Performance Test and Comparison of MEMS Accelerometers for Vibration Monitoring of High-Rise Building
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摘要: 为开展MEMS型加速度传感器在超高层建筑振动监测应用中的性能对比测试，选取4种不同类型MEMS型加速度传感器与G1B型力平衡式加速度传感器，将其安装在地王大厦相同测点，对MEMS型、G1B型加速度传感器记录的结构环境振动数据进行时程、频谱和模态频率对比分析，并对其记录的结构地震响应进行时域及频域对比。研究结果表明，不同类型MEMS型加速度传感器仪器噪声均大于G1B型加速度传感器，其中MEMS-I型加速度传感器噪声水平相对较小，与G1B型加速度传感器模态频率识别结果及地震响应监测数据吻合较好，验证了MEMS-I型加速度传感器可较准确地记录到结构强振动响应，适用于超高层建筑日常结构振动监测。
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关键词:
- 超高层建筑 /
- MEMS型加速度传感器 /
- 结构振动 /
- 模态频率 /
- 性能测试
Abstract: Field test was carried out in this study to evaluate and compare the performance of MEMS accelerometers for the vibration monitoring of high-rise buildings. Four MEMS accelerometers of different types and a G1B-type force-balanced accelerometer were selected and installed on the same location in Diwang Building for testing. The ambient vibration of the building recorded by the MEMS accelerometers was compared with the G1B accelerometer in terms of time history, Fourier spectrum and modal frequency identification. In addition, the earthquake-induced structural responses of the high-rise building recorded by MEMS-I type accelerometer was compared with those recorded by the G1B accelerometer in both time and frequency domain. The test results show that the noise levels of the four MEMS accelerometers are all higher than that of the G1B accelerometer. Among them, the MEMS-I type accelerometer has a relatively lower level of sensor noise and achieves a good match with the G1B accelerometer in terms of modal frequency identification and earthquake-induced structural vibration measurement, which proves that the MEMS-I type accelerometer can record the strong structural vibration with acceptable accuracy and is feasible for daily vibration monitoring of the high-rise buildings.-
Key words:
- High-rise building /
- MEMS accelerometer /
- Structural vibration /
- Modal frequency /
- Field test
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引言
关于场地地震反应的分析已有大量研究成果，研究表明土壤在地震作用下会表现出材料非线性效应ADDIN EN.CITE.DATA（Joyner等，1975；Huang等，2001；Arslan等，2006；Hosseini等，2012）。等效线性化方法ADDIN EN.CITE.DATA（Schnabel等，1972；Idriss等，1992；Bardet等，2000；王笃国等，2016）是一种频域方法，通过在不同土体应变条件下选择等效阻尼比和剪切模量，将非线性问题转化为线性问题。当采用材料非线性本构模型描述土体非线性时，需采用时间积分算法求解非线性动力有限元方程。时间积分算法可分为隐式方法和显式方法。隐式算法每时刻需求解线性代数方程组，计算效率相对较低，如Wilson-θ法和Newmark法等。显式算法无需求解线性代数方程组，适合于强非线性和自由度数目较大的问题。研究者已提出多种显式时间积分算法ADDIN EN.CITE.DATA（Chung等，1994；王进廷等，2002；Belytschko等，2014）。作者近期提出一种二阶精度的单步显式算法，该算法适合变时步问题，在线弹性范围内稳定性较好。本文将该算法推广至求解非线性动力有限元方程中，并将其应用于地震波垂直入射时非线性地震反应分析。
1. 非线性动力有限元方程的显式时间积分算法
设已知非线性体系第${t_i}$时步的受力状态，求解第${t_{i + 1}}$时步的非线性结构动力学方程：
