DAMPING RATIO IDENTIFICATION OF BEAM BRIDGE BASED ON VEHICLE BRIDGE COUPLING THEORY UNDER SINGLE VEHICLE
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摘要: 提出一种在单辆通行车激励的情况下, 基于车桥耦合理论识别桥梁阻尼比的方法. 首先, 将集成化检测车设计为单自由度体系, 利用集成在检测车上的传感器采集信号并滤波获取桥梁一阶响应信号, 通过相邻峰值点做比值并取平均得到桥梁一阶阻尼比, 理论推导出检测车停靠在桥面任一位置均可识别出桥梁一阶阻尼比. 其次, 考虑通行车车速、检测车停靠位置、环境噪音、桥梁阻尼比和通行车阻尼对识别结果的影响, 通过数值模拟验证该方法的有效性, 并与工程上经典的随机子空间法和文献方法进行对比. 最后, 在梁式桥实桥试验进行初步验证, 说明本文方法的工程适用性. 研究结果表明: 所提方法对检测车在桥面上的停靠位置没有具体要求, 避免了路面粗糙度对阻尼比识别结果的影响, 通行车的阻尼和车速在一定范围内对阻尼比识别无明显影响, 并且在信噪比30 dB以上可识别出桥梁一阶阻尼比, 具有较好的鲁棒性, 操作简单且不需要封桥, 可有效促推在实际工程中进行桥梁一阶阻尼比识别.Abstract: This paper comes up with a new method for identifying the bridge damping ratio based on the vehicle bridge coupling theory under the excitation of a single passing vehicle. Firstly, the integrated inspection vehicle is designed as a single degree of freedom system. The accelerometers integrated on the inspection vehicle are used to collect signals and filter to obtain the bridge frequency response signal of the first mode. The ratio of adjacent peak points is calculated and averaged to obtain the bridge damping ratio of the first mode. Theoretical derivation shows that the inspection vehicle can identify the bridge damping ratio of the first mode when the inspection vehicle parked at any position on the bridge deck. Secondly, considering the influence of the passing vehicle speed, parking position of the inspection vehicle, environmental noise, bridge damping ratio of the first mode and the damping of passing vehicle on the identification results, the effectiveness of this method is verified by the numerical simulation. And comparative studies were also conducted on the classical stochastic subspace method and literature method. Finally, the field test of beam bridge was used to demonstrate the practical applicability of the proposed new method. The research results indicate that the proposed method has no specific requirement on the parking position of the inspection vehicle, avoids the influence of the road roughness on the result of the damping ratio identification, and the passing vehicle damping and speed have no significant influence on the identification of the damping ratio within a certain range, besides, the proposed approach can identify the bridge damping ratio of the first mode with SNR ≥ 30 dB. It has good robustness, simple operation, and does not require bridge closures. It can significantly promote the identification of bridge damping ratio of the first mode in practical engineering.
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Keywords:
- passing vehicle excitation /
- vehicle bridge coupling /
- damping ratio /
- beam bridge
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引 言
阻尼比是桥梁结构动力特性中的重要参数[1], 表征结构能量耗散和振动衰减程度, 同时也是评价桥梁安全性能的重要指标[2]. 工程应用中, 由于客观条件和测试机理、阻尼模型及实测技术等方面落后于其他研究的影响, 导致阻尼比识别精度不高, 限制其在桥梁结构动力参数识别方面的应用[3]. 因此, 研究识别桥梁阻尼比的理论和方法很有必要, 且对桥梁安全评估有着重要意义[4-5].
目前获取桥梁阻尼比主要包括直接和间接测量二方面[6]. 传统的直接测量法是将传感器直接安装在桥梁结构待测点处, 通过主动或者被动激励桥梁, 获取桥梁结构动力响应, 进而对信号进行分析得到模态信息[7]. Kim等[8]结合自然激励技术(NExT)和特征实现算法(ERA), 基于主跨安装的大量传感器数据获取了桥梁运营状态下一阶阻尼比. Xiao等[9]将15个加速度传感器沿桥梁长度放置在中心线上, 同步采集后采用自由衰减法计算桥梁动力特性, 并使用希尔伯特−黄变换和快速傅里叶变换识别桥梁固有频率和一阶阻尼比. 封周权等[10]推导出负对数似然函数的Hessian矩阵解析式, 提取和计算变异系数来量化模态参数最佳估计的不确定性, 基于某斜拉桥斜拉索上布置加速度计采集响应信号, 采用该方法在适当情况下获取该索一阶阻尼比最佳估计. 孙倩等[11]构建桥梁结构响应功率谱传递比函数, 识别了特定谐波激励下的模型一阶阻尼比初步算例. 直接测量法能在一段时间内监测桥梁状态的发展与变化, 对于重要桥梁监测十分必要. 但对于量大面广的中小跨径桥梁集群监测, 也存在安装众多传感器、调试设备费时费力及人力消耗成本等问题[12-15].
