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A novel three-point method using a grating eddy current absolute position sensor (GECS) for bridge deflection estimation is proposed in this paper. Real spatial positions of the measuring points along the span axis are directly used as relative reference points of each other rather than using any other auxiliary static reference points for measuring devices in a conventional method. Every three adjacent measuring points are defined as a measuring unit and a straight connecting bar with a GECS fixed on the center section of it links the two endpoints. In each measuring unit, the displacement of the mid-measuring point relative to the connecting bar measured by the GECS is defined as the relative deflection. Absolute deflections of each measuring point can be calculated from the relative deflections of all the measuring units directly without any correcting approaches. Principles of the three-point method and displacement measurement of the GECS are introduced in detail. Both static and dynamic experiments have been carried out on a simple beam bridge model, which demonstrate that the three-point deflection estimation method using the GECS is effective and offers a reliable way for bridge deflection estimation, especially for long-term monitoring.

After a bridge is put into use, gradual deterioration is inevitable because of loading, temperature changes or other environmental factors. In order to guarantee the safety and durability of those bridges which are expensive and closely related with people's livelihood, long-term and continuous structural health monitoring is an essential part of the maintenance management. Among the various structural performance evaluations, vertical deflection is an important parameter that can directly and effectively indicate a bridge's behavior.

In terms of instrumentation for deflection estimation, there are contact and non-contact deflection estimation methods. Traditional displacement sensors such as mechanical dial gauges or linear variable differential transducers (LVDTs) are used in contact measurement, through which static or real-time displacement values can be obtained directly or fed into a computer for processing and displaying via a data cable. This method, however, requires access under the bridge and installation of a temporary supporting system to mount sensors, which is time consuming and not very efficienct. In addition, it might even be unavailable when bridges are over rivers, highways or have high clearance. Another contact sensor is the fiber optic Bragg-grating (FBG) sensor through which the deflection is calculated from the measured strain data and displacement-strain relationship [

To cope with those inconveniences in contact measuring methods, various non-contact approaches have been proposed. Based on the detection of the Doppler shift of the laser light, a laser Doppler vibrometer (LDV) equipped with displacement and velocity signal decoders can measure both bridge deflection and vibration simultaneously [

To avoid those deficiencies in conventional estimating deflection methods mentioned above, a novel three-point deflection estimation method is presented in this paper. Measuring points along lines paralleling the bridge span axis are chosen equidistantly. Among these measuring points, every three adjacent measuring points are defined as a measuring unit in which a straight connecting bar linking the two endpoints is taken as a relative reference line. Relative deflection of the mid-measuring point relative to the intermediate point of the connecting bar on which a displacement sensor is fixed can be measured, and thus the absolute deflection of each measuring point can be calculated from the relative deflections of all the measuring units. Compared with the contact and non-contact methods mentioned above, only real spatial positions of the measuring points are taken as relative references without any other static reference points. Moreover, the selected displacement sensor is the grating eddy current absolute position sensor (GECS) which is different from traditional eddy current sensors based on vertical characteristics [

In this paper, both the principles of the three-point method and displacement measurement of the GECS are presented. Then, this three-point method for deflection estimation is verified in a simply supported girder bridge model in the laboratory. Comprehensive static and dynamic experiment results on the laboratory tests demonstrate this method is effective and offers an alternative way for bridge deflection estimation.

The schematic diagram of the three-point method is shown in _{i}, y_{i}_{-1} and _{i}_{+1}, respectively. A straight connecting bar is used to link the measuring points _{i}_{i}_{i}_{i}_{i}_{-1}, _{i}_{+1} and _{i}_{i}_{i}_{i}_{i}

The three-point deflection estimation method is elaborated on a laboratory simple beam bridge model, on which deflection estimation is accomplished by making use of the real spatial positions of the measuring points for relative reference points of each other rather than using any other auxiliary static reference points. Suppose the length of the beam is

Due to the mid-measuring point in one measuring unit is used for installation of the GECS and meanwhile, it is the endpoint of the previous or next measuring unit used for connecting bar installation, which makes choice of the measuring points and connecting bars installation should be along two lines paralleled to the span axis, respectively. A complete instrumentation plan is shown in _{i}

The matrix expressions of _{i}_{i}_{1}, _{2}, …, _{n-2}, _{n-1}]^{T}, Y_{1}, _{2}, …, _{n-2}, _{n-1}]^{T}

In this paper, the measuring points are chosen equidistantly, thus _{i}

Therefore, the absolute deflection _{0} = _{n}

Furthermore, the deflection estimating is not only limited in those selected measuring point, which means one or more auxiliary measuring points can also be selected according to actual requirements. In each measuring unit, more displacement sensors can be installed on the connecting bar corresponding to the auxiliary measuring points, and from _{i}

As mentioned earlier, since the relative deflection of each measuring unit is actual an absolute displacement value measured by the GECS, from which absolute deflection of each measuring point can be calculated directly without any correcting approaches. In addition, the GECS is not susceptible to the velocity of displacement variation, and therefore, both static and dynamic deflection estimations can be realized. The selected GECS consists of a measuring board and reflective conductors. The measuring board fixed on the intermediate point of the connecting bar vertically includes four planar coils, measuring circuit and other functional modules. Coils, measuring circuit and reflective conductors can all be fabricated by the printed circuit board (PCB) technique. Grating reflective conductors usually called measuring track are two parallel series of equidistributed reflective conductors of which the spacing distance named measuring wavelength is

As shown in

The measuring board mainly consists of four parts:

Signal conversion circuit: converts inductance of the coils into electrical signal. Through an analog multiplexer, the four coils are connected to a LC oscillator circuit in turn, and outputs sinusoidal signal.

