THEORETICAL ANALYSIS ON THE CRITICAL FLOW VELOCITY AND VIBRATION MODE OF A TWIN-CHANNEL ROTATING PIPE
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摘要: 旋转叶片是航空发动机重要零件之一, 服役条件十分恶劣, 常常因振动过量导致其失效. 为了合理设计含冷却通道的叶片, 保证其可靠性与安全性, 需对含冷却通道的叶片的振动特性进行研究. 基于Euler-Bernoulli梁理论, 将叶片简化为含两通道的悬臂旋转输流管, 考虑了通道轴线偏移量对流体动能的影响, 采用Lagrange原理结合假设模态法建立包含双陀螺效应的运动控制方程, 采用降阶扩维的方法求解系统特征值. 研究两通道模型的流速比、转速和长细比等对前3阶特征根曲线影响. 将文章模型退化为简支单通道输流管, 与文献报道结果进行对比, 部分验证建模方法的正确性. 研究发现: 在相同的管道截面积下, 两通道模型的临界流速值大于单通道模型的; 旋转运动引入的陀螺效应会使得第2, 3阶特征根轨迹发生绕圈现象, 并多次穿越虚轴; 随着长细比的增大, 系统会表现出类似非旋转的悬臂输流管的动力学行为; 系统的横向位移模态响应呈现出行波特性, 且在不同参数组合下, 阻尼因子对前3阶模态产生不同的增强或减弱作用.Abstract: Rotating blade is an essential part of aero-engine. It serves in harsh conditions. Its failure is often caused by excessive vibration. To design the blade properly and to ensure the reliability and safety, the vibration characteristics of the blade need to be revealed. The blade is simplified as a cantilever rotating pipe with double cooling channels based on the Euler-Bernoulli beam theory. The influences of channel axis offset on fluid kinetic energy are considered in the present study. The motion governing equation of the blade is established including the bi-gyroscopic effects with the combination of Lagrange principle and assumed mode method. The method of order reduction and dimension expansion is applied to solve the eigenvalue of the system. The influences of the fluid velocity ratio, rotating speed, slenderness et al. on the first three order eigenvalue curves are studied. The present model degenerates into a simply supported pipe conveying fluid with a single channel to compare with the results reported in literature. The correctness of the present modeling method is verified, partly. The velocity ratio of two channels has great influence on the first three order critical flow velocity values. For a given value of the cross-section area of the cooling passage, the critical flow velocity of the twin-channel model is higher than the single-channel model. A circling phenomenon is introduced to on the second and the third eigenvalue curves by the gyroscopic effect due to the rotating motion. The second and the third eigenvalue curves travel through the imaginary axis several times. With the increase of the slenderness ratio, the system’s dynamic behaviors are similar to the non-rotating cantilever pipe. Moreover, due to the gyroscopic effect, the modal response of the lateral displacement presents a traveling wave property. And the damping factor has different enhancement or weakening effects on the first three modes under different parameter conditions.
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引 言
航空发动机是飞机的心脏, 是国家工业基础、科技水平和国防实力的重要标志. 涡轮叶片是航空发动机中最重要的关键部件之一,其可靠性直接影响发动机能否正常. 已有研究表明[1-4], 航空发动机故障原因中62%以上是由于涡轮叶片振动过量, 且由于高温、高压工况下通过蛇形冷却通道进行降温的设计, 叶片通道内部受到科氏力、离心力等因素的相互作用, 具有复杂的流固耦合效应. 因此, 研究含蛇形冷却通道的旋转叶片动力学特性, 对发动机叶片的合理化设计与减振降噪具有重要意义.
由于旋转叶片结构在工程中的广泛应用, 尤其是在航空发动机领域的核心地位, 从上世纪四50年代以来, 旋转叶片的动力学行为研究吸引了国内外学者的广泛关注. 如Carnegie[5]在1959年首次研究了旋转叶片的自振频率. 张伟等[1]将叶片简化为功能梯度材料薄壁梁, 通过伽辽金法研究了系统在空气热弹性载荷下的动力学行为. 此外, 许多学者[6-11]将叶片简化为大变形柔性薄板. 如赵飞云等[12]基于连续介质力学理论和Jourdain速度变分原理, 研究高速旋转下的柔性矩形薄板的耦合动力学模型. 郑彤等[13]、蒋建平等[14], 利用Langrage方程法得到系统的一次近似耦合动力学方程.
以上研究, 将旋转叶片简化为旋转的悬臂梁或旋转的大变形柔性薄板, 但均未考虑内部流体对叶片动力学特性的影响. 由前人的研究[15-21]可知内部流体会对叶片的振动产生复杂的影响, 将叶片简化为旋转悬臂输液管模型可以从宏观上更深入地研究流体的某一特性对旋转叶片的动力学影响. Chen[22]在1971年对输送脉动流体的管道振动进行研究, 并通过数值方法确定稳定性边界. Paidoussis[23]在1987年对输液管的振动作了详细的论述, 并指出发散失稳和颤振失稳两种值得深入研究的现象. 后续有Rousselet等[24]、Lundgren等[25-27]和徐鉴等[28-29]先后导出了悬臂输液管的非线性运动方程.