为了改进传统直接量测法成本高、耗时长等特点, 近年来基于图像摄影、激光扫描等非接触式间接量测法受到了大量研究人员的关注. Kim等[16]基于视觉的无人机位移测量系统, 将摄像机与目标之间的距离最大设置为2.5 m, 并与传统的激光测距传感器进行对比, 结果表明两者识别出的阻尼比相似, 但该方法也需要消除无人机本身运动带来的影响, 需更进一步研究. 朱天煦等[17]针对激光扫描测试中的噪声问题, 提出模态峰值汉克尔奇异值分解的阻尼比识别方法, 在多次累加迭代恢复模态信息后, 有效识别出结构的一阶阻尼比. Yang等[18]设计出双轴移动试验车, 在车上配备一定数量且均匀布置的加速度计和激光传感器, 利用两个独立扫描点之间时间引起的衰减特性, 使用希尔伯特变换识别出桥梁一阶阻尼比, 但该方法需要布设多个传感器保证测试精度, 实验经济性和便利性不足. 阳洋等[19]通过假定的桥梁阻尼比值提取假定的桥梁第一阶振型, 通过不断试算循环判断该第一阶振型最大值点是否居中来识别桥梁真实阻尼比, 并基于数值算例和实桥试验进行验证. Xu等[20]推导出单对称梁垂直、横向和扭转振动响应的封闭解析解, 通过车体信号反算接触点响应来分离桥梁的垂直频率和弯扭频率进而识别薄壁桥梁一阶阻尼比, 并通过数值模拟进行分析并得到验证. 曲春绪等[21]基于频域分解法正交模态分离来识别桥梁阻尼比, 当各阶模态分离的越彻底识别结果越精准. Yang等[22]设计椭圆滤波器从接触点响应获得窄带信号从而改进传统的间接量测法, 结果表明采用两辆质量更大的桥梁激励车识别出的阻尼比更准确. 上述识别桥梁阻尼比的方法已经取得了丰硕的程度, 然而路面粗糙度、环境噪音等因素依旧没有较好地解决, 难以实现对桥梁阻尼比的高精度识别[23-26].
有鉴于此, 本文提出一种基于车桥耦合技术识别桥梁阻尼比新方法, 以一辆静止检测车停靠在桥面除支座外的任一位置, 基于单辆通行车激励, 采集桥梁响应信号进而计算得到桥梁一阶阻尼比. 从而改进直接测量法需要布置大量传感器费时费力的不足, 解决车桥耦合间接量测法路面粗糙度参数影响模态识别等问题. 本文在既有车桥耦合理论[27-30]的基础上, 推导单车−桥耦合接触点响应解析解与桥梁阻尼比的关系, 从理论上说明本文方法的可行性. 然后采用数值模型分别考虑通行车车速、检测车停靠位置、环境噪音、桥梁阻尼比和通行车阻尼比等参数对阻尼比识别的影响, 并与文献[19]基于假定的振型识别阻尼比方法和模态识别常用的随机子空间法[31-33]识别结果进行对比, 进一步说明本文方法的有效性. 最后在实桥试验中对该方法进行验证, 说明本文方法的工程适用性.
1. 理论基础
1.1 单辆通行车激励下的桥梁阻尼比识别
运营中的桥梁激励源主要以随机交通载荷为主, 建立如图1所示的静止检测车简化模型, 车辆被建模为单自由度弹簧−质量系统[34]. 单辆通行车质量为${m_v}$, 刚度为${k_v}$, 阻尼为${c_v}$, ${q_v}$为通行车的绝对竖向位移, 以恒定速度$v$行驶在桥面,${f_c}\left( t \right)$为通行车随时间变化作用于桥梁上的载荷. 检测车的质量、刚度和阻尼分别为${m_{v1}},$ ${k_{v1}}$和${c_{v1}}$, 距离桥梁左端的距离为${d_1}$, ${q_{v1}}$为检测车的绝对竖向位移. 桥梁被简化为欧拉简支梁, 桥长为$L$, 抗弯刚度为$EI$, 单位长度质量为$\bar m $, 阻尼为$c$, 桥面的粗糙度用$r\left( x \right)$表示, ${u_b}$为桥梁的绝对竖向位移. 假定静止采集的检测车与桥梁是永久接触的, 检测车相较于桥梁的质量可以忽略不计[35].
