Zernike annular polynomials and atmospheric turbulence

Guang-ming Dai; Virendra N. Mahajan

doi:10.1364/JOSAA.24.000139

Zernike annular polynomials and atmospheric turbulence

Guang-ming Dai and Virendra N. Mahajan

Journal of the Optical Society of America A
Vol. 24,
Issue 1,
pp. 139-155
(2007)
•https://doi.org/10.1364/JOSAA.24.000139

Get Citation
Copy Citation Text
Guang-ming Dai and Virendra N. Mahajan, "Zernike annular polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 24, 139-155 (2007)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (334 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Active or adaptive optics (010.1080)
- Atmospheric turbulence (010.1330)
- Aberration compensation (220.1000)
- Aberrations (global) (220.1010)

About this Article
History
- Original Manuscript: June 7, 2006
- Manuscript Accepted: July 28, 2006

Not Accessible

Your account may give you access

Abstract

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.

Full Article | PDF Article

More Like This

Zernike annular polynomials for imaging systems with annular pupils

Virendra N. Mahajan
J. Opt. Soc. Am. 71(1) 75-85 (1981)

Imaging characteristics of Zernike and annular polynomial aberrations

Virendra N. Mahajan and José Antonio Díaz
Appl. Opt. 52(10) 2062-2074 (2013)

Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Loève functions

Guang-ming Dai
J. Opt. Soc. Am. A 12(10) 2182-2193 (1995)

References

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Figures (13)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Tables (7)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Equations (62)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Metrics

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

n	$n^{'}$	m	$g_{n^{'}} (n, m; ∊)$
0	0	0	1
1	1	1	$\frac{1}{\sqrt{1 + ∊^{2}}}$
2	0	0	$- \frac{∊^{2}}{1 - ∊^{2}}$
2	2	0	$\frac{1}{1 - ∊^{2}}$
2	2	2	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4}}}$
3	1	1	$- \frac{2 ∊^{4}}{(1 - ∊^{2}) \sqrt{1 + 5 ∊^{2} + 5 ∊^{4} + ∊^{6}}}$
3	3	1	$\frac{1 + ∊^{2}}{(1 - ∊^{2}) \sqrt{1 + 5 ∊^{2} + 5 ∊^{4} + ∊^{6}}}$
3	3	3	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6}}}$
4	0	0	$\frac{∊^{2} (1 + ∊^{2})}{{(1 - ∊^{2})}^{2}}$
4	2	0	$- \frac{3 ∊^{2}}{{(1 - ∊^{2})}^{2}}$
4	4	0	$\frac{1}{{(1 - ∊^{2})}^{2}}$
4	4	0	$- \frac{3 ∊^{6}}{1 - ∊^{6}} \sqrt{\frac{1 + ∊^{2} + ∊^{4}}{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}}$
4	4	2	$\frac{1}{1 - ∊^{2}} \sqrt{\frac{1 + ∊^{2} + ∊^{4}}{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}}$
4	4	4	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}}$
5	1	1	$\frac{∊^{4} (3 + 4 ∊^{2} + 3 ∊^{4})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 46 ∊^{4} + 46 ∊^{6} + 13 ∊^{8} + ∊^{10}}}$
5	3	1	$- \frac{4 ∊^{4} (3 + ∊^{2})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 46 ∊^{4} + 46 ∊^{6} + 13 ∊^{8} + ∊^{10}}}$
5	5	1	$\frac{1}{{(1 - ∊^{2})}^{2}} \sqrt{\frac{1 + 4 ∊^{2} + ∊^{4}}{1 + 9 ∊^{2} + 9 ∊^{4} + ∊^{6}}}$
5	3	3	$- \frac{4 ∊^{8}}{1 - ∊^{8}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 10 ∊^{8} + 4 ∊^{10} + ∊^{12}}}$
5	5	3	$\frac{1}{1 - ∊^{2}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 10 ∊^{8} + 4 ∊^{10} + ∊^{12}}}$
5	5	5	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10}}}$
6	0	0	$- \frac{∊^{2} (1 + 3 ∊^{2} + ∊^{4})}{{(1 - ∊^{2})}^{3}}$
6	2	0	$\frac{3 ∊^{2} (1 + 2 ∊^{2})}{{(1 - ∊^{2})}^{3}}$
6	4	0	$- \frac{5 ∊^{2}}{{(1 - ∊^{2})}^{3}}$
6	6	0	$\frac{1}{{(1 - ∊^{2})}^{3}}$
6	2	2	$\frac{3 ∊^{6} (2 + 3 ∊^{2} + 3 ∊^{4} + 2 ∊^{6})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 91 ∊^{4} + 339 ∊^{6} + 792 ∊^{8} + 1028 ∊^{10} + 792 ∊^{12} + 339 ∊^{14} + 91 ∊^{16} + 13 ∊^{18} + ∊^{20}}}$
6	4	2	$- \frac{5 ∊^{6} (6 + 3 ∊^{2} + ∊^{4})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 91 ∊^{4} + 339 ∊^{6} + 792 ∊^{8} + 1028 ∊^{10} + 792 ∊^{12} + 339 ∊^{14} + 91 ∊^{16} + 13 ∊^{18} + ∊^{20}}}$
6	6	2	$\frac{1}{{(1 - ∊^{2})}^{2}} \sqrt{\frac{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}{1 + 9 ∊^{2} + 45 ∊^{4} + 65 ∊^{6} + 45 ∊^{8} + 9 ∊^{10} + ∊^{12}}}$
6	4	4	$- \frac{5 ∊^{10}}{1 - ∊^{10}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 35 ∊^{8} + 20 ∊^{10} + 10 ∊^{12} + 4 ∊^{14} + ∊^{16}}}$
6	6	4	$\frac{1}{1 - ∊^{2}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 35 ∊^{8} + 20 ∊^{10} + 10 ∊^{12} + 4 ∊^{14} + ∊^{16}}}$
6	6	6	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10} + ∊^{12}}}$

