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 哈尔滨工程大学学报  2020, Vol. 41 Issue (7): 966-972  DOI: 10.11990/jheu.201901016 0

引用本文

PANG Fuzhen, HUO Ruidong, LI Haichao, et al. Analysis of the vibration characteristics of moderately thick laminated sector plates under complicated boundary conditions[J]. Journal of Harbin Engineering University, 2020, 41(7): 966-972. DOI: 10.11990/jheu.201901016.

文章历史

1. 哈尔滨工程大学 船舶工程学院, 黑龙江 哈尔滨 150001;
2. 北京航天计量测试技术研究所, 北京 100076;
3. 中国人民解放军 92578部队, 北京 100161

Analysis of the vibration characteristics of moderately thick laminated sector plates under complicated boundary conditions
PANG Fuzhen 1, HUO Ruidong 2, LI Haichao 1, YE Kaifu 1,3, WANG Xueren 1,3
1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China;
2. Beijing Aerospace Institute for Metrology and Measurement, Beijing 100076, China;
3. Naval Academy, Beijing 100161, China
Abstract: In this study, a semi-analytical method is proposed based on the first-order shear deformation theory to evaluate the free vibration associated with the moderately thick composite sector plates under general boundary conditions. A moderately thick laminated sector plate model with a local coordinate system is established, and the displacement and rotation components of its boundary conditions are expressed using a triangular series with an auxiliary function. The Raleigh-Ritz method is used for obtaining the modal parameters of the sector plate structure, and the vibration characteristics of the laminated sector plate are numerically evaluated under several complex boundary conditions. The obtained results are compared with the results obtained from relevant literature. Finally, the influence of the characteristic parameters of the sector plate on the natural frequency of the sector plate is discussed. The proposed method overcomes the discontinuity of the triangle series at the boundary and exhibits better convergence and higher accuracy. The frequency parameter of the sector plate decreases with the increasing thickness to diameter ratio and the lamination angle, whereas it increases with the increasing orthotropic anisotropy ratio.
Keywords: composite material    moderately thick plate    laminated sector plates    complex boundary condition    semi-analytical method    spring stiffness method    vibration characteristic analysis    auxiliary function

1 层合扇形板自由振动理论推导 1.1 模型描述

 Download: 图 1 中厚层合扇形板几何模型示意 Fig. 1 The geometric model of laminated sector plates
1.2 位移关系和应力-应变关系

 $\left\{ {\begin{array}{*{20}{l}} {\bar u(r,\theta ,z,t) = {u_0}(r,\theta ,t) + z{\psi _r}(r,\theta ,t)}\\ {\bar v(r,\theta ,z,t) = {v_0}(r,\theta ,t) + z{\psi _\theta }(r,\theta ,t)}\\ {\bar w(r,\theta ,z,t) = {w_0}(r,\theta ,t)} \end{array}} \right.$ (1)

 $\begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varepsilon _r} = \varepsilon _r^0 + z{\kappa _r},\;\:{\varepsilon _\theta } = \varepsilon _\theta ^0 + z{\kappa _\theta }}\\ {{\gamma _{r\theta }} = \gamma _{r\theta }^0 + z{\kappa _{r\theta }},\;\:{\gamma _{rz}} = \gamma _{rz}^0,{\gamma _{\theta z}} = \gamma _{\theta z}^0} \end{array}$ (2)

 $\left[ {\begin{array}{*{20}{l}} {{\sigma _r}}\\ {{\sigma _\theta }}\\ {{\tau _{r\theta }}}\\ {{\tau _{rz}}}\\ {{\tau _{\theta z}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\bar Q_{11}^k}&{\bar Q_{12}^k}&{\bar Q_{16}^k}&0&0\\ {\bar Q_{12}^k}&{\bar Q_{22}^k}&{\bar Q_{26}^k}&0&0\\ {\bar Q_{16}^k}&{\bar Q_{26}^k}&{\bar Q_{66}^k}&0&0\\ 0&0&0&{\bar Q_{44}^k}&{\bar Q_{45}^k}\\ 0&0&0&{\bar Q_{45}^k}&{\bar Q_{55}^k} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\varepsilon _r}}\\ {{\varepsilon _\theta }}\\ {{\gamma _{r\theta }}}\\ {{\gamma _{rz}}}\\ {{\gamma _{\theta z}}} \end{array}} \right]$ (3)

