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    Journals >Infrared and Laser Engineering >Volume 49 >Issue 7 >Page 20190469 > Article
    • Infrared and Laser Engineering
    • Vol. 49, Issue 7, 20190469 (2020)
    Structural design optimization of space gravitational wave telescope primary mirror system
    DOI: 10.3788/IRLA20190469 Cite this Article
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    Abstract

    The structure and supporting components of the primary mirror system of space gravitational wave telescope were designed and optimized. The side three-point support was used to restrain the mirror body with the primary mirror, and the selection and layout of support points were studied. The mirror adopted a semi-enclosed structure with back drill holes, which can achieve large bending stiffness. The lightweight structure of the mirror was optimized by finite element calculation combined with multi-objective genetic algorithm. The lightweight ratio of the mirror structure reached 74% without reducing the surface accuracy. An adjustable bi-axis bipod flexure hinge structure composed of two non-barrier series flexibility elements was designed, which can compensate the surface error of the mirror. The mathematical model of the flexible hinge parallel mechanism acting on the mirror was established, and the parameters were analyzed based on MATLAB. The parameters were corrected by finite element method. Finally, the analysis of mirror shape under space thermal load was carried out. The results show that the error of mirror shape is better than λ/60, which meets the design requirements.
    ${\delta _{{\rm{quilting}}}} = \dfrac{{12{\varphi _{{t}}}PB\left( {1 + {\mu ^2}} \right)}}{{E{t_{{F}}}^3}}$(1)

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    $\left\{\begin{aligned} & {{}_{{o_1}}^{{o_r}}t = {{\left( {0,0,{{ - {l_1}} / 2} - \sum\limits_{i = 2}^{n = 5} {{l_i}} } \right)}^{\rm T}}} \\ & {{}_{{o_2}}^{{o_r}}t = {{\left( {0,0,{{ - {l_2}} / 2} - \sum\limits_{i = 3}^{n = 5} {{l_i}} } \right)}^{\rm T}}} \\ & {{}_{{o_3}}^{{o_r}}t = {{\left( {0,0,{{ - {l_3}} / 2} - \sum\limits_{i = 4}^{n = 5} {{l_i}} } \right)}^{\rm T}}} \\ &{{}_{{o_4}}^{{o_r}}t = {{\left( {0,0,{{ - {l_4}} / 2} - {l_5}} \right)}^{\rm T}}} \end{aligned}\right.$(2)

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    $\left\{ {\begin{array}{*{20}{l}} {{}_{{o_1}}^{{o_r}}R = {}_{{o_3}}^{{o_r}}R = I} \\ {{}_{{o_2}}^{{o_r}}R = {}_{{o_4}}^{{o_r}}R = {R_{{\rm{Zr}}}}\left( {{\pi / 2}} \right)} \end{array}} \right.$(3)

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    $A{d_{si}} = \left[ {\begin{array}{*{20}{c}} {{}_{{o_i}}^{{o_r}}R}&0 \\ {{}_{{o_i}}^{{o_r}}\hat t \cdot {}_{{o_i}}^{{o_r}}R}&{{}_{{o_i}}^{{o_r}}R} \end{array}} \right]$(4)

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    ${}_{{o_i}}^{{o_r}}\hat t = \left[ {\begin{array}{*{20}{c}} 0&{ - {z_{{o_r}}}}&{{y_{{o_r}}}} \\ {{z_{{o_r}}}}&0&{ - {x_{{o_r}}}} \\ { - {y_{{o_r}}}}&{{x_{{o_r}}}}&0 \end{array}} \right]$(5)

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    ${C_{{\rm{si}}}} = {\rm{diag}}\left( {C_{{\rm{si}}}^{{\rm{11}}},C_{{\rm{si}}}^{22},C_{{\rm{si}}}^{33},C_{{\rm{si}}}^{44},C_{{\rm{si}}}^{55},C_{{\rm{si}}}^{66}} \right)$(6)

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    $\begin{split} {C_{\rm{leg}{A_1}}} =& \sum\limits_{i = 1}^{m = 4} {A{d_{{\rm{si}}}}{C_{{\rm{si}}}}Ad_{{\rm{si}}}^{\rm{T}}}= \\ \quad \quad \, & \left[ {\begin{array}{*{20}{c}} {C_{l{A_1}}^{{\rm{11}}}}&0&0&0&{C_{l{A_1}}^{{\rm{15}}}}&0 \\ 0&{C_{l{A_1}}^{22}}&0&{C_{l{A_1}}^{24}}&0&0 \\ 0&0&{C_{l{A_1}}^{33}}&0&0&0 \\ 0&{C_{l{A_1}}^{42}}&0&{C_{l{A_1}}^{44}}&0&0 \\ {C_{l{A_1}}^{{\rm{51}}}}&0&0&0&{C_{l{A_1}}^{55}}&0 \\ 0&0&0&0&0&{C_{l{A_1}}^{66}} \end{array}} \right] \\ \end{split} $(7)

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    ${C_{{\rm{si}}}} = {\rm{diag}}\left( {\dfrac{{{l_i}}}{{EI_x^i}},\dfrac{{{l_i}}}{{EI_y^i}},\dfrac{{{l_i}}}{{G{J_i}}},\dfrac{{{l_i}^3}}{{12EI_y^i}},\dfrac{{{l_i}^3}}{{12EI_x^i}},\dfrac{{{l_i}}}{{E{A_i}}}} \right)\\[-20pt]$(8)

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    $I_x^i = \dfrac{{{w_i}^3{t_i}}}{{12}},I_y^i = \dfrac{{{w_i}{t_i}^3}}{{12}},{A_i} = {w_i}{t_i},{J_i} = I_x^i + I_y^i$()