n	$n^{″}$	$h_{n^{″}} (n; ∊)$
0	0	1
1	1	ϵ
2	0	$- (1 - ∊^{2})$
2	2	$∊^{2}$
3	1	$- 2 ∊ (1 - ∊^{2})$
3	3	$∊^{3}$
4	0	$(1 - ∊^{2}) (1 - 2 ∊^{2})$
4	2	$- 3 ∊^{2} (1 - ∊^{2})$
4	4	$∊^{4}$
5	1	$∊ (1 - ∊^{2}) (3 - 5 ∊^{2})$
5	3	$- 4 ∊^{3} (1 - ∊^{2})$
5	5	$∊^{5}$
6	0	$- (1 - ∊^{2}) (1 - 5 ∊^{2} + 5 ∊^{4})$
6	2	$3 ∊^{2} (1 - ∊^{2}) (2 - 3 ∊^{2})$
6	4	$- 5 ∊^{4} (1 - ∊^{2})$
6	6	$∊^{6}$

n	m	$T_{n}^{m} (κ; ∊)$
0	0	$\frac{J_{1} (2 π κ) - ∊ J_{1} (2 π ∊ κ)}{κ}$
1	1	$- \frac{i [J_{2} (2 π κ) - ∊^{2} J_{2} (2 π ∊ κ)]}{κ \sqrt{1 + ∊^{2}}}$
2	0	$- \frac{J_{3} (2 π κ) - ∊^{3} J_{3} (2 π ∊ κ) + ∊ [∊ J_{1} (2 π κ) - J_{1} (2 π ∊ κ)]}{κ (1 - ∊^{2})}$
2	2	$- \frac{J_{3} (2 π κ) - ∊^{3} J_{3} (2 π ∊ κ) - ∊ (1 - ∊^{2}) J_{1} (2 π ∊ κ)}{κ \sqrt{1 + ∊^{2} + ∊^{4}}}$
3	1	$- \frac{i {(1 + ∊^{2}) [J_{4} (2 π κ) - ∊^{4} J_{4} (2 π ∊ κ)] + 2 ∊^{2} [∊^{2} J_{2} (2 π κ) - J_{2} (2 π ∊ κ)]}}{κ (1 - ∊^{2}) \sqrt{1 + 5 ∊^{2} + 5 ∊^{4} + ∊^{6}}}$
3	3	$\frac{i {[J_{4} (2 π κ) - ∊^{4} J_{4} (2 π ∊ κ)] - 2 ∊^{2} (1 - ∊^{2}) J_{2} (2 π ∊ κ)}}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6}}}$
4	0	$\frac{J_{5} (2 π κ) - ∊^{5} J_{5} (2 π ∊ κ) + 3 ∊^{2} [J_{3} (2 π κ) - ∊ J_{3} (2 π ∊ κ)] + ∊ (1 + ∊^{2}) [∊ J_{1} (2 π κ) - J_{1} (2 π ∊ κ)]}{κ {(1 - ∊^{2})}^{2}}$
4	2	$\frac{J_{5} (2 π κ) - ∊^{5} J_{5} (2 π ∊ κ) - 3 ∊^{3} [J_{3} (2 π κ) - ∊^{2} J_{3} (2 π ∊ κ)] - ∊ (1 - 3 ∊^{2} + 2 ∊^{4}) J_{1} (2 π ∊ κ)}{κ (1 - ∊^{2})} \sqrt{\frac{1 + ∊^{2} + ∊^{4}}{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}}$
4	4	$\frac{J_{5} (2 π κ) - ∊^{5} J_{5} (2 π ∊ κ) - 3 ∊^{3} (1 - ∊^{2}) J_{3} (2 π ∊ κ) - ∊ (1 - 3 ∊^{2} + 2 ∊^{4}) J_{1} (2 π ∊ κ)}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}}$
5	1	$- \frac{i {(1 + 4 ∊^{2} + ∊^{4}) [J_{6} (2 π κ) - ∊^{6} J_{6} (2 π ∊ κ)] + 4 ∊^{4} [(3 + ∊^{2}) J_{4} (2 π κ) - (1 + 3 ∊^{2}) J_{4} (2 π ∊ κ)] + ∊^{2} (3 + 4 ∊^{2} + 3 ∊^{4}) [∊^{2} J_{2} (2 π κ) - J_{2} (2 π ∊ κ)]}}{κ {(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 46 ∊^{4} + 46 ∊^{6} + 13 ∊^{8} + ∊^{10}}}$