桥梁振动微分方程为
$$ \begin{split} & \bar m {{\ddot u}_b}\left( {x,t} \right) + c{{\dot u}_b}\left( {x,t} \right) + EIu''''_b\left( {x,t} \right)= \\ &\qquad {f_c}\left( t \right)\delta \left( {x - vt} \right) \end{split} $$ (1) 通行车振动微分方程为
$$ \begin{split} & {m_v}{{\ddot q}_v}(t) + {c_v}({{\dot q}_v}(t) - {{\dot u}_b}(x,t)\left| {_{x = vt}} \right. - \dot r(x)\left| {_{x = vt}} \right.) + \\ &\qquad {k_v}({q_v}(t) - {u_b}(x,t)\left| {_{x = vt}} \right. - r(x)\left| {_{x = vt}} \right.) = 0 \end{split} $$ (2) 式中, $(\cdot)={\mathrm{d}}()/{\mathrm{d}}t$表示对时间t的一阶导数, $(\cdot\cdot)= {\mathrm{d}}()/{\mathrm{d}}t$对时间$ t $的二阶导数, $ u_b'''' $为桥梁位移对位置x的4次微分, $\delta $为狄利克雷函数. 其中
$$ \begin{split} & {{{f}}_c}(t) = {k_v}({q_v}(t) - {u_b}(x,t)\left| {_{x = vt}} \right. - r(x)\left| {_{x = vt}} \right.)+ \\ &\qquad {c_v}({{\dot q}_v}(t) - {{\dot u}_b}(x,t)\left| {_{x = vt}} \right. - \dot r(x)\left| {_{x = vt}} \right.) - {m_v}g \end{split} $$ (3) 由振型叠加法知, 桥梁竖向位移可表示为
$$ {u_b}\left( {x,t} \right) = \sum\limits_{j = 1}^\infty {{\phi _j}\left( x \right)} {q_j}\left( t \right) $$ (4) 式中, $ {\phi _j}\left( x \right) = {\text{sin}}\left( {j{\text{π}}x/L} \right) $为桥梁第$ j $阶模态振型, ${q_j}\left( t \right)$是第$ j $阶模态振型对应的广义坐标. 把式(4)代入式(1), 等式两边分别乘以$ {\text{sin}}\left( {j{\text{π}}x/L} \right) $, 并沿着桥梁跨度方向积分[36], 利用振型正交性可得
$$ \begin{split} & {{\ddot q}_j}\left( t \right) + 2{\xi _j}{\omega _j}{{\dot q}_j}\left( t \right) + \omega _j^2{q_j}\left( t \right) = \\ &\qquad \dfrac{{{f_c}\left( t \right)\displaystyle\int_0^L {\delta \left( {x - vt} \right){\text{sin}}\dfrac{{j{\text{π}}x}}{L}{\mathrm{d}}x} }}{{\displaystyle\int_0^L {\bar m {\text{si}}{{\text{n}}^2}\dfrac{{j{\text{π}}x}}{L}{\mathrm{d}}x} }} \end{split} $$ (5) 式中, ${\omega _j}$为桥梁第$ j $阶模态频率, ${\xi _j}$为对应的桥梁第$ j $阶模态阻尼比. 狄利克雷函数$\delta $具有挑选性, 即$\displaystyle\int_{ - \infty }^{ + \infty } {\delta \left( {x - vt} \right){\text{sin}}} \dfrac{{j{\text{π}}x}}{L}{\mathrm{d}}x = {\text{sin}}\dfrac{{j{\text{π}}vt}}{L}$, 把式(3)代入式(5)可得
$$ \begin{split} & {{\ddot q}_j}\left( t \right) + 2{\xi _j}{\omega _j}{{\dot q}_j}\left( t \right) + \omega _j^2{q_j}\left( t \right)= 2{\text{sin}}\left( {\frac{{j{\text{π}}vt}}{L}} \right)\frac{1}{{\bar m L}} \cdot \\ &\qquad \left[ {k_v}\left( {q_v}\left( t \right) - {u_b}\left( {x,t} \right)\left| {_{x = vt}} - r\left( x \right)\right| {_{x = vt}} \right) +\right. \\ &\qquad \left. {c_v}\left( {{\dot q}_v}\left( t \right) - {{\dot u}_b}\left( {x,t} \right)\left| {_{x = vt}} - \dot r\left( x \right)\right| {_{x = vt}} \right) - {m_v}g \right] \end{split} $$ (6) 结合式(2), 不考虑车辆惯性力的贡献, 即车辆的加速度远小于重力加速度g, 可以做出以下简化
$$\begin{split} & \frac{{{k_v}\left( {{q_v}\left( t \right) - {u_b}\left( {x,t} \right)\left| {_{x = vt}} \right. - r\left( x \right)\left| {_{x = vt}} \right.} \right)}}{{\bar m L}}+ \\ &\qquad \frac{{{c_v}\left( {{{\dot q}_v}\left( t \right) - {{\dot u}_b}\left( {x,t} \right)\left| {_{x = vt}} \right. - \dot r\left( x \right)\left| {_{x = vt}} \right.} \right)}}{{\bar m L}} = - \frac{{{m_v}{{\ddot q}_v}}}{{\bar m L}} \approx 0 \end{split} $$ (7) 将式(7)代入到式(6), 则式(6)简化为