 $\begin{array}{*{20}{l}} {\left[ {\begin{array}{*{20}{c}} {{N_r}}\\ {{N_\theta }}\\ {{N_{r\theta }}} \end{array}} \right] = b\int_{ - h/2}^{h/2} {\left[ {\begin{array}{*{20}{c}} {{\sigma _r}}\\ {{\sigma _\theta }}\\ {{\tau _{r\theta }}} \end{array}} \right]} {\rm{d}}z,\left[ {\begin{array}{*{20}{c}} {{M_r}}\\ {{M_\theta }}\\ {{M_{r\theta }}} \end{array}} \right] = b\int_{ - h/2}^{h/2} {\left[ {\begin{array}{*{20}{c}} {{\sigma _r}}\\ {{\sigma _\theta }}\\ {{\tau _{r\theta }}} \end{array}} \right]} z{\rm{d}}z,}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} {{Q_r}}\\ {{Q_\theta }} \end{array}} \right] = \kappa b\int_{ - h/2}^{h/2} {\left[ {\begin{array}{*{20}{c}} {{\tau _{rz}}}\\ {{\tau _{\theta z}}} \end{array}} \right]} {\rm{d}}z} \end{array}$ (4)

1.3 扇形板中的能量关系

 $\begin{array}{l} {U_s} = \frac{1}{2}\int {\int {\int_V ( } } {N_r}\varepsilon _r^0 + {N_\theta }\varepsilon _\theta ^0 + {N_{r\theta }}\gamma _{r\theta }^0 + {M_r}{\kappa _r} + {M_\theta }{\kappa _\theta } + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {M_{r\theta }}{\kappa _{r\theta }} + {Q_r}\gamma _{rz}^0 + {Q_\theta }\gamma _{\theta z}^0)(s + a){\rm{d}}V \end{array}$ (5)

 $T = \frac{{\rho (s + a)}}{2}\int {\int {\int_V {\left[ {{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial v}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial w}}{{\partial t}}} \right)}^2}} \right]{\rm{d}}V} } }$ (6)

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T = \frac{1}{2}\int_0^R {\int_0^\varphi {\left\{ {{I_0}\left[ {{{\left( {\frac{{\partial {u_0}}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial {v_0}}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial {w_0}}}{{\partial t}}} \right)}^2}} \right] + } \right.} } \\ 2{I_1}\left[ {\left( {\frac{{\partial {u_0}}}{{\partial t}}} \right)\left( {\frac{{\partial {\psi _r}}}{{\partial t}}} \right) + \left( {\frac{{\partial {v_0}}}{{\partial t}}} \right)\left( {\frac{{\partial {\psi _\theta }}}{{\partial t}}} \right)} \right] + {I_2}\left[ {{{\left( {\frac{{\partial {\psi _r}}}{{\partial t}}} \right)}^2} + } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\left. {{{\left( {\frac{{\partial {\psi _\theta }}}{{\partial t}}} \right)}^2}} \right]} \right\}(s + a){\rm{d}}s{\rm{d}}\theta \end{array}$ (7)

 $\left[ {\begin{array}{*{20}{l}} {{I_0}}&{{I_1}}&{{I_2}} \end{array}} \right] = \int_{ - h/2}^{h/2} \rho \left[ {\begin{array}{*{20}{l}} 1&z&{{z^2}} \end{array}} \right]{\rm{d}}z$ (8)