5	3	$- \frac{i {(1 + ∊^{2} + ∊^{4} + ∊^{6}) [J_{6} (2 π κ) - ∊^{6} J_{6} (2 π ∊ κ)] + 4 ∊^{4} [∊^{4} J_{4} (2 π κ) - J_{4} (2 π ∊ κ)] - ∊^{2} (3 - 5 ∊^{2} + 5 ∊^{8} - 3 ∊^{10}) J_{2} (2 π ∊ κ)} \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6}}}{κ (1 - ∊^{8}) \sqrt{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 10 ∊^{8} + 4 ∊^{10} + ∊^{12}}}$
5	5	$- \frac{i [J_{6} (2 π κ) - ∊^{6} J_{6} (2 π ∊ κ) - 4 ∊^{4} (1 - ∊^{2}) J_{4} (2 π ∊ κ) - ∊^{2} (3 - 8 ∊^{2} + 5 ∊^{4}) J_{2} (2 π ∊ κ)]}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10}}}$
6	0	$- \frac{J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ) + 5 ∊^{2} [J_{5} (2 π κ) - ∊^{3} J_{5} (2 π ∊ κ)] + 3 ∊^{2} [(1 + 2 ∊^{2}) J_{3} (2 π κ) - ∊ (2 + ∊^{2}) J_{3} (2 π ∊ κ)] + ∊ (1 + 3 ∊^{2} + ∊^{4}) [∊ J_{1} (2 π κ) - J_{1} (2 π ∊ κ)]}{κ {(1 - ∊^{2})}^{3}}$
6	2	$- \frac{(1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}) [J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ)] + 5 ∊^{5} [∊ (6 + 3 ∊^{2} + ∊^{4}) J_{5} (2 π κ) - (1 + 3 ∊^{2} + 6 ∊^{4}) J_{5} (2 π ∊ κ)] + (6 ∊^{3} + 9 ∊^{5} + 9 ∊^{7} + 6 ∊^{9}) [∊^{3} J_{3} (2 π κ) - J_{3} (2 π ∊ κ)] - {(1 - ∊^{2})}^{5} (∊ + 3 ∊^{3} + ∊^{5}) J_{1} (2 π ∊ κ)}{κ {(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 91 ∊^{4} + 339 ∊^{6} + 792 ∊^{8} + 1028 ∊^{10} + 792 ∊^{12} + 339 ∊^{14} + 91 ∊^{16} + 13 ∊^{18} + ∊^{20}}}$
6	4	$- \frac{{(1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}) [J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ)] + 5 ∊^{5} [∊^{5} J_{5} (2 π κ) - J_{5} (2 π ∊ κ)] - 3 ∊^{3} (2 - 3 ∊^{2} + 3 ∊^{10} - 2 ∊^{12}) J_{3} (2 π ∊ κ) - ∊ {(1 - ∊^{2})}^{3} (1 - 2 ∊^{2} - 4 ∊^{4} - 5 ∊^{6} - 5 ∊^{8}) J_{1} (2 π ∊ κ)} \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}}{κ (1 - ∊^{10}) \sqrt{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 35 ∊^{8} + 20 ∊^{10} + 10 ∊^{12} + 4 ∊^{14} + ∊^{16}}}$
6	6	$- \frac{J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ) - 5 ∊^{5} (1 - ∊^{2}) J_{5} (2 π ∊ κ) - 3 ∊^{3} (2 - 5 ∊^{2} + 3 ∊^{4}) J_{3} (2 π ∊ κ) - ∊ (1 - 6 ∊^{2} + 10 ∊^{4} - 5 ∊^{6}) J_{1} (2 π ∊ κ)}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10} + ∊^{12}}}$

