The piezoelectric layer is segmented into infinitesimal elements so that microscopic distributed sensing signals can be explored. In this paper, the deformable mirror is modeled as a pre-tensioned plate and membrane respectively and sensing signal distributions of the two models are compared. Different pre-tensioning forces are also applied to reveal the tension effects on the mode shape and sensing signals of the mirror. Analytical results in this study could be used as guideline of optimal sensor/actuator placement for deformable space mirrors. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Circular thin plates are commonly used as key components for optical deformable mirror systems in space telescopes [1– 5]. To further reduce the areal density and total mass of the optical system in space, the concept of thin film optics was proposed in recent years and series of membrane deformable mirror prototypes have been developed and tested [6–11]. These high-precision deformable plate/membrane mirrors used in space are usually lightweight and flexible and they often exhibit high flexibility, poor stiffness and low damping properties. Thus, undesirable external induced vibration can become a serious problem during operations, which lead to surface inaccuracy and even jeopardize relevant working performance. Precision sensing and control are therefore necessary to these high-performance optical mirror systems. This study is (1) to investigate spatially distributed microscopic sensing signals of lightweight circular plate/membrane mirrors, and (2) to evaluate the tensioning effects on different signal components of different modes. Sensor and actuator design criteria can be summarized from these results. Due to the distinct direct and converse piezoelectric effects, piezoelectric materials are widely used in sensor and actuator applications where high accuracy and precision are needed [12]. Polyvinylidene fluoride (PVDF) polymeric materials, as an outstanding representative of thin-film piezoelectric materials, are lightweight, dynamically sensitive, and can be easily seg⇑ Corresponding author. E-mail address: [email protected] (H. Yue). http://dx.doi.org/10.1016/j.ymssp.2017.04.028 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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mented and shaped for distributed sensing and control of flexible structures. Accordingly, the distributed piezoelectric layer can serve as distributed sensors and actuators in sensing and control of optical deformable plate/membrane mirrors. Distributed sensing and control of various structures using piezoelectric materials have been investigated over the years [13–17]. Electromechanical sensing properties of flexible rings laminated with distributed piezoelectric sensors were investigated [18]. Static and dynamic control of a simply-supported non-linear circular plate with bimorph fully-covered piezoelectric layers were studied [19]. Modal sensing and actuating signals of thin cylindrical shells and panels were evaluated [20– 23]. Sensing and active control of piezoelectric elastic spherical shells were explored and micro sensing signals were obtained [24–26]. Micro-sensing characteristics and modal voltages of linear and nonlinear toroidal shells were achieved [27]. Distributed sensing and control of piezoelectric laminated conical shells with different actuator locations were analyzed [28,29]. Distributed modal sensing and actuation of thin-walled rotating paraboloidal shells with different boundary conditions were also studied [30–33]. Neural sensing signals and microscopic actuation of precision parabolic cylindrical shell panels were discussed [34,35]. However, investigation of distributed sensing signals on thin plates and membranes is insufficient. In this study, dynamic governing equations of circular plate and membrane mirrors are presented first. Transverse mode shape functions of the pretensioned plate and membrane mirrors with clamped boundary conditions are derived respectively. Then, the microscopic signal generations of infinitesimal piezoelectric sensor distributed on two different models, i.e., the plate and membrane mirror models, are investigated. Details are described in following sections. 2. Modeling of a flexible thin deformable mirror Based on the Kirchhoff-Love linear thin shell assumptions, the fundamental dynamic equations for a generic doublecurvature thin shell are [12,36]
@ @ @A1 @A2 Q €1 ðN11 A2 Þ þ ðN12 A1 Þ þ N12 N22 þ A1 A2 13 ¼ qhA1 A2 u @ a1 @ a2 @ a2 @ a1 R1
ð1Þ
@ @ @A2 @A1 Q €2 ðN12 A2 Þ þ ðN22 A1 Þ þ N12 N11 þ A1 A2 23 ¼ qhA1 A2 u @ a1 @ a2 @ a1 @ a2 R2
ð2Þ
@ @ N11 N22 €3 ¼ qhA1 A2 u ðQ 13 A2 Þ þ ðQ 23 A1 Þ A1 A2 þ @ a1 @ a2 R1 R2
ð3Þ
Q13 and Q23 are defined as
Q 13 A1 A2 ¼
@ @ @A1 @A2 ðM 11 A2 Þ þ ðM21 A1 Þ þ M 12 M 22 @ a1 @ a2 @ a2 @ a1
ð4Þ
Q 23 A1 A2 ¼
@ @ @A2 @A1 ðM 12 A2 Þ þ ðM22 A1 Þ þ M 21 M 11 @ a1 @ a2 @ a1 @ a2
ð5Þ
Then the vibration equations of a circular thin plate/membrane deformable mirror can be simplified from Eqs. (1)–(5) with specified two Lamé parameters and two radii of curvature. The geometry of the mirror is defined in a polar coordinate system which is shown in Fig. 1, where r and h respectively define the radial and angular directions of the neutral surface and a3 defines the transverse direction of the mirror. The thickness of the mirror is h and the radius is a. The two Lamé parameters of the mirror are A1 = Ar = 1 and A2 = Ah = r; two radii of curvature are R1 = Rr = 1 and R2 = Rh = 1. With the four parameters, one can simplify the double-curvature shell dynamic equations to the dynamic equations of the mirror.