$${\boldsymbol{M}}{{\boldsymbol{\ddot u}}_{i + 1}}{\boldsymbol{ + C}}{{\boldsymbol{\dot u}}_{i + 1}} + {\boldsymbol{f}}_{i + 1}^S{\boldsymbol{ = }}{{\boldsymbol{f}}_{i + 1}}$$ (1) 式中M、C、${{\boldsymbol{f}}^S}$和${\boldsymbol{f}}$分别表示非线性体系的质量矩阵、阻尼矩阵、内力向量和外荷载向量；u表示位移，点号对时间t求导，i+1表示第${t_{i + 1}}$时刻。第i+1时刻时间步长为：
$${\boldsymbol{\Delta }}{t_i} = {t_{i + 1}} - {t_i}$$ (2) 文献显式方法求解非线性方程（1）的过程如下，第i+1时刻位移${{\boldsymbol{u}}_{i + 1}}$为：
$${{\boldsymbol{u}}_{i + 1}} = {{\boldsymbol{u}}_i} + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{t_i}{{\boldsymbol{\dot u}}_i} + \frac{{\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{t_i}^2}}{2}{{\boldsymbol{\ddot u}}_i}$$ (3) 第i+1时刻位移增量$\mathit{\Delta }{{\boldsymbol{u}}_i}$、内力增量$\mathit{\Delta }{\boldsymbol{f}}_i^S$和内力全量${\boldsymbol{f}}_{i + 1}^S$分别为：
$$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{{\boldsymbol{u}}_i} = {{\boldsymbol{u}}_{i + 1}} - {{\boldsymbol{u}}_i}$$ (4) $$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{f}}_i^S = {\boldsymbol{f}}(\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{{\boldsymbol{u}}_i})$$ (5) $${\boldsymbol{f}}_{i + 1}^S = {\boldsymbol{f}}_i^S + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{f}}_i^S$$ (6) 第i+1时刻预估速度${{\boldsymbol{\dot {\tilde u}}}_{i + 1}}$、预估加速度${{\boldsymbol{\ddot {\tilde u}}}_{i + 1}}$、速度${{\boldsymbol{\dot u}}_{i + 1}}$和加速度${{\boldsymbol{\ddot u}}_{i + 1}}$分别为
$${{\boldsymbol{\dot {\tilde u}}}_{i + 1}} = {{\boldsymbol{\dot u}}_i} + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{t_i}{{\boldsymbol{\ddot u}}_i}$$ (7) $${{\boldsymbol{\ddot {\tilde u}}}_{i + 1}} = {{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{f}}_{i + 1}} - {\boldsymbol{C\dot {\tilde u}}}_{i + 1}^{} - {\boldsymbol{f}}_{i + 1}^S)$$ (8) $${{\boldsymbol{\dot u}}_{i + 1}} = {{\boldsymbol{\dot u}}_i} + \frac{{\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{t_i}}}{2}({{\boldsymbol{\ddot u}}_i} + {{\boldsymbol{\ddot {\tilde u}}}_{i + 1}})$$ (9) $${{\boldsymbol{\ddot u}}_{i + 1}} = {{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{f}}_{i + 1}} - {\boldsymbol{C\dot u}}_{i + 1}^{} - {\boldsymbol{f}}_{i + 1}^S)$$ (10) 式（3）—式（10）为求解式（1）的显式算法。算法中需由位移增量计算内力增量，目前常用的应力计算方法包括向前欧拉法、向后欧拉法和完全隐式计算法等ADDIN EN.CITE.DATA（Sloan等，1992；2001；Ahadi等，2003）。下面给出式（5）由位移增量计算内力增量的过程，即一种带误差控制的修正欧拉算法。
对于每个有限单元，由位移增量$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{u}}_i^e$计算应变增量$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_i^e$的表达式为：
$$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_i^e = {{\boldsymbol{B}}^e}\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{u}}_i^e$$ (11) 式中Be为应变矩阵。将ti时刻单元应变增量$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_i^e$赋值给子步应变增量$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e$，ti时刻单元应力${\boldsymbol{ \pmb{\mathit{ σ}} }}_i^e$赋值给${\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e$，初始化子步应变增量和应力状态分别为：