ϵ	0	0.05	0.1	0.15	0.2	0.25	0.35	0.5	0.75
$⟨ a_{2}^{2} ⟩, ⟨ a_{3}^{2} ⟩$	0.44888	0.45000	0.45336	0.45893	0.46670	0.47660	0.50260	0.55610	0.67917
$⟨ a_{4}^{2} ⟩$	0.02322	0.02302	0.02245	0.02153	0.02033	0.01891	0.01560	0.01028	0.00292
$⟨ a_{5}^{2} ⟩, ⟨ a_{6}^{2} ⟩$	0.02322	0.02328	0.02345	0.02375	0.02418	0.02475	0.02635	0.02997	0.03888
$⟨ a_{7}^{2} ⟩, ⟨ a_{8}^{2} ⟩$	0.00619	0.00621	0.00624	0.00628	0.00630	0.00626	0.00593	0.00468	0.00162
$⟨ a_{9}^{2} ⟩, ⟨ a_{10}^{2} ⟩$	0.00619	0.00621	0.00625	0.00633	0.00645	0.00660	0.00705	0.00815	0.01132
$⟨ a_{11}^{2} ⟩$	0.00245	0.00241	0.00227	0.00207	0.00183	0.00158	0.00109	0.00052	0.00007
$⟨ a_{12}^{2} ⟩, ⟨ a_{13}^{2} ⟩$	0.00245	0.00246	0.00248	0.00251	0.00255	0.00260	0.00266	0.00247	0.00108
$⟨ a_{14}^{2} ⟩, ⟨ a_{15}^{2} ⟩$	0.00245	0.00246	0.00248	0.00251	0.00256	0.00262	0.00280	0.00326	0.00480

