Abstract

Proposed designs for the next generation of large optical telescopes favor a tripod or quadrupod secondary support, and a primary supported from the back, but it is not yet clear whether the elevation axis should be in front of the primary or behind it. A study is described of the effect of elevation-axis location on key performance parameters (fundamental frequency, blockage, and wind-induced secondary decenter) for a 30-m Cassegrain telescope with a mount configuration that is typical of the new designs. For a fast (e.g., f/1) primary, the best location for the elevation axis is behind the primary. The penalty for moving the elevation axis in front of the primary is roughly a 40% decrease in fundamental frequency and a corresponding reduction in the control bandwidth for pointing and optical alignment.

© 2004 Optical Society of America

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References

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  1. J. E. Nelson, T. S. Mast, S. M. Faber, eds., “The design of the Keck Observatory and Telescope,” Keck Observatory Rep. 90 (Keck Observatory, Kamuela, Hawaii, 1985).
  2. M. Quattri, F. Dimichino, G. Marchiori, E. Piccinini, “VLT 8m unit telescope main structure: design solutions and performance calculations,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 986–996 (1994).
    [CrossRef]
  3. K. Miyawaki, N. Itoh, R. Sugiyama, M. Sawa, “Mechanical structure for the Subaru Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 754–761 (1994).
    [CrossRef]
  4. S. M. Gunnels, “Design of the Magellan Project 6.5 meter Telescope: telescope structure and mechanical systems,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 414–427 (1994).
    [CrossRef]
  5. K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
    [CrossRef]
  6. S. Strom, L. M. Stepp, M. Mountain, B. Gregory, “Giant Segmented Mirror Telescope: a point design based on science drivers,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2003).
    [CrossRef]
  7. T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
    [CrossRef]
  8. J. E. Nelson, T. S. Mast, eds., “Conceptual design for a thirty-meter telescope,” CELT Rep. 34 (University of California, Santa Cruz, Santa Cruz, Calif., 2002).
  9. J. Nelson, T. Mast, G. Chanan, “Aberration correction in a telescope with a segmented primary,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 241–257 (1989).
    [CrossRef]
  10. D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 2000), p. 132.
  11. P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), p. 344.
  12. W. B. Davison, N. J. Woolf, J. R. P. Angel, “Design and analysis of 20m track mounted and 30m telescopes,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 533–540 (2003).
    [CrossRef]
  13. R. H. Brown, A. C. B. Lovell, The Exploration of Space by Radio (Chapman & Hall, London, 1957), p. 194.
  14. R. J. Roark, W. C. Young, Formulas for Stress and Strain (McGraw-Hill, New York, 1975), pp. 96, 534.
  15. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), p. 57.
  16. R. Richards, Solid Mechanics (CRC Press, Boca Raton, Fla., 2001), p. 150.

Andersen, T.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Angel, J. R. P.

W. B. Davison, N. J. Woolf, J. R. P. Angel, “Design and analysis of 20m track mounted and 30m telescopes,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 533–540 (2003).
[CrossRef]

Ardeberg, A. L.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Beckers, J.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Brown, R. H.

R. H. Brown, A. C. B. Lovell, The Exploration of Space by Radio (Chapman & Hall, London, 1957), p. 194.

Chanan, G.

J. Nelson, T. Mast, G. Chanan, “Aberration correction in a telescope with a segmented primary,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 241–257 (1989).
[CrossRef]

Cohen, I. M.

P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), p. 344.

Davison, W. B.

W. B. Davison, N. J. Woolf, J. R. P. Angel, “Design and analysis of 20m track mounted and 30m telescopes,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 533–540 (2003).
[CrossRef]

Dimichino, F.

M. Quattri, F. Dimichino, G. Marchiori, E. Piccinini, “VLT 8m unit telescope main structure: design solutions and performance calculations,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 986–996 (1994).
[CrossRef]

Gillett, P.

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

Goncharov, A.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Gregory, B.

S. Strom, L. M. Stepp, M. Mountain, B. Gregory, “Giant Segmented Mirror Telescope: a point design based on science drivers,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2003).
[CrossRef]

Gunnels, S. M.

S. M. Gunnels, “Design of the Magellan Project 6.5 meter Telescope: telescope structure and mechanical systems,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 414–427 (1994).
[CrossRef]

Hatton, P.