@Nrr 1 @Nrh Nrr Nhh €r þ þ ¼ qhu r @h @r r
ð6Þ
@Nrh 1 @Nhh 2 €h þ þ Nrh ¼ qhu r @h r @r
ð7Þ
3
a
r
h
(Not to scale) Fig. 1. Geometry of a thin deformable mirror.
Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
@Q r3 1 @Q h3 Q r3 1 @ Nhh @u3 @u3 1 @ @u3 @u3 €3 þ ¼ qhu rN rr þ þ þ þ Nrh þ Nrh r @h r @h r @h r @r @r r @r @r @h
395
ð8Þ
Qr3 and Qh3 are defined as
Q r3 ¼
@M rr 1 @Mrh M rr M hh þ þ r @h @r r
Q h3 ¼
@M rh 1 @Mhh 2 þ þ Mrh r @h r @r
ð9Þ ð10Þ
where Nij is the membrane force per unit length; Mij is the bending moment per unit length; Qij is the shear force per unit length; q is the mass density; h is the mirror thickness; ui is the displacements in the i-th direction; and the dot denotes the partial derivation with respect to time. According to the Kirchhoff-Love thin shell assumptions, the transverse shear strains Sr3 and Sh3 and the transverse normal strain S33 are negligible. Radial, angular and in-plane twist strains are respectively defined as
Srr ¼ S0rr þ a3 krr
ð11Þ
Shh ¼ S0hh þ a3 khh
ð12Þ
Srh ¼ S0rh þ a3 krh
ð13Þ
where the membrane strains S0ij are defined as
S0rr ¼
@ur @r
ð14Þ
S0hh ¼
1 @uh ur þ r @h r
ð15Þ
S0rh ¼
@uh uh 1 @ur þ r @h @r r
ð16Þ
The bending strains kij are defined as
krr ¼
@ 2 u3 @r2
ð17Þ
khh ¼
1 @ 2 u3 1 @u3 r 2 @h2 r @r
ð18Þ
krh ¼
2 @u3 2 @ 2 u3 r2 @h r @r@h
ð19Þ
Then, the membrane forces Nij and bending moments Mij of the mirror can be defined as
Nrr ¼ KðS0rr þ lS0hh Þ
ð20Þ
Nhh ¼ KðS0hh þ lS0rr Þ
ð21Þ
Nrh ¼ Nhr ¼
Kð1 lÞ 0 Srh 2
ð22Þ
Mrr ¼ Dðkrr þ lkhh Þ
ð23Þ
Mhh ¼ Dðkhh þ lkrr Þ
ð24Þ
Mrh ¼ Mhr ¼
Dð1 lÞ krh 2
ð25Þ
where K = Yh/(1 l2) is the membrane stiffness; D = Yh3/12(1 l2) is the bending stiffness; Y is Young’s modulus; and l is Poisson’s ratio. Substituting these forces and moments into Eqs. (6)–(10) yields the final dynamic equations of the mirror. Modal vibration analysis of the mirror is presented below.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
3. Free vibration analysis of a thin deformable mirror Note that no external mechanical and electrical excitations are applied to the mirror during the free vibration analysis. The thin deformable mirror is pretensioned and clamped on the boundary. Based on the dynamic equations obtained previously, the vibration behavior of the mirror can be studied. The deformable mirror is modeled as a circular thin plate firstly and the mode shape function for the transverse harmonic oscillation is derived. Then with membrane assumption, the plate mirror model is simplified to a circular membrane model and the mode shape function is also obtained with similar procedure. Its application in sensing signal analysis is investigated later. 3.1. Free vibration of plate mirror model The deformable mirror is modeled as a circular thin plate in this section. It has been demonstrated that for thin shells and plates, the transverse component u3 is usually dominant among the three displacements [12,36]. In previous studies, only the transverse deflection u3 is considered for the explorations of dynamics and piezoelectric sensing/actuation behaviors of different shell structures [21,22,32,34]. Thus only mode shape function in transverse direction is derived here. The transverse dynamic equation of the mirror is
@ 2 u3 1 @u3 1 @ 2 u3 € 3 ¼ Nrr Dr u3 þ qhu þ N þ 2 hh r @r r @h2 @r2
!