ϵ	0	0.05	0.1	0.15	0.2	0.25	0.35	0.5	0.75
$⟨ a_{2} a_{8} ⟩, ⟨ a_{3} a_{7} ⟩$	$- 0.01416$	$- 0.01420$	$- 0.01430$	$- 0.01444$	$- 0.01460$	$- 0.01474$	$- 0.01481$	$- 0.01392$	$- 0.00885$
$⟨ a_{4} a_{11} ⟩$	$- 0.00388$	$- 0.00381$	$- 0.00363$	$- 0.00334$	$- 0.00300$	$- 0.00262$	$- 0.00186$	$- 0.00091$	$- 0.00012$
$⟨ a_{4} a_{22} ⟩$	0.00032	0.00030	0.00024	0.00017	0.00009	0.00002	$- 0.00008$	$- 0.00011$	$- 0.00003$
$⟨ a_{5} a_{13} ⟩, ⟨ a_{6} a_{12} ⟩$	$- 0.00388$	$- 0.00389$	$- 0.00392$	$- 0.00397$	$- 0.00404$	$- 0.00412$	$- 0.00433$	$- 0.00451$	$- 0.00341$
$⟨ a_{7} a_{17} ⟩, ⟨ a_{8} a_{16} ⟩$	$- 0.00156$	$- 0.00158$	$- 0.00164$	$- 0.00172$	$- 0.00178$	$- 0.00173$	$- 0.00095$	0.00365	0.01180
$⟨ a_{11} a_{22} ⟩$	$- 0.00076$	$- 0.00073$	$- 0.00067$	$- 0.00058$	$- 0.00048$	$- 0.00038$	$- 0.00022$	$- 0.00007$	0.00000
$⟨ a_{9} a_{19} ⟩, ⟨ a_{10} a_{18} ⟩$	$- 0.00156$	$- 0.00156$	$- 0.00157$	$- 0.00159$	$- 0.00162$	$- 0.00166$	$- 0.00176$	$- 0.00196$	$- 0.00177$
$⟨ a_{14} a_{26} ⟩, ⟨ a_{15} a_{25} ⟩$	$- 0.00076$	$- 0.00076$	$- 0.00077$	$- 0.00078$	$- 0.00079$	$- 0.00081$	$- 0.00086$	$- 0.00099$	$- 0.00104$

ϵ	0	0.25	0.5	0.75
$Δ_{1}$	1.0324	1.0870	1.2461	1.5010
$Δ_{2}$	0.5835	0.6104	0.6900	0.8218
$Δ_{3}$	0.1347	0.1338	0.1339	0.1426
$Δ_{4}$	0.1115	0.1149	0.1236	0.1397
$Δ_{5}$	0.0882	0.0902	0.0937	0.1008
$Δ_{6}$	0.0650	0.0654	0.0637	0.0620
$Δ_{7}$	0.0588	0.0591	0.0590	0.0603
$Δ_{8}$	0.0526	0.0529	0.0543	0.0587
$Δ_{9}$	0.0464	0.0463	0.0462	0.0474
$Δ_{10}$	0.0402	0.0397	0.0380	0.0361
$Δ_{11}$	0.0378	0.0381	0.0375	0.0360
$Δ_{12}$	0.0353	0.0355	0.0350	0.0349
$Δ_{13}$	0.0329	0.0329	0.0325	0.0338
$Δ_{14}$	0.0304	0.0303	0.0293	0.0290
$Δ_{15}$	0.0280	0.0277	0.0260	0.0242
$Δ_{16}$	0.0268	0.0264	0.0244	0.0218
$Δ_{17}$	0.0256	0.0251	0.0229	0.0193
$Δ_{18}$	0.0244	0.0239	0.0215	0.0186
$Δ_{19}$	0.0232	0.0226	0.0201	0.0178
$Δ_{20}$	0.0220	0.0213	0.0185	0.0153
$Δ_{21}$	0.0208	0.0200	0.0169	0.0129
$Δ_{22}$	0.0202	0.0197	0.0168	0.0129
$Δ_{23}$	0.0195	0.0190	0.0159	0.0115
$Δ_{24}$	0.0189	0.0183	0.0151	0.0101
$Δ_{25}$	0.0182	0.0176	0.0142	0.0095
$Δ_{26}$	0.0176	0.0169	0.0134	0.0089
$Δ_{27}$	0.0169	0.0162	0.0125	0.0075
$Δ_{28}$	0.0162	0.0155	0.0117	0.0061

