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Microscopic control actions of LaSMP actuators on hemispherical shells
Mechanical Systems and Signal Processing 142 (2020) 106744
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Microscopic control actions of LaSMP actuators on hemispherical shells Dan Wang, Mu Fan, Zhu Su, Hornsen Tzou ⇑ Key Laboratory of Mechanics and Control of Mechanical Structures, Interdisciplinary Research Institute, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 211100 Nanjing, PR China
a r t i c l e
i n f o
Article history: Received 4 September 2019 Received in revised form 12 February 2020 Accepted 15 February 2020
Keywords: Light activated shape memory polymer Hemispherical shell Control actions The normalized control forces
a b s t r a c t Light activated shape memory polymer (LaSMP) is one of novel shape memory polymers with dynamic strain and Young’s modulus when exposed to UV lights. This study investigates distributed microscopic control actions of LaSMP patches on hemispherical shells with free boundary. Based on strain models established in the experiments, governing equations of the spherical shell are defined and control forces in the meridional, circumferential and transverse directions are derived respectively. In case studies of LaSMP patch laminated on the shell by covering specific wavelength at each mode in the circumferential direction, the meridional and circumferential forces turn to zero. Then, the dominant transverse control forces and its components resulting from the bending moments and membrane forces are compared at different modes. Parametric analyses are carried out to evaluate the microscopic actuation effects of actuator position, thickness and shell radius, LaSMP thickness, etc. LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. The analyses indicate that the transverse control forces increase with increasing of LaSMP patch thickness and decreasing of hemispherical shell thickness and radius. Then, the normalized control forces are derived by dividing LaSMP patch size. The parametric analyses confirm that the normalized control forces caused by membrane forces dominate the control effect and the best control position is near the boundary of hemispherical shell. This work provides an analysis procedure on estimating LaSMP vibration control actions and gives theoretical predictions for idealized LaSMP vibration control of hemispherical shells. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction At present, shape memory materials as a kind of functional material are widely used in engineering, aerospace, electronics, bio-medical devices etc. [1–3]. For example, shape memory alloy (SMA) is served as a new type of foldable structure in the space-borne antenna [4] as well as artificial bone, heart stents and surgical suture in bionic structure [5,6]. Different from SMAs activated by thermos elasticity, a variety of shape memory polymers (SMP) are compounded and applied in many fields. Recently, a new category of novel shape memory polymers is compounded which is activated by light, i.e. light activated shape memory polymer (LaSMP). Young’s modulus and strain of LaSMP can dynamically change when exposed to UV lights and realize non-contact actuation and control [7–10]. By putting optical active substance Spiropyran into Ethylene vinyl acetate, Li, et al. in the StrucTronics and Control Laboratory has synthesized a kind of LaSMP [11]. The relation between ⇑ Corresponding author. https://doi.org/10.1016/j.ymssp.2020.106744 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
Young’s modulus, strains and light intensities have been established and proved by the laboratory experiments [12]. With dynamic Young’s modulus and strain, this kind of shape memory polymers can be able to work as vibration actuator and controller [13]. Structures and components of hemispherical shells are very common in engineering systems, for example pressure vessels, storage tanks and signal receivers, etc. The vibration and natural frequency of hemispherical shells have been studied by theoretical methods and experiments over the years [14–17]. The control effect of piezoelectric actuators on hemispherical shells was also investigated [18]. Based on dynamic strain model of LaSMP, the control effect of LaSMP/hemispherical shell systems on microscopic control actions are studied in this paper. In this study, the LaSMP/shell governing equations are derived by using the classical bending approximation theory based on the LaSMP strain forward/backward reaction models proved in the laboratory experiments. By considering LaSMP patch on the hemispherical shell with free boundary, the control forces are derived as a function of LaSMP induced strains. In the case study of LaSMP patch laminated on the shell by covering specific wavelength at each mode in the circumferential direction, the components of the bending and membrane forces in the meridional and circumferential direction are compared at different modes, followed by discussion of actuator position, shell thickness and radius effect. LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. The normalized control forces are analyzed by considering patch sizes and the results are evaluated. 2. Dynamic equations with LaSMP control forces In this section, LaSMP constitutive behaviors are discussed first, followed by LaSMP control forces and moments defined for an arbitrary actuator patch. Dynamic control equations of a hemispherical shell coupled with LaSMP control forces/moments are then defined. Distributed LaSMP actuations are presented in the next section. 2.1. LaSMP actuation strain model The optical active substance Spiropyran has two kinds of isomerism forms, namely a ring-closed form (SP) and a ringopened merocyanine form (MC). In a reverse reaction, SP turns into MC with decreased Young’s modulus when exposed to 365 nm UV lights. In a forward reaction, MC transforms to SP with increased Young’s modulus when exposed to 540 to 560 nm UV lights. In the reverse and forward reactions, both strain and Young’s modulus change with time. With Spiropyran added into Ethylene vinyl acetate (EVA) and processed, a kind of LaSMP (Sp2/EVA_4) [11] was composed in the StrucTronics and Control Laboratory. After testing and validation, this LaSMP is used to evaluate the control actions imposed on flexible structures [12]. To focus on control forces, the strain models including the forward and backward reactions are introduced while Young’s modulus is assumed unchanged. For photochemical series k = 1, the forward reaction exhibits the strain variation:
Sa ¼ Samax Samax
t1 t P0 t þ 1 esr þ e sd sr Pmax
ð1Þ
where Sa is the LaSMP actuation strain, Samax is the LaSMP maximum strain, P 0 is the initial concentration of the optical active Spiropyran substance in the reverse reaction, Pmax is the maximum concentration of Spiropyran; sr and sd are respectively the time constants in the forward and reverse reactions, t 1 is the forward reaction time. However, the reverse reaction exhibits strain variation:
Sa ¼ Samax Samax
P0 st e d Pmax
ð2Þ
The forward and backward reaction models and their strain variations are validated with experimental data [12]. 2.2. Dynamic control equations of hemispherical shells with LaSMP actuators A hemispherical shell with an arbitrary LaSMP actuator is illustrated in Fig. 1. As shown in Fig. 1, the LaSMP patches are circumferentially laminated and an arbitrary patch is defined from ð/1 ; w1 Þ to ð/2 ; w2 Þ in the spherical coordinate system ð/; w; u3 Þ, i.e. the meridional, circumferential and transverse directions. The shell radius and thickness are R and h respeca tively. The LaSMP thickness is h and its Young’s modulus is Y a . Assuming that the LaSMP induced strain is Saij ði; j ¼ /; wÞ; thus, LaSMP induced membrane control forces and control moments of a selected actuator defined by /1 ! /2 ; w1 ! w2 are defined as
a Na// ¼ Y a h Sa// us / /1 us / /2 us w w1 us w w2
ð3aÞ
a Naww ¼ Y a h Saww us / /1 us / /2 us w w1 us w w2
ð3bÞ
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
3
Fig. 1. The LaSMP patch on hemisphere shell.
Ma// ¼
a hþh a Y a h Sa// us / /1 us / /2 us w w1 us w w2 2
Maww ¼
a hþh a Y a h Saww us / /1 us / /2 us w w1 us w w2 2
ð3cÞ
ð3dÞ
where the two step functions us , i.e. us / /1 us / /2 and us w w1 us w w2 , define the location of an LaSMP patch on the shell. Substituting control membrane forces and moments (Eq.(3)) into the dynamic system equations, using the bending approximation theory, i.e. N // ¼ N ww ¼ N /w ¼ 0, one can simplify the governing equations and degenerate to [19]:
@ Na// sin/ @/ @ Naww @w
€ // ¼ R sin /F / Naww cos / Q /3 sin / þ R sin /qhu
€ ww ¼ R sin /F w Q w3 sin / þ R sin /qhu
@ Q /3 sin / @ Q w3 € 3 ¼ R sin /F 3 Na// þ Naww sin / þ R sin /qhu @/ @w
ð4Þ
ð5Þ
ð6Þ
where:
Q /3
Q w3
2 3 a @ Mw/ 1 4@ M // M// sin / a þ M ww Mww cos /5 ¼ @/ @w R sin / 2 3 @ M ww Maww 1 4@ M w/ sin / ¼ þ M w/ cos /5 þ R sin / @w @/
ð7Þ
ð8Þ
Note that although elastic membrane forces are neglected under the bending approximation, the LaSMP actuator induced membrane control forces still remain for distributed control. Their influences on control effects would appear in later parametric analyses of microscopic actuations. From Eqs. (4–8), it is abvious that the control forces and moments caused by LaSMP exist in all governing equations. And the control moments mainly affect the transverse shearing forces in Eqs. (7) and (8). Detailed microscopic control actions are defined next. To focus on LaSMP control effect, the effect of LaSMP patch on the mode shape functions and natural frequencies of hemipherical shell is neglected here.
