Responses of SDOF Structures With SPIS-Ⅱ Dampers Under Random Seismic Excitation
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摘要:
提出了基于Clough-Penzien谱的混联Ⅱ型惯容结构地震动响应封闭解的计算方法，并基于所提方法研究了单自由度混联Ⅱ型惯容结构的减震性能及影响因素。首先，建立混联Ⅱ型惯容结构的运动方程，获得了结构位移与惯容出力等结构响应频域解的统一表达式。其次，基于有理式分解与留数定理分别获得了频率响应特征值函数与Clough-Penzien谱的二次正交式，进而获得了结构响应功率谱的二次正交式。最后，得到了结构与惯容随机地震动响应0 ~ 2阶谱矩的简明封闭解。利用所提方法和虚拟激励法分别对一算例进行分析，验证了所提方法的正确性；同时，利用所提方法分析了惯容各参数对结构抗震性能的影响。研究表明：该文方法获得的响应解为封闭解，在计算效率与计算准确性上都优于虚拟激励法。当μω不变时，随着μm与μξ增大，减震性能增强；当μω=1时，对减震性能的影响最大。
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关键词:
- 惯容 /
- Clough-Penzien谱 /
- 二次正交化 /
- 谱矩 /
- 封闭解
Abstract:A closed-form solution of responses of SDOF structures with SPIS-Ⅱ dampers under seismic excitation modeled with the Clough-Pezien spectrum was proposed, and the shock absorption performance and influential factors of this system were studied based on the proposed method. Firstly, the motion equation for the SPIS-Ⅱ damper was established, and the unified expressions of frequency domain solutions of structural responses, such as the structural displacement and the inerter force, were obtained. Secondly, based on the rational expression decomposition and the residue theorem, the quadratic orthogonal equations of the frequency response eigenvalue function and the Clough-Pezien spectrum were obtained respectively, and in turn the quadratic orthogonal equation of the structural response power spectrum was deduced. Thirdly, the concise closed-form solutions of the 0~2nd-order spectral moments of the structural responses were acquired. The proposed method and the virtual excitation method were used to analyze a case respectively, which verifies the correctness of the proposed method. Finally, the proposed method was used to analyze the effects of the inerter parameters on the seismic performances of the structure. The research shows that, the proposed method gives closed-form solutions better than those given by the virtual excitation method in terms of computation efficiency and accuracy. The damping performance will improve with the increase of μm and μξ for a constant μω and the damping performance will reach the optimum for μω=1.
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Key words:
- inerter /
- Clough-Penzien spectrum /
- quadratic orthogonalization /
- spectral moment /
- closed-form solution
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图 4 惯容质量
${m_{{\text{in}}}}$ 对结构位移的影响Figure 4. Effects of inerter coefficient
${m_{{\text{in}}}}$ on structural displacements
图 6 惯容阻尼
${c_{\text{d}}}$ 对位移的影响Figure 6. Effects of inerter damping
${c_{\text{d}}}$ on displacements
图 5 惯容刚度
${k_{\text{s}}}$ 对结构位移的影响Figure 5. Effects of inerter stiffness
${k_{\text{s}}}$ on structural displacements表 1 谱矩对比
Table 1. Comparison of spectral moments between the presented method and the virtual excitation method
calculation method integration step $\varDelta/( { {\text{rad/s} } } )$ spectral moment ${\alpha _{x,0} }/( {10^{ - 5} }\;{ {\text{m} }^2})$ ${\alpha _{x,1} }/( {10^{ - 4} }\;{ {\text{m} }^{\text{2} } }/{\text{s} })$ ${\alpha _{x,2}}$ $/( { { {\text{m} }^{\text{2} } }/{ {\text{s} }^2} } )$ present method – 7.369214319 1.721475154 0.040331515 PEM 0.01 7.369214317 1.721475145 0.040331509 0.1 7.247097750 1.692932655 0.039664397 0.5 1.465220153 3.434894652 0.080641421 
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表 2 惯容出力对比
Table 2. Comparison of inerter forces between the virtual excitation method and this method
calculation method integration interval integration step inerter force error present method – – 2.466×102 kN – PEM [0, 500] 0.01 rad/s 2.466×102 kN 0 0.05 rad/s 2.466×102 kN 0 0.5 rad/s 2.314×102 kN 4.6%
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