Abstract

In this paper, an elastic mode method of deformable mirror is proposed to decompose arbitrary wave-front errors of adaptive optics system. The elastic modes are derived with an analytical method of linear piezoelectricity based on a bimorph piezoelectric deformable mirror (BPDM), and the three-dimensional formulas of elastic modes are presented. Here a BPDM with an aperture of 165 mm as an example is numerically studied. Two different kinds of dynamic boundary conditions are considered, and the dependence of the elastic modes aberrations upon the orders and rotational symmetries is evaluated. Besides, a comparative study for elastic mode and Zernike polynomials is presented in the numerical analysis. The results have demonstrated that the elastic mode method can be not only used instead of Zernike polynomials, but also more effective to decompose arbitrary wave-front errors of a deformable mirror. Furthermore, finite element analysis method is used to validate the analytic method. The conclusions have shown reasonably consistent results between the two methods.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  6. Y. Ning, W. Jiang, N. Ling, and C. Rao, “Response function calculation and sensitivity comparison analysis of various bimorph deformable mirrors,” Opt. Express 15(19), 12030–12038 (2007).
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  7. H. Wang, Y. Hu, and J. Wang, “On the nonlinear behavior of a multilayer circular piezoelectric plate-like transformer operating near resonance,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 752–757 (2013).
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  8. H. Wang, M. Hu, and Z. Li, “Modelling and analysis of circular bimorph piezoelectric actuator for deformable mirror,” Appl. Math. Mech. 37(5), 639–646 (2016).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  22. H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
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    [Crossref]

2018 (2)

H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

2017 (1)

2016 (4)

H. Wang, M. Hu, and Z. Li, “Modelling and analysis of circular bimorph piezoelectric actuator for deformable mirror,” Appl. Math. Mech. 37(5), 639–646 (2016).
[Crossref]

H. Wang and S. Yang, “Modeling and analysis of the thermal effects of a circular bimorph piezoelectric actuator,” Appl. Opt. 55(4), 873–878 (2016).
[Crossref] [PubMed]

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

2015 (1)

H. Wang, “Analytical analysis of a beam flexural-mode piezoelectric actuator for deformable mirrors,” J. Astron. Telesc. Instrum. Syst. 1(4), 49001 (2015).
[Crossref]

2014 (3)

H. Wang, X. Xie, Y. Hu, and J. Wang, “Nonlinear analysis of a 5-layer beam-like piezoelectric transformer near resonance,” Acta Mechanica Solida Sinica 27(2), 195–201 (2014).
[Crossref]

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

H. R. Wang, X. Xie, Y. T. Hu, and J. Wang, “Weakly nonlinear characteristics of a three-layer circular piezoelectric plate-like power harvester near resonance,” J. Mech. 30(01), 97–102 (2014).
[Crossref]

2013 (3)

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

H. Wang, H. Hu, J. Yang, and Y. Hu, “Spiral piezoelectric transducer in torsional motion as low-frequency power harvester,” Appl. Math. Mech. 34(5), 589–596 (2013).
[Crossref]

H. Wang, Y. Hu, and J. Wang, “On the nonlinear behavior of a multilayer circular piezoelectric plate-like transformer operating near resonance,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 752–757 (2013).
[Crossref] [PubMed]

2012 (1)

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

2007 (2)

Y. Ning, W. Jiang, N. Ling, and C. Rao, “Response function calculation and sensitivity comparison analysis of various bimorph deformable mirrors,” Opt. Express 15(19), 12030–12038 (2007).
[Crossref] [PubMed]

H. Xue, Y. Hu, Q.-M. Wang, and J. Yang, “Analysis of temperature compensation in a plate thickness mode bulk acoustic wave resonator,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(9), 1826–1833 (2007).
[Crossref] [PubMed]

1993 (1)

N. Hubin and L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262(5138), 1390–1394 (1993).
[Crossref] [PubMed]

1991 (1)

L. Noethe, “Use of minimum-energy modes for modal-active optics corrections of thin meniscus mirrors,” J. Mod. Opt. 38(6), 1043–1066 (1991).
[Crossref]

1987 (1)

R. Wilson, F. Franza, and L. Noethe, “Active Optics,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Chen, X.

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

Chen, Z.