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

Itoh, N.

K. Miyawaki, N. Itoh, R. Sugiyama, M. Sawa, “Mechanical structure for the Subaru Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 754–761 (1994).
[CrossRef]

Kundu, P. K.

P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), p. 344.

Lovell, A. C. B.

R. H. Brown, A. C. B. Lovell, The Exploration of Space by Radio (Chapman & Hall, London, 1957), p. 194.

Marchiori, G.

M. Quattri, F. Dimichino, G. Marchiori, E. Piccinini, “VLT 8m unit telescope main structure: design solutions and performance calculations,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 986–996 (1994).
[CrossRef]

Mast, T.

J. Nelson, T. Mast, G. Chanan, “Aberration correction in a telescope with a segmented primary,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 241–257 (1989).
[CrossRef]

Miyawaki, K.

K. Miyawaki, N. Itoh, R. Sugiyama, M. Sawa, “Mechanical structure for the Subaru Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 754–761 (1994).
[CrossRef]

Mountain, M.

S. Strom, L. M. Stepp, M. Mountain, B. Gregory, “Giant Segmented Mirror Telescope: a point design based on science drivers,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2003).
[CrossRef]

Nelson, J.

J. Nelson, T. Mast, G. Chanan, “Aberration correction in a telescope with a segmented primary,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 241–257 (1989).
[CrossRef]

Owner-Petersen, M.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Pentland, G.

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

Piccinini, E.

M. Quattri, F. Dimichino, G. Marchiori, E. Piccinini, “VLT 8m unit telescope main structure: design solutions and performance calculations,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 986–996 (1994).
[CrossRef]

Quattri, M.

M. Quattri, F. Dimichino, G. Marchiori, E. Piccinini, “VLT 8m unit telescope main structure: design solutions and performance calculations,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 986–996 (1994).
[CrossRef]

Raybould, K.

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

Richards, R.

R. Richards, Solid Mechanics (CRC Press, Boca Raton, Fla., 2001), p. 150.

Riewaldt, H.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Roark, R. J.

R. J. Roark, W. C. Young, Formulas for Stress and Strain (McGraw-Hill, New York, 1975), pp. 96, 534.

Sawa, M.

K. Miyawaki, N. Itoh, R. Sugiyama, M. Sawa, “Mechanical structure for the Subaru Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 754–761 (1994).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 2000), p. 132.

Sheehan, M.

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

Snel, R.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Stepp, L. M.

S. Strom, L. M. Stepp, M. Mountain, B. Gregory, “Giant Segmented Mirror Telescope: a point design based on science drivers,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2003).
[CrossRef]

Strom, S.

S. Strom, L. M. Stepp, M. Mountain, B. Gregory, “Giant Segmented Mirror Telescope: a point design based on science drivers,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2003).
[CrossRef]

Sugiyama, R.

K. Miyawaki, N. Itoh, R. Sugiyama, M. Sawa, “Mechanical structure for the Subaru Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 754–761 (1994).
[CrossRef]

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), p. 57.

Walker, D.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

Warner, M.

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), p. 57.

Woolf, N. J.

W. B. Davison, N. J. Woolf, J. R. P. Angel, “Design and analysis of 20m track mounted and 30m telescopes,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 533–540 (2003).
[CrossRef]

Young, W. C.

R. J. Roark, W. C. Young, Formulas for Stress and Strain (McGraw-Hill, New York, 1975), pp. 96, 534.

Other (16)

J. E. Nelson, T. S. Mast, S. M. Faber, eds., “The design of the Keck Observatory and Telescope,” Keck Observatory Rep. 90 (Keck Observatory, Kamuela, Hawaii, 1985).

M. Quattri, F. Dimichino, G. Marchiori, E. Piccinini, “VLT 8m unit telescope main structure: design solutions and performance calculations,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 986–996 (1994).
[CrossRef]

K. Miyawaki, N. Itoh, R. Sugiyama, M. Sawa, “Mechanical structure for the Subaru Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 754–761 (1994).
[CrossRef]

S. M. Gunnels, “Design of the Magellan Project 6.5 meter Telescope: telescope structure and mechanical systems,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 414–427 (1994).
[CrossRef]

K. Raybould, P. Gillett, P. Hatton, G. Pentland, M. Sheehan, M. Warner, “Gemini Telescope structure design,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 376–393 (1994).
[CrossRef]

S. Strom, L. M. Stepp, M. Mountain, B. Gregory, “Giant Segmented Mirror Telescope: a point design based on science drivers,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2003).
[CrossRef]

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2003).
[CrossRef]

J. E. Nelson, T. S. Mast, eds., “Conceptual design for a thirty-meter telescope,” CELT Rep. 34 (University of California, Santa Cruz, Santa Cruz, Calif., 2002).