4
þ 2Nrh
1 @ 2 u3 1 @u3 r @r@h r 2 @h
!
ð26Þ
For a circular thin plate model with uniform pretension N⁄, we have Nrr = Nhh = N⁄ and Nrh = 0. Thus, the equation of motion for free vibration becomes
€3 ¼ 0 Dr4 u3 N r2 u3 þ qhu
ð27Þ
where the biharmonic operator in polar coordinates is
r ¼r r ¼ 4
2
2
@2 1 @ 1 @2 þ þ @r 2 r @r r2 @h2
!2 ð28Þ
Assume that all points on the mirror vibrate at a natural frequency x, the harmonic oscillatory displacement is
u3 ðr; h; tÞ ¼ U 3 ðr; hÞejxt
ð29Þ
where U3 is the mode shape function in the transverse direction. Substituting Eq. (29) into Eq. (27) yields
Dr4 U 3 N r2 U 3 qhx2 U 3 ¼ 0 Let
k21
ð30Þ
N ¼ 2D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4qhDx2 1þ 1 2 ðN Þ
ð31Þ
N 2D
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4qhDx2 1þ þ 1 2 ðN Þ
ð32Þ
k22 ¼
Eq. (30) can then be written as
ðr2 þ k21 Þðr2 k22 ÞU 3 ¼ 0
ð33Þ
For axisymmetric boundary conditions, it is possible to use the separation of the variables r and h to represent U3 in terms of R(r) and H(h)
U 3 ðr; hÞ ¼ RðrÞHðhÞ
ð34Þ
The solutions of H and R can be derived as
H ¼ A cos nðh uÞ
ð35Þ
R ¼ BJ n ðk1 rÞ þ CIn ðk2 rÞ
ð36Þ
where Jn and In are the Bessel function and the modified Bessel function of the first kind of order n, respectively. The boundary conditions, as mentioned above, are, at the boundary radius r = a,
u3 ða; h; tÞ ¼ 0
ð37Þ
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424 Table 1 Values for (k1a)mn and (k2a)mn. N N cr
m
n 0
1
2
k1a
k2a
k1a
k2a
k1a
k2a
0
0 1 2
3.196 6.306 9.440
3.196 6.306 9.440
4.611 7.799 10.958
4.611 7.799 10.958
5.906 9.197 12.402
5.906 9.197 12.402
1
0 1 2
2.954 6.226 9.401
4.840 7.312 10.153
4.478 7.745 10.929
5.895 8.642 11.582
5.822 9.158 12.380
6.971 9.928 12.960
2
0 1 2
2.840 6.165 9.367
6.121 8.210 10.824
4.392 7.700 10.903
6.978 9.418 12.177
5.758 9.123 12.359
7.909 10.613 13.496
Table 2 Values for (ka)mn [36]. m
0 1 2 3
n 0
1
2
3
2.404 3.832 5.135 6.379
5.520 7.016 8.417 9.760
8.654 10.173 11.620 13.017
11.792 13.323 14.796 16.224
Segmented piezoelectric sensors
z r
hs h
(Not to scale)
Plate/membrane mirror
Fig. 2. A plate/membrane mirror laminated with segmented piezoelectric sensors.
Table 3 Properties and dimensions of the mirror sensing model. Parameters
Symbol
Mirror substrate
PVDF layer
Units
Young’s modules Poisson’s ratio Mass density Thickness Radius Piezoelectric strain constant Dielectric constant
Y, Yp m, mp q, qp h, hs a h31, h32
1.013 106 0.497 1020 0.0015 0.1 / /
2 109 0.29 1800 5 105 / 9.6 103 8.85 1011
N/m2 / kg/m3 m m C/m2 F/m
@u3 ða; h; tÞ ¼ 0 @r
33
ð38Þ
This translates into
RðaÞ ¼ 0
ð39Þ
dR ðaÞ ¼ 0 dr
ð40Þ
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Substituting Eq. (36) in these conditions gives
"
J n ðk1 aÞ
In ðk2 aÞ
dJn dr
dIn dr
ðk1 aÞ
# B
ðk2 aÞ
C
¼0
ð41Þ
Eq. (41) is satisfied in a meaningful way only if the determinant is 0 and this gives the frequency equation
J n ðk1 aÞ
dIn dJ ðk2 aÞ In ðk2 aÞ n ðk1 aÞ ¼ 0 dr dr
ð42Þ
Combining Eqs. (31) and (32) yields
ðk2 aÞ2 ðk1 aÞ2 ¼
a2 N D
ð43Þ
(Φ )bend,θ
4
0
2
s
s
(Φ )bend,r
5
-5 2
0.1
2
0
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
5
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
0 -5 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
(c)
0.05 θ (rad)
r(m)
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 3. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Searching for roots k1a and k2a of the simultaneous equations Eqs. (42) and (43), labeled successively m = 0, 1, 2, . . . for each n = 0, 1, 2, . . ., gives the natural frequencies. Values of the roots k1a and k2a are calculated and collected in Table 1, where the critical load Ncr = 14.7D/a2. The natural frequencies are related to these roots by
xmn
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDk21 þ N Þk21 ðDk22 N Þk22 ¼ ¼ qh qh
ð44Þ
5 (Φ )bend,θ
(Φ )bend,r
10
0
s
s
0 -10 2
-5 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 4. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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To find the mode shapes, one can formulate from Eq. (41)
C J ðk1 aÞ ¼ n B In ðk2 aÞ
ð45Þ
Substituting Eqs. (35), (36) and (45) into Eq. (34) gives the mode shape function
J ðk1 aÞ U 3mn ¼ Amn J n ðk1 rÞ n In ðk2 rÞ cos nðh uÞ In ðk2 aÞ
ð46Þ
20 (Φ )bend,θ
(Φ )bend,r
20
10 0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
20
U3
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0 y(m)
0 -0.1 -0.1
x(m)
(e) Mode shape Fig. 5. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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where m, n denote the mode numbers in the radial and angular directions, respectively; Amn is the arbitrary modal amplitude of mode (m, n). 3.2. Free vibration of membrane mirror model Mode shape function of a clamped circular thin plate mirror model has been derived above. However, for some space deformable mirrors made of Kapton [6] or Room Temperature Vulcanizing (RTV) Silicone [7] or Linear Low Density Polyethylene (LLDPE) materials [8], the mirror substrates are such thin and flexible that may be modeled as circular membranes. In classical membrane theories, membranes are considered as pretensioned plates without bending stiffness. For circular membrane model with pretension N⁄, the transverse dynamic equation for free vibration is