Mode	Shape Function
$d_{2}, d_{3}$	$8 ν^{2}$
$d_{4}$	$48 ν^{2} {(1 - ν)}^{2}$
$d_{5}, d_{6}$	$24 ν^{2} {(1 - ν)}^{2}$
$d_{7}, d_{8}$	$16 ν^{2} (39 ν^{4} - 84 ν^{3} + 66 ν^{2} - 24 ν + 4)$
$d_{9}, d_{10}$	$16 ν^{2} [ν^{4} + 3 {(1 - ν)}^{4}]$
$d_{11}$	$240 ν^{2} {(1 - ν)}^{2} (34 ν^{4} - 56 ν^{3} + 32 ν^{2} - 8 ν + 1)$
$d_{12}, d_{13}$	$(40 / 3) ν^{2} {(1 - ν)}^{2} (298 ν^{4} - 488 ν^{3} + 280 ν^{2} - 72 ν + 9)$
$d_{14}, d_{15}$	$80 ν^{2} {(1 - ν)}^{2} [2 ν^{4} + {(1 - ν)}^{4}]$
$d_{16}, d_{17}$	$8 ν^{2} (7265 ν^{8} - 31,220 ν^{7} + 57,750 ν^{6} - 60,020 ν^{5} + 38,087 ν^{4} - 14,892 ν^{3} + 3438 ν^{2} - 432 ν + 27)$
$d_{18}, d_{19}$	$2 ν^{2} (10,135 ν^{8} - 47,530 ν^{7} + 98,745 ν^{6} - 116,380 ν^{5} + 83,053 ν^{4} - 35,994 ν^{3} + 9231 ν^{2} - 1344 ν + 96)$
$d_{20}, d_{21}$	$8 ν^{2} [3 ν^{8} + 100 ν^{4} {(1 - ν)}^{4} + 15 {(1 - ν)}^{8}]$
$d_{22}$	$(56 / 9) ν^{2} {(1 - ν)}^{2} (144,925 ν^{8} - 500,500 ν^{7} + 747,240 ν^{6} - 636,940 ν^{5} + 338,245 ν^{4} - 111,960 ν^{3} + 21,690 ν^{2} - 2160 ν + 108)$
$d_{23}, d_{24}$	$(7 / 3) ν^{2} {(1 - ν)}^{2} (208,755 ν^{8} - 725,580 ν^{7} + 1,082,970 ν^{6} - 914,220 ν^{5} + 476,435 ν^{4} - 153,760 ν^{3} + 29,000 ν^{2} - 2880 ν + 144)$
$d_{25}, d_{26}$	$(56 / 5) ν^{2} {(1 - ν)}^{2} (9702 ν^{8} - 33,552 ν^{7} + 55,240 ν^{6} - 56,000 ν^{5} + 36,070 ν^{4} - 14,152 ν^{3} + 3192 ν^{2} - 400 ν + 25)$
$d_{27}, d_{28}$	$56 ν^{2} {(1 - ν)}^{2} [9 ν^{8} + 50 ν^{4} {(1 - ν)}^{4} + 3 {(1 - ν)}^{8}]$
$c_{2, 8}, c_{3, 7}$	$8 \sqrt{2} ν^{2} (6 ν^{2} - 6 ν + 1)$
$c_{2, 16}, c_{3, 17}$	$8 \sqrt{3} ν^{2} (45 ν^{4} - 90 ν^{3} + 61 ν^{2} - 16 ν + 1)$
$c_{5, 13}, c_{6, 12}$	$8 \sqrt{15} ν^{2} {(1 - ν)}^{2} (10 ν^{2} - 8 ν + 1)$
$c_{5, 23}, c_{6, 24}$	$8 \sqrt{21} ν^{2} {(1 - ν)}^{2} (95 ν^{4} - 160 ν^{3} + 90 ν^{2} - 20 ν + 1)$
$c_{9, 19}, c_{10, 18}$	$8 \sqrt{6} ν^{2} (60 ν^{6} - 230 ν^{5} + 399 ν^{4} - 372 ν^{3} + 183 ν^{2} - 42 ν + 3)$
$c_{11, 22}$	$8 \sqrt{35} ν^{2} {(1 - ν)}^{2} (1799 ν^{6} - 4564 ν^{5} + 4690 ν^{4} - 2560 ν^{3} + 785 ν^{2} - 120 ν + 3)$
$c_{12, 24}, c_{13, 23}$	$(4 / 3) \sqrt{35} ν^{2} {(1 - ν)}^{2} (5607 ν^{6} - 14,292 ν^{5} + 14,660 ν^{4} - 7900 ν^{3} + 2375 ν^{2} - 360 ν + 18)$
$c_{14, 26}, c_{15, 25}$	$16 \sqrt{35} ν^{2} {(1 - ν)}^{2} (56 ν^{6} - 136 ν^{5} + 185 ν^{4} - 160 ν^{3} + 75 ν^{2} - 16 ν + 1)$

Abstract

References

Cited By

Figures (13)

Tables (7)

Equations (62)

Metrics

Login or Create Account