J. Nelson, T. Mast, G. Chanan, “Aberration correction in a telescope with a segmented primary,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 241–257 (1989).
[CrossRef]

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 2000), p. 132.

P. K. Kundu, I. M. Cohen, Fluid Mechanics (Academic, San Diego, Calif., 2002), p. 344.

W. B. Davison, N. J. Woolf, J. R. P. Angel, “Design and analysis of 20m track mounted and 30m telescopes,” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 533–540 (2003).
[CrossRef]

R. H. Brown, A. C. B. Lovell, The Exploration of Space by Radio (Chapman & Hall, London, 1957), p. 194.

R. J. Roark, W. C. Young, Formulas for Stress and Strain (McGraw-Hill, New York, 1975), pp. 96, 534.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), p. 57.

R. Richards, Solid Mechanics (CRC Press, Boca Raton, Fla., 2001), p. 150.

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Figures (11)

Fig. 1
Fig. 1

Telescope with a tube and a fork mount. Filled circles, centers of mass of the telescope components.

Fig. 2
Fig. 2

Lumped-element, coupled-oscillator model of the telescope in Fig. 1: x 1, x 2, and x 3, lateral displacements of the forks and of the bottom and the top tubes respectively; x 4, rotation of the entire tube about the elevation axis; k, spring constant (e.g., in units of N m-1); τ, torque stiffness (e.g., in units of N m rad-1); m, mass.

Fig. 3
Fig. 3

(a) Fundamental frequency, (b) blockage, (c) mass contributions, and (d) secondary decenter in a 5-m s-1 wind; (e) tube stiffness for end loading and (f) tube wall thickness for a 30-m telescope as in Fig. 1; h, distance of the elevation axis in front of the primary. In all the figures except (c) the curves represent different primary mirror focal ratios: f/1 (solid curves), f/1.5 (dashed curves), and f/2 (dotted curves). The mass contributions in (c) are m f (thinner solid curve), m t (dashed curve), m b (dotted curve), and total mass (thicker solid curve), all for an f/1.5 primary. In (e) the upper curve near h = 0 is for the bottom tube, and in (f) the upper curve is for the top tube. Blockages in (b) are for the spider (roughly horizontal curves) and the secondary mirror (curves running from top left to bottom right). Telescope parameters are given in Table 1.

Fig. 4
Fig. 4

Telescope with a quadrupod and a c-ring mount.

Fig. 5
Fig. 5

Lumped-element, coupled-oscillator model of the telescope in Fig. 5: x, angular displacement; τ, torque stiffness (e.g., in units of N m rad-1); J, moment of inertia (e.g., in units of kg m2).

Fig. 6
Fig. 6

(a) Fundamental frequency, (b) blockage, (c) mass contributions, and (d) secondary decenter in a 5-m s-1 wind; (e) quadrupod stiffness for uniform loading and (f) quadrupod tube diameter for a 30-m telescope as in Fig. 5, with c-rings of diameter D 1; h, distance of the elevation axis in front of the primary. In all the figures except (c) the curves represent different primary mirror focal ratios: f/1 (solid curves), f/1.5 (dashed curves), and f/2 (dotted curves). The mass contributions in (c) are m w + m r + m x (thinner solid curve), m q (dashed curve), m c + m p (dotted curve), and total mass (thicker solid curve), all for an f/1.5 primary. Blockages in (b) are for the secondary mirror (lower curves) and the quadrupod legs (upper curves). Telescope parameters are given in Table 1.

Fig. 7
Fig. 7

Same as Fig. 4(a) but with R = D 1/4 and m w = 245 × 103 kg.

Fig. 8
Fig. 8

Space-frame beam made from layers of tetrahedrons and sheets of hexagons. Filled triangles, fixed points for FEA.