10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
50 Φ
0 0
-50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 6. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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€3 ¼ 0 N r2 u3 qhu
ð47Þ
Similarly, substituting Eq. (29) in the equation of motion gives
N r2 U 3 þ qhx2 U 3 ¼ 0
ð48Þ
Defining
k2 ¼
qhx2
ð49Þ
N
10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 7. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
Eq. (48) can then be written as
ðr2 þ k2 ÞU 3 ¼ 0
ð50Þ
Also, the mode shape function U3 is possible to be separated as Eq. (34). The solutions of H and R for each n can be derived
H ¼ A cos nðh uÞ
ð51Þ
R ¼ BJn ðkrÞ þ CY n ðkrÞ
ð52Þ
(Φ )bend,θ
40
0
20 0
s
s
(Φ )bend,r
50
-50 2
-20 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0 y(m)
0 -0.1 -0.1
x(m)
(e) Mode shape Fig. 8. Sensing signal distributions and mode shape of plate mirror model, N* = 0, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
where Jn is the Bessel function of the first kind of order n; Yn is the Bessel function of the second kind of order n.Note that
Yð0Þ ¼ 1
ð53Þ
Thus, since it is physically impossible for the membrane to have an infinite deflection at its center, it follows that C = 0. The boundary condition for the membrane model at r = a is
u3 ða; h; tÞ ¼ 0
ð54Þ
which implies that
RðaÞ ¼ 0
ð55Þ
4 (Φ )bend,θ
0 -5
2
s
s
(Φ )bend,r
5
-10 2
2
0.1
0
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0 y(m)
0 -0.1 -0.1
x(m)
(e) Mode shape Fig. 9. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
or that
J n ðkaÞ ¼ 0
ð56Þ
For a giving n, Eq. (56) has an infinite number of roots (ka)mn identified by m = 0, 1, 2, . . . in ascending order. With similar procedure of solving the plate model above, the natural frequencies of the membrane are given by
xmn
ðkaÞmn ¼ a
sffiffiffiffiffiffi N qh
ð57Þ
The mode shape function is derived as
U 3mn ¼ Amn J n ðkmn rÞ cos nðh uÞ
ð58Þ
5 (Φ )bend,θ
(Φ )bend,r
10
0
s
s
0 -10 2
-5 2
0.1
0.1 0.05
0.05 θ (rad)
θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 10. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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A few values for (ka)mn are listed in Table 2. Eq. (56) indicates that for the membrane model, the pretension N⁄ has no influence on the values of (ka)mn. In other words, the pretension N⁄ does not affect the natural frequencies and mode shape function of the membrane mirror model. Application of the mode shape functions on distributed sensing signal analysis of the plate/membrane mirror fully covered with infinitesimal piezoelectric sensors are discussed in the next section. 4. Distributed sensing signals of a thin deformable mirror A piezoelectric layer is fully laminated on the non-reflective side of the mirror in order to evaluate the distributed sensing signals. To demonstrate spatially distributed sensing signals, the sensor layer is segmented into a series of infinitesimal ele-
20 (Φ )bend,θ
(Φ )bend,r
20
10 0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
20
U3
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 11. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
(b) ( s)bend,
1
0
U3
s
50 Φ
r(m)
0 0
-50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 12. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
ments to study the microscopic spatial distribution of modal sensing signals responding to modal oscillations of the plate/ membrane mirror (see Fig. 2). The thickness of the piezoelectric sensing layer hs is thin as compared with the plate/membrane mirror thickness h, usually less than 1/10 and in this study hs/h = 1/30. Thus, it is assumed that: (1) mass density and elastic stiffness of the sensing layer are neglected; (2) the neutral surface of the mirror is not influenced by the sensing layer; and (3) sensors are perfectly bonded on the mirror so that they exhibit the elastic strain consistent to the mirror substrate [26–34]. It is worth noting that as the distributed sensor, only the direct piezoelectric effect is considered. So one can define an open-circuit voltage /s in the transverse direction as
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1 U3
0
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 13. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system. s
/s ¼
h Se
Z Z h
r
ðh31 Srr þ h32 Shh Þrdrdh
ð59Þ
