Abstract

The strong stray light has huge interference on the detection of weak and small optical signals, and is difficult to suppress. In this paper, a miniaturized baffle with angled vanes was proposed and a rapid optimization model of strong light elimination was built, which has better suppression of the stray lights than the conventional vanes and can optimize the positions of the vanes efficiently and accurately. Furthermore, the light energy distribution model was built based on the light projection at a specific angle, and the light propagation models of the vanes and sidewalls were built based on the Lambert scattering, both of which act as the bias of a calculation method of stray light. Moreover, the Monte-Carlo method was employed to realize the Point Source Transmittance (PST) simulation, and the simulation result indicated that it was consistent with the calculation result based on our models, and the PST could be improved by 2-3 times at the small incident angles for the baffle designed by the new method. Meanwhile, the simulation result was verified by laboratory tests, and the new model with derived analytical expressions which can reduce the simulation time significantly.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [PubMed]
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    [PubMed]
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  6. X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.
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2017 (2)

2016 (1)

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

2014 (1)

2013 (1)

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

2012 (2)

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

2007 (1)

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

2005 (1)

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

2004 (1)

S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).

2003 (2)

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

2002 (1)

C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).

1996 (1)

S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

1995 (1)

C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).

1989 (1)

R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).

1977 (2)

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).

Ahmed, N.

Averbuch, Y.

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

Bao, J.

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Breault, R. P.

R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).

R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).

Buffington, A.

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

Cady, E.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Chen, X.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Chunnilall, C. J.

Devasenathipathy, S.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Fidler, S.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Fox, N.

Gibbs, D.

Harris, O. B.

He, L.

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Hick, P. P.

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

Hollinger, A.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Howlett, G.

Isbrucker, V.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Jackson, B. V.

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

Jensen, B.

Kasdin, N. J.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Kendrick, S.

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

Kovacs, R.

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

Kundu, K.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Laurin, D.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Liebe, C. C.

C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).

C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).

Liu, Z.

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Ma, H.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Macintosh, B.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Marchen, L.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Martin, S.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Meier, S. R.

S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).

Meinhart, C. D.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Mole, R.

Natan, M. J.

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Penn, S. G.

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Pompea, S. M.

S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

Reveles, J. R.

Santiago, J. G.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Savransky, D.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Schechner, Y. Y.

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

Shaklan, S.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Shang, N.

Stauder, J.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Sun, T.

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Takehara, K.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Taylor, R.

Theocharous, E.

Urtasun, R.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Vaglia, A. L.

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

van Breukelen, E.

E. van Breukelen, “Facet nano, a modular star tracker concept for highly miniaturized spacecraft,” in Proc. of the 60th International Astronautical Congress (2009), pp. 12–16.

Vanderbei, R. J.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Wang, G.

Wei, M.

Wereley, S. T.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Xing, F.

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

You, Z.

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Zhang, Z.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Zhao, J.

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

Appl. Opt. (1)

Chin. Opt. Lett. (1)

Curr. Opin. Chem. Biol. (1)

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Exp. Fluids (1)

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

IEEE Aerosp. Electron. Syst. Mag. (1)

C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).

IEEE Trans. Aerosp. Electron. Syst. (1)

C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

Opt. Express (1)

Proc. SPIE (8)

S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).

R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

Sensors (Basel) (2)

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

Other (6)

P. R. Lawson and J. A. Dooley, “Technology Plan for the Terrestrial Planet Finder Interferometer,” NASA Sti/recon Technical Report N 6 (2005).

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

E. van Breukelen, “Facet nano, a modular star tracker concept for highly miniaturized spacecraft,” in Proc. of the 60th International Astronautical Congress (2009), pp. 12–16.

R. D. Seals and M. B. Mcintosh, “Advanced baffles: knife-edged diffuse-absorptive and dual reflective baffles,” in San Diego (1993).

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

J. Haghshenas, “A theoretical method for vanes profile design in star sensor baffle,” in SPIE Asia Pacific Remote Sensing (2014), p. 92641R.

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Figures (11)

Fig. 1
Fig. 1

Cutaway view of the imaging system and the baffle with angled vanes.

Fig. 2
Fig. 2

The projection area in the light incident direction.

Fig. 3
Fig. 3

The energy ratios when the vane in different positions with 36° incident light.

Fig. 4
Fig. 4

The incident light energy distribution on different (a) sidewalls and (b) vanes.

Fig. 5
Fig. 5

The light propagation plane model between the visible area and (a) the vane, (b) the sidewall.

Fig. 6
Fig. 6

The PST curves of the baffle with 36° incident angle.

Fig. 7
Fig. 7

The simulation results of the energy entering into the lens with 36° incident angle.

Fig. 8
Fig. 8

The calculation results of two-time scattering energy (a) entering into the lens and (b) from different cavities at 36°.

Fig. 9
Fig. 9

PST simulation performance of different baffles.

Fig. 10
Fig. 10

(a) The cutaway view and (b) the installation of the baffle in the star tracker.

Fig. 11
Fig. 11

(a) The PST test of the baffle with angled vanes, (b) the normalized PST result.

Tables (1)

Tables Icon

Table 1 The design parameters of the baffle

Equations (14)

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E 1 = E in (1η) 2 s 1 e s 1 r p 1 .
E 2 = E in (1η) 2 ( v 2 e v 2 r+ s 2 e s 2 r) p 2 .
E 3 = E in (1η) 2 ( v 3 e v 3 r+ s 3 e s 3 r) p 3 .
E 4 = E in (1η) 2 ( s 1 e s 1 r_opp+ s 2 e s 2 r_opp+ s 3 e s 3 r_opp) p 3 .
E out = E 1 + E 2 + E 3 + E 4 .
x 2 (D/2) 2 + y 2 (Dcosφ/2) 2 =1.
x 2 ( D V /2) 2 + (y+ M V ) 2 ( D V cosφ/2) 2 =1.
Φ p =LdA θ=0 θ=φ sinθdθ= I N (1cosφ).
{ Φ 1 = A=0 A=l Φ p dA cosφ= A+mcos φ 1 [ (A+mcos φ 1 ) 2 + (msin φ 1 ) 2 ] 1 2 .
{ Φ 2 = A=0 A=l Φ p dA cosφ= A+mcos φ 1 +scosβ [ (A+mcos φ 1 +scosβ) 2 + (msin φ 1 ssinβ) 2 ] 1 2 .
v x r= Φ 1 Φ 2 Φ =( Φ 1 Φ 2 )/ A=0 A=l 2 I N dA .
{ Φ 3 = A= J s A=J Φ p dA cosφ= A+ l 1 cosα [ (A+ l 1 cosα) 2 + ( l 1 sinα) 2 ] 1 2 .
{ Φ 4 = A= J s A=J Φ p dA cosφ= A+ l 1 cosα+scosα [ (A+ l 1 cosα+scosα) 2 + ( l 1 sinα+ssinα) 2 ] 1 2 .
s x r= Φ 4 Φ 3 Φ = Φ 4 Φ 3 A= J s A=J 2 I N dA .

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