$$ {{\ddot q}_j}\left( t \right) + 2{\xi _j}{\omega _j}{{\dot q}_j}\left( t \right) + \omega _j^2{q_j}\left( t \right) = - 2{\text{sin}}\left( {\frac{{j{\text{π}}vt}}{L}} \right)\frac{{{m_v}g}}{{\bar m L}} $$ (8) 考虑到通行车在上桥之前, 桥梁是未受到激励的, 因此假设初始条件${\dot q_j}\left( t \right) = {q_j}\left( t \right) = 0$, 从而求解式(8)二阶非齐次线性微分方程得到桥梁模态坐标响应为
$$\begin{split} &{q_j}\left( t \right) = {\varDelta _{1,j}}{\varDelta _{2,j}}\left\{ {F_{{\omega _d},j}}\left( t \right) +\right.\\ &\qquad \left.{{\mathrm{e}}^{ - {\xi _j}{\omega _j}t}}\left[ 2{\xi _j}{\beta _j}{\text{cos}}\left( {{\omega _{D,j}}t} \right) + {\varDelta _{3,j}}{\text{sin}}\left( {{\omega _{D,j}}t} \right) \right] \right\}\end{split} $$ (9) 其中
$$\left.\begin{split} &{\varDelta _{1,j}} = - \frac{{2{m_v}g{L^3}}}{{EI{j^4}{{\text{π}}^4}}} \\ &{\varDelta _{2,j}} = \frac{1}{{\sqrt {{{\left( {1 - {\beta _j}} \right)}^2} + {{\left( {2{\xi _j}{\beta _j}} \right)}^2}} }} \\ &{\varDelta _{3,j}} = \frac{{{\beta _j}}}{{\sqrt {1 - \xi _j^2} }}\left( {2\xi _j^2 + \beta _j^2 - 1} \right) \end{split}\right\}$$ (10) 式(9)中${F_{{\omega _{d,j}}}}\left( t \right)$是与通行车频率${\omega _{d,j}}$有关的响应, 表达式为
$$ {F_{{\omega _{d,j}}}}\left( t \right) = \left( {1 - \beta _j^2} \right){\text{sin}}\left( {{\omega _{d,j}}t} \right) - 2{\xi _j}{\beta _j}{\text{cos}}\left( {{\omega _{d,j}}t} \right) $$ (11) 式(11)中${\beta _j} = {\omega _{d,j}}/{\omega _{D,j}}$为频率比, ${\omega _{d,j}} = j{\text{π}}v/L$为驱车频率, ${\omega _{D,j}} = {\omega _j}\sqrt {1 - \xi _j^2} $为桥梁第$j$阶模态频率, 以主梁振动为主的桥梁阻尼比在0.03以下[37-38], 因此简化计算${\omega _{D,j}} = {\omega _j}$, 式(9)可以表示为
$$\begin{split} &{q_j}\left( t \right) = {\varDelta _{1,j}}{\varDelta _{2,j}}\left\{ {F_{{\omega _d},j}}\left( t \right) +\right.\\ &\qquad \left.{{\mathrm{e}}^{ - {\xi _j}{\omega _j}t}}\left[ 2{\xi _j}{\beta _j}{\text{cos}}\left( {{\omega _j}t} \right) + {\varDelta _{3,j}}{\text{sin}}\left( {{\omega _j}t} \right) \right] \right\}\end{split} $$ (12) 式(12)对时间$t$求二阶导, 可得
$$\begin{split} &{\ddot q_j}\left( t \right) = {\varDelta _{1,j}}{\varDelta _{2,j}}\left\{ {{\ddot F}_{{\omega _d},j}}\left( t \right) + \right.\\ &\qquad \left.{{\mathrm{e}}^{ - {\xi _j}{\omega _j}t}}\left[ {\varDelta _{4,j}}{\text{sin}}\left( {{\omega _j}t} \right) - {\varDelta _{5,j}}{\text{cos}}\left( {{\omega _j}t} \right) \right] \right\}\end{split} $$ (13) 其中
$$\left.\begin{split} &{\varDelta _{4,j}} = 2{\omega _j}{\xi _j}{\beta _j} + {\varDelta _{3,j}}{\omega _j}{\xi _j} + 2\omega _j^2\xi _j^2{\beta _j} - {\varDelta _{3,j}}\omega _j^2 \\ &{\varDelta _{5,j}} = - 2\xi _j^2{\omega _j}{\beta _j} + {\varDelta _{3,j}}{\xi _j}\omega _j^2 + 2{\xi _j}\omega _j^2{\beta _j} + {\varDelta _{3,j}}{\omega _j} \end{split}\right\}$$ (14) 根据式(4)和式(13)可知, 桥梁加速度响应为
$$\begin{split} & {{\ddot u}_b}\left( {x,t} \right) = \sum\limits_{j = 1}^\infty {\varDelta _{1,j}}{\varDelta _{2,j}}\left\{ {{\ddot F}_{{\omega _d},j}}\left( t \right) +\right.\\ &\qquad \left.{{\mathrm{e}}^{ - {\xi _j}{\omega _j}t}}\left[ {\varDelta _{4,j}}{\text{sin}}\left( {{\omega _j}t} \right) - {\varDelta _{5,j}}{\text{cos}}\left( {{\omega _j}t} \right) \right] \right\}\cdot \\ & \qquad {\text{sin}}\left( {\frac{{j{\text{π}}x}}{L}} \right) \end{split} $$ (15) 式(1) ~ 式(15)即为桥梁竖向加速度响应的理论推导. 检测车与桥梁接触位置即为接触点, 从图1中可以看出, 检测车停靠在$x = {d_1}$处即为车桥接触点. 现场测试时桥梁第一阶模态振型易识别且较为精准, 因此, 取一阶振型对应的接触点响应为