3. Microscopic LaSMP actuations Dynamic control equations of the hemispherical shell coupled with LaSMP control forces/moments were defined above. Detailed distributed LaSMP microscopic actuations and their key contributing components are derived in this section.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
3.1. Definition of LaSMP control forces on hemispherical shells According to the modal expansion method [19], the displacement ui ð/; w; t Þ can be written as 1 P gk U ik ð/; wÞi ¼ ð/; w; 3Þ, where gk is the mode participation factor (or modal coordinate) and U ik ð/; wÞ are ui ð/; w; t Þ ¼ k¼1
the mode shape functions. For a hemispherical shell with free boundary, the mode shape functions U ik ð/; wÞ;i ¼ ð/; w; 3Þ are
k / U /k ¼ A tan sin kw 2
ð9Þ
k / U wk ¼ B sin / tan cos kw 2
ð10Þ
k / cos kw U 3k ¼ CRðk þ cos /Þ tan 2
ð11Þ
where A, B, C are the modal amplitudes. Substituting mode shape functions into the dynamic equations, imposing the modal orthogonality and simplifying gives the modal response equation of the hemispherical shell m
a
g€ k þ 2fk xk g_ k þ x2k gk ¼ F^k þ F^k
ð12Þ
Here, for mode k, fk is the damping ratio, xk is the natural frequency, F^k is the mechanical excitation and F^k is the LaSMP m induced control action. To focus on the LaSMP effect, the mechanical excitation F^ is neglected and the modal control forces m
a
k
a F^k can be written as:
a F^k ¼
1 qhNk R R
where N k ¼
/
w
Z Z /
3 P
i¼1
3 X
w
! Lai U ik R2 sin /d/dw
ð13Þ
i¼1
U 2ik R2 sin/d/dw, and Lai ;i ¼ ð/; w; 3Þ are the Love control operators defined as:
a a 1 @N// 1 @M// La/ ¼ 2 R @/ R @/
Law ¼
La3
ð14Þ
a @Maww 1 @Nww 1 2 R sin / @w R sin / @w
ð15Þ
8 2 9 3 ! < @ @ M a// sin / = @Maww 1 @ a a a 4 Mww cos /5 þ ¼ 2 R sin / N// þ Nww ; @/ sin / @w @w R sin / :@/ 1
ð16Þ
The modal control force, i.e., Eq.(13), can be written in a form of three possible control force components i ¼ ð/; w; 3Þ: a F^k ¼
1 qhNk
Z Z a La/ U /k þ Law U wk þ La3 U 3k R2 sin/d/dw ¼ F^k /
þ F^k
a
merd
þ F^k
a
cir
ð17Þ
trans
w
a a a a Here, F^k denotes the overall actuation action and F^k merd , F^k cir and F^k trans are the control forces in the meridional, circumferential and transverse directions respectively. It is noted that LaSMP used in this study is an isotropic material, which means the strains in the meridional and circumferential directions are the same, i.e. Sa// ¼ Saww , and this bi-axial behavior is experimentally validated (see Appendix). Then, the control membrane force and moment components both are equal, i.e., N a// ¼ N aww and M a// ¼ Maww . a
Substituting Eqs. (9)–(11) and (14)–(16) into Eq. (13) yields the meridional and circumferential control forces F^k F^k
a
cir :
a F^k
merd
k Z Z Z p=2 Z 2p a a 1 1 / Y a S// h h þh La/ U /k R2 sin /d/dw ¼ A tan sin kw sin / R þ 2 2 qhNk / w qhNk 0 R2 0 d / /1 d / /2 us w w1 us w w2 d/dw # " k k a a Y a S// h h þh A / / Rþ tan 1 sin /1 tan 2 sin /2 cos kw1 cos kw2 ¼ 2 k qhNk 2 2
merd
and
¼
ð18Þ
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744 a F^k
k a a / Y a S// h h þh B sin / tan cos kw R þ 2 2 R2 0 / w 0 k Z a a /2 Y a S// h h þh B / us / /1 us / /2 d w w1 d w w2 d/dw ¼ cos kw1 cos kw2 Rþ sin / tan d/ 2 k 2 qhNk /1 cir
¼
Z Z
1 qhN k
Law U wk R2 sin /d/dw ¼
1 qhN k
5
Z p=2 Z 2p
ð19Þ
Here, dð/Þ is the Dirac delta function with definition as:
dð/ / Þ ¼
1 / ¼ /
ð20Þ
/–/
0
And dð/Þ has following mathematical properties:
Z
/2
dð/ / Þd/ ¼ 1/1 < / < /2
ð21Þ
/1
Z
þ1
1
0
d0 ðx aÞfd/ ¼ f ðaÞ
ð22Þ
For the control force in the transverse direction, considering N a// ¼ N aww and M a// ¼ Maww :
F^k
a trans
8 2 3 ! Z Z < a @ M a// sin / 1 @ 1 @ @Mww a 2 a 4 5 M ww cos / þ L3 U 3k R sin /d/dw ¼ @/ sin / @w qhNk / w :@/ @w / w )" # k / CRðk þ cos /Þ tan R sin / Na// þ Naww cos kw d/dw ¼ 2 8 ! ) a Z Z < a @Ma// 1 @ @ M// 1 @ @M ww þ cos / þ sin / Rsin / Na// þ Naww @/ @/ sin / @w qhNk / w : @/ @w " # k / CRðk þ cos /Þ tan cos kw d/dw 2
1 ¼ qhNk
Z Z
The transverse actuation force can be divided into: a F^k
trans
a ¼ F^k
trans
/;bend
a þ F^k
trans
w;bend
a þ F^k
trans
/;mem
a þ F^k
trans
w;mem
ð23Þ
ð24Þ
a a Here, F^k trans and F^k trans are the bending control moment actions in transverse direction resulting from M a// a /;bend a w;bend and M aww . F^k trans and F^k trans are the membrane control force actions in transverse direction resulting from N a// /;mem
w;mem
and N aww . The specific expression of each action is defined next and evaluated in case studies. For the first term in Eq. (23), it can be integrated as: 1 qhN
R R /
k
w
@ sin/ @/
@ ðM a// Þ @/
h
i k CRðk þ cos/Þ tan /2 coskw d/dw
k R p=2 R 2p ðhþha Þ 1 CRðk þ cos/Þ tan /2 coskwsin/Y a S// ha 2 ¼ qhN 0 0 k @ d / /1 d / /2 us w w1 us w w2 d/dw @/ h 2 /2 k /1 k ðhþha Þ Y a ha S 2 sinkw sinkw þ 2kcos/ þ cos2/ k þ 2kcos/ þ cos2/ tan tan k ¼ qhN // 2 CR 1 2 2 2 1 1 2 2 k
ð25Þ
k
For the second term in Eq. (23), the integration is
k Z Z Z p=2 @M a// 1 / 1 cos/ coskwd/dw ¼ CRðk þ cos/Þ tan 2 qhNk / w @/ qhNk 0 k Z 2p / CRðk þ cos/Þ tan coskwcos/Y a S// ha 2 0 a hþh d / /1 d / /2 us w w1 us w w2 d/dw 2 " # k k a a Y a h S// h þ h CR /1 /2 cos/1 k þ cos/2 tan cos/2 ¼ sinkw1 sinkw2 k þ cos/1 tan k qhNk 2 2 2
ð26Þ
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Equations (25) and (26) make up the meridional bending control action F^k trol moment.
a For the third term, it is the circumferential bending control action F^k
trans
trans
w;bend
/;bend
resulting from the meridional con-
resulting from the circumferential control
moment:
! k Z Z Z p=2 a 1 1 @ @M ww / 1 CRðk þ cos /Þ tan cos kwd/dw ¼ qhNk / w sin / @w @w qhNk 0 2 k Z 2p / 1 Y a S// ha CRðk þ cos /Þ tan cos kw 2 sin / 0 a a @ hþh Y a S// h us / /1 us / /2 d w w1 d w w2 d/dw ¼ 2 @w qhNk k Z / a 2 hþh a / 1 d/ ¼ F^k trans ðk þ cos /Þ tan CRk sin kw2 sin kw1 w;bend 2 sin / 2 /1
a The meridional and circumferential membrane control actions F^k
trans
/;mem
a and F^k
brane control forces in the meridional and circumferential directions are equal due to
ð27Þ
trans
N a//
resulting from the mem-
w;mem ¼ N aww .
k Z Z Z p=2 1 / 1 Rsin/Na// CRðk þ cos/Þ tan cos kwd/dw ¼ 2 qhNk / w qhNk 0 k Z 2p a / Y a S// h CR2 ðk þ cos /Þ tan cos kwsin/Y a S// ha us / /1 us / /2 us w w1 us w w2 d/dw ¼ 2 qhNk 0 k Z / 2 2 CR / ðk þ cos /Þ tan sin/d/ sinkw2 sinkw1 2 k /1 a a ¼ F^ ¼ F^ k trans
/;mem
k trans
w;mem
ð28Þ In this section, the total actuation control forces and its detailed micro-control actions in meridium, circumferential and transverse direaction is defined in Eq.(18), (19) and (23). The simplified microscopic contributing components in transverse direction from meridional/circumferential bending and membrane control forces are derived in Eqs.(25–28). Mathematical expressions of actuation forces indicate that the actuation magnitude is related with vibration mode k, shell radius R and a thickness h, the LaSMP’s thichness h and actuator position /1 /2 ; w1 w2 . Microscopic LaSMP membrane and bending control actions on the hemispherical shells related to various design parameters are evaluated next.