Fig. 9
Fig. 9

Stiffness of the space-frame beam in Fig. 8 from FEA, divided by the stiffness predicted by Eq. (A1). The beam dimensions are W = 5 m, H = 4.6 m, and d = 8.2 m, with 1-m tetrahedrons; frame members are steel tubes 10 mm in diameter × 1 mm wall (cross), 20 mm × 2 mm (circle), and 40 mm × 5 mm (asterisk).

Fig. 10
Fig. 10

Space-frame tube.

Fig. 11
Fig. 11

Quadrupod.

Tables (1)

Tables Icon

Table 1 Telescope Parameters for Figs. 3 and 6 (below)

Equations (62)

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

msd-h+mtd-h2=mbh2+mph+mch+t2,
mt=πD1atd-hηtρ
mb=πD1abhηbρ
mc=πD1/22tηcρ
mskt+3mt8kt=3mb8kb+mpkb+mckb,
kt=ηt33πED13at8d-h3,
kb=ηb33πED13ab8h3,
k1=2kd+1kf-1,
kf=ηf39ED1464L3
L=h+t2+D1/221/2.
mf=3D12Lηfρ8.
k2kbmp+mc+mbmp+mc+3mb/8,
m2=mb+mp+mc;
k3ktms+mtms+3mt/8,
m3=mt+ms.
0=m1x¨1+k1x1+k2hx4-x1+x2+k3x3-x1-d-hx4,0=m2x¨2+k2hx4-x1+x2-τ4x4/h,0=m3x¨3+k3x3-x1-d-hx4+τ4x4/d-h,0=Jx¨4+τ4x4+k3x3-x1-d-hx4d-h-k2hx4-x1+x2h,
J=msd-h2+mtd-h23+mbh23+mph2+mch+t22
0=k1-k2-k3-ω2m1k2k3k2h-k3d-h-ω2m2k2-ω2m20k2h-τ4/d2-ω2m30k3-ω2m3-k3d-h+τ4/d-h0-k2hk3d-hτ4-k2h2-k3d-h2-Jω2q=Aq.
ζ=D2D12+8wπD12D1-D2,
D2=d+eD1f+2αd,
At12atd-h+πD1at.
AsD22/β,
Ax2wD1-D2D2/β.
δ12 ρairv238 At+As+Ax1kt,
σ=mleggl8Z,
mleg=πaq2ηqlρ4
l=D124+d21/2
Z=π32 aq3ηq2-ηq
aq>Sgρl2Y2-ηq.
Fcr=π2EIl2,
I=π64 aq4ηq2-ηq
aq2>16Sgρl3π2E2-ηq,
msd-h+mqd2-h-mph-mch+t2-mxh+t2 =mr4R3π+mwR,
mq=4mleg=πaq2ηqlρ
mx3R2ρηx-h-t
mr=34 πR3ρηr
τ1=2kd+3R4EηrD12-h-t3-112D12-h-t2.
J13R2R3D12-3R23+πR22D12-4R3π2-2Rh+tD12-h+t22ρηr+mwD12-R2.
kc=ηc316πEt331-ν5+νD1/22,
τ214 kcD122=ηc38πEt331-ν5+ν.
τ2ηc32Et3.
J2=mp+mcD142.
kq38πED12aq2ηq2d3
τ3=83 kqd22πED12aq2ηq4d.
J3=mqd23+msd2.
0=J1x¨1+τ1x1+τ2x2-x1,0=J2x¨2+τ2x2-x1-τ3x3-x2,0=J3x¨3+τ3x3-x2,
0=τ1+τ2-ω2J1-τ20-τ2τ2+τ3-ω2J2-τ30-τ3k3-ω2J3q=Aq.
ζ=D2D12+8aqπD12D1-D2,
δ12 ρairv238 Aq+As1kq,
Aq=4aql
kq3ED12m16ρd4,
Δq64aqρd53ED12m.
kt=ED12m8ρd4,
Δt96atρd5ED12m.
ΔtΔq9at2aq.
k=η33EId3,
k=η33πED3a8d3,
Iz2 πa2η41-zdD22,
Mz=-Qad22-dz+z22,
δz= MzEIzdz4Qd3πED2aη1-zd.
δ=0d δzdz2Qd4πED2aη,
k=38Qδ38πED2a2η2d3.

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