$$\begin{split} & {{\ddot u}_{b,1}}(x,t)\left| {_{x = {d_1}} = {\varDelta _{1,1}}} \right.{\varDelta _{2,1}}\Big\{ {{\mathrm{e}}^{ - {\xi _1}{\omega _1}t}}\left[ {\varDelta _{4,1}}\sin \left( {{\omega _1}t} \right) -\right. \\ &\qquad \left. {\varDelta _{5,1}}\cos \left( {{\omega _1}t} \right) \right] \Big\} \cdot \sin \left( {\frac{{{\text{π}}{d_1}}}{L}} \right) \end{split} $$ (16) 从式(16)可以看出接触点响应信号仅包含桥梁模态信息, 这意味着在车桥耦合系统中提取接触点响应即可获得桥梁模态信息. 因此, 在桥梁模态测试中, 首先把传感器安装在检测车轮轴上收集车体响应信号, 进一步利用车体响应反算得到接触点响应. 检测车的运动微分方程为
$$ \begin{split} & {m_{v1}}{{\ddot q}_{v1}}\left( t \right) + {c_{v1}}({{\dot q}_{v1}}(t) - {{\dot u}_b}(x,t)\left| {_{x = {d_1}}} \right. - \dot r(x)\left| {_{x = {d_1}}} \right.)+ \\ &\qquad {k_{v1}}({q_{v1}}(t) - {u_b}(x,t)\left| {_{x = {d_1}}} \right. - r(x)\left| {_{x = {d_1}}} \right.) = 0 \end{split} $$ (17) 由式(17)可以看出, 在静止采集的情况下, 停靠位置$x$为定值, 而与路面粗糙度有关的变量$ \dot{r}\left(x\right) 与r\left(x\right) $只与检测车停靠位置$x$有关, 故$ \dot{r}\left(x\right)与r\left(x\right) $在式(17)中为定值, 与动采时的$ \dot{r}(x)|{}_{x = vt}和r(x)|{}_{x = vt} $相比, 是不随时间变化而改变的, 即式(17)可以进一步简化为
$$ \begin{split} & {m_{v1}}{{\ddot q}_{v1}}\left( t \right) + {c_{v1}}\left( {{{\dot q}_{v1}}\left( t \right) - {{\dot u}_b}\left( {{d_1},t} \right)} \right)+ \\ &\qquad {k_{v1}}\left( {{q_{v1}}\left( t \right) - {u_b}\left( {{d_1},t} \right)} \right)= \\ &\qquad {k_{v1}}r\left( {{d_1}} \right) + {c_{v1}}\dot r\left( {{d_1}} \right) = {k_{v1}}r\left( {{d_1}} \right) = A \end{split}$$ (18) 由式(18)易知, 当停靠位置不变时, 等式右边可以视作常数$A$, 即检测车静止时采集到的信号是不含粗糙度动态变化的, 因此本文所提出的方法是不受粗糙度影响的[39].
在现场桥梁测试中, 车体加速度响应信号${\ddot q_{v1}}$可由加速度传感器获取, 进一步采用${\mathrm{Newmark}} - \beta $法和中心差分法等常用的数值计算方法即可求得接触点响应.
有别于经典的自由衰减法计算结构阻尼比, 本文方法的桥梁响应信号衰减段是单辆通行车在桥面行驶时的时域曲线, 而不是利用自由衰减法处理通行车出桥面后自由衰减段的时域曲线, 即不需要封桥, 可快速便捷实现桥梁阻尼比的识别. 如图1所示, 把检测车停靠在桥梁任意位置$x = {d_1}$处, 利用三角函数对式(16)进行简化, 可以得出
$$ \begin{split} & {{\ddot u}_{b,1}}(x,t)\left| {_{x = {d_1}} = {\varDelta _{1,1}}} \right.{\varDelta _{2,1}}\left\{ {{{\text{e}}^{ - {\xi _1}{\omega _1}t}}\left[ {\sin \left( {{\omega _1}t + {\varphi _1}} \right)} \right]} \right\}\cdot \\ & \qquad \sin \left( {\frac{{{\text{π}}{d_1}}}{L}} \right) \end{split} $$ (19) 其中${\varphi _1} = {\text{ta}}{{\text{n}}^{ - 1}}\left( { - {\varDelta _{5,1}}/{\varDelta _{4,1}}} \right)$. 可以得到在$x = {d_1}$处, 任意两个相邻振动峰值之比为
$$ \frac{{{R_{b,t}}}}{{{R_{b,\left( {t + \Delta t} \right)}}}} = {{\mathrm{e}}^{{\xi _1}{\omega _1}\Delta t}} = B $$ (20) 其中$\Delta t = 2{\text{π /}}{\omega _1}$, 为桥梁基频对应的一个周期时长. 从式(20)可得, 任意两个相邻振动峰值之比为常数$B$, 桥梁基频和对应的周期时长都可以通过傅里叶变换得到. 因此, 阻尼比可由相邻振动峰值比的对数衰减率得到. 在小阻尼体系下, 可以求得阻尼比如下
$$ {\xi _1} = \frac{{{\text{ln}}B}}{{{\omega _1}\Delta t}} $$ (21) 在实际操作中, 为了获得更高的精度, 采用振动峰值衰减至50%所需要的次数来计算阻尼比. 这意味着, 通过把传感器和无线传输设备集成到检测车上, 形成快速自动模块化巡检设备, 即可在桥面任意位置求得阻尼比, 可实现中小跨径桥梁集群快速检测.
具体而言, 本文方法的流程如图2所示.