Signal acquisition circuit: output signals of the conversion circuit and fundamental frequency signal produced by a 12 MHz oscillator are connected to the two multi-cycle synchronous counters built-in the MCU respectively for frequency measurement and displacement calculation.

Power management unit: the conversion circuit and signal acquisition circuit are powered by LDO combined with the MCU through intermittent power supply mode.

MCU and its interface circuits: transmit the displacement value to PC or other data display modules.

Suppose the frequencies corresponding to the four coils are _{1}, _{2}, _{3} and _{4}, the differential frequencies of these two sets of coils are _{12} = _{1} − _{2} and _{34} = _{3} − _{4} respectively. These two differential frequency curves approach to the characteristic of sin or cosine curves, thus the differential frequencies can be presented by the following expressions:

Since there is a distance of _{12}, thus _{34} is:

Suppose the phase is

The phase curve of two measuring wavelength is shown in

The accuracy of this sensor is ±0.01 mm within a measurement range which is equal to the wavelength _{2} = 4.88 mm. Applications of the GECS can be continually extended in other fields such as digital caliper [

A laboratory simply bridge model is shown in

The estimated deflection curves of static loads with different weight is shown in

Multiple estimating results of which loads with various weights are placed at different positions show that the deflection estimated by the three-point method is credible and the relative errors are less than 5%.

While in dynamic estimations, dynamic estimated data is recorded in a time history and then it will be analyzed in time or frequency domain, which will provide the calculation basis for subsequent engineering design. The choice of the estimated data type should take the dynamic response of the object model into account. For bridge models, due to the low acceleration value and low natural frequency (often within 0.05 Hz−2 Hz), the priority estimating object selection is the displacement estimation. In this paper, two dynamic experiments include vehicle moving tests and vibration tests are performed.

In the simulated vehicle moving tests, time history curves of the dynamic deflection under different running speed are estimated for further analyses. During the laboratory tests, a moving load simulated the vehicle running in different speed is developed on the beam. As shown in

In the vibration tests, frequency components of the bridge model are very complex, and among them, vibration displacement under the natural vibration frequency is the key research object rather than that of the high frequency vibration. Thus, the type of the estimating object in the bridge vibration is also the dynamic displacement, that is to say, the proposed three-point method is also ideal for dynamic vibration tests. For the laboratory model in this paper, a random vibration force is imposed on the beam, and the measuring point 2 of which the dynamic deflection variation range is more obvious than the others is taken as the observation point.

It also can be seen that the two curves are fitted well, which indicates that the proposed three-point method is also appropriate for dynamic estimations. Plenty of static and dynamic experimental results demonstrate that the three-point method presents comprehensive reliability.

A reliable three-point deflection estimation method using the GECS is proposed in this paper. This method takes the real spatial positions of the measuring points as relative reference points of each other directly without using any other auxiliary referenced points, which offers great convenience for bridge deflection estimation. The accurate relative deflection estimated by the GECS of each measuring unit gives security for the calculation of absolute deflections. The selected displacement sensor GECS is not susceptible to the velocity of displacement variation, based on which both static and dynamic deflections can be calculated from the relative deflections directly without any correction approaches. As an inductive sensor, moreover, the selected GECS is waterproof and dustproof in principle, thus it can work under bad weather conditions, which makes it very suitable for long-term monitoring. Both static and dynamic experiments have been carried out on a simple beam laboratory bridge model, and the experimental results indicate that the proposed method is adequate for deflection estimation with great precision, which show the effectiveness of the three-point method and offers an alternative way for bridge deflection estimation. For the purpose of enhancing its efficiency, wireless data transmission technology will be introduced in this estimating system in the future.

This work was supported by the Ministry of Railways plan for Science and Technology Research and Development, under the 2009G026 project. The authors also appreciate the anonymous reviewers for their valuable comments on the work.

Schematic diagram of the three-point method.

Instrumentation plan of a simple beam.

Schematic diagram of the GECS.

Relation curve of phase and displacement.

Laboratory simply beam bridge model: (

Diagram of static experiment.

Deflection curves in different loads.

Diagram of simulated vehicle moving test.

Results of simulated vehicle moving test: (

Results of vibration test: (

Static deflection estimating results.

Measuring point | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

Relative deflection (mm) | 0.13 | 0.21 | 0.07 | 0.14 | 0.30 | 0.10 | 0.21 | 0.38 | 0.11 |

Calculated deflection (mm) | 0.44 | 0.62 | 0.38 | 0.56 | 0.84 | 0.52 | 0.75 | 1.08 | 0.65 |

Dial gauges (mm) | 0.46 | 0.62 | 0.39 | 0.58 | 0.82 | 0.51 | 0.78 | 1.08 | 0.68 |

Relative error (%) | −4.3 | 0 | −2.6 | −3.4 | 2.4 | 1.9 | −3.8 | 0 | −4.4 |