Paidoussis等[30]、黄玉盈等[31]和Ibrahim[32-33]分别在1993年、1998年和2010年对输液管模型动力学模型的研究现状进行了详细的综述. 王乙坤等[34]研究了脉动内流作用下的输液管振动问题. Wang等[35]用牛顿法和龙格-库塔法对波纹输液管的动力学行为进行理论和数值分析,发现波纹总数和波纹幅度对输液管的动力学行为有较大影响. 宫亚飞等[36]利用哈密顿原理和伽辽金法推导了含有初始弯曲的功能梯度输液管的非线性动力学方程, 分析超临界功能梯度输液管的自由振动特性. Oh等[37]利用瑞丽利兹法推导含冷却通道的旋转叶片的热弹动力耦合模型, 研究冷却效应下系统的固有频率与拉伸特性. 张博等[38]基于Langrage原理和假设模态法研究不同端部集中质量、转速对旋转输液管临界流速的影响及特定参数组合下的内共振现象.
通过文献调研不难发现, 在内冷叶片研究领域, 考虑内流作用对系统振动特性影响的报道较少, 文献[38]中虽开展了相关研究, 但只局限于单冷却通道模型. 为了更贴合包含多条冷却通道且通有流体介质的先进内冷叶片工程实例[39], 本文在文献[38]基础上, 引入叶片换热特性研究[40]中对内部通道的简化方式, 将叶片简化为含双输液通道的旋转悬臂输液管, 采用能量法建立其动力学模型, 揭示包含多条冷却通道的旋转叶片结构中, 冷却通道内流体流动对叶片动力学特性的影响规律.
1. 动力学方程推导
本文将含蛇形冷却通道的涡轮叶片简化为如图1所示长为L的两通道旋转悬臂输流管, 固接在半径r的中空圆柱上以常数Ω旋转, 截面高为h, 宽为b, 单位长度质量为m, 两通道半径均为r0, 两通道轴线与输流管轴线距离均为d, 两通道内流体的单位长度质量分别为M1, M2, 相对通道的流速分别为常数U1与U2. 以中空圆柱中心O点为原点, 建立全局坐标系XYZ, 单位方向向量分别为i, j, k. 以输流管中轴线与圆柱侧面的交点为o点, 沿轴线方向为x轴, 建立随转坐标系xyz, 单位方向向量分别为i′, j′, k′,轴线上任意一点在t时刻的x, y方向位移分量分别用w1(x, t)和w2(x, t)表示, 由变形引起的转角为θ. 为了简化分析, 引入以下假设: (1)忽略输流管剪切变形和转动惯量的影响; (2)忽略输流管在z方向的变形, 即振动发生在XY平面内; (3)两通道内的流体均为为定常不可压缩的无黏流体; (4)输流管是均匀、各向同性的线弹性材料.
在全局坐标系下, 输流管上任意一点的速度矢量为
$$ \begin{split} & {\boldsymbol{v}} = \left( {{{\dot w}_1} - y{{\dot w}_{2,x}} - \varOmega {w_2}} \right){\boldsymbol{i}} + \\ &\qquad \left[ {{{\dot w}_2} + \varOmega \left( {{w_1} - y{w_{2,x}} + x + r} \right)} \right]{\boldsymbol{j}} \end{split} $$ (1) 令y = 0可表示输流管中轴线上任意一点的速度矢量为
$$ {{{\boldsymbol{v}}}_{_{{{P}}}}} = \left( {{{\dot w}_1} - \varOmega {w_2}} \right){\boldsymbol{i}} + {\kern 1pt} \left[ {{{\dot w}_2} + \varOmega\left( {{w_1} + x + r} \right)} \right]{\boldsymbol{j}} $$ (2) 则输流管的动能为
$$ {T_p} = \frac{m}{2}\int_0^L {{{{\boldsymbol{v}}_{\boldsymbol{p}}}} \cdot {{{\boldsymbol{v}}_{\boldsymbol{p}}}}{\text{d}}x} $$ (3) 输流管因旋转运动而产生的轴向收缩势能[41]为
$$ {V_s} = \frac{{{\varOmega ^2}}}{4}\int_0^L {{w_{2,x}}^2} \left[ {m({L^2} - {x^2})} + {2mr(L - x)} \right]{\rm{d}}x $$ (4) 输流管的变形势能为
$$ {V_p} = \frac{1}{2}\int_0^L {\left( {EA_{p}{w_{1,x}}^2 + IE{w_{2,xx}}^2} \right)} {\text{d}}x $$ (5) 式中, EI为输流管的抗弯刚度, Ap为截面面积. 在全局坐标系下, 两通道内流体的速度分别为
$$ {{\boldsymbol{v}}_{{{f}}{1}}} = {\left. {\boldsymbol{v}} \right|_{y = d}} + {U_1}\left( {{{\cos }^2}\theta {\boldsymbol{i}} + \theta {\boldsymbol{j}}} \right) $$ (6) $$ {{\boldsymbol{v}}_{{{f}}{2}}} = {\left. {\boldsymbol{v}} \right|_{y = - d}} + {U_2}\left( {{{\cos }^2}\theta {\boldsymbol{i}} + \theta {\boldsymbol{j}}} \right) $$ (7) 两通道内流体的动能为
$$ {T_{f1}} = \frac{1}{2}{M_1}\int_0^L {{{\boldsymbol{v}}_{{{f1}}}} \cdot {{\boldsymbol{v}}_{{{f1}}}}{\text{d}}x} $$ (8) $$ {T_{f2}} = \frac{1}{2}{M_2}\int_0^L {{{\boldsymbol{v}}_{{{f2}}}} \cdot {{\boldsymbol{v}}_{{{f2}}}}{\text{d}}x} $$ (9) 根据Euler-Bernoulli梁理论, 在随转坐标系下, 通道1(y = d)和通道2(y = −d)中轴线上任意一点的位置矢量为