H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

Cheng, J.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

Franza, F.

R. Wilson, F. Franza, and L. Noethe, “Active Optics,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Hu, H.

H. Wang, H. Hu, J. Yang, and Y. Hu, “Spiral piezoelectric transducer in torsional motion as low-frequency power harvester,” Appl. Math. Mech. 34(5), 589–596 (2013).
[Crossref]

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

Hu, L.

H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

Hu, M.

H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

H. Wang, M. Hu, and Z. Li, “Modelling and analysis of circular bimorph piezoelectric actuator for deformable mirror,” Appl. Math. Mech. 37(5), 639–646 (2016).
[Crossref]

Hu, Y.

H. Wang, X. Xie, Y. Hu, and J. Wang, “Nonlinear analysis of a 5-layer beam-like piezoelectric transformer near resonance,” Acta Mechanica Solida Sinica 27(2), 195–201 (2014).
[Crossref]

H. Wang, H. Hu, J. Yang, and Y. Hu, “Spiral piezoelectric transducer in torsional motion as low-frequency power harvester,” Appl. Math. Mech. 34(5), 589–596 (2013).
[Crossref]

H. Wang, Y. Hu, and J. Wang, “On the nonlinear behavior of a multilayer circular piezoelectric plate-like transformer operating near resonance,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 752–757 (2013).
[Crossref] [PubMed]

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

H. Xue, Y. Hu, Q.-M. Wang, and J. Yang, “Analysis of temperature compensation in a plate thickness mode bulk acoustic wave resonator,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(9), 1826–1833 (2007).
[Crossref] [PubMed]

Hu, Y. T.

H. R. Wang, X. Xie, Y. T. Hu, and J. Wang, “Weakly nonlinear characteristics of a three-layer circular piezoelectric plate-like power harvester near resonance,” J. Mech. 30(01), 97–102 (2014).
[Crossref]

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

Hubin, N.

N. Hubin and L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262(5138), 1390–1394 (1993).
[Crossref] [PubMed]

Jiang, W.

Li, Z.

H. Wang, M. Hu, and Z. Li, “Modelling and analysis of circular bimorph piezoelectric actuator for deformable mirror,” Appl. Math. Mech. 37(5), 639–646 (2016).
[Crossref]

Liang, M.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

Ling, N.

Lou, Z.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

Luo, B.

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

Ning, Y.

Noethe, L.

N. Hubin and L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262(5138), 1390–1394 (1993).
[Crossref] [PubMed]

L. Noethe, “Use of minimum-energy modes for modal-active optics corrections of thin meniscus mirrors,” J. Mod. Opt. 38(6), 1043–1066 (1991).
[Crossref]

R. Wilson, F. Franza, and L. Noethe, “Active Optics,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Qian, Y.

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

Rao, C.

Wang, H.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

H. Wang, “Research on a bimorph piezoelectric deformable mirror for adaptive optics in optical telescope,” Opt. Express 25(7), 8115–8122 (2017).
[Crossref] [PubMed]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

H. Wang and S. Yang, “Modeling and analysis of the thermal effects of a circular bimorph piezoelectric actuator,” Appl. Opt. 55(4), 873–878 (2016).
[Crossref] [PubMed]

H. Wang, M. Hu, and Z. Li, “Modelling and analysis of circular bimorph piezoelectric actuator for deformable mirror,” Appl. Math. Mech. 37(5), 639–646 (2016).
[Crossref]

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

H. Wang, “Analytical analysis of a beam flexural-mode piezoelectric actuator for deformable mirrors,” J. Astron. Telesc. Instrum. Syst. 1(4), 49001 (2015).
[Crossref]

H. Wang, X. Xie, Y. Hu, and J. Wang, “Nonlinear analysis of a 5-layer beam-like piezoelectric transformer near resonance,” Acta Mechanica Solida Sinica 27(2), 195–201 (2014).
[Crossref]

H. Wang, H. Hu, J. Yang, and Y. Hu, “Spiral piezoelectric transducer in torsional motion as low-frequency power harvester,” Appl. Math. Mech. 34(5), 589–596 (2013).
[Crossref]

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

H. Wang, Y. Hu, and J. Wang, “On the nonlinear behavior of a multilayer circular piezoelectric plate-like transformer operating near resonance,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 752–757 (2013).
[Crossref] [PubMed]

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

Wang, H. R.

