A TSHBM method and its application in response calculation of dry friction damped blades
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摘要:
建立了以时间谱形式的谐波平衡法为核心的新型非线性强迫振动响应高效计算方法，并应用于含接触界面的干摩擦阻尼叶片结构响应分析。基于时间谱形式的谐波平衡法的原理和特点，建立非线性强迫振动响应通用求解方案；根据接触非线性的特性，提出了干摩擦力及解析雅可比矩阵计算的适应性处理方案，形成了含接触界面的叶片结构新型非线性振动响应高效预测方法。数值仿真结果表明：在带燕尾型榫根的叶片单扇区有限元模型中，分别保留1阶和3阶谐波阶次时，该方法的非线性振动响应计算时间消耗相较于传统的多谐波平衡法分别削减37%和46%，因此该方法在易用性、可推广性和计算效率方面具有独特的优势。
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关键词:
- 响应预测 /
- 叶片结构 /
- 干摩擦阻尼 /
- 多谐波平衡法 /
- 时间谱形式的谐波平衡法
Abstract:A novel nonlinear forced vibration response calculation method based on the time-spectral form of harmonic balance method (TSHBM) was established and applied to the response analysis of dry friction damped blades structure with contact interfaces. A general framework for the nonlinear forced response computation was established based on the TSHBM, of which the principles and characteristics were emphasized. In order to appropriately account for the contact nonlinearity, the computational schemes for dry friction force and the construction of an analytic Jacobian matrix were proposed. The novel computational method for nonlinear forced responses of blades with contact interfaces was built. The numerical simulation results showed that for the FE model of a bladed disk sector with dovetail joint, when the 1st and 3rd harmonic orders were respectively reserved, the nonlinear vibration response calculation time of this method was reduced by 37% and 46%, respectively, compared with traditional multi-harmonic balance method. So this method has unique advantages in terms of ease of use, generalizability and computational efficiency.
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图 2 带干摩擦单元的悬臂梁模型及正切函数描述的干摩擦力曲线
Figure 2. Cantilever beam with dry friction element and dry friction force curve described by tangent function
图 3
${\tilde {\boldsymbol{U}}_{{\mathrm{tip}}}}$ 的伪时间域历程曲线Figure 3. Pseudo-time history of
${\tilde {\boldsymbol{U}}_{{\mathrm{tip}}}}$ 
图 5 不同方法计算的带干摩擦单元的悬臂梁模型的幅频曲线
Figure 5. Amplitude-frequency curve of cantilever beam with dry friction element by different methods
图 6 不同方法结合弧长延拓法计算的杜芬振子的幅频曲线
Figure 6. Amplitude-frequency curve of duffing oscillator by different methods with arc-length continuation
图 8 不同方法计算的带燕尾型榫根的叶片单扇区模型的幅频曲线
Figure 8. Amplitude-frequency curve of bladed disk sector with dovetail joint by different methods
表 1 带干摩擦单元的悬臂梁模型的幅频曲线的归一化计算时间
Table 1. Normalized computational time of amplitude-frequency curve of cantilever beam with dry friction element
计算方法 保留谐波阶次 ${N_{\rm{h}}} = 1$ ${N_{\rm{h}}} = 3$ ${N_{\rm{h}}} = 5$ ${N_{\rm{h}}} = 7$ MHBM 1.000 4.247 15.493 66.795 MHBM ACC. 0.139 0.371 0.746 1.559 TSHBM-NR 0.038 0.082 0.171 0.309 TSHBM-PTM 0.043 0.085 0.184 0.361 
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表 2 杜芬振子系统的幅频曲线的归一化计算时间
Table 2. Normalized computational time of amplitude-frequency curve of duffing oscillator
计算方法 保留谐波阶次 ${N_{\rm{h}}} = 1$ ${N_{\rm{h}}} = 3$ ${N_{\rm{h}}} = 5$ ${N_{\rm{h}}} = 7$ MHBM 1.000 1.713 2.631 4.233 MHBM ACC. 0.719 0.984 1.161 1.454 TSHBM-NR 0.539 0.650 0.681 0.779
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表 3 主要模型参数
Table 3. Model parameters
参数 数值 弹性模量E/1011 Pa 2.1 泊松比μe 0.3 材料密度ρ/（kg/m3） 7980 β-阻尼系数/10−5 1 切向接触刚度kt/105 （N/m） 1 法向接触刚度kn/105 （N/m） 1 摩擦因数μ 0.3
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表 4 带燕尾型榫根的叶片单扇区模型的幅频曲线的归一化计算时间
Table 4. Normalized computational time of amplitude-frequency curve of bladed disk sector with dovetail joint
计算方法 保留谐波阶次 ${N_{\rm{h}}} = 1$ ${N_{\rm{h}}} = 3$ MHBM ACC. 1.000 6.085 TSHBM-NR 0.926 5.404 TSHBM-PTM 0.628 3.304
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