$$ {{\boldsymbol{R}}_1} = \left( {{w_1} - d{w_{2,x}} + x} \right){\boldsymbol{i}}' + {w_2}{\boldsymbol{j}}' $$ (10) $$ {{\boldsymbol{R}}_2} = \left( {{w_1}{\text{ + }}d{w_{2,x}} + x} \right){\boldsymbol{i}}' + {w_2}{\boldsymbol{j}}' $$ (11) 在输流管开放系统中, 输流管做无穷小变形运动下, 流体对输流管做的功可由文献[42]得
$$ {W_{n1}} = {M_1}{U_1}^2\left( {L - {R_1}_x} \right) - {M_1}{U_1}\left( {{{\dot R}_{1y}} + {U_1}\theta } \right){R_{1y}} $$ (12) $$ {W_{n2}} = {M_2}{U_2}^2\left( {L - {R_2}_x} \right) - {M_2}{U_2}\left( {{{\dot R}_{2y}} + {U_2}\theta } \right){R_{2y}} $$ (13) 则旋转输流管开放系统的Lagrange函数为
$$ L = {T_P} + {T_{f1}} + {T_{f2}} - {V_P} - {V_s} + {W_{n1}} + {W_{n2}} $$ (14) 本文两通道旋转悬臂输流管的边界条件为
$$ \left.\begin{split} & x = 0,{w_1} = {w_2} = {w_{2,x}} = 0 \\ & x = L,{w_{1,x}} = {w_{2,xx}} = {w_{2,xxx}} = 0 \end{split}\right\} $$ (15) 应用假设模态法离散系统的两个变形分量
$$ \left.\begin{split} & {w_1}(x,t) = \sum\limits_l^{{N_1}} {{\phi _{1l}}\left( x \right)} \cdot {q_{1l}}\left( t \right) \\[-3pt] & {w_2}(x,t) = \sum\limits_l^{{N_2}} {{\phi _{2l}}\left( x \right)} \cdot {q_{2l}}\left( t \right) \end{split}\right\} $$ (16) 式中, qi(t)代表系统在相应方向的广义位移, N1, N2为选取的模态函数个数, ϕi(x)为满足系统位移边界条件的模态函数
$$ {\phi _{1l}}\left( x \right) = \sin \frac{{(2l - 1)\text{π} x}}{{2L}},l = 1,2, \cdots ,{N_1} $$ (17) $$ \begin{split} & {\phi _{2l}}\left( x \right) = \frac{{\cos \left( {{\beta _l}L} \right) + {{\rm{ch}}} \left( {{\beta _l}x} \right)}}{{\sin \left( {{\beta _l}x} \right) + {{\rm{sh}}} \left( {{\beta _l}x} \right)}}\left[ {{{\rm{sh}}} \left( {{\beta _l}x} \right) - \sin \left( {{\beta _l}x} \right)} \right] + \\ &\qquad \cos \left( {{\beta _l}x} \right) - {{\rm{ch}}} \left( {{\beta _l}x} \right),l = 1,2, \cdots ,{N_1} \end{split} $$ (18) 其中参数βl为频率方程
$\cos \left( {{\beta _l}L} \right){{\rm{ch}}} \left( {{\beta _l}L} \right) = - 1$ 的根.根据Lagrange原理, 得到系统矩阵形式的动力学方程
$$ \begin{split} & \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{M}}^{{11}}}}&{{{\boldsymbol{M}}^{{12}}}} \\ {{{\boldsymbol{M}}^{{21}}}}&{{{\boldsymbol{M}}^{{22}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\ddot q}}}_{1}}} \\ {{{{\boldsymbol{\ddot q}}}_{2}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {0}&{{{\boldsymbol{G}}^{{12}}}} \\ {{{\boldsymbol{G}}^{{21}}}}&{{{\boldsymbol{C}}^{{22}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\dot q}}}_{1}}} \\ {{{{\boldsymbol{\dot q}}}_{2}}} \end{array}} \right] + \\ &\qquad {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{K}}^{{11}}}}&{{{\boldsymbol{K}}^{{12}}}} \\ {{{\boldsymbol{K}}^{{21}}}}&{{{\boldsymbol{K}}^{{22}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{q}}_{1}}} \\ {{{\boldsymbol{q}}_{2}}} \end{array}} \right]{\boldsymbol{ = }}\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{Q}}_{1}}} \\ {{{\boldsymbol{Q}}_{2}}} \end{array}} \right] \end{split} $$ (19) 为了使研究结果具有普遍性, 定义以下无量纲参数