 $\begin{array}{l} \begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {U_{sp}} = \frac{1}{2}\int_0^\varphi \{ a[k_{r\_0}^uu_0^2 + k_{r\_0}^vv_0^2 + k_{r\_0}^w{w^2} + K_{r\_0}^R\psi _r^2 + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} K_{r\_0}^\theta \psi _\theta ^2{]_{s = 0}} + b[k_{r\_1}^uu_0^2 + k_{r\_1}^vv_0^2 + k_{r\_1}^w{w^2} + K_{r\_1}^R\psi _r^2 + } \end{array}\\ K_{r\_1}^\theta \psi _\theta ^2{]_{s = R}}\} {\rm{d}}\theta + \frac{1}{2}\int_0^R \{ [k_{deg \_0}^uu_0^2 + k_{deg\_0}^uv_0^2 + k_{deg \_0}^uw_0^2 + \\ \begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} K_{deg \_0}^R\psi _r^2 + K_{deg \_0}^\theta \psi _\theta ^2{]_{\theta = 0}} + [k_{deg \_1}^uu_0^2 + k_{deg \_1}^vv_0^2 + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k_{deg \_1}^w{w^2} + K_{deg \_1}^R\psi _r^2 + K_{deg \_1}^\theta \psi _\theta ^2{]_{\theta = \varphi }}\} {\rm{d}}s} \end{array} \end{array}$ (9)
1.4 控制方程和边界条件

 ${L = T - {U_s} - {U_b} - {U_{bs}} - {U_{sp}}}$ (10)

 ${\delta \int_0^t {(T - {U_s} - {U_{sp}} - {U_{cp}})} {\rm{d}}t = 0}$ (11)

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {\frac{{\partial {N_r}}}{{\partial r}} + \frac{{\partial {N_{r\theta }}}}{{r\partial \theta }} - \frac{{{N_\theta }}}{r} = {I_0}{{\left( {\frac{{\partial {u_0}}}{{\partial t}}} \right)}^2} + {I_1}{{\left( {\frac{{\partial {\psi _r}}}{{\partial t}}} \right)}^2}}\\ {\frac{{\partial {N_\theta }}}{{r\partial \theta }} + \frac{{\partial {N_{r\theta }}}}{{\partial r}} + \frac{{{N_{r\theta }}}}{r} = {I_0}{{\left( {\frac{{\partial {v_0}}}{{\partial t}}} \right)}^2} + {I_1}{{\left( {\frac{{\partial {\psi _\theta }}}{{\partial t}}} \right)}^2}} \end{array}\\ \frac{{\partial {Q_r}}}{{\partial r}} + \frac{{\partial {Q_\theta }}}{{r\partial \theta }} = {I_0}{\left( {\frac{{\partial w}}{{\partial t}}} \right)^2}\\ \begin{array}{*{20}{l}} {\frac{{\partial {M_r}}}{{\partial r}} + \frac{{\partial {M_{r\theta }}}}{{r\partial \theta }} - \frac{{{M_\theta }}}{r} - {Q_r} = {I_1}{{\left( {\frac{{\partial {u_0}}}{{\partial t}}} \right)}^2} + {I_2}{{\left( {\frac{{\partial {\psi _r}}}{{\partial t}}} \right)}^2}}\\ {\frac{{\partial {M_\theta }}}{{r\partial \theta }} + \frac{{\partial {M_{r\theta }}}}{{\partial r}} + \frac{{{M_{r\theta }}}}{r} - {Q_\theta } = {I_1}{{\left( {\frac{{\partial {v_0}}}{{\partial t}}} \right)}^2} + {I_2}{{\left( {\frac{{\partial {\psi _\theta }}}{{\partial t}}} \right)}^2}} \end{array} \end{array} \right.$ (12)