2. 数值模拟
2.1 数值模型
以重庆市某大桥为基础模型, 该桥为5跨, 跨长均为30 m的简支梁桥. 基于该桥加固完成的第3跨, 重点开展现场试验. 如图3所示为桥梁示意图. 第3跨长度为30 m, 截面面积为4.621 m², 截面惯性矩为3.606 8 m4, 桥梁材料的混凝土强度采用C50, 其弹性模量为3.45 × 104 MPa. 激励源来自桥面上的一辆通行车, 如图1所示, 结合实桥试验检测车设备, 检测车模型简化为单自由度弹簧质量模型, 质量为1470 kg, 车辆刚度为524076 N/m, 阻尼为100 N·s/m, 检测车的一阶频率为3 Hz. 由式(16)可知, 通行车的参数对识别桥梁一阶阻尼比无影响. 因此, 把通行车同样简化为单自由度的弹簧质量模型, 质量为5000 kg, 车辆刚度为524 076 N/m, 阻尼为100 N·s/m, 可得通行车的一阶频率为1.63 Hz. 将该桥第3跨等间距划分为15个单元, 每个单元长度为2 m. 如图4所示, 通行车按照一定速度前行激起桥梁振动, 集成在检测车上的传感器采集车体响应信号, 采样频率设置为100 Hz, 采样时间为30 s. 其中, 数字1 ~ 16代表单元节点处的编号, 带圈数字1 ~ 15代表桥梁单元编号. 将检测车停靠在节点处, 待数据采集完成, 即可按照图2流程识别桥梁一阶阻尼比.
2.2 通行车车速分析
为研究不同车速对该方法识别效果的影响, 设置通行车车速分别为1, 2, 5和2 ~ 4 m/s, 其中非匀速行驶是从2 m/s增加到4 m/s, 加速度为0.4 m/s2, 再从4 m/s减少至2 m/s, 加速度为−0.4 m/s2. 检测车停靠在节点8处, 桥梁阻尼比设置为0.01, 车桥其他参数与2.1节一致. 当得到车桥接触点响应信号后, 采用图2流程识别出桥梁一阶阻尼比. 如图5 ~ 图8所示, 分别为不同车速下滤波后的接触点加速度响应, 阻尼比识别结果如表1所示.
表 1 不同车速下的阻尼比识别结果Table 1. Damping ratio identification results under different vehicle speedsVehicle speed/
(m·s−1)Identification
valueTheoretical
valueAbsolute
errorRelative
error1 0.0098 0.01 0.0002 2% 2 0.0098 0.01 0.0002 2% 5 0.0097 0.01 0.0003 3% 2 ~ 4 0.0097 0.01 0.0003 3% 从表1可知, 在1, 2, 5和2 ~ 4 m/s不同车速下均可识别出桥梁阻尼比, 相对误差分别为2%, 2%, 3%和3%. 因此, 在不同车速下, 均可识别出桥梁一阶阻尼比.
2.3 检测车停靠位置影响分析
上一节已经验证在桥梁节点8位置处可以识别出阻尼比, 本节选择2, 5和13这3个节点来说明检测车停靠桥梁不同位置对阻尼比识别结果的影响. 除了节点位置不一样, 其他参数均与2.1节一致, 以通行车车速1 m/s为例, 按照图2所示流程识别桥梁阻尼比. 图9 ~ 图11分别为不同停靠位置下滤波后的接触点加速度响应, 阻尼比的识别结果如表2所示.
从表2可知, 检测车停靠在不同位置处均可识别出桥梁阻尼比, 且相对误差均为2%. 这意味着, 在现场测试时, 检测车停靠在除了支座之外任意位置均可识别出阻尼比, 无须考虑设置检测车停放位置, 更加省时高效.
表 2 检测车停靠不同位置处的阻尼比识别结果Table 2. Damping ratio identification results for the inspection vehicle parked at different locationsParking location Identification value Theoretical value Absolute error Relative error node 2 0.0098 0.01 0.0002 2% node 5 0.0098 0.01 0.0002 2% node 13 0.0098 0.01 0.0002 2% 2.4 噪音影响分析
通过在车体响应上添加白噪声, 分别选择信噪比为40, 30和20 dB的环境噪音下识别桥梁阻尼比. 选择检测车停靠在节点8处, 以桥梁阻尼比0.01为例, 行车速度1 m/s, 其他车桥参数与2.1节一致. 按照图2流程, 不同信噪比下滤波后的接触点加速度响应如图12 ~ 图14所示, 识别出的阻尼比如表3所示.
表 3 不同信噪比下的阻尼比识别结果Table 3. Damping ratio identification results under different signal-to-noise ratiosSNR/dB Identification value Theoretical value Absolute error Relative error 40 0.0096 0.01 0.0004 4% 30 0.0095 0.01 0.0005 5% 20 0.0093 0.01 0.0007 7% 从表3可知, 在信噪比为40和30 dB时, 本文方法均可以识别出桥梁阻尼比, 且误差在5%以内. 当信噪比为20 dB时, 识别的桥梁阻尼比虽然相对误差为7%, 但其结果仍具有一定的参考值. 结果表明, 本文所提方法在识别桥梁阻尼比时, 具有一定的噪音抗干扰能力, 有助于在现场测试中应用.
2.5 桥梁阻尼比影响分析
在实际情况中, 不同桥梁的一阶阻尼比可能会有变化. 因此, 分别选择阻尼比0.015, 0.02和0.025, 通行车速度为1 m/s, 其他车桥参数与2.1节一致, 在信噪比为30 dB下的识别结果, 图15 ~ 图17分别为不同桥梁阻尼比下滤波后的接触点加速度响应, 阻尼比的识别结果如表4所示.