where hs is the distributed sensor thickness; Se is the effective sensor electrode area; h3i is the piezoelectric constant indicating a signal generation in the transverse direction due to the strain in the ith direction; Substituting Eqs. (14), (15) and Eqs. (17), (18) into Eqs. (11), (12) gives the radial and angular strains of the plate/membrane mirror. Then substituting Eqs. (11) and (12) into the electric signal equation Eq. (59) yields the signal generation of a sensor patch due to the transverse deflection: s
h / ¼ e S s
!# ! Z Z " @ 2 u3 1 @ 2 u3 1 @u3 s s rdrdh h31 r 2 þ h32 r 2 r @h2 r @r @r h r
ð60Þ
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
40 (Φ )bend,θ
(Φ )bend,r
50
20 0
s
s
0 -50 2
-20 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0 -1 2
0.1
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 14. Sensing signal distributions and mode shape of plate mirror model, N* = Ncr, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
where rs denotes the distance between the neutral surface of the mirror and the mid-surface of the piezoelectric sensor. For a sensor layer with constant thickness, the bending arm rs=(h + hs)/2 h/2, when h hs. Furthermore, when reducing the sensor patch area to an infinitesimal element, the localized microscopic sensing signal at an arbitrary point (r, h) on the mirror can be described as
" s
s
/ ðr; hÞ ¼ h h31 r
s
@ 2 u3 2 @r
! þ h32 r
s
1 @ 2 u3 1 @u3 2 r @h2 r @r
!# ð61Þ
This distributed microscopic sensing signal can be divided into two contributing components: (1) the bending strain contribution in the radial direction (/s)bend,r; (2) the bending strain contribution in the angular direction (/s)bend,h.
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4 (Φ )bend,θ
0 -5
s
s
(Φ )bend,r
5
-10 2
2 0 -2 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
10 Φ
0 0
-10 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 15. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
/s ðr; hÞ ¼ ð/s Þbend;r þ ð/s Þbend;h
ð62Þ
Substituting Eq. (29) into Eq. (61) and (62) gives the expressions of the two components s
ð/s Þbend;r ¼ h h31 r s ð1Þ s
ð/s Þbend;h ¼ h h32 r s ð1Þ
@ 2 U 3mn @r2 1 @ 2 U 3mn 1 @U 3mn þ r2 @h2 r @r
ð63Þ ! ð64Þ
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
5 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-5 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0 s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 16. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
For the two different mirror models discussed in Section 3, i.e., the plate mirror model and the membrane mirror model, the mode shape functions U3mn are different and thus lead to the different sensing signal /s. Substituting mode shape functions of the two mirror model, Eqs. (46) and (58), into expressions of the two sensing signal components respectively gives (1) For plate mirror model s
ð/ Þbend;r
" # @ 2 J n ðwrÞ J n ðwaÞ @ 2 In ð/rÞ cos nh ¼ h h31 r ð1ÞAmn @r 2 In ð/aÞ @r 2 s
s
ð65Þ
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20 (Φ )bend,θ
(Φ )bend,r
20
10 0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1 U3
20
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 17. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
1 J n ðwaÞ 1 @J n ðwrÞ Jn ðwaÞ @In ð/rÞ 2 J ðwrÞ ð/rÞ ðn Þ þ cos nh I n r2 n In ð/aÞ r @r In ð/aÞ @r
s
ð/s Þbend;h ¼ h h31 r s ð1ÞAmn
ð66Þ
(2) For membrane mirror model
@ 2 J n ðkrÞ cos nh @r 2 1 1 @J n ðkrÞ s cos nh ¼ h h31 r s ð1ÞAmn 2 J n ðkrÞðn2 Þ þ r r @r s
ð/s Þbend;r ¼ h h31 r s ð1ÞAmn
ð67Þ
ð/s Þbend;h
ð68Þ
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10 (Φ )bend,θ
(Φ )bend,r
50
0
s
s
0 -50 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
50 Φ
0 0
-50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 18. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
In order to simplify the expression, the coordinate axes are coincide with the symmetry axes of the mode shape and thus u = 0. Since only the transverse displacement is considered and u3 has no contribution to the membrane strains S0ij , there is no signal component related to the membrane strain. The two bending signal components will be evaluated and compared to determine their respective contributions in future design applications. In the following section, specific material properties and mirror dimensions will be given and spatial modal sensing signals of the two models will be explored as well as the tension effect. 5. Evaluation of modal sensing signals of different models In this section, spatial distributions of modal sensing signals of two mirror models are investigated. The mirror is clamped on the boundary with pretension N⁄ and is discussed in two conditions: (1) the plate mirror model; and (2) the membrane
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
(a) ( s)bend,r
1
0
U3
s
r(m)
(b) ( s)bend,
20 Φ
0 0
-20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
θ (rad)
r(m)
0 0
(c)
0.05