4. Evaluation of distributed microscopic actuations The in-plane vibrations in the meridional and circumferential directions are usually small, as compared with the out-ofa plane transverse vibration of thin shells. Thus, the transverse LaSMP control force F^k trans is focused and its microscopic meridional and circumferential contributing components induced by LaSMP membrane forces and bending moments are compared. The geometric and material parameters used in case studies are listed in Table 1. Again, the biaxial LaSMP actuation was demonstrated in laboratory experiments and data are provided in Appendix. 4.1. Control effect of LaSMP actuator placements Fig. 2 illustrates the LaSMP actuators laminated on the hemispherical shell at 00 100 , 100 200 , 200 300 , . . . , 800 900 in the meridional direction / and divided at every ð2n 1Þp=2k in the circumferential direction w, where n is the number of divisions and n ¼ 1; 2; :::; 2k, where k is the wave number in the circumferential direction and vibration mode as well. In this actuator layout, as w1 ¼ ð2n 1Þp=2k, w2 ¼ ð2n þ 1Þp=2k, there are cosðkw1 Þ ¼ 0 and cosðkw2 Þ ¼ 0 in Eqs.(18) a a and (19), accordingly the in-plane forces F^ and F^ both become zero. Thus, the total control force is equal to the transk merd
k cir
a verse control force F^k trans when LaSMP actuators laminated on the shell as shown in Fig. 2. Accordingly, the total transverse a control force F^ and its contributing meridional/circumferential membrane/bending components are analyzed in details k trans
next.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744 Table 1 Material and geometric parameters of shell and LaSMP. Parameter
Value
Unit
Young modulus of shell (polypropylene) Shell density Damping ratio Shell thickness The radius of shell Photochemical series k The maximum concentration of the optical active substance P max The initial concentration of the optical active substance P 0
1556 907 0.01 1 103 1 1 3% 0.64% 0.95%
MPa Kg/m3 = m m = = = =
24.8 52 25 10.6 40.2 1 104 0.001
S S S S MPa m =
The maximum strain induced by LaSMP S0max Time of reverse reaction t 1 Time of forward reaction t 2 Time constant in the forward reaction sr Time constant in the reverse reaction sd LaSMP Young’s modulus Thickness of LaSMP Strain of LaSMP Saij ði; j ¼ /; wÞ
Fig. 2. The distribution of LaSMP on the hemisphere shell.
a Fig. 3 illustrates the total transverse control force F^k
trans
at different meridional locations ð/1 ! /2 Þ at every 10° with shell a
modes k ¼ 2; 3; 4. (Note that the dots are plotted at the half-way.) Furthermore, for each mode k ¼ 2; 3; 4, the total F^k trans and its microscopic contributing components in the meridional and circumferential directions resulting from the membrane control forces or bending control moments are plotted respectively in Figs. 4 to 6. For reference, the value of denominator N k , i.e., 3 R R P U 2ik R2 sin/d/dw; is also drawn in Fig. 7. Nk ¼ / w i¼1
a Fig. 3. The total control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 2; 3; 4
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 4. The components of the control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 2
a Fig. 5. The components of the control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 3
a Fig. 6. The components of the control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 4
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
9
Fig. 7. The value of N k at different ð/1 ; /2 Þ with mode k ¼ 2; 3; 4
a From Figs. 3 to 6, it is obvious that the total control force F^k trans and its components increase when the actuator patch moves from the pole to the free edge and all reach the maximums at the ð80; 90Þ segment in the meridional direction /, which means the LaSMP control effect reaches the best when located near the free boundary (or free perimeter) of the shell. a Fig. 3 also indicates that the overall control forces F^k trans decrease at higher natural modes. The four microscopic contributing a a a , F^k trans component actions of F^k trans are plotted in Figs. 4 to 6, from which the membrane control actions F^k trans /;mem a w;mem resulting from the membrane control forces dominating the total control force and the bending control actions F^k trans , /;bend a induced by the bending control moments are negligibly small. Fig. 7 suggests that the modal constant of N k is F^k trans w;bend
independent of LaSMP patch position, but it increases with increasing of mode k, which leads to decreasing of total control force and its components since N k is on the denominator of the control force definition. Other design parameters influencing the LaSMP control actions and effects are evaluated next. 4.2. Effect of shell thickness and radius a As the control force F^k trans and its component actions are functions of shell thickness h and radius R, the effect will be investigated in this section. The thickness of shell is assumed to be changing from 1 to 10 mm with constant radius of 1 m and the thickness of LaSMP is still 0.1 mm. The radius of hemispherical shell changes from 0.6 to 1.6 m with a constant shell thickness of 1 mm. The thickness effects are drawn in Figs. 8 to 11 and the radius effects are presented in Figs. 12 to 15.
a Fig. 8. The value of the total control force F^k
trans
with different shell thickness at mode k ¼ 2; 3; 4
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 9. The components of F^k
trans
with different shell thickness at mode k ¼ 2
a Fig. 10. The components of F^k
trans
with different shell thickness at mode k ¼ 3
a Fig. 11. The components of F^k
trans
with different shell thickness at mode k ¼ 4
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 12. The values of total control force F^k
trans
with different shell radius at mode k ¼ 2; 3; 4
a Fig. 13. The components of F^k
trans
with different shell radius at mode k ¼ 2
a Fig. 14. The components of F^k
trans
with different shell radius at mode k ¼ 3
11
12
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 15. The components of F^k
trans
with different shell radius at mode k ¼ 4
Figs. 8 and 12 are the total transverse control forces; Figs. 9–11 (i.e., shell thickness) and 13–15 (i.e., shell radius) are respectively their (i.e., k = 2, k = 3 and k = 4 modes) corresponding contributing meridional/circumferential membrane/bending control actions. The distribution of modal constant N k with shell radius is drawn in Fig. 16. a In Fig. 9, the total control force F^ decreases with increasing of shell thickness h, a same trend exists for four microk trans
scopic component actions shown in Figs. 10 to 12. Fig. 9 also demonstrates the decreasing of total controlling forces at higher natural modes again and micro-membrane control actions dominate the overall control action in Figs. 10 to 12. Since shell’s rigidity is a cubic function of shell thickness, increasing shell thickness makes the shell harder to actuate and control generally. LaSMP control effect of shell’s radius is presented in Figs. 13–17 whose physical significances are discussed next. a decreases with increasing shell radius R or at higher natural mode. In Fig. 13, the value of the total control force F^ k trans
Figs. 13 to 15 indicate that two micro-membrane control actions domain the total control forces and all component actions 3 R R P U 2ik R2 sin/d/dw; at each modes decrease with increasing of shell radius R as well. Fig. 16 shows that N k , i.e., N k ¼ / w i¼1
Fig. 16. The value of N k at different shell radius with mode k ¼ 2; 3; 4
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a significantly increases with the shell radius R, which causes reduction of F^k trans and its components in the denominator. Analysis in this section proposes that the control forces of LaSMP are larger for shell with smaller radius, and the control effect would be better.
