Abstract

A new type of microactuators based on optothermal (OT) expansion is introduced. The mechanism of the OT expansion is theoretically analyzed, and comprehensive models for OT expansion and bi-direction microactuator are presented in this paper. An expansion arm and a microswitch-like OT microactuator with 1200µm-length are fabricated by an excimer laser micromachining system using single layer material. A laser diode (650nm) is employed as the external power source to activate the arm and the microactuator. Experimental results indicate that the OT expansion increment is approximately linear with the laser power irradiating the expansion arm, coinciding with theoretical predictions quite well. As to the switch-like microactuator, an enlarged bi-direction deflection has been obviously observed. The OT expansion and deflection amplitude that can reach micron scale is generally large enough for most microsystems. The new technique of OT microactuators can be widely applied in those fields where simple structure, easy fabrication, large displacement, wireless and remote controlling are required.

© 2008 Optical Society of America

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References

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  1. J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).
  2. J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000).
    [CrossRef]
  3. D. L. Devoe and A. P. Pisano, "Modeling and optimal design of piezoelectric cantilever microactuators," Journal of Microelectromechanical Systems 6,266-270(1997).
    [CrossRef]
  4. X. Huang and H. Shen, "Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators," International Journal of Mechanical Sciences 47,187-208 (2005).
    [CrossRef]
  5. Y. W. Park and D. Y. Kim, "Development of a magnetostrictive microactuator," Journal of Magnetism and Magnetic Materials E 1765-1766,272-276 (2004).
  6. C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006).
    [CrossRef]
  7. T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
    [CrossRef]
  8. L. J. Li and D. Uttamchandani, "Modified asymmetric microelectrothermal actuator: analysis and experimentation," J. Micromech. Microeng. 14, 1734-1741 (2004).
    [CrossRef]
  9. M. F. Pai, and N. C. Tien, "Low voltage electrothermal vibromotor for silicon optical bench applications," Sensors Actuators A 83, 237-243 (2000).
    [CrossRef]
  10. J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
    [CrossRef]
  11. S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
    [CrossRef]
  12. Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005).
    [CrossRef]
  13. J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999).
    [CrossRef]
  14. L. Li and D. Uttamchandani, "Modified asymmetric micro-electrothermal actuator," J. Micromech. Microeng. 14, 1734-1741(2004).
    [CrossRef]

2006 (1)

C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006).
[CrossRef]

2005 (4)

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

X. Huang and H. Shen, "Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators," International Journal of Mechanical Sciences 47,187-208 (2005).
[CrossRef]

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005).
[CrossRef]

2004 (3)

L. Li and D. Uttamchandani, "Modified asymmetric micro-electrothermal actuator," J. Micromech. Microeng. 14, 1734-1741(2004).
[CrossRef]

Y. W. Park and D. Y. Kim, "Development of a magnetostrictive microactuator," Journal of Magnetism and Magnetic Materials E 1765-1766,272-276 (2004).

L. J. Li and D. Uttamchandani, "Modified asymmetric microelectrothermal actuator: analysis and experimentation," J. Micromech. Microeng. 14, 1734-1741 (2004).
[CrossRef]

2000 (2)

M. F. Pai, and N. C. Tien, "Low voltage electrothermal vibromotor for silicon optical bench applications," Sensors Actuators A 83, 237-243 (2000).
[CrossRef]

J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000).
[CrossRef]

1999 (1)

J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999).
[CrossRef]

1998 (1)

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

1997 (1)

D. L. Devoe and A. P. Pisano, "Modeling and optimal design of piezoelectric cantilever microactuators," Journal of Microelectromechanical Systems 6,266-270(1997).
[CrossRef]

Bhaumik, S. K.

C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006).
[CrossRef]

Bright, V. M.

J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999).
[CrossRef]

Butler, J. T.

J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999).
[CrossRef]

Chen, S. C.

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

Chlpík, J.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Cowan, W. D.

J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999).
[CrossRef]

Devoe, D. L.

D. L. Devoe and A. P. Pisano, "Modeling and optimal design of piezoelectric cantilever microactuators," Journal of Microelectromechanical Systems 6,266-270(1997).
[CrossRef]

Držík, M.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Fleck, N. A.

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

Flewitt, A. J.

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

Fukuma, M.

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).

Grigoropoulos, C. P.

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

Haš??ík, Š.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

He, Y. L.

Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005).
[CrossRef]

Huang, X.

X. Huang and H. Shen, "Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators," International Journal of Mechanical Sciences 47,187-208 (2005).
[CrossRef]

Hui, C. Y.

J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000).
[CrossRef]

Kerstens, P.

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

Kim, D. Y.

Y. W. Park and D. Y. Kim, "Development of a magnetostrictive microactuator," Journal of Magnetism and Magnetic Materials E 1765-1766,272-276 (2004).

