Review of Indirect Bridge Monitoring Using Passing Vehicles
Theoretical Background for
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2. Theoretical Background for
Indirect Bridge Monitoring The concept of an indirect approach utilizing an instru- mented vehicle, sometimes also referred to as “drive-by bridge health monitoring,” is illustrated by Figure 1 . The vehicle is fitted with sensors, most commonly accelerometers and most commonly on its axles. Therefore, the vehicle passing over the bridge is effectively used as a “moving sensor.” Figure 2 shows an example of a simplified VBI model used for a theoretical study of such an approach, a so-called quarter-car model. This simple model can be used to briefly explain the theoretical background supporting the concept. In Figure 2 , the parameters 𝑚 𝑠 and 𝑚 𝑢 refer to the sprung and unsprung masses, representing the vehicle body and tire assemblies, respectively. 𝐾 𝑠 and 𝐾 𝑡 are the suspension and tire stiffnesses, respectively, while 𝐶 𝑠 corresponds to the suspen- sion damping. 𝑦 𝑠 and 𝑦 𝑢 are the time dependent vertical dis- placements of the sprung and unsprung masses, respectively. The vehicle is assumed to travel at constant velocity 𝑐 here. As a vehicle crosses a bridge, both vehicle and bridge vibrate and there is dynamic interaction between them. Therefore, the vehicle response is influenced by the bridge response. From the principle underlying SHM, it follows that if the bridge is damaged, for example, due to bridge bashing or concrete cracking, the stiffness, damping, and/or mass of the bridge change due to this damage and its vibration characteristics will also change. Shock and Vibration 3 0.1 1 1 1 10 100 0.004 0.000 0.008 0.012 0.016 Frequency (Hz) FEM Analytical 4 4 2 2 3 3 Vehicle acceleration spectrum Am p li tude (m/s 2 ) 𝜔 2𝜋/L 𝜔 b − 𝜋/L 𝜔 b + 𝜋/L (a) 0.1 1 10 100 0.000 0.002 0.004 0.006 0.008 0.010 Frequency (Hz) Bridge acceleration spectrum FEM Analytical Am p li tude (m/s 2 ) 𝜔 b 𝜋/L (b) Figure 3: Vertical acceleration spectrum of (a) vehicle and (b) bridge midpoint. (Speed, V, is 10 m/s, 𝜔 𝑏 is bridge natural frequency, and 𝜔 V is vehicle frequency.) After [ 10 ]. Equation (1) gives the equations of motion for the sprung and unsprung masses: 𝑚 𝑠 ̈𝑦 𝑠 + 𝐶 𝑠 ( ̇𝑦 𝑠 − ̇𝑦 𝑢 ) + 𝐾 𝑠 (𝑦 𝑠 − 𝑦 𝑢 ) = 0, 𝑚 𝑢 ̈𝑦 𝑢 − 𝐶 𝑠 ( ̇𝑦 𝑠 − ̇𝑦 𝑢 ) − 𝐾 𝑠 (𝑦 𝑠 − 𝑦 𝑢 ) + 𝐾 𝑡 (𝑦 𝑢 − 𝑦 𝑏 − 𝑟) = 0, (1) where 𝑟 and 𝑦 𝑏 are the road profile displacement and bridge displacement, respectively. It can be seen from these equations that change due to damage can be detected in the vehicle response via the presence of the term 𝑦 𝑏 , the bridge displacement under the wheel of the vehicle. Therefore it is theoretically feasible to detect damage by using measure- ments in the vehicle alone, without using sensors on the bridge or the need to stop the vehicle. There is an added advantage of an indirect approach in that the moving sensor passes over all cross sections of the bridge, unlike sensors at fixed positions. This can provide greater spatial information compared to direct SHM [ 18 , 19 ]. The theoretical study of the indirect monitoring concept can be extended to a range of VBI models of varying complexity incorporating, for example, vehicle pitching and rolling motions and inertial and centrifugal forces. A variety of models which allow for simplification of the problem have been used in the papers reviewed here. A comprehensive survey of the most commonly used VBI models is given by [ 20 ]. Download 1.91 Mb. Do'stlaringiz bilan baham: |
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