a
4.3. Effect of LaSMP thickness h
The LaSMP thickness will affect the control forces reflected in Eqs. (23), (27) and (28). As LaSMP is a very thin membrane, the assumption is made that the strain is uniformly distributed in the thickness direction in the modeling process. The LaSMP thickness usually is around 104 m under the limitation to this assumption. The actuation effect of actuator thickness h is shown in Figs. 17 to 20. a a In Fig. 17, the total control force F^ exhibits a linear variation with LaSMP thickness h , and this behavior appears in all a
k trans
microscopic control actuations of different modes in Figs. 18 to 20. The figures suggest, again, that the primary contributing control actions are contributed by two meridional and circumferential membrane control actions while the bending control actions are minimal. The two membrane control actions reduce in magnitudes when the mode increases. Fig. 20 indicates that the control effect is better for a hemispherical shell laminated with thicker LaSMP actuator patches, which is in accordance to physics as the control forces in LaSMP will increase. 4.4. Normalized LaSMP control forces Recall that Fig. 1 illustrates the actuator placement on the hemispherical shell and it shows the actuator size actually increases when it moves from the pole to the free boundary, i.e., the size is not a constant. In this section, the effect of LaSMP
a Fig. 17. The values of the total control force F^k
a Fig. 18. The components of F^k
trans
trans
at different LaSMP thickness with mode k ¼ 2; 3; 4
at different LaSMP thickness with mode k ¼ 2
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 19. The components of F^k
trans
at different LaSMP thickness with mode k ¼ 3
a Fig. 20. The components of F^k
trans
at different LaSMP thickness with mode k ¼ 4
Fig. 21. The LaSMP size at different position in the meridional direction at mode k ¼ 2; 3; 4
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
15
Fig. 22. The LaSMP size with different shell radius at mode k ¼ 2; 3; 4
ea Fig. 23. Normalized control forces F^ k
trans
at various meridional locations (i.e., 0: pole; 90: free boundary) at mode k ¼ 2; 3; 4
patch size is evaluated. When a LaSMP actuator distributes as /1 /2 ; w1 w2 in the meridional and circumferential directions, the true LaSMP actuator size is
Z A¼
/2
/1
Z
w2
w1
R2 sin/d/dw ¼ R2 cos/1 cos/2 w2 w1 :
ð29Þ
The variation of LaSMP size A with different meridional angle /j ; j ¼ 1; 2 and shell radius R is shown in Figs. 21 and 22. Figs. 21 and 22 indicate that the LaSMP size increases when meridional degree / and shell radius R increase. Accordingly, to objectively estimate control effectiveness on the shell, earlier calculations, i.e., with respect to /, h and R in Figs. 3, 8, 12, of ea the total LaSMP control forces, i.e., F^ k trans ; k ¼ 2; 3; 4, need to be normalized with respect to the true actuator size A and these three normalized results are presented in Figs. 23-25. Fig. 23 indicates that the normalized control forces also reach the maximums at the free boundary of hemispherical shell, which suggest the best control location is near the boundary. The normalized control force also decreases with increasing of shell thickness as the patch size is independent of shell thickness as shown in Eq. (29) (Fig. 24). And the trend between the normalized control force and shell radius in Fig. 25 is the same with that of control force and shell radius in Fig. 12, but normalized control force decreases much faster with shell radius as the patch size increases with shell radius.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
ea Fig. 24. Normalized control forces F^ k
trans
ea Fig. 25. Normalized control forces F^ k
versus shell thickness h at mode k ¼ 2; 3; 4
trans
versus shell radius R at mode k ¼ 2; 3; 4
5. Conclusions LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated in detail in this study. The shell governing equations with LaSMP control forces/moments are derived based on the bending approximation theory. The specific expressions of control forces of LaSMP patch on a hemisphere shell with free boundary are derived by using the Love control operators, followed by detailed analyses of modal control forces and their microscopic control actions. In case studies, with LaSMP patch laminated on ð2n 1Þp=2k;k ¼ 2; 3; 4 in the circumferential direction, the control forces are defined by material, geometric design parameters, e.g., actuator location, shell thickness/radius, LaSMP thickness, etc. LaSMP induced modal (k ¼ 2; 3; 4) meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. With thorough analyses of various parameters, following conclusions can be drawn: 1. In-plane meridional and circumferential control forces disappear in the above LaSMP lamination and the transverse control force dominants the total control forces. 2. Among the four meridional and circumferential microscopic membrane/bending control actions the actuation components resulting from the membrane forces are the main control actions and those resulting from the bending control moments are very small. 3. The total control force and its components decrease with increasing of shell thickness and radius. The total control force and its component actions would also increase with the increase of LaSMP thickness. 4. At last, for the total control force and the normalized total control force, they reach the maximums near the free boundary of the hemispherical shell, which means the boundary area is the most effective control location for the shell.
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
17
The work in this study would provide a universal analysis procedure and give a general guideline for choosing idealized design parameters for vibration control of LaSMP laminated hemispherical shells. CRediT authorship contribution statement Dan Wang: Conceptualization, Investigation, Writing - original draft, Writing - review & editing, Funding acquisition, Resources. Mu Fan: Investigation, Funding acquisition, Resources. Zhu Su: Investigation, Supervision. Hornsen Tzou: Conceptualization, Writing - review & editing, Funding acquisition, Supervision. Acknowledgements This research is supported by Natural Science Foundation of China (Nos. 11872206, 11902151), Natural Science Foundation of Jiangsu Province (No. BK20180411, BK20180429, BK20170773), State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-IZD1900201907). Appendix: Experimental verification of LaSMP Young’s modulus in orthotropic direction For LaSMP used as vibration controller in two-dimensioning spherical shell, Young’s modulus in orthotropic direction need be verified by the experiment. In laboratory, a cross shaped LaSMP membrane is fabricated to validate Young’s modulus [11]. As shown in Fig. A1, according to the requirement of biaxial tensile test, the area of LaSMP specimen is 118 mm*118 mm with the thickness of 0.32 mm, and the rectangular area in the center is 36 mm*36 mm.
Fig. A1. LaSMP sample for biaxial tension experiments [11].
Fig. A2. Biaxial tension result of the LaSMP sample under UV lights [11].
18
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
The loading and unloading in biaxial direction were taken six times under UV light exposures with wavelength at 365 nm and the tensile curves were recorded in Fig. A2, which shows the tension curves in biaxial directions almost overlap. Thus, Young’s modulus keep the same in orthotropic direction as the orientation of sample was chosen arbitrarily. Accordingly, the lab fabricated LaSMP is suitable in biaxial actuations and two dimensional vibration control. References [1] H.S. Tzou, Piezoelectric Shells: Distributed Sensing and Control of Continua, Springer Science & Business Media, Boston, 2012. [2] T.J. Johnson, R.L. Brown, D.E. Adams, M. Schiefer, Distributed structural health monitoring with a smart sensor array, Mech. Syst. Signal Pr. 18 (2004) 555–572. [3] M.D. Hager, S. Bode, C.W. Schubert, Shape memory polymers: past, present and future developments, Prog. Polm. Sci. 49 (2015) 3–33. [4] H.J. Wang, B. Fan, M. Yi, Inflatable antenna for space-borne microwave remote sensing, IEEE. Antenn. Propag. M. 54 (2012) 58–70. [5] Y. Luo, M. Higa, S. Amae, Preclinical development of SMA artificial anal sphincters, Minim. Invasiv. Ther. 15 (2006) 241–245. [6] G. Sourav, S.K. Mishra, K. Roy, Stochastic seismic response of building with super-elastic damper, Mech. Syst. Signal Pr. 72 (2016) 642–659. [7] A. Lendlein, H. Jiang, O. Jünger, R. Langer, Light-induced shape-memory polymers, Nature 434 (2005) 879. [8] J.S. Sodhi, P.R. Cruz, I.J. Rao, Inhomogeneous deformations of light activated shape memory polymers, Int. J. Eng. Sci. 89 (2015) 1–17. [9] D.C. Sun, L.Y. Tong, Theoretical investigation on wireless vibration control of thin beams using photostrictive actuators, J. Sound Vib. 312 (2008) 182– 194. [10] X. Zhang, Q. Zhou, H. Liu, H. Liu, UV light induced plasticization and light activated shape memory of spiropyran doped ethylene-vinyl acetate copolymers, Soft Matter 10 (2014) 3748–3754. [11] H. Li, D. Guo, H.S. Tzou, Light-induced vibration control of parabolic cylindrical shell panels, J. Intel. Mat. Syst. Str. 28 (2017) 2947–2958. [12] H.Y. Li, H. Li, H.S. Tzou, Frequency control of beams and cylindrical shells with light-activated shape memory polymers, J. Vib. Acoust. 137 (2015) 011006. [13] D. Wang, M. Fan, Z. Su, H.S. Tzou, Frequency control of thin plates with light-activated shape-memory polymers, AIAA. J. (2019), online. [14] H.S. Tzou, P. Smithmaitrie, J.H. Ding, Micro-sensor electromechanics and distributed signal analysis of piezo(electric)-elastic spherical shells, Mech. Syst. Signal Pr. 16 (2002) 185–199. [15] C. Hwang, Some experiments on the vibration of a hemispherical shell, J. Appl. Mech. 33 (1966) 817–824. [16] H. Yue, Y. Lu, Z. Deng, H.S. Tzou, Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Pr. 82 (2017) 279– 295. [17] K.A. Kuo, H.E.M. Hunt, J.R. Lister, Small oscillations of a pressurized, elastic, spherical shell: model and experiments, J. Sound Vib. 359 (2015) 168–178. [18] P. Smithmaitrie, H.S. Tzou, Micro-control actions of actuator patches laminated on hemispherical shells, J. Sound Vib. 277 (2004) 691–710. [19] H. Li, S. Hu, H.S. Tzou, Modal signal analysis of conical shells with flexoelectric sensors, J. Intel. Mat. Syst. Str. 26 (2015) 1551–1563.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Microscopic control actions of LaSMP actuators on hemispherical shells Dan Wang, Mu Fan, Zhu Su, Hornsen Tzou ⇑ Key Laboratory of Mechanics and Control of Mechanical Structures, Interdisciplinary Research Institute, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 211100 Nanjing, PR China