Kosti??, I.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Krná??, M.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Kyokane, J.

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).

Lalinský, T.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Li, L. J.

L. J. Li and D. Uttamchandani, "Modified asymmetric microelectrothermal actuator: analysis and experimentation," J. Micromech. Microeng. 14, 1734-1741 (2004).
[CrossRef]

Luo, J. K.

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

Milne, W. I.

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

Mozolová, Ž.

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Pai, M. F.

M. F. Pai, and N. C. Tien, "Low voltage electrothermal vibromotor for silicon optical bench applications," Sensors Actuators A 83, 237-243 (2000).
[CrossRef]

Park, H. K.

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

Park, Y. W.

Y. W. Park and D. Y. Kim, "Development of a magnetostrictive microactuator," Journal of Magnetism and Magnetic Materials E 1765-1766,272-276 (2004).

Pisano, A. P.

D. L. Devoe and A. P. Pisano, "Modeling and optimal design of piezoelectric cantilever microactuators," Journal of Microelectromechanical Systems 6,266-270(1997).
[CrossRef]

Ramaiah, K. V.

C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006).
[CrossRef]

Saikrishna, C. N.

C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006).
[CrossRef]

Shen, H.

X. Huang and H. Shen, "Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators," International Journal of Mechanical Sciences 47,187-208 (2005).
[CrossRef]

Spearing, S. M.

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

Tam, A. C.

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

Tien, N. C.

M. F. Pai, and N. C. Tien, "Low voltage electrothermal vibromotor for silicon optical bench applications," Sensors Actuators A 83, 237-243 (2000).
[CrossRef]

J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000).
[CrossRef]

Tsujimoto, K.

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).

Ueda, T.

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).

Uttamchandani, D.

L. J. Li and D. Uttamchandani, "Modified asymmetric microelectrothermal actuator: analysis and experimentation," J. Micromech. Microeng. 14, 1734-1741 (2004).
[CrossRef]

Yanagisawa, Y.

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).

Yeh, J. L.

J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000).
[CrossRef]

Zhang, D. X.

Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005).
[CrossRef]

Zhang, H. J.

Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005).
[CrossRef]

Appl. Phy. Lett. (1)

S. C. Chen, C. P. Grigoropoulos, H. K. Park, P. Kerstens, and A. C. Tam, "Photothermal displacement measurement of transient melting and surface deformation during pulsed laser heating," Appl. Phy. Lett. 73,2093-2095 (1998).
[CrossRef]

International Journal of Mechanical Sciences (1)

X. Huang and H. Shen, "Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators," International Journal of Mechanical Sciences 47,187-208 (2005).
[CrossRef]

J. Micromech. Microeng. (4)

L. J. Li and D. Uttamchandani, "Modified asymmetric microelectrothermal actuator: analysis and experimentation," J. Micromech. Microeng. 14, 1734-1741 (2004).
[CrossRef]

Y. L. He, H. J. Zhang, and D. X. Zhang, "Theoretical and experimental study of photo-thermal expansion using an atomic force microscope," J. Micromech. Microeng. 15,1637-1640 (2005).
[CrossRef]

J. K. Luo, A. J. Flewitt, S. M. Spearing, N. A. Fleck, and W. I. Milne, "Comparison of microtweezers based on three lateral thermal actuator configurations," J. Micromech. Microeng. 15,1294-1302 (2005).
[CrossRef]

L. Li and D. Uttamchandani, "Modified asymmetric micro-electrothermal actuator," J. Micromech. Microeng. 14, 1734-1741(2004).
[CrossRef]

Journal of Magnetism and Magnetic Materials E (1)

Y. W. Park and D. Y. Kim, "Development of a magnetostrictive microactuator," Journal of Magnetism and Magnetic Materials E 1765-1766,272-276 (2004).

Journal of Microelectromechanical Systems (2)

J. L. Yeh, C. Y. Hui, and N. C. Tien, "Electrostatic model for an asymmetric comb drive," Journal of Microelectromechanical Systems 9,126-130 (2000).
[CrossRef]

D. L. Devoe and A. P. Pisano, "Modeling and optimal design of piezoelectric cantilever microactuators," Journal of Microelectromechanical Systems 6,266-270(1997).
[CrossRef]

Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing (1)

C. N. Saikrishna, K. V. Ramaiah, and S. K. Bhaumik, "Effects of thermo-mechanical cycling on the strain response of Ni-Ti-Cu shape memory alloy wire actuator," Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 428,217-224 (2006).
[CrossRef]

Sensors Actuators A (1)

M. F. Pai, and N. C. Tien, "Low voltage electrothermal vibromotor for silicon optical bench applications," Sensors Actuators A 83, 237-243 (2000).
[CrossRef]

Sensors and Actuators (1)