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 19. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
mirror model. Analyzing results of the sensing signal distributions would provide a design criterion to identify the optimal sensor locations to maximize the monitoring and feedback control effects. It has been derived above that the sensing signal consists of two contributing components: (1) the signal component induced by the bending strain in the radial direction (/s)bend,r; and (2) the signal component induced by the bending strain in the angular direction (/s)bend,h.The total sensing signal of the mirror and its signal components are evaluated and plotted together with corresponding mode shapes. Note that the mirror sensing signal distribution greatly depends on the transverse mode shape function U3mn. For the membrane mirror model, as discussed previously, the pretension N⁄ does not affect the U3mn and therefore has no influence on the sensing signal distributions. However, for the plate mirror model, U3mn is influenced by the eigenvalues k1a and k2a according to Eq. (46), which is determined by the pretension N⁄ in a given boundary condition (see Table 1). Thus, different pretension forces (N⁄= 0, Ncr, 2Ncr) are applied to the mirror to evaluate the tension effects on modal sensing signals. Material
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Y. Lu et al. / Mechanical Systems and Signal Processing 96 (2017) 393–424
(Φ )bend,θ
40
0
20 0
s
s
(Φ )bend,r
50
-50 2
-20 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 20. Sensing signal distributions and mode shape of plate mirror model, N* = 2Ncr, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
properties and some dimensions of the mirror and the piezoelectric sensors are summarized in Table 3. The material parameters of piezoelectric polyvinylidene fluoride (PVDF) are used for piezoelectric sensors [12] and properties of the AFIT thin deformable mirror are used for mirror substrate [2]. Without loss of generality, the arbitrary modal amplitude Amn is assumed to be unity. 5.1. The plate mirror model Sensing signals of the plate mirror model are investigated in this section. Spatial distributions of mirror’s modal sensing signals and their signal components under different pretension forces are evaluated and their maximal signal magnitudes are identified respectively.
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Table 4 Maximum signal magnitudes of (/s)bend,r of the plate mirror model. Sensing signal
ð/s Þbend;r
Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude
Trend
N* = 0
N* = 1 Ncr
N* = 2 Ncr
3.617 14.950 33.414 8.266 18.828 37.003 11.688 23.279 36.689
3.182 14.555 33.140 9.252 19.251 36.797 12.635 24.134 37.480
2.987 15.293 32.902 10.217 20.124 36.611 13.536 24.942 38.230
N/A
Table 5 Maximum signal magnitudes of (/s)bend,h of the plate mirror model. Sensing signal
ð/s Þbend;h
Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude
Trend
N* = 0
N* = 1 Ncr
N* = 2 Ncr
3.617 14.950 33.414 2.765 8.219 16.165 6.574 15.857 28.840
3.182 14.555 33.140 2.629 8.103 16.075 6.371 15.724 28.737
2.987 14.262 32.902 2.546 8.008 15.995 6.224 15.605 28.640
Table 6 Maximum signal magnitudes of /s of the plate mirror model. Sensing signal
/s
Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude
Trend
N* = 0
N* = 1 Ncr
N* = 2 Ncr
7.234 29.900 66.828 8.945 26.509 52.291 12.398 30.850 56.085
6.363 29.109 66.280 9.251 26.121 52.029 12.636 30.566 55.894
5.974 28.524 65.803 10.218 25.828 51.794 13.537 30.312 55.712
5.1.1. Case 1: pretension N⁄=0 In this case, the pretension force applied on the boundary of the mirror is set to zero. Figs. 3–6 show the distribution of mirror modal sensing signals with modes (m, n) from (0,0) to (1,1) and Figs. 7 and 8 illustrate that generated in higher natural modes, i.e., (0,2) and (2,0). In each figure, the top-left (a) denotes the signal component resulting from the radial bending stain (/s)bend,r; the top-right (b) denotes the signal component resulting from the angular bending strain(/s)bend,h; the bottom-left (c) denotes the summation of both contributing components, i.e., total signal /s; and the bottom-right (d) denotes the mode shape plotted in the polar coordinate system. 5.1.2. Case 2: pretension N⁄=Ncr Pretension force N⁄ = Ncr (Ncr = 14.7D/a2) is applied to the mirror in this case. Modal sensing signal distributions are presented in Figs. 9–14 below. Also, figure (a), (b), (c) and (d) denote (/s)bend,r, (/s)bend,h, total signal /s and the mode shape, respectively. 5.1.3. Case 3: pretension N⁄=2Ncr In this case, pretension force N⁄ increases to 2Ncr (Ncr = 14.7D/a2) and is applied to the mirror. Modal sensing signal distributions are presented in Figs. 15–20 below. Also, figure (a), (b), (c) and (d) denote (/s)bend,r, (/s)bend,h, total signal /s and the mode shape correspondingly.