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s = 0:\left\{ {\begin{array}{*{20}{l}} {{N_r} + k_{r\_0}^u{u_0} = 0,\;\:{N_{r\theta }} + k_{r\_0}^v{v_0} = 0}\\ {{M_r} + K_{r\_0}^R{\psi _r} = 0,\;\:{M_{r\theta }} + K_{r\_0}^\theta {\psi _\theta } = 0}\\ {{Q_r} + k_{r\_0}^ww = 0,} \end{array}} \right.\\ \theta = 0:\left\{ {\begin{array}{*{20}{l}} {{N_{r\theta }} + k_{deg\_0}^u{u_0} = 0,\;\:{N_\theta } + k_{deg\_0}^v{v_0} = 0}\\ {{M_{r\theta }} + K_{deg\_0}^R{\psi _r} = 0,\;\:{M_\theta } + K_{deg\_0}^\theta {\psi _\theta } = 0}\\ {{Q_\theta } + k_{deg\_0}^ww = 0} \end{array}} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s = R:\left\{ {\begin{array}{*{20}{l}} {{N_r} - k_{r\_1}^u{u_0} = 0,}&{{N_{r\theta }} - k_{r\_1}^vv = 0}\\ {{M_r} - K_{r\_1}^R{\psi _r} = 0,}&{{M_{r\theta }} - K_{r\_1}^\theta {\psi _\theta } = 0}\\ {{Q_r} - k_{r\_1}^ww = 0}&{} \end{array}} \right.\\ \theta = \varphi :\left\{ {\begin{array}{*{20}{l}} {{N_{r\theta }} - k_{deg\_1}^u{u_0} = 0,\;\:{N_\theta } - k_{deg\_1}^v{v_0} = 0}\\ {{M_{r\theta }} - K_{deg\_1}^R{\psi _r} = 0,\;\:{M_\theta } - K_{deg\_1}^\theta {\psi _\theta } = 0}\\ {{Q_\theta } - k_{deg\_1}^ww = 0} \end{array}} \right. \end{array}$ (13)
1.5 位移形函数与方程求解

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {u_0}(s,\theta ,t) = (\sum\limits_{m = 0}^\infty {\sum\limits_{n = 0}^\infty {{A_{mn}}} } {\rm{cos}}({\lambda _{Rm}}s){\rm{cos}}({\lambda _{\varphi n}}\theta ) + \\ \sum\limits_{l = 1}^2 {\zeta _b^l} (\theta )\sum\limits_{m = 0}^\infty {a_m^l} {\rm{cos}}({\lambda _{Rm}}s) + \sum\limits_{l = 1}^2 {\zeta _a^l} (s)\sum\limits_{m = 0}^\infty {b_n^l} {\rm{cos}}({\lambda _{\varphi n}}\theta )){{\rm{e}}^{j\omega t}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {v_0}(s,\theta ,t) = (\sum\limits_{m = 0}^\infty {\sum\limits_{n = 0}^\infty {{B_{mn}}} } {\rm{cos}}({\lambda _{Rm}}s){\rm{cos}}({\lambda _{\varphi n}}\theta ) + \\ \sum\limits_{l = 1}^2 {\zeta _b^l} (\theta )\sum\limits_{m = 0}^\infty {c_m^l} {\rm{cos}}({\lambda _{Rm}}s) + \sum\limits_{l = 1}^2 {\zeta _a^l} (s)\sum\limits_{n = 0}^\infty {d_n^l} {\rm{cos}}({\lambda _{\varphi n}}\theta )){{\rm{e}}^{j\omega t}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} w(s,\theta ,t) = (\sum\limits_{m = 0}^\infty {\sum\limits_{n = 0}^\infty {{C_{mn}}} } {\rm{cos}}({\lambda _{Rm}}s){\rm{cos}}({\lambda _{\varphi n}}\theta ) + \\ \sum\limits_{l = 1}^2 {\zeta _b^l} (\theta )\sum\limits_{m = 0}^\infty {e_m^l} {\rm{cos}}({\lambda _{Rm}}s) + \sum\limits_{l = 1}^2 {\zeta _a^l} (s)\sum\limits_{n = 0}^\infty {f_n^b} {\rm{cos}}({\lambda _{\varphi n}}\theta )){{\rm{e}}^{j\omega t}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\psi _r}(s,\theta ,t) = (\sum\limits_{m = 0}^\infty {\sum\limits_{n = 0}^\infty {{D_{mn}}} } {\rm{cos}}({\lambda _{Rm}}s){\rm{cos}}({\lambda _{\varphi n}}\theta ) + \\ \sum\limits_{l = 1}^2 {\zeta _b^l} (\theta )\sum\limits_{m = 0}^\infty {g_m^l} {\rm{cos}}({\lambda _{Rm}}s) + \sum\limits_{l = 1}^2 {\zeta _a^l} (s)\sum\limits_{n = 0}^\infty {h_n^l} {\rm{cos}}({\lambda _{\varphi n}}\theta )){{\rm{e}}^{j\omega t}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\psi _\theta }(s,\theta ,t) = (\sum\limits_{m = 0} {\sum\limits_{n = 0} {{E_{mn}}} } {\rm{cos}}({\lambda _{Rm}}s){\rm{cos}}({\lambda _{\varphi n}}\theta ) + \\ \sum\limits_{l = 1}^2 {\zeta _b^l} (\theta )\sum\limits_{m = 0}^\infty {k_m^l} {\rm{cos}}({\lambda _{Rm}}s) + \sum\limits_{l = 1}^2 {\zeta _a^l} (s)\sum\limits_{n = 0}^\infty {q_n^l} {\rm{cos}}({\lambda _{\varphi n}}\theta )){{\rm{e}}^{j\omega t}} \end{array}$ (14)