a r t i c l e
i n f o
Article history: Received 4 September 2019 Received in revised form 12 February 2020 Accepted 15 February 2020
Keywords: Light activated shape memory polymer Hemispherical shell Control actions The normalized control forces
a b s t r a c t Light activated shape memory polymer (LaSMP) is one of novel shape memory polymers with dynamic strain and Young’s modulus when exposed to UV lights. This study investigates distributed microscopic control actions of LaSMP patches on hemispherical shells with free boundary. Based on strain models established in the experiments, governing equations of the spherical shell are defined and control forces in the meridional, circumferential and transverse directions are derived respectively. In case studies of LaSMP patch laminated on the shell by covering specific wavelength at each mode in the circumferential direction, the meridional and circumferential forces turn to zero. Then, the dominant transverse control forces and its components resulting from the bending moments and membrane forces are compared at different modes. Parametric analyses are carried out to evaluate the microscopic actuation effects of actuator position, thickness and shell radius, LaSMP thickness, etc. LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. The analyses indicate that the transverse control forces increase with increasing of LaSMP patch thickness and decreasing of hemispherical shell thickness and radius. Then, the normalized control forces are derived by dividing LaSMP patch size. The parametric analyses confirm that the normalized control forces caused by membrane forces dominate the control effect and the best control position is near the boundary of hemispherical shell. This work provides an analysis procedure on estimating LaSMP vibration control actions and gives theoretical predictions for idealized LaSMP vibration control of hemispherical shells. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction At present, shape memory materials as a kind of functional material are widely used in engineering, aerospace, electronics, bio-medical devices etc. [1–3]. For example, shape memory alloy (SMA) is served as a new type of foldable structure in the space-borne antenna [4] as well as artificial bone, heart stents and surgical suture in bionic structure [5,6]. Different from SMAs activated by thermos elasticity, a variety of shape memory polymers (SMP) are compounded and applied in many fields. Recently, a new category of novel shape memory polymers is compounded which is activated by light, i.e. light activated shape memory polymer (LaSMP). Young’s modulus and strain of LaSMP can dynamically change when exposed to UV lights and realize non-contact actuation and control [7–10]. By putting optical active substance Spiropyran into Ethylene vinyl acetate, Li, et al. in the StrucTronics and Control Laboratory has synthesized a kind of LaSMP [11]. The relation between ⇑ Corresponding author. https://doi.org/10.1016/j.ymssp.2020.106744 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
Young’s modulus, strains and light intensities have been established and proved by the laboratory experiments [12]. With dynamic Young’s modulus and strain, this kind of shape memory polymers can be able to work as vibration actuator and controller [13]. Structures and components of hemispherical shells are very common in engineering systems, for example pressure vessels, storage tanks and signal receivers, etc. The vibration and natural frequency of hemispherical shells have been studied by theoretical methods and experiments over the years [14–17]. The control effect of piezoelectric actuators on hemispherical shells was also investigated [18]. Based on dynamic strain model of LaSMP, the control effect of LaSMP/hemispherical shell systems on microscopic control actions are studied in this paper. In this study, the LaSMP/shell governing equations are derived by using the classical bending approximation theory based on the LaSMP strain forward/backward reaction models proved in the laboratory experiments. By considering LaSMP patch on the hemispherical shell with free boundary, the control forces are derived as a function of LaSMP induced strains. In the case study of LaSMP patch laminated on the shell by covering specific wavelength at each mode in the circumferential direction, the components of the bending and membrane forces in the meridional and circumferential direction are compared at different modes, followed by discussion of actuator position, shell thickness and radius effect. LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. The normalized control forces are analyzed by considering patch sizes and the results are evaluated. 2. Dynamic equations with LaSMP control forces In this section, LaSMP constitutive behaviors are discussed first, followed by LaSMP control forces and moments defined for an arbitrary actuator patch. Dynamic control equations of a hemispherical shell coupled with LaSMP control forces/moments are then defined. Distributed LaSMP actuations are presented in the next section. 2.1. LaSMP actuation strain model The optical active substance Spiropyran has two kinds of isomerism forms, namely a ring-closed form (SP) and a ringopened merocyanine form (MC). In a reverse reaction, SP turns into MC with decreased Young’s modulus when exposed to 365 nm UV lights. In a forward reaction, MC transforms to SP with increased Young’s modulus when exposed to 540 to 560 nm UV lights. In the reverse and forward reactions, both strain and Young’s modulus change with time. With Spiropyran added into Ethylene vinyl acetate (EVA) and processed, a kind of LaSMP (Sp2/EVA_4) [11] was composed in the StrucTronics and Control Laboratory. After testing and validation, this LaSMP is used to evaluate the control actions imposed on flexible structures [12]. To focus on control forces, the strain models including the forward and backward reactions are introduced while Young’s modulus is assumed unchanged. For photochemical series k = 1, the forward reaction exhibits the strain variation:
Sa ¼ Samax Samax
t1 t P0 t þ 1 esr þ e sd sr Pmax
ð1Þ
where Sa is the LaSMP actuation strain, Samax is the LaSMP maximum strain, P 0 is the initial concentration of the optical active Spiropyran substance in the reverse reaction, Pmax is the maximum concentration of Spiropyran; sr and sd are respectively the time constants in the forward and reverse reactions, t 1 is the forward reaction time. However, the reverse reaction exhibits strain variation:
Sa ¼ Samax Samax
P0 st e d Pmax
ð2Þ
The forward and backward reaction models and their strain variations are validated with experimental data [12]. 2.2. Dynamic control equations of hemispherical shells with LaSMP actuators A hemispherical shell with an arbitrary LaSMP actuator is illustrated in Fig. 1. As shown in Fig. 1, the LaSMP patches are circumferentially laminated and an arbitrary patch is defined from ð/1 ; w1 Þ to ð/2 ; w2 Þ in the spherical coordinate system ð/; w; u3 Þ, i.e. the meridional, circumferential and transverse directions. The shell radius and thickness are R and h respeca tively. The LaSMP thickness is h and its Young’s modulus is Y a . Assuming that the LaSMP induced strain is Saij ði; j ¼ /; wÞ; thus, LaSMP induced membrane control forces and control moments of a selected actuator defined by /1 ! /2 ; w1 ! w2 are defined as
a Na// ¼ Y a h Sa// us / /1 us / /2 us w w1 us w w2
ð3aÞ
a Naww ¼ Y a h Saww us / /1 us / /2 us w w1 us w w2
ð3bÞ
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
3
Fig. 1. The LaSMP patch on hemisphere shell.
Ma// ¼
a hþh a Y a h Sa// us / /1 us / /2 us w w1 us w w2 2
Maww ¼
a hþh a Y a h Saww us / /1 us / /2 us w w1 us w w2 2
ð3cÞ
ð3dÞ
where the two step functions us , i.e. us / /1 us / /2 and us w w1 us w w2 , define the location of an LaSMP patch on the shell. Substituting control membrane forces and moments (Eq.(3)) into the dynamic system equations, using the bending approximation theory, i.e. N // ¼ N ww ¼ N /w ¼ 0, one can simplify the governing equations and degenerate to [19]:
@ Na// sin/ @/ @ Naww @w
€ // ¼ R sin /F / Naww cos / Q /3 sin / þ R sin /qhu
€ ww ¼ R sin /F w Q w3 sin / þ R sin /qhu
@ Q /3 sin / @ Q w3 € 3 ¼ R sin /F 3 Na// þ Naww sin / þ R sin /qhu @/ @w
ð4Þ
ð5Þ
ð6Þ
where:
Q /3
Q w3
2 3 a @ Mw/ 1 4@ M // M// sin / a þ M ww Mww cos /5 ¼ @/ @w R sin / 2 3 @ M ww Maww 1 4@ M w/ sin / ¼ þ M w/ cos /5 þ R sin / @w @/
ð7Þ
ð8Þ
Note that although elastic membrane forces are neglected under the bending approximation, the LaSMP actuator induced membrane control forces still remain for distributed control. Their influences on control effects would appear in later parametric analyses of microscopic actuations. From Eqs. (4–8), it is abvious that the control forces and moments caused by LaSMP exist in all governing equations. And the control moments mainly affect the transverse shearing forces in Eqs. (7) and (8). Detailed microscopic control actions are defined next. To focus on LaSMP control effect, the effect of LaSMP patch on the mode shape functions and natural frequencies of hemipherical shell is neglected here.