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3 (Φ )bend,θ
2 0
s
s
(Φ )bend,r
4
-2 2
2
0.1
2 1 0
0.1 0.05
0.05 θ (rad)
0 0
θ (rad)
r(m)
5
1
0
0
-5 2
0.1
-1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 21. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (0,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
5.1.4. Modal sensing signal analysis From Figs. 3–20 one can find that the sensing signal distribution changes slightly when the pretension N⁄ increases. In order to evaluate the sensing signals of the mirror under different pretension forces of different modes, modal signal magnitudes of various components of each mode (m, n) are summarized in Tables 4–6. Due to the clamped condition, the mode shape of the mirror reduces to zero at the boundary r = a. Also, the signal component (/s)bend,h = 0 of all modes at the bound ary r = a. This can be demonstrated by the definition of (/s)bend,h, referring to Eq. (64), that @U@r3mn r¼a ¼ 0 achieves according to 2 U 3mn the clamped boundary condition. With Eq. (46) one can derive that @ @h ¼ 0, which leads to ð/s Þbend;h ¼ 0. From Tables 2 r¼a
r¼a
4–6 it can be observed that with the increasing of mode m, all sensing signal components are also enhanced. However, this is
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(Φ )bend,θ
2
0
0
s
s
(Φ )bend,r
5
-5 2
-2 2
0.1
0.1
0.05 θ (rad)
0.05 θ (rad)
r(m)
0 0
10
1
0
0
-10 2
0.1
-1 2
0.1
0.05 θ (rad)
(c)
0.05 θ (rad)
r(m)
0 0
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 22. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (0,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
not always applicable with mode n. When n = 0, the maximal signal magnitude of (/s)bend,r and (/s)bend,h are the same. While with the growing of mode n = 2, (/s)bend,r component dominates obviously. However when n = 3, the gap of the two components narrows. Furthermore, when the pretension force N⁄ augments, only the (/s)bend,h component decreases in all modes. For most modes, the total sensing signal /s decreases, while mode (0,1) and (0,2) are the opposite. Variation tendency of each sensing signal component with time is also concluded in Tables 4–6.
5.2. The membrane mirror model For a membrane mirror model with arbitrary pretension N⁄, Figs. 21–26 present the spatial distribution of modal sensing signals and the corresponding mode shapes. In these figures, plots of component signals, total signal and mode
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(Φ )bend,θ
20
10
10 0
s
0
s
(Φ )bend,r
20
-10 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
20
U3
s
r(m)
(b) ( s)bend,
40 Φ
0 0
0 -20 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 23. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (1,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
shapes are arranged the same as before. Also, the maximum sensing signal magnitude of each mode is presented in Table 7. It can also be observed from Figs. 21–26 that owing to the clamped boundary condition, the mode shapes of the mirror maintain zero as well as the total sensing signals at r = a. This can be derived from Eq. (48) that when r = a, it is obvious that U3mn = 0 and therefore r2U3mn = 0. Referring to the definition of total sensing signal in Eq. (62), we have /s = 0, which is different from the plate mirror model where (/s)bend,h = 0. Additionally, it is easy to find that for all modes, the trends of /s and the mode shape consistent with each other qualitatively. Similarly with the plate mirror model, it can be concluded from Table 7 that all sensing signal components increase with the growing of mode m while mode n does not have the same effect. Moreover, the (/s)bend,r component starts to dominate when n = 2 and increases little when n = 3, which is the same with
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10 (Φ )bend,θ
(Φ )bend,r
20
0
s
s
0 -20 2
-10 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
50
1
0
0
-50 2
0.1
-1 2
0.1
0.05 θ (rad)
0 0
(c)
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
0.05 θ (rad)
r(m)
s
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 24. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (1,1). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
plate mirror model. Comparing Table 7 with Tables 4–6 one can find that the sensing signal magnitudes of the membrane mirror model are lower than that of the plate mirror model for each component and all modes. Moreover, from the mathematical procedure of solving U3mn of the two models one can find that for the membrane model, the bending stiffness D is neglected as a simplification, which leads to a weaker membrane boundary condition, i.e., Eq. (54), as compared with the boundary conditions, Eqs. (37) and (38), of the plate model. As a consequence, pretension N⁄ will not affect the transverse mode shape and sensing signal of the mirror. Actually, this is to some extent physically unreasonable since the bending stiffness should never be zero even though it can be quite small for an ultra-thin and soft space membrane mirror.