$$ \begin{split} &\xi = {x \mathord{\left/ {\vphantom {x L}} \right. } L},\gamma = {r \mathord{\left/ {\vphantom {r L}} \right. } L},\delta = {d \mathord{\left/ {\vphantom {d L}} \right. } L},\eta = {q \mathord{\left/ {\vphantom {q L}} \right. } L},{\bar w_2} ={{{w_2}} \mathord{\left/ {\vphantom {{{w_2}} L}} \right. } L}\\ & {\alpha _0} = \frac{m}{{m + {M_1} + {M_2}}},{\alpha _1} = \frac{{{M_1}}}{{m + {M_1} + {M_2}}}\\ &{\alpha _2} = \frac{{{M_2}}}{{m + {M_1} + {M_2}}},\;\;\;\tau = \sqrt {\frac{{EI}}{{m + {M_1} + {M_2}}}} \frac{t}{{{L^2}}}\\ &\kappa = \sqrt {\frac{{{A_p}{L^2}}}{I}} ,{\varOmega ^ * } = \varOmega \sqrt {\frac{{\left( {m + {M_1} + {M_2}} \right){L^4}}}{{EI}}}\\ &{u_1} = {U_1}L\sqrt {\frac{{m + {M_1} + {M_2}}}{{EI}}} ,{u_2} = {U_2}L\sqrt {\frac{{m + {M_1} + {M_2}}}{{EI}}}\end{split} $$ 其中, α0, α1, α2分别表示输流管及两通道内流体的单位长度质量权重; τ, κ, Ω*分别表示无量纲时间、长细比与无量纲转速; u1和u2分别表示两通道中流体的无量纲流速.
得到无量纲化后的系统动力学方程为
$$ {{\bar {\boldsymbol{M}}\ddot {\boldsymbol{\eta}} }} + \left( {{{\bar {\boldsymbol{C}}}} + {{\bar {\boldsymbol{G}}}}} \right){{\dot {\boldsymbol{\eta}} }} + {{\bar {\boldsymbol{K\eta}} }} = {{\bar {\boldsymbol{Q}}}} $$ (20) 其中质量子阵、阻尼子阵、陀螺子阵、刚度子阵和广义力子阵见附录.
2. 复模态分析
采用降阶扩维的方法求解系统的特征值,为简化方程, 引入以下向量与矩阵
$$ \left.\begin{split} &{\boldsymbol{y}} = {\left[ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\dot \eta }}}_{1}}^{\text{T}}}&{{{{\boldsymbol{\dot \eta }}}_{2}}^{\text{T}}}&{{{\boldsymbol{\eta }}_{1}}^{\text{T}}}&{{{\boldsymbol{\eta }}_{2}}^{\text{T}}} \end{array}} \right]^{\text{T}}}\\ &{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {0}&{\overline {\boldsymbol{M}} } \\ {\overline {\boldsymbol{M}} }&{\overline {\boldsymbol{C}} + \overline {\boldsymbol{G}} } \end{array}} \right],{{{{\boldsymbol{B}}}}} = \left[ {\begin{array}{*{20}{c}} { - \overline {\boldsymbol{M}} }&{0} \\ {0}&{\overline {\boldsymbol{K}} } \end{array}} \right] \end{split}\right\}$$ (21) 由于本文研究的是自由振动问题, 方程可简化为
$$ {\boldsymbol{A\dot y}} + {\boldsymbol{By}} = {{\boldsymbol{0}}} $$ (22) 设其通解为
$$ {\boldsymbol{y}} = {\left\{ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\dot \eta }}}^{\text{T}}}} \\ {{{\boldsymbol{\eta }}^{\text{T}}}} \end{array}} \right\}_{(4N \times 1)}} = \left\{ {\begin{array}{*{20}{l}} {{{\boldsymbol{\psi }}_{(2N \times 1)}}\lambda } \\ {{{\boldsymbol{\psi }}_{(2N \times 1)}}} \end{array}} \right\}{{\rm{e}}^{\lambda \tau }} $$ (23) 将其代入式(22), 得到关于实值矩阵的一般特征值问题
$$ \left| {{\boldsymbol{A}}\lambda + {\boldsymbol{B}}} \right| = 0 $$ (24) 求解得到系统呈复共轭出现的特征根和特征向量