3. Microscopic LaSMP actuations Dynamic control equations of the hemispherical shell coupled with LaSMP control forces/moments were defined above. Detailed distributed LaSMP microscopic actuations and their key contributing components are derived in this section.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
3.1. Definition of LaSMP control forces on hemispherical shells According to the modal expansion method [19], the displacement ui ð/; w; t Þ can be written as 1 P gk U ik ð/; wÞi ¼ ð/; w; 3Þ, where gk is the mode participation factor (or modal coordinate) and U ik ð/; wÞ are ui ð/; w; t Þ ¼ k¼1
the mode shape functions. For a hemispherical shell with free boundary, the mode shape functions U ik ð/; wÞ;i ¼ ð/; w; 3Þ are
k / U /k ¼ A tan sin kw 2
ð9Þ
k / U wk ¼ B sin / tan cos kw 2
ð10Þ
k / cos kw U 3k ¼ CRðk þ cos /Þ tan 2
ð11Þ
where A, B, C are the modal amplitudes. Substituting mode shape functions into the dynamic equations, imposing the modal orthogonality and simplifying gives the modal response equation of the hemispherical shell m
a
g€ k þ 2fk xk g_ k þ x2k gk ¼ F^k þ F^k
ð12Þ
Here, for mode k, fk is the damping ratio, xk is the natural frequency, F^k is the mechanical excitation and F^k is the LaSMP m induced control action. To focus on the LaSMP effect, the mechanical excitation F^ is neglected and the modal control forces m
a
k
a F^k can be written as:
a F^k ¼
1 qhNk R R
where N k ¼
/
w
Z Z /
3 P
i¼1
3 X
w
! Lai U ik R2 sin /d/dw
ð13Þ
i¼1
U 2ik R2 sin/d/dw, and Lai ;i ¼ ð/; w; 3Þ are the Love control operators defined as:
a a 1 @N// 1 @M// La/ ¼ 2 R @/ R @/
Law ¼
La3
ð14Þ
a @Maww 1 @Nww 1 2 R sin / @w R sin / @w
ð15Þ
8 2 9 3 ! < @ @ M a// sin / = @Maww 1 @ a a a 4 Mww cos /5 þ ¼ 2 R sin / N// þ Nww ; @/ sin / @w @w R sin / :@/ 1
ð16Þ
The modal control force, i.e., Eq.(13), can be written in a form of three possible control force components i ¼ ð/; w; 3Þ: a F^k ¼
1 qhNk
Z Z a La/ U /k þ Law U wk þ La3 U 3k R2 sin/d/dw ¼ F^k /
þ F^k
a
merd
þ F^k
a
cir
ð17Þ
trans
w
a a a a Here, F^k denotes the overall actuation action and F^k merd , F^k cir and F^k trans are the control forces in the meridional, circumferential and transverse directions respectively. It is noted that LaSMP used in this study is an isotropic material, which means the strains in the meridional and circumferential directions are the same, i.e. Sa// ¼ Saww , and this bi-axial behavior is experimentally validated (see Appendix). Then, the control membrane force and moment components both are equal, i.e., N a// ¼ N aww and M a// ¼ Maww . a
Substituting Eqs. (9)–(11) and (14)–(16) into Eq. (13) yields the meridional and circumferential control forces F^k F^k
a
cir :
a F^k
merd
k Z Z Z p=2 Z 2p a a 1 1 / Y a S// h h þh La/ U /k R2 sin /d/dw ¼ A tan sin kw sin / R þ 2 2 qhNk / w qhNk 0 R2 0 d / /1 d / /2 us w w1 us w w2 d/dw # " k k a a Y a S// h h þh A / / Rþ tan 1 sin /1 tan 2 sin /2 cos kw1 cos kw2 ¼ 2 k qhNk 2 2
merd
and
¼
ð18Þ
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744 a F^k
k a a / Y a S// h h þh B sin / tan cos kw R þ 2 2 R2 0 / w 0 k Z a a /2 Y a S// h h þh B / us / /1 us / /2 d w w1 d w w2 d/dw ¼ cos kw1 cos kw2 Rþ sin / tan d/ 2 k 2 qhNk /1 cir
¼
Z Z
1 qhN k
Law U wk R2 sin /d/dw ¼
1 qhN k
5
Z p=2 Z 2p
ð19Þ
Here, dð/Þ is the Dirac delta function with definition as:
dð/ / Þ ¼
1 / ¼ /
ð20Þ
/–/
0
And dð/Þ has following mathematical properties:
Z
/2
dð/ / Þd/ ¼ 1/1 < / < /2
ð21Þ
/1
Z
þ1
1
0
d0 ðx aÞfd/ ¼ f ðaÞ
ð22Þ
For the control force in the transverse direction, considering N a// ¼ N aww and M a// ¼ Maww :
F^k
a trans
8 2 3 ! Z Z < a @ M a// sin / 1 @ 1 @ @Mww a 2 a 4 5 M ww cos / þ L3 U 3k R sin /d/dw ¼ @/ sin / @w qhNk / w :@/ @w / w )" # k / CRðk þ cos /Þ tan R sin / Na// þ Naww cos kw d/dw ¼ 2 8 ! ) a Z Z < a @Ma// 1 @ @ M// 1 @ @M ww þ cos / þ sin / Rsin / Na// þ Naww @/ @/ sin / @w qhNk / w : @/ @w " # k / CRðk þ cos /Þ tan cos kw d/dw 2
1 ¼ qhNk
Z Z
The transverse actuation force can be divided into: a F^k
trans
a ¼ F^k
trans
/;bend
a þ F^k
trans
w;bend
a þ F^k
trans
/;mem
a þ F^k
trans
w;mem
ð23Þ
ð24Þ
a a Here, F^k trans and F^k trans are the bending control moment actions in transverse direction resulting from M a// a /;bend a w;bend and M aww . F^k trans and F^k trans are the membrane control force actions in transverse direction resulting from N a// /;mem
w;mem
and N aww . The specific expression of each action is defined next and evaluated in case studies. For the first term in Eq. (23), it can be integrated as: 1 qhN
R R /
k
w
@ sin/ @/
@ ðM a// Þ @/
h
i k CRðk þ cos/Þ tan /2 coskw d/dw
k R p=2 R 2p ðhþha Þ 1 CRðk þ cos/Þ tan /2 coskwsin/Y a S// ha 2 ¼ qhN 0 0 k @ d / /1 d / /2 us w w1 us w w2 d/dw @/ h 2 /2 k /1 k ðhþha Þ Y a ha S 2 sinkw sinkw þ 2kcos/ þ cos2/ k þ 2kcos/ þ cos2/ tan tan k ¼ qhN // 2 CR 1 2 2 2 1 1 2 2 k
ð25Þ
k
For the second term in Eq. (23), the integration is
k Z Z Z p=2 @M a// 1 / 1 cos/ coskwd/dw ¼ CRðk þ cos/Þ tan 2 qhNk / w @/ qhNk 0 k Z 2p / CRðk þ cos/Þ tan coskwcos/Y a S// ha 2 0 a hþh d / /1 d / /2 us w w1 us w w2 d/dw 2 " # k k a a Y a h S// h þ h CR /1 /2 cos/1 k þ cos/2 tan cos/2 ¼ sinkw1 sinkw2 k þ cos/1 tan k qhNk 2 2 2
ð26Þ
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Equations (25) and (26) make up the meridional bending control action F^k trol moment.
a For the third term, it is the circumferential bending control action F^k
trans
trans
w;bend
/;bend
resulting from the meridional con-
resulting from the circumferential control
moment:
! k Z Z Z p=2 a 1 1 @ @M ww / 1 CRðk þ cos /Þ tan cos kwd/dw ¼ qhNk / w sin / @w @w qhNk 0 2 k Z 2p / 1 Y a S// ha CRðk þ cos /Þ tan cos kw 2 sin / 0 a a @ hþh Y a S// h us / /1 us / /2 d w w1 d w w2 d/dw ¼ 2 @w qhNk k Z / a 2 hþh a / 1 d/ ¼ F^k trans ðk þ cos /Þ tan CRk sin kw2 sin kw1 w;bend 2 sin / 2 /1
a The meridional and circumferential membrane control actions F^k
trans
/;mem
a and F^k
brane control forces in the meridional and circumferential directions are equal due to
ð27Þ
trans
N a//
resulting from the mem-
w;mem ¼ N aww .
k Z Z Z p=2 1 / 1 Rsin/Na// CRðk þ cos/Þ tan cos kwd/dw ¼ 2 qhNk / w qhNk 0 k Z 2p a / Y a S// h CR2 ðk þ cos /Þ tan cos kwsin/Y a S// ha us / /1 us / /2 us w w1 us w w2 d/dw ¼ 2 qhNk 0 k Z / 2 2 CR / ðk þ cos /Þ tan sin/d/ sinkw2 sinkw1 2 k /1 a a ¼ F^ ¼ F^ k trans
/;mem
k trans
w;mem
ð28Þ In this section, the total actuation control forces and its detailed micro-control actions in meridium, circumferential and transverse direaction is defined in Eq.(18), (19) and (23). The simplified microscopic contributing components in transverse direction from meridional/circumferential bending and membrane control forces are derived in Eqs.(25–28). Mathematical expressions of actuation forces indicate that the actuation magnitude is related with vibration mode k, shell radius R and a thickness h, the LaSMP’s thichness h and actuator position /1 /2 ; w1 w2 . Microscopic LaSMP membrane and bending control actions on the hemispherical shells related to various design parameters are evaluated next.