H. R. Wang, X. Xie, Y. T. Hu, and J. Wang, “Weakly nonlinear characteristics of a three-layer circular piezoelectric plate-like power harvester near resonance,” J. Mech. 30(01), 97–102 (2014).
[Crossref]

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

Wang, J.

H. Wang, X. Xie, Y. Hu, and J. Wang, “Nonlinear analysis of a 5-layer beam-like piezoelectric transformer near resonance,” Acta Mechanica Solida Sinica 27(2), 195–201 (2014).
[Crossref]

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

H. R. Wang, X. Xie, Y. T. Hu, and J. Wang, “Weakly nonlinear characteristics of a three-layer circular piezoelectric plate-like power harvester near resonance,” J. Mech. 30(01), 97–102 (2014).
[Crossref]

H. Wang, Y. Hu, and J. Wang, “On the nonlinear behavior of a multilayer circular piezoelectric plate-like transformer operating near resonance,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 752–757 (2013).
[Crossref] [PubMed]

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

Wang, Q.-M.

H. Xue, Y. Hu, Q.-M. Wang, and J. Yang, “Analysis of temperature compensation in a plate thickness mode bulk acoustic wave resonator,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(9), 1826–1833 (2007).
[Crossref] [PubMed]

Wilson, R.

R. Wilson, F. Franza, and L. Noethe, “Active Optics,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Xie, J. M.

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

Xie, X.

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

H. Wang, X. Xie, Y. Hu, and J. Wang, “Nonlinear analysis of a 5-layer beam-like piezoelectric transformer near resonance,” Acta Mechanica Solida Sinica 27(2), 195–201 (2014).
[Crossref]

H. R. Wang, X. Xie, Y. T. Hu, and J. Wang, “Weakly nonlinear characteristics of a three-layer circular piezoelectric plate-like power harvester near resonance,” J. Mech. 30(01), 97–102 (2014).
[Crossref]

Xue, H.

H. Xue, Y. Hu, Q.-M. Wang, and J. Yang, “Analysis of temperature compensation in a plate thickness mode bulk acoustic wave resonator,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(9), 1826–1833 (2007).
[Crossref] [PubMed]

Yang, J.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

H. Wang, H. Hu, J. Yang, and Y. Hu, “Spiral piezoelectric transducer in torsional motion as low-frequency power harvester,” Appl. Math. Mech. 34(5), 589–596 (2013).
[Crossref]

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

H. Xue, Y. Hu, Q.-M. Wang, and J. Yang, “Analysis of temperature compensation in a plate thickness mode bulk acoustic wave resonator,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(9), 1826–1833 (2007).
[Crossref] [PubMed]

Yang, S.

H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

H. Wang and S. Yang, “Modeling and analysis of the thermal effects of a circular bimorph piezoelectric actuator,” Appl. Opt. 55(4), 873–878 (2016).
[Crossref] [PubMed]

Zheng, X.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

Zuo, Y.

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016).
[Crossref] [PubMed]

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

Acta Mechanica Solida Sinica (1)

H. Wang, X. Xie, Y. Hu, and J. Wang, “Nonlinear analysis of a 5-layer beam-like piezoelectric transformer near resonance,” Acta Mechanica Solida Sinica 27(2), 195–201 (2014).
[Crossref]

Appl. Math. Mech. (3)

H. R. Wang, J. M. Xie, X. Xie, Y. T. Hu, and J. Wang, “Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode,” Appl. Math. Mech. 35(2), 229–236 (2014).
[Crossref]

H. Wang, M. Hu, and Z. Li, “Modelling and analysis of circular bimorph piezoelectric actuator for deformable mirror,” Appl. Math. Mech. 37(5), 639–646 (2016).
[Crossref]

H. Wang, H. Hu, J. Yang, and Y. Hu, “Spiral piezoelectric transducer in torsional motion as low-frequency power harvester,” Appl. Math. Mech. 34(5), 589–596 (2013).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (3)