$$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e \leftarrow \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_i^e$$ (12) $${\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e \leftarrow {\boldsymbol{ \pmb{\mathit{ σ}} }}_i^e$$ (13) 每个子步中应力增量计算思路见图 1，具体计算公式如下：
$${\boldsymbol{D}}_1^e = {\boldsymbol{D}}({\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e)$$ (14) $$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_1^e = {\boldsymbol{D}}_1^e\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e$$ (15) $${\boldsymbol{D}}_2^e = {\boldsymbol{D}}({\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_1^e)$$ (16) $$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_2^e = {\boldsymbol{D}}_2^e\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e$$ (17) $$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_s^e = \frac{{\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_1^e + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_2^e}}{2}$$ (18) 式中${{\boldsymbol{D}}^e}$为单元应力-应变关系矩阵。判断每个子步中应力增量$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{{\boldsymbol{ \pmb{\mathit{ σ}} }}_s}$是否符合精度要求的误差判断式为：
$${e_r} = \frac{{\left\| {\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_1^e - \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_2^e} \right\|}}{{\left\| {{\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_s^e} \right\|}}$$ (19) 判断误差er是否小于预先给定的判断值st，条件不满足时，缩小子步应变增量为：
$$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e \leftarrow A\sqrt {{{{s_t}} \mathord{\left/ {\vphantom {{{s_t}} {{e_r}}}} \right. } {{e_r}}}} \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e$$ (20) 式中A为误差峰值系数。采用缩小的子步应变增量重新进行式（14）—式（19）的计算与判断，循环直至满足精度要求，更新剩余应变增量和应力状态分别为：
$$\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_i^e \leftarrow \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_i^e - \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ ε}} }}_s^e$$ (21) $${\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e \leftarrow {\boldsymbol{ \pmb{\mathit{ σ}} }}_{i + 1}^e + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}{\boldsymbol{ \pmb{\mathit{ σ}} }}_s^e$$ (22) 利用更新剩余应变增量和应力状态循环执行式（14）—式（20），直至剩余应变增量小于等于零结束。
利用求得的第i+1时刻单元应力可得到单元应力增量和内力增量分别为：
$$ \Delta \boldsymbol{\sigma }_i^e = \boldsymbol{\sigma }_{i + 1}^e - \boldsymbol{\sigma }_i^e $$ (23) $$ \Delta {\boldsymbol{f}}_i^S{\rm{ = }}\sum\limits_e {\int {{{\boldsymbol{B}}^{e{\rm{T}}}}\boldsymbol{\Delta }{\boldsymbol{\sigma }}_i^e{\bf{d}}A} } $$ (24) 2. 地震波垂直入射时场地非线性地震反应分析