$$ \left.\begin{split} &\left[ {\begin{array}{*{20}{c}} {\boldsymbol{\varLambda }}&{0} \\ {0}&{{{\boldsymbol{\varLambda }}^{\boldsymbol{*}}}} \end{array}} \right] =\\ &\qquad {\rm{diag}}\left( {{\lambda _r},\lambda _r^ * } \right) = {\rm{diag}}\left( { - {\sigma _r} + {\rm{j}}{\omega _r}, - {\sigma _r} - {\rm{j}}{\omega _r}} \right) \\ & {{{\boldsymbol{\varPsi '}}}_r} = {\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{\varPsi }}_r}{\lambda _r}} \\ {{{\boldsymbol{\varPsi }}_r}} \end{array}} \right\}_{(4N \times 1)}}\\ &{{{\boldsymbol{\varPsi '}}}_r}^ * = {\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\varPsi }}_r^ * \lambda _r^ * } \\ {{\boldsymbol{\varPsi }}_r^ * } \end{array}} \right\}_{(4N \times 1)}}\\ &\qquad r = 1,2, \cdots ,2N \end{split}\right\} $$ (25) 式中, σr为阻尼因子, ωr为固有频率.因此, 特征矢量矩阵为
$$ \begin{split} & {\boldsymbol{\varPsi '}}{ = }\left[ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\varPsi '}}}_{1}}} & {{{{\boldsymbol{\varPsi '}}}_{2}}} & \cdots & {{{{\boldsymbol{\varPsi '}}}_{{2}N}}} & {{{{\boldsymbol{\varPsi '}}}_{1}}^ * } & {{{{\boldsymbol{\varPsi '}}}_{2}}^ * } & \cdots & {{{{\boldsymbol{\varPsi '}}}^ * }_{{2}N}} \end{array}} \right]= \\ & \qquad {\left[ {\begin{array}{*{20}{c}} {{\boldsymbol{\varPsi \varLambda }}}&{{{\boldsymbol{\varPsi }}^ * }{{\boldsymbol{\varLambda }}^ * }} \\ {\boldsymbol{\varPsi }}&{{{\boldsymbol{\varPsi }}^ * }} \end{array}} \right]_{(4N \times 4N)}}\\[-12pt] \end{split} $$ (26) 将
${\boldsymbol{y}} = {\boldsymbol{\varPsi 'y'}}$ 代入式(23), 并左乘Ψ′T, 解得$$ {\boldsymbol{y'}} = {\rm{diag}}\left( {{{\rm{e}}^{{\lambda _r}\tau }},{{\rm{e}}^{\lambda _r^*\tau }}} \right){\left[ {\begin{array}{*{20}{c}} {{{\{ y\} }_0}}&{{{\{ {y^*}\} }_0}} \end{array}} \right]^{\text{T}}} $$ (27) 则有
$$ \begin{split} & {{\boldsymbol{\eta }}_r} = {\boldsymbol{\varPsi }}{\rm{diag}}({{\rm{e}}^{{\lambda _r}\tau }}){{\{ }y{\} }_0}{ + }{{\boldsymbol{\varPsi }}^{\boldsymbol{*}}}{\rm{diag}}({{\rm{e}}^{\lambda _r^*\tau }}){{\{ }{y^*}{\} }_0}, \\ &\qquad r = 1,2, \cdots ,2N \end{split} $$ (28) 其中
$$ {\eta _{2l}}(\tau ) = \sum\limits_{r = 1}^{2N} {{\psi _{lr}}} {{\rm{e}}^{{\lambda _r}\tau }}{y_{r0}} + \sum\limits_{r = 1}^{2N} {\psi _{lr}^*} {{\rm{e}}^{\lambda _r^*\tau }}y_{r0}^*,\;\;l = 1,2, \cdots ,N $$ (29) 式中的复数可表示为
$$ {\psi _{lr}} = {\zeta _{lr}}{{\rm{e}}^{{\rm{j}}{\beta _{lr}}}},{y_{r0}} = T{}_r{{\rm{e}}^{{\rm{j}}{\theta _r}}},{\lambda _r} = - {\sigma _r} + {\rm{j}}{\omega _r} $$ (30) 由式(16)、式(29)与式(30)可得输液管横向位移的第r阶模态响应为
$$ \begin{split} & {{\bar w}_{2r}}(\xi ,\tau ) = 2{T_r}\underline {{{\rm{e}}^{ - {\sigma _r}\tau }}} \sum\limits_{l = 1}^N {{\varphi _{2l}}\left( \xi \right)} \cdot {\zeta _{lr}} \cdot \\ &\qquad \underline{\underline {\cos ({\omega _r}\tau + {\beta _{lr}} + {\theta _r})}} \;,\;\;r = 1,2, \cdots ,2N \end{split} $$ (31) 式中单划线与双划线部分分别代表了阻尼因子和相位变化对输流管横向位移模态响应的影响, 为表述方便, 下文使用符号
$ \bar w_{2 r}^A $ 和$ \bar w_{2 r}^\theta $ 分别表示上述两因素归一化模态.3. 收敛性研究与对比验证
本文的系统参数选取参考文献[38], 在后文中, 若无特别说明, 系统的无量纲参数设置为: α0 = 0.675, α1 = 0.1625, α2 = 0.1625, γ = 0.5, κ = 61.94, Ω* = 5, δ = 0.012.