H. Wang, Y. Hu, and J. Wang, “On the nonlinear behavior of a multilayer circular piezoelectric plate-like transformer operating near resonance,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 752–757 (2013).
[Crossref] [PubMed]

H. Hu, L. Hu, J. Yang, H. Wang, and X. Chen, “A piezoelectric spring-mass system as a low-frequency energy harvester,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(4), 846–850 (2013).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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H. Wang, Z. Chen, S. Yang, L. Hu, and M. Hu, “Analysis of a discrete-layout bimorph disk elements piezoelectric deformable mirror,” J. Astron. Telesc. Instrum. Syst. 4(02), 1 (2018).
[Crossref]

H. Wang, “Analytical analysis of a beam flexural-mode piezoelectric actuator for deformable mirrors,” J. Astron. Telesc. Instrum. Syst. 1(4), 49001 (2015).
[Crossref]

J. Mech. (1)

H. R. Wang, X. Xie, Y. T. Hu, and J. Wang, “Weakly nonlinear characteristics of a three-layer circular piezoelectric plate-like power harvester near resonance,” J. Mech. 30(01), 97–102 (2014).
[Crossref]

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[Crossref]

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[Crossref]

Opt. Eng. (1)

H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 35105 (2016).
[Crossref]

Opt. Express (2)

Optik (1)

H. Wang, J. Cheng, Z. Lou, M. Liang, X. Zheng, Y. Zuo, and J. Yang, “A comparative study of the thermal performance of primary mirror at the four typical sites,” Optik 174, 727–738 (2018).
[Crossref]

Science (1)

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[Crossref] [PubMed]

Smart Mater. Struct. (1)

J. Wang, H. Wang, H. Hu, B. Luo, Y. Hu, and J. Wang, “On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters,” Smart Mater. Struct. 21(1), 015006 (2012).
[Crossref]

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D. Fang, J. Wang, and W. Chen, Analysis of piezoelectric structures and devices (Walter de Gruyter, 2013).

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Figures (9)

Fig. 1
Fig. 1 The configuration of a bimorph piezoelectric deformable mirror
Fig. 2
Fig. 2 Deformation function u m n ( r ) versus radius r for m = 1. (a) The edge of the BPDM is considered free; (b) The edge of the BPDM is simply supported.
Fig. 3
Fig. 3 The 3D deformation function u m n ( r , θ ) when the edge of the BPDM is considered free. (a) The u m n ( r , θ ) of m = 1 and n = 0; (b) The u m n ( r , θ ) of m = 2 and n = 0; (c) The u m n ( r , θ ) of m = 1 and n = 1; (d) The u m n ( r , θ ) of m = 1 and n = 2; (e) The u m n ( r , θ ) of m = 1 and n = 3; (f) The u m n ( r , θ ) of m = 1and n = 4.
Fig. 4
Fig. 4 The 3D deformation function u m n ( r , θ ) when the edge of the BPDM is simply supported. (a) The u m n ( r , θ ) of m = 1 and n = 0; (b) The u m n ( r , θ ) of m = 2 and n = 0; (c) The u m n ( r , θ ) of m = 1 and n = 1; (d) The u m n ( r , θ ) of m = 1and n = 2; (e) The u m n ( r , θ ) of m = 1 and n = 3; (f) The u m n ( r , θ ) of m = 1and n = 4.
Fig. 5
Fig. 5 The comparisons of the radial functions of the Zernike polynomial and the deformation functions of the elastic modes.
Fig. 6
Fig. 6 The finite element model of BPDM.
Fig. 7
Fig. 7 The results of the elastic modes of finite element analysis when the edge of the BPDM is considered free. (a) The u m n ( r , θ ) of m = 1 and n = 0; (b) The u m n ( r , θ ) of m = 2 and n = 0; (c) The u m n ( r , θ ) of m = 1 and n = 1; (d) The u m n ( r , θ ) of m = 1 and n = 2; (e) The u m n ( r , θ ) of m = 1 and n = 3; (f) The u m n ( r , θ ) of m = 1 and n = 4.
Fig. 8
Fig. 8 The mode shapes for three different radii, 2a = 165, 148.5, 140.25 mm of the free boundary condition
Fig. 9
Fig. 9 The comparisons of the deformation functions of the elastic modes for three different radii, 2a = 165, 148.5, 140.25 mm of the free boundary condition

Tables (1)

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Table 1 The values of λ and corresponding natural frequency f for different n and m.