本节将上述非线性有限元方程的显式时间积分算法应用于地震波垂直入射时场地非线性地震反应分析中。假定基岩为线弹性半空间，考虑基岩上覆土层的材料非线性，不考虑土体阻尼。在土层下部设置黏性边界条件模拟半空间基岩的辐射阻尼，并在该处以等效结点力的方式实现地震动输入。
计算模型见图 2，选取A点作为观测点。土体非线性材料本构模型选取邓肯-张模型，土体线弹性参数见表 1，未给出配套的非线性参数，故算例中的非线性参数参考实际情况选取，后续研究中将使用更真实表现土体非线性行为的本构模型及真实工程场地参数。算例中的大气压参数取100kPa，内摩擦角增量取0°。入射地震动分别选取狄拉克脉冲和实测地震动（Gilroy Array #3，Coyote Lake, 1979）。入射狄拉克脉冲见图 3，观测点结果见图 4，实测地震动见图 5，观测点结果见图 6。图 4、图 6中给出采用中心差分法的计算结果作为参考解，由图 4、图 6可知，本文算法与中心差分法计算结果吻合较好，说明本文算法的有效性。
图 2 大开车站沿线土层纵断面构造Figure 2. Site condition of the Daikai subway station in vertical direction表 1 土层参数Table 1. Parameters of soils土质 深度/
m$\rho $/
（g/cm3）cs /
（m/s）v
-EN
-Rf
-c/
（MPa）θ/（°） D
-F
-人工填土 0—1.0 1.9 140 0.33 0.33 0.758 0.084 26.9 1.06 0.021 全新世砂土 1.0—5.1 1.9 140 0.32 0.33 0.758 0.084 26.9 1.06 0.021 全新世砂土 5.1—8.3 1.9 170 0.32 0.36 0.768 0.120 31.0 1.11 0.015 更新世粘土 8.3—11.4 1.9 190 0.40 0.44 0.822 0.188 28.4 1.01 0.012 更新世粘土 11.4—17.2 1.9 240 0.30 0.44 0.822 0.188 28.4 1.01 0.012 更新世砂土 17.2—22.2 2.0 330 0.26 0.51 0.840 0.300 30.0 1.02 0.011 基岩 ＞22.2 2.0 330 0.26 - - - - - - 
图 3 狄拉克脉冲速度和加速度时程图Figure 3. Velocity and acceleration time history of the Dirac pulse
图 4 狄拉克脉冲入射时场地反应分析结果Figure 4. Results of site analysis under the incident of Dirac pulse
图 5 实测地震动速度和加速度时程图Figure 5. Velocity and acceleration time history of the seismic motion
图 6 实测地震动入射时场地反应分析结果Figure 6. Results of site reaction analysis under the incident of the seismic motion表 1中ρ、cs、v、EN、Rf、c、θ为模型参数，分别表示密度、剪切波速、泊松比、无量纲幂次、破坏比、土的内聚力、土的摩擦角。D、F为试验常数。
3. 结论
本文发展一种求解材料非线性结构动力学方程的显式时间积分算法，并应用于地震波竖直入射时非线性地震反应分析中，通过算例验证了该方法的有效性。该显式算法具有无需对角阻尼矩阵、单步、稳定性良好等优点。本文考虑了邓肯-张非线性弹性本构模型，下步研究可考虑将该显式算法扩展到弹塑性本构模型及更能反映土层真实变形的本构模型中。
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图 3 不同类型加速度传感器加速度时程曲线
Figure 3. Acceleration time histories of different sensors in x and y directions
图 4 不同传感器x、y向加速度傅里叶谱
Figure 4. Fourier spectrum of acceleration measurement of different sensors in x and y directions
图 5 不同传感器x、y向加速度傅里叶谱幅值相对误差对比
Figure 5. Relative errors of peak Fourier spectrum amplitudes of different sensors
图 6 基于频域分解法的模态频率识别流程
Figure 6. Flowchart of the FDD-based modal frequency identification method
图 7 不同传感器监测数据时频能量云图
Figure 7. Time-frequency domain color map of the vibration energy of different sensor measurement
图 8 G1B型、MEMS-I型加速度传感器监测的结构地震响应时程曲线
Figure 8. Comparison of earthquake-induced structural responses measured by the G1B and MEMS-I accelerometers
图 9 G1B型、MEMS-I型加速度传感器监测的结构地震响应局部波形
Figure 9. Detailed comparison of earthquake-induced structural responses measured by the G1B and MEMS-I accelerometers
图 10 G1B型、MEMS-I型加速度传感器监测的结构地震响应频谱曲线
Figure 10. Comparison of Fourier spectrum of earthquake-induced structural responses measured by the G1B and MEMS-I accelerometers
表 1 传感器技术参数对比