图2研究了假设模态法的收敛性, 在不同试探函数个数下, 系统的特征根轨迹图相似, 临界流速数值存在微小差别. 当试探函数个数取15时, 临界流速收敛. 在后文的数值算例中均取N1 = N2 = 15. 由图可知, 当流体流速较小时, 流体引起了系统各阶模态的阻尼效应. 随着流速的增加, 第1和3阶模态轨迹相继穿越横轴, 预示着输流管1阶和3阶模态将发生颤振失稳, 但第2阶模态始终未发生失稳.
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图 2 不同试探函数个数下前3阶特征根轨迹曲线 (Ω* = 0)Figure 2. The trajectories of the first three eigenvalues for different trail function numbers (Ω* = 0)以文献[43]中两端简支的单通道输流管系统为研究对象,得到该系统的前两阶特征根轨迹曲线, 如图3所示. 将所得的前两阶临界流速与文献[43]的结果相比较, 如表1所示, 两者结果吻合, 验证了本文建模方法的正确性.
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图 3 两端简支的单通道输流管系统的前两阶特征根轨迹Figure 3. The trajectories of the first two eigenvalues for simple supported flow pipe表 1 两端简支单通道输流管系统的前两阶临界流速对比Table 1. Comparison of the first two critical flow velocities for simply supported flow pipeCritical flow vewary Result Ref. [43] first 3.14 3.14 second 6.28 6.28 4. 结果分析与讨论
采用双通道模型更贴合含多条冷却通道的先进内冷叶片工程实际, 在此基础上开展的动力学特性研究更具有参考价值. 图4对比了单位时间流量相同时, 中心单通道模型与对称双通道模型的系统临界流速值. 其中, 单通道的截面半径为
$\sqrt 2 {r_0}$ (r0为双通道模型的截面半径), 两模型的长细比分别为κsingle = 59.75, κtwin = 74.05, 其余参数与前文一致. 由图可见, 在不同转速下, 双通道模型的临界流速值均大于单通道模型, 且随着转速的增加, 双通道模型的临界流速值也比单通道模型增长得更快.![]()
图 4 单、双通道模型的临界流速随转速的变化曲线Figure 4. The curves of critical flow velocity with rotating speed for the single/twin-channel models图5研究了无量纲流速u1和u2的比例分别为1:1, 1:2, 1:3, 1:4时系统的特征根轨迹图, 其中图中所标识的流速代表通道2中流体流速. 可以发现在不同流速比值下, 得到的系统特征根轨迹图曲线形状大致相同, 随着比值减小, 第1阶模态和第3阶模态在发生失稳时的临界流速值增大, 第2阶模态始终未发生失稳.
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图 5 不同流速比下前3阶特征根轨迹(ucr = u2, Ω* = 5)Figure 5. The trajectories of the first three eigenvalues for different velocity ratios (ucr = u2, Ω* = 5)从系统的动力学方程不难发现, 通道内流体的流动与管道整体的旋转运动均会给系统引入陀螺效应, 即本文建立的动力学模型为典型的双陀螺系统. 图6研究了转速对系统特征根轨迹的影响. 由图6可见, 转速第1阶模态的特征根轨迹影响较小, 并始终只存在一个临界流速. 而随着转速的提高, 系统的第2, 3阶特征根轨迹绕圈现象越来越显著, 并会多次穿越虚轴, 预示流速变化过程中系统的第2, 3阶模态将经历失稳→稳定→失稳的复杂动力学历程, 产生该现象是由于旋转运动引入的陀螺效应对系统响应的影响. 为了进一步研究这一现象, 控制转速不变, 在图7中绘制了系统特征根轨迹随长细比的演变规律. 可以发现, 随着κ增大, 系统第2, 3阶模态特征根轨迹的绕圈现象逐渐消失, 当κ足够大时, 第2阶模态不再发生失稳, 第3阶特征根轨迹只穿越一次虚轴, 只存在一个临界流速, 类似于经典的悬臂输流管模型[15]. 实际上, 随着κ增大, 旋转运动引起的管道轴向与横向间的陀螺耦合效应逐渐减弱[44], 系统双陀螺效应中通道内流体的流动引起的陀螺效应成为主导因素, 因而系统表现出类似非旋转的悬臂输流管的动力学行为.
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图 6 不同转速下前3阶特征根轨迹曲线(κ = 35)Figure 6. The trajectories of the first three eigenvalues for different rotating speeds (κ = 35)![]()
图 7 不同长细比下前3阶特征根轨迹曲线(Ω* = 5)Figure 7. The trajectories of the first three eigenvalues for different slenderness ratios (Ω* = 5)图8揭示了相位变化对输流管横向位移前3阶模态响应的影响. 由图可见, 陀螺效应使得系统不同位置的响应出现相位差. 与实模态系统振动呈现驻波特性不同, 复模态系统振动呈现出行波特性, 即出现“节点”移动现象.