Equations (23)

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ε r = z w z , r r , ε θ = z r w z , r z r 2 w z , θ θ , γ r θ = 2 ( 1 r w z , θ ) , r .
σ r s = E s 1 ν s 2 ( ε r + ν s ε θ ) , σ θ s = E s 1 ν s 2 ( ε θ + ν s ε r ) , τ r θ s = E s 2 ( 1 + ν s ) γ r θ .
σ r p = E p 1 ν p 2 ( ε r + ν p ε θ ) , σ θ p = E p 1 ν p 2 ( ε θ + ν p ε r ) , τ r θ p = E p 2 ( 1 + ν p ) γ r θ .
h m 1 = E s h s 2 + E P [ ( h s + h p ) 2 h s 2 ] 2 ( E s h s + E P h p ) , h m 2 = E p h p 2 + E s [ ( h p + h s ) 2 h p 2 ] 2 ( E s h s + E P h p ) .
M r = h m 2 h m 1 σ r z d z = D s [ w z , r r + ν s ( 1 r 2 w z , θ θ + 1 r w z , r ) ] D p [ w z , r r + ν p ( 1 r 2 w z , θ θ + 1 r w z , r ) ] , M θ = h m 2 h m 1 σ θ z d z = D s [ w z , θ θ + ν s w z , r r + 1 r w z , r ) ] D p [ w z , θ θ + ν p w z , r r + 1 r w z , r ) ] , M θ r = h m 2 h m 1 τ r θ z d z = D s ( 1 ν s ) ( 1 r w z , θ ) , r D p ( 1 ν p ) ( 1 r w z , θ ) , r .
D s = E s h s [ E s 2 h s 4 + 2 E s E P h s 3 h p + E p 2 h p 2 ( 4 h s 2 + 6 h s h p + 3 h p 2 ) ] ) 12 ( E s h s + E p h p ) 2 ( 1 ν s 2 ) . D p = E p h p [ E p 2 h p 4 + 2 E s E P h s h p 3 + E s 2 h s 2 ( 4 h p 2 + 6 h s h p + 3 h s 2 ) ] ) 12 ( E s h s + E p h p ) 2 ( 1 ν p 2 )
Q r = M r , r + 1 r M r θ , θ + M r M θ r , Q θ = M r θ , r + 1 r M θ , θ + 2 M r θ r .
V r = Q r + M r θ r θ , V θ = Q θ + M r θ r .
Q r r + Q r , r + Q θ , θ r = ξ w ¨ z .
D 2 2 w z = ξ w ¨ z
w z ( r , θ , t ) = Re { W z ( r , θ ) exp ( i ω t ) } .
D 2 2 W z ω 2 ξ W z = 0
W z ( r , θ ) = u n ( r ) cos ( n θ ) n=rotational symmetry
D r 2 r 2 u n ( r ) ω 2 m u n ( r ) = 0
u n ( r ) = A 1 , n J n ( λ r ) + A 2 , n Y n ( λ r ) + A 3 , n I n ( λ r ) + A 4 , n K n ( λ r ) .
u n ( r ) = A 1 , n J n ( λ r ) + A 3 , n I n ( λ r ) .
M r ( a ) = 0 , V r ( a ) = 0
u n ( a ) = 0 , M r ( a ) = 0.
( b 11 ( λ ) b 21 ( λ ) b 12 ( λ ) b 21 ( λ ) ) ( A 1 , n A 3 , n ) = ( 0 0 )
| b 11 ( λ ) b 21 ( λ ) b 12 ( λ ) b 22 ( λ ) | = 0
u m n ( r ) = K n m u ¯ n m ( r )
K n m = a 2 β 0 a u ¯ n m ( r ) d r , β = { 2 n=0 1 n>0
Γ ( r , θ ) = n = 0 m = 1 c n m u n m ( r ) cos ( n θ ) .

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