J. T. Butler, V. M. Bright, and W. D. Cowan, "Average power control and positioning of polysilicon thermal actuators," Sensors and Actuators 72, 88-97 (1999).
[CrossRef]

Sensors and Actuators A (1)

T. Lalinský, M. Držík, J. Chlpík, M. Krná�?, Š. Haš�?ík, Ž. Mozolová, and I. Kosti�?, "Thermo-mechanical characterization of micromachined GaAs-based thermal converter using contactless optical methods," Sensors and Actuators A 123-124,99-105 (2005).
[CrossRef]

Other (1)

J. Kyokane, K. Tsujimoto, Y. Yanagisawa, T. Ueda, and M. Fukuma, "Actuator using electrostriction effect of fullerenol-doped polyurethane elastomer (PUE) films," IEICE Transactions on Electronics E 87C, 136-141 (2004).

Supplementary Material (4)

» Media 1: MOV (1699 KB)     
» Media 2: MOV (2413 KB)     
» Media 3: MOV (1740 KB)     
» Media 4: MOV (1950 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

The basic concept of OT expansion, (a) longitudinal expansion of OT arm, (b) lateral deflection of bi-direction OT microactuator

Fig. 2.
Fig. 2.

The schematic model of the slender arm, (a) one-dimensional coordinate system, (b) an arbitrary infinitesimal OT transmission element within the arm

Fig. 3.
Fig. 3.

Theoretical relationship between the OT expansion increment and the laser power

Fig. 4.
Fig. 4.

Simplified model of the bi-direction deflection of OT microactuator

Fig. 5.
Fig. 5.

Microscopic images captured from videos showing the different expansions when the power of irradiating laser is (a) 0 mW; (b) 6 mW (Media 1); (c) 10 mW (Media 2)

Fig. 6.
Fig. 6.

The experimental results showing the relationship between expansion increment and the laser power

Fig. 7.
Fig. 7.

SEM and microscopic images captured from videos showing the bi-directional deflection, (a) SEM image; (b) original state; (c) laser (3mW) irradiates the upper arm (Media 3); (d) irradiates the lower arm (Media 4)

Tables (1)

Tables Icon

Table 1. Parameters of the Laser and the OT Arm

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

Q ( x ) = { ρ A q 0 L 1 R x L 1 + R 0 0 x < L 1 R , L 1 + R < x L
KT ( x ) DW + Q ( x ) W · dx = KT ( x + dx ) DW + [ T ( x ) T 0 ] · ( 2 Wh w · dx + 2 Dh D · dx )
d 2 T ( x ) d x 2 2 ( h D + h W W ) KDW · Δ T ( x ) = Q ( x ) KD
Δ T ( x ) = C 1 · ch ( bx ) + C 2 · sh ( bx ) + 1 b x 0 x f ( ξ ) sh ( b ( x ξ ) ) d ξ
b 2 = 2 ( h D D + h W W ) KDW , f ( x ) = Q ( x ) KD
K . d T dx x = 0 h 1 Δ T ( 0 ) = 0 K . d T dx x = L + h 2 Δ T ( L ) = 0
Δ T ( x ) = { 2 ρ A q 0 sh ( bR ) b 2 KDC 3 [ Kbch ( bL 2 ) + h 2 sh ( bL 2 ) ] [ Kbch ( bx ) + h 1 sh ( bx ) ] 0 x 1 L 1 R 2 ρ A q 0 sh ( bR ) b 2 KDC 3 [ Kbch ( bL 2 ) + h 2 sh ( bL 2 ) ] [ Kbch ( bx ) + h 1 sh ( bx ) ] + ρ A q 0 1 ch ( b ( x + R L 1 ) ) KD b 2 , L 1 R x L 1 + R 2 ρ A q 0 sh ( bR ) b 2 KDC 3 [ Kbch ( bL 2 ) + h 2 sh ( bL 2 ) ] [ Kbch ( bx ) + h 1 sh ( bx ) ] ρ A q 0 2 sh ( bR ) sh ( b ( x L 1 ) ) KD b 2 , L 1 + R x L
C 3 = ( h 1 h 2 + K 2 b 2 ) sh ( bL ) + Kb ( h 1 + h 2 ) · ch ( bL )
Δ L = α 0 L Δ T ( x ) dx
Δ L = α 2 ρ A q 0 R KD b 2
2 α ρ A q 0 h 1 h 2 · sh ( bR ) · ( sh ( bL 1 ) + sh ( bL 2 ) ) + Kb · sh ( bR ) · ( h 2 ch ( bL 1 ) + h 1 ch ( bL 2 ) ) b 3 KD [ ( h 1 h 2 + K 2 b 2 ) sh ( bL ) + K ( h 1 + h 2 ) b · ch ( bL ) ]
d AO sin θ
tg θ = BC CO = [ ( L + Δ L ) L ] S
d AO cos θ × Δ L S
d Δ L × L S

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