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图 8 相位变化对前3阶模态响应的影响(u1 = u2 = 8, Ω* = 10)Figure 8. The curves of the first three modal responses for different phase changes (u1 = u2 = 8, Ω* = 10)图9(a) ~ 图9(c)给出了3组不同流速和转速组合下, 阻尼因子对输流管横向位移前3阶模态响应的影响. 当u1 = u2 = 6, Ω* = 15时, 前3阶模态响应的幅值均随时间减弱; 当u1 = u2 = 10, Ω* = 12时, 前3阶模态响应的幅值随时间分别增强、减弱、减弱; 当u1 = u2 = 15, Ω* = 15时, 前3阶模态响应的幅值随时间分别增强、增强、减弱. 这表明在不同的流速和转速组合下, 系统不同阶模态响应会出现不同的增强或减弱现象.
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图 9 不同流速和转速下阻尼因子对前3阶模态响应的影响Figure 9. The effects of damping factor on the first three mode responses for different flow rates and rotating speeds5. 结论
本文通过能量法建立了含两通道的旋转输流管动力学控制方程, 通过与文献结果对比验证了本文方法的正确性, 研究了流体单位长度质量、不同流速比和输流管长细比对系统临界流速的影响, 并分别分析了相位变化与阻尼因子对前3阶模态响应的影响, 得出了以下结论.
(1)在单位时间流量相同时, 双通道模型的临界流速值大于单通道模型的, 且随着转速的增大, 双通道模型的临界流速值增长得更快;
(2)随着两通道内流速比值减小, 第1阶模态和第3阶模态在发生失稳时的临界流速值增大, 且第2阶模态始终未发生失稳;
(3)旋转运动引入的陀螺效应对系统第1阶特征根轨迹影响较小, 但会显著影响第2, 3阶特征根轨迹, 使其发生绕圈现象, 并多次穿越虚轴, 从而导致第2, 3阶模态振动存在多个临界流速.
(4)输流管横向位移模态响应出现相位差, 呈现出行波特性, 即出现"节点"移动现象, 在不同的流速和转速组合下, 系统不同阶模态响应会出现不同的增强或减弱现象.
本文首次研究了含两通道的旋转输流管动力学特性, 为含复杂冷却通道布局的旋转叶片的设计提供一定理论参考. 但本文中只考虑两通道为简单的对称式布局, 输流管内部复杂不对称的通道布局等情况有待进一步的研究与讨论.
附录
$\; $ $$ \overline {M_{ij}^{11}} = \int_0^1 {{\varphi _{1i}}\left( \xi \right){\varphi _{1j}}\left( \xi \right){\text{d}}\xi } $$ $$ \overline {M_{ij}^{12}} = \delta \left( {{\alpha _2} - {\alpha _1}} \right)\int_0^1 {{\varphi _{1i}}\left( \xi \right){{\varphi '}_{2j}}\left( \xi \right){\text{d}}\xi } $$ $$ \overline {M_{ij}^{21}} = \overline {M_{ji}^{12}} $$ $$ \overline {M_{ij}^{22}} = \int_0^1 {{\varphi _{2i}}\left( \xi \right){\varphi _{2j}}\left( \xi \right){\text{d}}\xi } + {\delta ^2}\left( {{\alpha _1} + {\alpha _2}} \right)\int_0^1 {{{\varphi '}_{2i}}\left( \xi \right){{\varphi '}_{2j}}\left( \xi \right){\text{d}}\xi } $$ $$ \overline {G_{ij}^{12}} = - 2{\varOmega ^ * }\int_0^1 {{\varphi _{1i}}\left( \xi \right){\varphi _{2j}}\left( \xi \right){\text{d}}\xi }$$ $$ \overline {G_{ij}^{21}} = - \overline {G_{ij}^{12}} $$ $$ \overline {C_{ij}^{22}} = \left( {{\alpha _1}{u_1} + {\alpha _2}{u_2}} \right)\int_0^1 {{\varphi _{2i}}(\xi ){{\varphi '}_{2j}}(\xi )} {\text{d}}\xi $$ $$ \begin{split} & \overline{K_{ij}^{22}}=\frac{{{\alpha }_{0}}{{\varOmega }^{*}}^{2}}{2}\int_{0}^{1}{{{{{\varphi }'}}_{2i}}(\xi ){{{{\varphi }'}}_{2j}}(\xi )\left( 1-\xi \right)\left( 1+2\gamma +\xi \right)\text{d}\xi }- \\ &\qquad {{\varOmega }^{*}}^{2}\int_{0}^{1}{{{\varphi }_{2i}}(\xi ){{\varphi }_{2j}}(\xi )}\text{d}\xi +\int_{0}^{1}{{{{{\varphi }''}}_{2i}}(\xi ){{{{\varphi }''}}_{2j}}(\xi )\text{d}\xi }+ \\ &\qquad \left[ 2{{\varOmega }^{*}}\delta \left( {{\alpha }_{1}}{{u}_{1}}-{{\alpha }_{2}}{{u}_{2}} \right)-{{\varOmega }^{*}}^{2}{{\delta }^{2}}({{\alpha }_{1}}+{{\alpha }_{2}})-({{\alpha }_{1}}{{u}_{1}}^{2}+{{\alpha }_{2}}{{u}_{2}}^{2}) \right]\cdot \\ &\qquad \int_{0}^{L}{{{{{\varphi }'}}_{2i}}(\xi ){{{{\varphi }'}}_{2j}}(\xi )}\text{d}\xi +2\left( {{\alpha }_{1}}{{u}_{1}}^{2}+{{\alpha }_{2}}{{u}_{2}}^{2} \right){{\left. {{\varphi }_{2i}}(1)\frac{\text{d}}{\text{d}\xi }{{\varphi }_{2j}}(\xi ) \right|}_{\xi =1}} \end{split}$$ $$ \overline{K_{ij}^{12}}={{\varOmega }^{*}}\left[ {{\alpha }_{1}}\left( \delta {{\varOmega }^{*}}-{{u}_{1}} \right)-{{\alpha }_{2}}\left( \delta {{\varOmega }^{*}}+{{u}_{2}} \right) \right]\int_{0}^{1}{{{\varphi }_{1i}}(\xi ){{{{\varphi }'}}_{2j}}(\xi )}\text{d}\xi $$ $$ \overline{K_{ij}^{21}}=\overline{K_{ji}^{12}}$$ $$ \overline{{{\boldsymbol{Q}}_{1}}}={{\varOmega }^{*}}^{2}\int_{0}^{1}{\left( \gamma +\xi \right)}{{\mathbf{\varphi }}_{1}}\left( \xi \right)\text{d}\xi -{{\mathbf{\varphi }}_{1}}\left( \xi \right)\left[ {{\alpha }_{1}}{{u}_{1}}^{2}+{{\alpha }_{2}}{{u}_{2}}^{2} \right]$$ $$ \begin{split} & \overline {{{\boldsymbol{Q}}_2}} = - {\alpha _1}{\varOmega ^ * }\int_0^1 {\left[ {\left( {\gamma + \xi } \right)\left( {\delta {\varOmega ^ * } - {u_1}} \right)\frac{{\text{d}}}{{{\text{d}}\xi }}{{\boldsymbol{\varphi }}_2}\left( \xi \right) + {u_1}{{\boldsymbol{\varphi }}_2}\left( \xi \right)} \right]} {\text{d}}\xi - \\ &\qquad {\alpha _2}{\varOmega ^ * }\int_0^1 {\left[ { - \left( {\gamma + \xi } \right)\left( {\delta {\varOmega ^ * } + {u_2}} \right)\frac{{\text{d}}}{{{\text{d}}\xi }}{{\boldsymbol{\varphi }}_2}\left( \xi \right) + {u_2}{{\boldsymbol{\varphi }}_2}\left( \xi \right)} \right]} {\text{d}}\xi + \\ &\qquad \delta \left( {{\alpha _1}{u_1}^2 - {\alpha _2}{u_2}^2} \right){\left. {\frac{{\text{d}}}{{{\text{d}}\xi }}{{\boldsymbol{\varphi }}_2}\left( \xi \right)} \right|_{\xi = 1}} \end{split} $$ -
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图 2 不同试探函数个数下前3阶特征根轨迹曲线 (Ω* = 0)
Figure 2. The trajectories of the first three eigenvalues for different trail function numbers (Ω* = 0)
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图 3 两端简支的单通道输流管系统的前两阶特征根轨迹
Figure 3. The trajectories of the first two eigenvalues for simple supported flow pipe
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图 4 单、双通道模型的临界流速随转速的变化曲线
Figure 4. The curves of critical flow velocity with rotating speed for the single/twin-channel models
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图 5 不同流速比下前3阶特征根轨迹(ucr = u2, Ω* = 5)
Figure 5. The trajectories of the first three eigenvalues for different velocity ratios (ucr = u2, Ω* = 5)
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图 6 不同转速下前3阶特征根轨迹曲线(κ = 35)
Figure 6. The trajectories of the first three eigenvalues for different rotating speeds (κ = 35)
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图 7 不同长细比下前3阶特征根轨迹曲线(Ω* = 5)
Figure 7. The trajectories of the first three eigenvalues for different slenderness ratios (Ω* = 5)
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图 8 相位变化对前3阶模态响应的影响(u1 = u2 = 8, Ω* = 10)
Figure 8. The curves of the first three modal responses for different phase changes (u1 = u2 = 8, Ω* = 10)
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图 9 不同流速和转速下阻尼因子对前3阶模态响应的影响
Figure 9. The effects of damping factor on the first three mode responses for different flow rates and rotating speeds
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1. 张博，孙东生，郑昊楷，史云帆，丁虎，陈立群. 旋转输流管拉弯扭耦合振动研究. 振动与冲击. 2025(04): 1-9+60 . 
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