4. Evaluation of distributed microscopic actuations The in-plane vibrations in the meridional and circumferential directions are usually small, as compared with the out-ofa plane transverse vibration of thin shells. Thus, the transverse LaSMP control force F^k trans is focused and its microscopic meridional and circumferential contributing components induced by LaSMP membrane forces and bending moments are compared. The geometric and material parameters used in case studies are listed in Table 1. Again, the biaxial LaSMP actuation was demonstrated in laboratory experiments and data are provided in Appendix. 4.1. Control effect of LaSMP actuator placements Fig. 2 illustrates the LaSMP actuators laminated on the hemispherical shell at 00 100 , 100 200 , 200 300 , . . . , 800 900 in the meridional direction / and divided at every ð2n 1Þp=2k in the circumferential direction w, where n is the number of divisions and n ¼ 1; 2; :::; 2k, where k is the wave number in the circumferential direction and vibration mode as well. In this actuator layout, as w1 ¼ ð2n 1Þp=2k, w2 ¼ ð2n þ 1Þp=2k, there are cosðkw1 Þ ¼ 0 and cosðkw2 Þ ¼ 0 in Eqs.(18) a a and (19), accordingly the in-plane forces F^ and F^ both become zero. Thus, the total control force is equal to the transk merd
k cir
a verse control force F^k trans when LaSMP actuators laminated on the shell as shown in Fig. 2. Accordingly, the total transverse a control force F^ and its contributing meridional/circumferential membrane/bending components are analyzed in details k trans
next.
7
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744 Table 1 Material and geometric parameters of shell and LaSMP. Parameter
Value
Unit
Young modulus of shell (polypropylene) Shell density Damping ratio Shell thickness The radius of shell Photochemical series k The maximum concentration of the optical active substance P max The initial concentration of the optical active substance P 0
1556 907 0.01 1 103 1 1 3% 0.64% 0.95%
MPa Kg/m3 = m m = = = =
24.8 52 25 10.6 40.2 1 104 0.001
S S S S MPa m =
The maximum strain induced by LaSMP S0max Time of reverse reaction t 1 Time of forward reaction t 2 Time constant in the forward reaction sr Time constant in the reverse reaction sd LaSMP Young’s modulus Thickness of LaSMP Strain of LaSMP Saij ði; j ¼ /; wÞ
Fig. 2. The distribution of LaSMP on the hemisphere shell.
a Fig. 3 illustrates the total transverse control force F^k
trans
at different meridional locations ð/1 ! /2 Þ at every 10° with shell a
modes k ¼ 2; 3; 4. (Note that the dots are plotted at the half-way.) Furthermore, for each mode k ¼ 2; 3; 4, the total F^k trans and its microscopic contributing components in the meridional and circumferential directions resulting from the membrane control forces or bending control moments are plotted respectively in Figs. 4 to 6. For reference, the value of denominator N k , i.e., 3 R R P U 2ik R2 sin/d/dw; is also drawn in Fig. 7. Nk ¼ / w i¼1
a Fig. 3. The total control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 2; 3; 4
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 4. The components of the control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 2
a Fig. 5. The components of the control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 3
a Fig. 6. The components of the control force F^k
trans
at different ð/1 ; /2 Þ with mode k ¼ 4
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
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Fig. 7. The value of N k at different ð/1 ; /2 Þ with mode k ¼ 2; 3; 4
a From Figs. 3 to 6, it is obvious that the total control force F^k trans and its components increase when the actuator patch moves from the pole to the free edge and all reach the maximums at the ð80; 90Þ segment in the meridional direction /, which means the LaSMP control effect reaches the best when located near the free boundary (or free perimeter) of the shell. a Fig. 3 also indicates that the overall control forces F^k trans decrease at higher natural modes. The four microscopic contributing a a a , F^k trans component actions of F^k trans are plotted in Figs. 4 to 6, from which the membrane control actions F^k trans /;mem a w;mem resulting from the membrane control forces dominating the total control force and the bending control actions F^k trans , /;bend a induced by the bending control moments are negligibly small. Fig. 7 suggests that the modal constant of N k is F^k trans w;bend
independent of LaSMP patch position, but it increases with increasing of mode k, which leads to decreasing of total control force and its components since N k is on the denominator of the control force definition. Other design parameters influencing the LaSMP control actions and effects are evaluated next. 4.2. Effect of shell thickness and radius a As the control force F^k trans and its component actions are functions of shell thickness h and radius R, the effect will be investigated in this section. The thickness of shell is assumed to be changing from 1 to 10 mm with constant radius of 1 m and the thickness of LaSMP is still 0.1 mm. The radius of hemispherical shell changes from 0.6 to 1.6 m with a constant shell thickness of 1 mm. The thickness effects are drawn in Figs. 8 to 11 and the radius effects are presented in Figs. 12 to 15.
a Fig. 8. The value of the total control force F^k
trans
with different shell thickness at mode k ¼ 2; 3; 4
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 9. The components of F^k
trans
with different shell thickness at mode k ¼ 2
a Fig. 10. The components of F^k
trans
with different shell thickness at mode k ¼ 3
a Fig. 11. The components of F^k
trans
with different shell thickness at mode k ¼ 4
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 12. The values of total control force F^k
trans
with different shell radius at mode k ¼ 2; 3; 4
a Fig. 13. The components of F^k
trans
with different shell radius at mode k ¼ 2
a Fig. 14. The components of F^k
trans
with different shell radius at mode k ¼ 3
11
12
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 15. The components of F^k
trans
with different shell radius at mode k ¼ 4
Figs. 8 and 12 are the total transverse control forces; Figs. 9–11 (i.e., shell thickness) and 13–15 (i.e., shell radius) are respectively their (i.e., k = 2, k = 3 and k = 4 modes) corresponding contributing meridional/circumferential membrane/bending control actions. The distribution of modal constant N k with shell radius is drawn in Fig. 16. a In Fig. 9, the total control force F^ decreases with increasing of shell thickness h, a same trend exists for four microk trans
scopic component actions shown in Figs. 10 to 12. Fig. 9 also demonstrates the decreasing of total controlling forces at higher natural modes again and micro-membrane control actions dominate the overall control action in Figs. 10 to 12. Since shell’s rigidity is a cubic function of shell thickness, increasing shell thickness makes the shell harder to actuate and control generally. LaSMP control effect of shell’s radius is presented in Figs. 13–17 whose physical significances are discussed next. a decreases with increasing shell radius R or at higher natural mode. In Fig. 13, the value of the total control force F^ k trans
Figs. 13 to 15 indicate that two micro-membrane control actions domain the total control forces and all component actions 3 R R P U 2ik R2 sin/d/dw; at each modes decrease with increasing of shell radius R as well. Fig. 16 shows that N k , i.e., N k ¼ / w i¼1
Fig. 16. The value of N k at different shell radius with mode k ¼ 2; 3; 4
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a significantly increases with the shell radius R, which causes reduction of F^k trans and its components in the denominator. Analysis in this section proposes that the control forces of LaSMP are larger for shell with smaller radius, and the control effect would be better.