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图 15 桥梁阻尼比为0.015时接触点响应Figure 15. Contact point response when bridge damping ratio is 0.015![]()
图 16 桥梁阻尼比为0.02时接触点响应Figure 16. Contact point response when bridge damping ratio is 0.02![]()
图 17 桥梁阻尼比为0.025时接触点响应Figure 17. Contact point response when bridge damping ratio is 0.025表 4 不同桥梁阻尼比下的识别结果Table 4. Identification results under different bridge damping ratiosSNR/dB Identification value Theoretical value Absolute error Relative error 30 0.0145 0.015 0.0005 3.3% 0.019 0.02 0.001 5% 0.026 0.025 0.001 4% 从表4可知, 在信噪比30 dB条件下, 均可识别出不同的桥梁阻尼比工况, 最大误差不超过5%. 由于式(17)是基于结构小阻尼的前提下近似推导得出的, 因此, 在现场测试时, 应尽量选择低阻尼的桥梁, 这样识别结果会更精准.
2.6 通行车阻尼影响分析
在实际情况中, 激励来源通行车的阻尼可能有变化. 因此, 分别选择阻尼100, 500和800 N·s/m, 行车车速为$ 1\;\mathrm{m}/\mathrm{s} $, 其他车桥参数与2.1节一致, 在信噪比为30 dB下的识别结果, 如图18 ~ 图20所示, 分别为不同行车阻尼下滤波后的接触点加速度响应, 阻尼比的识别结果如表5所示.
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图 18 通行车阻尼为100 N·s/m时接触点响应Figure 18. Contact point response when passing vehicle damping is 100 N·s/m![]()
图 19 通行车阻尼为500 N·s/m时接触点响应Figure 19. Contact point response when passing vehicle damping is 500 N·s/m![]()
图 20 通行车阻尼为800 N·s/m时接触点响应Figure 20. Contact point response when passing vehicle damping is 800 N·s/m表 5 不同通行车阻尼下的阻尼比识别结果Table 5. Damping ratio identification results under different passing vehicle dampingsVehicle damping/
(N·s·m−1)Identification value Theoretical value Absolute error Relative error 100 0.0098 0.01 0.0002 2% 500 0.0098 0.01 0.0002 2% 800 0.0098 0.01 0.0002 2% 从表5可知, 在信噪比30 dB条件及不同行车阻尼比的情况下, 桥梁阻尼比均可以识别出来, 相对误差均在2%. 结果表明, 通行车阻尼对本文所提方法几乎没有影响, 在实际应用中更具有普遍性.
2.7 方法对比
本文方法的主要优势体现在: 与文献[18]相比, 需要设计专用的双轴测试车, 且加速度计和激光传感器的安装要求比较高, 本文方法不需要布置较多传感器, 仅在检测车体上集成单个加速度传感器即可, 同时避免了路面粗糙度对识别结果的影响. 与文献[19]相比, 本文方法是基于单辆通行车激励桥梁, 不需要封桥, 更加高效便捷; 同时也不需要事先假设阻尼比参数重复修正振型直至第一阶振型的最大值处于测试跨跨中的繁琐程序, 且抗噪性较好.
为进一步说明本文方法的优势, 选择车速为1 m/s、桥梁阻尼比为0.01、信噪比为50 dB下的工况, 基于同样采集的单个加速度数据与文献[19]基于假定的振型识别阻尼比方法和随机子空间法[31]做对比分析, 识别精度如表6所示, 可以看出, 本文方法在识别精度上较优于其他两种方法. 更重要的是, 本文方法无需封桥, 同时避免了路面粗糙度对识别结果的影响, 检测车停靠在桥面除支座外任一位置均可识别出桥梁阻尼比, 更加经济高效, 适用于中小跨径桥梁集群的快速检测.
表 6 不同方法识别结果Table 6. Identification results using different methods3. 现场实验
试验选取的梁式桥为重庆市某大桥, 桥面全宽: 2.30 m(人行道 + 栏杆) + 16.00 m(车行道) + 2.30 m(人行道 + 栏杆)＝20.60 m, 桥面铺装为沥青混凝土, 栏杆为钢筋混凝土栏杆, 为简支梁桥. 其他资料可详见第3节数值模拟介绍. 图21为桥梁检测车, 图22为集成化检测车传感器的布置示意图.
在正式上桥之前, 预先准备3个试验. 首先, 将力锤作为激振器, 给检测车施加一个瞬态的冲击力, 进行动力特性测试获取检测车频率等模态信息. 图23为经过快速傅里叶变换之后检测车的加速度信号频谱图, 可以看出, 检测车车体基频为3 Hz. 其次, 在桥面布置加速度传感器获取大桥的基频, 如图24所示, 桥梁的基频为5.45 Hz. 最后, 把集成加速度传感器、采集仪和电源等设备的检测车停靠在桥面, 分析采集到的信号是否包括车体信息和桥梁信息, 如图25可以看出, 频谱图包含两个峰值, 分别为检测车车体基频3 Hz和桥梁基频5.45 Hz.