Table 1. Comparison of parameters of different sensors
传感器类型 测量范围/g 频响范围/Hz 动态范围/dB 噪声均方根/(${\text{μ}}{{\rm{g}}}/\sqrt{\rm{H}\rm{z} }$) 功耗/W G1B ±3 0～100 >130 0.5 3 MEMS-I ±2.5 0～80 >90 10.0 <2 Palert-Plus ±2 0～100 >100 25.0 2 AC217 ±4 0～100 >104 25.0 <1 Palert-Advance ±2 0～100 >90 25.0 3 
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表 2 不同传感器
$x $ 、$y $ 向加速度时程的均方根Table 2. RMS of acceleration measurement of different sensors in
${\boldsymbol{x}} $ and${\boldsymbol{y}} $ directions方向 传感器类型 G1B型 MEMS-I型 Palert-Plus型 AC217型 Palert-Advance型 x 0.015 0.037 0.040 0.048 0.052 y 0.018 0.038 0.042 0.048 0.059
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表 3 G1B型加速度传感器实测自振频率识别结果与已有研究结果对比
Table 3. Comparison of modal frequency identification results between G1B accelerometer and references
阶数 G1B型加速度传感器
实测自振频率/Hz郭西锐等（2016）
自振频率/Hz与郭西锐等（2016）
研究的相对误差/%徐枫等（2014）
自振频率/Hz与徐枫等（2014）
研究的相对误差/%1 0.168 6 0.169 7 0.65 0.168 9 0.18 2 0.198 4 0.198 9 0.25 0.199 3 0.45 3 0.276 9 0.277 8 0.32 0.278 2 0.47 4 0.540 2 0.539 4 0.15 0.538 3 0.35 5 0.648 5 0.649 4 0.14 0.642 2 0.98 6 0.676 7 0.677 2 0.07 — — 7 0.841 5 0.844 7 0.38 0.839 3 0.26 8 1.169 0 1.179 0 0.85 1.168 0 0.09 9 1.498 0 1.498 0 0.00 — — 10 1.582 0 1.591 0 0.57 — — 11 1.834 0 1.844 0 0.54 1.852 0 0.97 12 1.929 0 1.943 0 0.72 1.942 0 0.67 13 1.965 0 1.972 0 0.35 1.962 0 0.15
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表 4 不同类型加速度传感器监测数据的模态频率识别结果对比
Table 4. Comparison of modal frequencies identified from the measurement of different sensors
阶数 G1B型加速度传感器监测数据的模态频率/Hz MEMS-I型加速度传感器监测数据的模态频率/Hz MEMS-I型与G1B型加速度传感器监测数据的模态频率相对误差/% Palert-Plus型加速度传感器监测数据的模态频率频率/Hz Palert-Plus型与G1B型加速度传感器监测数据的模态频率相对误差/% AC217型加速度传感器监测数据的模态频率频率/Hz AC217型与G1B型加速度传感器监测数据的模态频率相对误差/% Palert-Advance型加速度传感器监测数据的模态频率频率/Hz Palert-Advance型与G1B型加速度传感器监测数据的模态频率相对误差/
%1 0.168 4 0.168 3 0.059 0.168 0 0.238 0.169 1 0.416 0.169 9 0.891 2 0.198 5 0.198 5 0.000 0.198 3 0.101 0.198 6 0.050 0.199 8 0.655 3 0.277 4 0.277 3 0.036 0.277 6 0.072 0.277 1 0.108 0.279 5 0.757 4 0.540 7 0.540 8 0.018 0.539 8 0.166 0.541 1 0.074 — — 5 0.647 8 0.647 8 0.000 0.648 4 0.093 0.648 1 0.046 0.647 9 0.015 6 0.675 9 0.675 9 0.000 0.676 3 0.059 0.675 8 0.015 0.676 9 0.148 7 0.843 3 0.843 1 0.024 0.843 1 0.024 0.842 9 0.047 0.843 4 0.012 8 1.171 8 1.169 8 0.171 1.170 3 0.128 1.170 2 0.137 1.168 6 0.273 9 1.493 7 1.491 2 0.167 1.491 5 0.147 1.491 1 0.174 1.491 0 0.181 10 1.583 1 1.581 4 0.107 1.581 7 0.088 1.583 5 0.025 1.584 2 0.069 11 1.839 4 — — — — — — — — 12 1.933 0 1.933 0 0.000 1.932 5 0.026 1.933 0 0.000 1.931 3 0.088 13 1.964 6 1.964 4 0.010 1.963 2 0.071 1.964 0 0.031 1.962 7 0.097
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