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5 (Φ )bend,θ
(Φ )bend,r
10
0
s
s
0 -10 2
-5 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
10
1
0
0
-10 2
0.1
-1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
r(m)
(b) ( s)bend,
U3
Φ
s
(a) ( s)bend,r
0 0
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 25. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (0,2). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
6. Conclusions In this paper, a mathematical model of a deformable plate/membrane mirror is established. Dynamic characteristic of plate and membrane models are studied respectively and transverse mode shape functions are derived and used for distributed signal analysis. Total micro-sensing signal and its two components of different mirror models are presented. Furthermore, effects of different pretension forces on mirror modal sensing behavior are investigated. Based on the detailed analysis, the following conclusions can be drawn:
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(Φ )bend,θ
40
0
20 0
s
s
(Φ )bend,r
50
-50 2
-20 2
0.1
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
(a) ( s)bend,r
1
50
U3
s
r(m)
(b) ( s)bend,
100 Φ
0 0
0 -50 2
0.1
0 -1 2
0.1
0.05 θ (rad)
0 0
0.05 θ (rad)
r(m)
s
(c)
0 0
r(m)
(d) Mode shape
U3
1 0
-1 0.1
0.1 0
0
y(m)
-0.1 -0.1
x(m)
(e) Mode shape Fig. 26. Sensing signal distributions and mode shape of membrane mirror model with arbitrary N*, mode (2,0). (a) Signal component (/s)bend,r; (b) signal component (/s)bend,h; (c) total signal /s; (d) mode shape in the polar coordinate system; (e) mode shape in the projected coordinate system.
(1) For thin deformable plate/membrane mirrors, the distributed microscopic sensing signal could be divided into two bending components, i.e., the (/s)bend,r and the (/s)bend,h. However, the membrane strains have no contribution to the sensing signals induced by transverse vibration. Note that only linear free vibration is considered in the study. When it comes to large amplitude nonlinear vibrations, the problem would be more complicated. (2) For the membrane mirror model, pretension N⁄ affects neither the mode shape nor the sensing signal distributions. While for the plate mirror model, pretension force will influence both. With the increasing of N⁄, sensing signal component (/s)bend,h decreases in all modes. Also, for most modes, the total sensing signal /s reduces. (3) For each microscopic sensing signal component and total signal /s, the signal magnitudes of all modes of the membrane mirror model are lower than that of the plate mirror model.
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Table 7 Maximum sensing signal magnitudes of membrane mirror model. Mode
(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)
Maximal signal magnitude ð/s Þbend;r
ð/s Þbend;h
/s
2.169 11.426 28.084 4.540 15.215 31.970 5.578 15.290 29.005
2.169 11.426 28.084 1.982 6.643 13.971 4.946 13.284 25.317
4.338 22.853 56.169 6.407 21.424 45.113 9.623 25.772 48.984
(4) For both plate and membrane mirror models, with the increasing of mode m, all sensing signal components also increase. However, when mode n increases, the sensing signal magnitudes show an irregular variation. (5) Due to the clamped boundary condition of the mirror, all mode shapes of two models reduce to zero at the boundary. Additionally, for plate mirror model, the signal component (/s)bend,h = 0 when r = a. However, for membrane mirror model, the total signal /s = 0 at the boundary. (6) The plate mirror model is physically more reasonable compared with the membrane mirror model since it take into account the bending stiffness and the pretension effect. Distributed sensing signals are location and modal dependent. Based on the mathematical model and signal analysis in this paper, one can get some design criteria for optimal sensor placement on lightweight deformable mirrors. These data can also be used to develop dynamic vibration control of large-scale space flexible structures. Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant No. 51175103) and Self-Planned Task of State Key Laboratory of Robotics and System (HIT) (Grant No. SKLRS201301B). Prof. Tzou would like to thank the ‘‘Ph. D. Supervisor Program” and the ‘‘111 Project” (B07018) at the Harbin Institute of Technology sponsored by the Chinese Ministry of Education. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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