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图 23 力锤敲击检测车的信号傅里叶变换Figure 23. Fourier transform of the signal from the impact of the hammer on the inspection vehicle![]()
图 24 传感器布置在桥面信号的傅里叶变换Figure 24. Fourier transform of the bridge deck signals with sensors arranged![]()
图 25 检测车停在桥面车体响应信号傅里叶变换Figure 25. Fourier transform of the vehicle response signal with the inspection vehicle parked on the bridge deck在凌晨时分进行试验, 当仅有单辆通行车经过时, 采集通过时的响应信号. 将检测车停靠在桥梁跨中位置, 采集到的检测车加速度响应以及接触点加速度响应如图26和图27所示.
基于衰减段的接触点响应信号, 采用本文方法得到的阻尼比为0.0086. 为了说明本文方法的可行性, 与文献[19]基于假定的振型识别阻尼比方法和随机子空间法[31]做对比分析. 如表7所示, 本文方法与随机子空间方法识别出的一阶阻尼比相对误差为7.5%, 文献[19]方法与随机子空间方法识别出的一阶阻尼比相对误差为8.75%. 结果表明本文方法具有一定的工程适用性.
表 7 不同方法识别结果Table 7. Identification results using different method4. 结 论
本文通过理论推导, 证明在单辆通行车激励下, 基于车桥耦合理论的车桥接触点响应包含桥梁阻尼比信息. 分析通行车车速、检测车停靠位置、环境噪音、桥梁阻尼比和通行车阻尼比等参数对识别桥梁一阶阻尼比的影响, 通过数值模拟初步验证本文方法的有效性. 最终通过实桥试验说明了本文方法的可行性, 并得出以下结论:
(1)本文方法在单辆通行车激励下, 适用于环境信噪比大于30 dB的条件;
(2)通行车阻尼在100 ~ 800 N·s/m及行车速度在1 ~ 5 m/s范围内对阻尼比识别结果无影响, 说明本文方法具有一定的适用性;
(3)检测车在桥面上的停靠位置对阻尼比识别结果无影响, 避免了路面粗糙度对阻尼比识别结果的影响;
(4)本文方法操作简单, 不需要封桥, 仅靠单辆通行车激励识别桥梁一阶阻尼比, 经济高效.
后续将进一步开展在多辆通行车激励下的桥梁阻尼比识别研究, 进一步推动该方法在实际工程中的应用.
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图 15 桥梁阻尼比为0.015时接触点响应
Figure 15. Contact point response when bridge damping ratio is 0.015
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图 16 桥梁阻尼比为0.02时接触点响应
Figure 16. Contact point response when bridge damping ratio is 0.02
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图 17 桥梁阻尼比为0.025时接触点响应
Figure 17. Contact point response when bridge damping ratio is 0.025
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图 18 通行车阻尼为100 N·s/m时接触点响应
Figure 18. Contact point response when passing vehicle damping is 100 N·s/m
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图 19 通行车阻尼为500 N·s/m时接触点响应
Figure 19. Contact point response when passing vehicle damping is 500 N·s/m
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图 20 通行车阻尼为800 N·s/m时接触点响应
Figure 20. Contact point response when passing vehicle damping is 800 N·s/m
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图 23 力锤敲击检测车的信号傅里叶变换
Figure 23. Fourier transform of the signal from the impact of the hammer on the inspection vehicle
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图 24 传感器布置在桥面信号的傅里叶变换
Figure 24. Fourier transform of the bridge deck signals with sensors arranged
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图 25 检测车停在桥面车体响应信号傅里叶变换
Figure 25. Fourier transform of the vehicle response signal with the inspection vehicle parked on the bridge deck
表 1 不同车速下的阻尼比识别结果
Table 1 Damping ratio identification results under different vehicle speeds
Vehicle speed/
(m·s−1)Identification
valueTheoretical
valueAbsolute
errorRelative
error1 0.0098 0.01 0.0002 2% 2 0.0098 0.01 0.0002 2% 5 0.0097 0.01 0.0003 3% 2 ~ 4 0.0097 0.01 0.0003 3% 
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表 2 检测车停靠不同位置处的阻尼比识别结果
Table 2 Damping ratio identification results for the inspection vehicle parked at different locations
Parking location Identification value Theoretical value Absolute error Relative error node 2 0.0098 0.01 0.0002 2% node 5 0.0098 0.01 0.0002 2% node 13 0.0098 0.01 0.0002 2%
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表 3 不同信噪比下的阻尼比识别结果
Table 3 Damping ratio identification results under different signal-to-noise ratios
SNR/dB Identification value Theoretical value Absolute error Relative error 40 0.0096 0.01 0.0004 4% 30 0.0095 0.01 0.0005 5% 20 0.0093 0.01 0.0007 7%
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表 4 不同桥梁阻尼比下的识别结果
Table 4 Identification results under different bridge damping ratios
SNR/dB Identification value Theoretical value Absolute error Relative error 30 0.0145 0.015 0.0005 3.3% 0.019 0.02 0.001 5% 0.026 0.025 0.001 4%
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表 5 不同通行车阻尼下的阻尼比识别结果
Table 5 Damping ratio identification results under different passing vehicle dampings
Vehicle damping/
(N·s·m−1)Identification value Theoretical value Absolute error Relative error 100 0.0098 0.01 0.0002 2% 500 0.0098 0.01 0.0002 2% 800 0.0098 0.01 0.0002 2%
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