 $\left\{ {\begin{array}{*{20}{l}} {\zeta _a^1(s) = \frac{R}{{2\pi }}{\rm{sin}}\left( {\frac{{\pi s}}{{2R}}} \right) + \frac{R}{{2\pi }}{\rm{sin}}\left( {\frac{{3\pi s}}{{2R}}} \right)}\\ {\zeta _a^2(s) = \frac{{ - R}}{{2\pi }}{\rm{cos}}\left( {\frac{{\pi s}}{{2R}}} \right) + \frac{R}{{2\pi }}{\rm{cos}}\left( {\frac{{3\pi s}}{{2R}}} \right)} \end{array}} \right.$ (15)

 $\left\{ {\begin{array}{*{20}{l}} {\zeta _a^1(0) = \zeta _a^1(R) = \zeta _a^{1\prime }(R) = 0,\;\:\zeta _a^{1\prime }(0) = 1}\\ {\zeta _a^2(0) = \zeta _a^2(R) = \zeta _a^{2\prime }(0) = 0,\;\:\zeta _a^{2\prime }(R) = 1} \end{array}} \right.$ (16)

 $(\mathit{\boldsymbol{K}} - {\omega ^2}\mathit{\boldsymbol{M}})\mathit{\boldsymbol{E}} = 0$ (17)

2 数值结果和讨论 2.1 收敛性研究 2.1.1 边界弹簧刚度的影响

 Download: 图 2 双层非对称层合扇形板[0°/90°]的频率参数ΔΩ随弹性约束参数Γλ的变化关系 Fig. 2 The frequency parameters ΔΩ versus the elastic restraint parameters Γλ for laminated sector plate[0°/90°]

2.1.2 截断数的影响

2.2 有效性验证及特性分析

 Download: 图 3 不同边界下层合扇形板前三阶模态阵型 Fig. 3 The lowest three mode shapes for a laminated sector plate in different boundary conditions

 Download: 图 4 2种边界下扇形板[0°/90°/90°/0°]前三阶模态阵型图 Fig. 4 The lowest three mode shapes for a [0°/90°/90°/0°] laminated sector plate with two boundary conditions
3 结论

1) 通过增加辅助函数有效解决了三角级数的连续性问题，计算结果通过与参考文献对比验证了本方法具有较好的准确性。

2) 弹簧刚度法在解决此类问题时可有效应用，使用适当取值的弹簧刚度可较好地模拟各种边界条件约束。

3) 计算结果随三角级数展开阶数的增加快速收敛，通常取较低的阶数就能保证足够的计算精度。

4) 无量纲频率随扇形板厚径比的增大而减小，随正交各向异性比的增大而增大，随层压角的增大而减小。

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