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    ${\eta _i} = {\left( {\dfrac{{{t_i}}}{{{w_i}}}} \right)^2},{\gamma _i} = {\left( {\dfrac{{{t_i}}}{{{l_i}}}} \right)^2},{\varepsilon _i} = \dfrac{{{J_i}}}{{I_y^i}},\xi = \dfrac{G}{E} = \dfrac{1}{{2\left( {1 + \mu } \right)}}$()

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    ${C_{{\rm{si}}}} = \dfrac{{{l_i}}}{{EI_y^i}}{\rm{diag}}\left( {{\eta _i},1,\dfrac{1}{{\xi {\varepsilon _i}}},\dfrac{{{l_i}^2}}{{12}},\dfrac{{{l_i}^2}}{{12}}{\eta _i},\dfrac{{{l_i}^2}}{{12}}{\gamma _i}} \right)$(9)

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    ${\varepsilon _i} = 12\left\{ {\dfrac{1}{3} - 0.21\dfrac{{{t_i}}}{{{w_i}}}\left[ {1 - \dfrac{1}{{12}}{{\left( {\dfrac{{{t_i}}}{{{w_i}}}} \right)}^4}} \right]} \right\}$(10)

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    $\dfrac{{{t_i}}}{{{w_i}}} \in \left[ {\dfrac{1}{{16}},\dfrac{1}{4}} \right],\dfrac{{{t_i}}}{{{l_i}}} \in \left[ {\dfrac{1}{{20}},\dfrac{1}{4}} \right]$()

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    $\begin{split} {C_{{Hinge_-}A}}= &{\left( {\sum\limits_{j = 1}^{n = 2} {{{\left( {A{d_{{\rm{A}}_j}}{C_{leg{\rm{A}}}}_{_j}Ad_{{\rm{A}}_j}^{\rm T}} \right)}^{ - 1}}} } \right)^{ - 1}}=\\ & \left[\begin{array}{*{20}{c}} {C_{hA}^{11}} & 0 & 0 & 0 & {C_{hA}^{15}} & 0 \\ 0 & {C_{hA}^{22}} & 0 & {C_{hA}^{24}} & 0 & 0 \\ 0 & 0 & {C_{hA}^{33}} & 0 & 0 & 0 \\ 0 & {C_{hA}^{42}} & 0 & {C_{hA}^{44}} & 0 & 0 \\ {C_{hA}^{51}} & 0 & 0 & 0 & {C_{hA}^{55}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {C_{hA}^{66}} \end{array}\right] \end{split}$(11)

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    $\begin{split} & {R_{{z_A} - {M_x}}} = \dfrac{{\delta _{{y_A}}^{{o_A}}}}{{\theta _{{x_A}}^{{o_A}}}} = \dfrac{{C_{hA}^{51}{M_{{x_A}}}}}{{C_{hA}^{11}{M_{{x_A}}}}} =\\ & \dfrac{{C_{l{A_1}}^{15}C_{l{A_1}}^{66}\cos {\theta _0}}}{{{{\left( {\sin {\theta _0}} \right)}^2}\left[ {C_{l{A_1}}^{11}C_{l{A_1}}^{55} - {{\left( {C_{l{A_1}}^{15}} \right)}^2}} \right] + {{\left( {\cos {\theta _0}} \right)}^2}C_{l{A_1}}^{11}C_{l{A_1}}^{66}}} \\ \end{split} $(12)

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    ${C_{Model}} = {\left[ \begin{array}{l} {\left( {{{{Ad}}_A}{C_{Hing{e_ - }A}}{{Ad}}_A^{\rm{T}}} \right)^{ - 1}} + \\ {\left( {{{{Ad}}_B}{C_{Hing{e_ - }B}}{{Ad}}_B^{\rm{T}}} \right)^{ - 1}} + \\ {\left( {{{{Ad}}_C}{C_{Hing{e_ - }C}}{{Ad}}_C^{\rm{T}}} \right)^{ - 1}} \\ \end{array} \right]^{ - 1}}$(13)

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    ${\zeta _0} = {C_{Model}}{F_0}$(14)

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    $\left\{ {\begin{array}{*{20}{l}} {C_{Model}^r\left( X \right) = \max \left\{ {{{\left[ {{{\left( {C_{Model}^x} \right)}^2} + {{\left( {C_{Model}^y} \right)}^2}} \right]}^{{1 / 2}}}} \right\}} \\ {C_{Model}^Z\left( X \right) = \min \left\{ {C_{Model}^Z} \right\}} \end{array}} \right.$(15)

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    $U\left( X \right) = \mathop {\min }\limits_{X \in D} \dfrac{{C_{Model}^Z}}{{C_{Model}^r}}$(16)

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    $\left\{ {\begin{array}{*{20}{c}} {C_{Model}^{z\;(n + 1)} < C_{Model}^{z\;(n)}} \\ {C_{Model}^{r\;(n + 1)} > C_{Model}^{r\;(n)}} \end{array}} \right.$(17)

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    $\min U\left( {{l_i},{w_i},{t_i},{\theta _0}} \right)\left\{ {\begin{array}{*{20}{l}} {2 \leqslant {l_i} \leqslant 16} \\ {4 \leqslant {w_i} \leqslant 10} \\ {0.4 \leqslant {t_i} \leqslant 2} \\ {{{25}^ \circ } \leqslant {\theta _0} \leqslant {{45}^ \circ }} \end{array}} \right.$(18)

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    Abstract
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