a
4.3. Effect of LaSMP thickness h
The LaSMP thickness will affect the control forces reflected in Eqs. (23), (27) and (28). As LaSMP is a very thin membrane, the assumption is made that the strain is uniformly distributed in the thickness direction in the modeling process. The LaSMP thickness usually is around 104 m under the limitation to this assumption. The actuation effect of actuator thickness h is shown in Figs. 17 to 20. a a In Fig. 17, the total control force F^ exhibits a linear variation with LaSMP thickness h , and this behavior appears in all a
k trans
microscopic control actuations of different modes in Figs. 18 to 20. The figures suggest, again, that the primary contributing control actions are contributed by two meridional and circumferential membrane control actions while the bending control actions are minimal. The two membrane control actions reduce in magnitudes when the mode increases. Fig. 20 indicates that the control effect is better for a hemispherical shell laminated with thicker LaSMP actuator patches, which is in accordance to physics as the control forces in LaSMP will increase. 4.4. Normalized LaSMP control forces Recall that Fig. 1 illustrates the actuator placement on the hemispherical shell and it shows the actuator size actually increases when it moves from the pole to the free boundary, i.e., the size is not a constant. In this section, the effect of LaSMP
a Fig. 17. The values of the total control force F^k
a Fig. 18. The components of F^k
trans
trans
at different LaSMP thickness with mode k ¼ 2; 3; 4
at different LaSMP thickness with mode k ¼ 2
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
a Fig. 19. The components of F^k
trans
at different LaSMP thickness with mode k ¼ 3
a Fig. 20. The components of F^k
trans
at different LaSMP thickness with mode k ¼ 4
Fig. 21. The LaSMP size at different position in the meridional direction at mode k ¼ 2; 3; 4
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
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Fig. 22. The LaSMP size with different shell radius at mode k ¼ 2; 3; 4
ea Fig. 23. Normalized control forces F^ k
trans
at various meridional locations (i.e., 0: pole; 90: free boundary) at mode k ¼ 2; 3; 4
patch size is evaluated. When a LaSMP actuator distributes as /1 /2 ; w1 w2 in the meridional and circumferential directions, the true LaSMP actuator size is
Z A¼
/2
/1
Z
w2
w1
R2 sin/d/dw ¼ R2 cos/1 cos/2 w2 w1 :
ð29Þ
The variation of LaSMP size A with different meridional angle /j ; j ¼ 1; 2 and shell radius R is shown in Figs. 21 and 22. Figs. 21 and 22 indicate that the LaSMP size increases when meridional degree / and shell radius R increase. Accordingly, to objectively estimate control effectiveness on the shell, earlier calculations, i.e., with respect to /, h and R in Figs. 3, 8, 12, of ea the total LaSMP control forces, i.e., F^ k trans ; k ¼ 2; 3; 4, need to be normalized with respect to the true actuator size A and these three normalized results are presented in Figs. 23-25. Fig. 23 indicates that the normalized control forces also reach the maximums at the free boundary of hemispherical shell, which suggest the best control location is near the boundary. The normalized control force also decreases with increasing of shell thickness as the patch size is independent of shell thickness as shown in Eq. (29) (Fig. 24). And the trend between the normalized control force and shell radius in Fig. 25 is the same with that of control force and shell radius in Fig. 12, but normalized control force decreases much faster with shell radius as the patch size increases with shell radius.
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
ea Fig. 24. Normalized control forces F^ k
trans
ea Fig. 25. Normalized control forces F^ k
versus shell thickness h at mode k ¼ 2; 3; 4
trans
versus shell radius R at mode k ¼ 2; 3; 4
5. Conclusions LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated in detail in this study. The shell governing equations with LaSMP control forces/moments are derived based on the bending approximation theory. The specific expressions of control forces of LaSMP patch on a hemisphere shell with free boundary are derived by using the Love control operators, followed by detailed analyses of modal control forces and their microscopic control actions. In case studies, with LaSMP patch laminated on ð2n 1Þp=2k;k ¼ 2; 3; 4 in the circumferential direction, the control forces are defined by material, geometric design parameters, e.g., actuator location, shell thickness/radius, LaSMP thickness, etc. LaSMP induced modal (k ¼ 2; 3; 4) meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. With thorough analyses of various parameters, following conclusions can be drawn: 1. In-plane meridional and circumferential control forces disappear in the above LaSMP lamination and the transverse control force dominants the total control forces. 2. Among the four meridional and circumferential microscopic membrane/bending control actions the actuation components resulting from the membrane forces are the main control actions and those resulting from the bending control moments are very small. 3. The total control force and its components decrease with increasing of shell thickness and radius. The total control force and its component actions would also increase with the increase of LaSMP thickness. 4. At last, for the total control force and the normalized total control force, they reach the maximums near the free boundary of the hemispherical shell, which means the boundary area is the most effective control location for the shell.
D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
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The work in this study would provide a universal analysis procedure and give a general guideline for choosing idealized design parameters for vibration control of LaSMP laminated hemispherical shells. CRediT authorship contribution statement Dan Wang: Conceptualization, Investigation, Writing - original draft, Writing - review & editing, Funding acquisition, Resources. Mu Fan: Investigation, Funding acquisition, Resources. Zhu Su: Investigation, Supervision. Hornsen Tzou: Conceptualization, Writing - review & editing, Funding acquisition, Supervision. Acknowledgements This research is supported by Natural Science Foundation of China (Nos. 11872206, 11902151), Natural Science Foundation of Jiangsu Province (No. BK20180411, BK20180429, BK20170773), State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-IZD1900201907). Appendix: Experimental verification of LaSMP Young’s modulus in orthotropic direction For LaSMP used as vibration controller in two-dimensioning spherical shell, Young’s modulus in orthotropic direction need be verified by the experiment. In laboratory, a cross shaped LaSMP membrane is fabricated to validate Young’s modulus [11]. As shown in Fig. A1, according to the requirement of biaxial tensile test, the area of LaSMP specimen is 118 mm*118 mm with the thickness of 0.32 mm, and the rectangular area in the center is 36 mm*36 mm.
Fig. A1. LaSMP sample for biaxial tension experiments [11].
Fig. A2. Biaxial tension result of the LaSMP sample under UV lights [11].
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D. Wang et al. / Mechanical Systems and Signal Processing 142 (2020) 106744
The loading and unloading in biaxial direction were taken six times under UV light exposures with wavelength at 365 nm and the tensile curves were recorded in Fig. A2, which shows the tension curves in biaxial directions almost overlap. Thus, Young’s modulus keep the same in orthotropic direction as the orientation of sample was chosen arbitrarily. Accordingly, the lab fabricated LaSMP is suitable in biaxial actuations and two dimensional vibration control. References [1] H.S. Tzou, Piezoelectric Shells: Distributed Sensing and Control of Continua, Springer Science & Business Media, Boston, 2012. [2] T.J. Johnson, R.L. Brown, D.E. Adams, M. Schiefer, Distributed structural health monitoring with a smart sensor array, Mech. Syst. Signal Pr. 18 (2004) 555–572. [3] M.D. Hager, S. Bode, C.W. Schubert, Shape memory polymers: past, present and future developments, Prog. Polm. Sci. 49 (2015) 3–33. [4] H.J. Wang, B. Fan, M. Yi, Inflatable antenna for space-borne microwave remote sensing, IEEE. Antenn. Propag. M. 54 (2012) 58–70. [5] Y. Luo, M. Higa, S. Amae, Preclinical development of SMA artificial anal sphincters, Minim. Invasiv. Ther. 15 (2006) 241–245. [6] G. Sourav, S.K. Mishra, K. Roy, Stochastic seismic response of building with super-elastic damper, Mech. Syst. Signal Pr. 72 (2016) 642–659. [7] A. Lendlein, H. Jiang, O. Jünger, R. Langer, Light-induced shape-memory polymers, Nature 434 (2005) 879. [8] J.S. Sodhi, P.R. Cruz, I.J. Rao, Inhomogeneous deformations of light activated shape memory polymers, Int. J. Eng. Sci. 89 (2015) 1–17. [9] D.C. Sun, L.Y. Tong, Theoretical investigation on wireless vibration control of thin beams using photostrictive actuators, J. Sound Vib. 312 (2008) 182– 194. [10] X. Zhang, Q. Zhou, H. Liu, H. Liu, UV light induced plasticization and light activated shape memory of spiropyran doped ethylene-vinyl acetate copolymers, Soft Matter 10 (2014) 3748–3754. [11] H. Li, D. Guo, H.S. Tzou, Light-induced vibration control of parabolic cylindrical shell panels, J. Intel. Mat. Syst. Str. 28 (2017) 2947–2958. [12] H.Y. Li, H. Li, H.S. Tzou, Frequency control of beams and cylindrical shells with light-activated shape memory polymers, J. Vib. Acoust. 137 (2015) 011006. [13] D. Wang, M. Fan, Z. Su, H.S. Tzou, Frequency control of thin plates with light-activated shape-memory polymers, AIAA. J. (2019), online. [14] H.S. Tzou, P. Smithmaitrie, J.H. Ding, Micro-sensor electromechanics and distributed signal analysis of piezo(electric)-elastic spherical shells, Mech. Syst. Signal Pr. 16 (2002) 185–199. [15] C. Hwang, Some experiments on the vibration of a hemispherical shell, J. Appl. Mech. 33 (1966) 817–824. [16] H. Yue, Y. Lu, Z. Deng, H.S. Tzou, Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Pr. 82 (2017) 279– 295. [17] K.A. Kuo, H.E.M. Hunt, J.R. Lister, Small oscillations of a pressurized, elastic, spherical shell: model and experiments, J. Sound Vib. 359 (2015) 168–178. [18] P. Smithmaitrie, H.S. Tzou, Micro-control actions of actuator patches laminated on hemispherical shells, J. Sound Vib. 277 (2004) 691–710. [19] H. Li, S. Hu, H.S. Tzou, Modal signal analysis of conical shells with flexoelectric sensors, J. Intel. Mat. Syst. Str. 26 (2015) 1551–1563.