Damped fiber optic low-frequency tiltmeter for real-time monitoring of structural displacements

Tiltmeters and inclinometers have been widely employed for monitoring structural movements and rotations. Example applications include monitoring movements of retaining walls, bridge piers and applications in monitoring of displacements and tilt in earth embankments and landfills. In general, design of the tiltmeters has been based on using either purely mechanical systems as in liquid bubble levels, or by employing electrical transducers in combination with mechanical systems [1]. In recent years, a number of tiltmeters have been developed based on fiber optic Bragg grating (FBG) sensors. In general, fiber optic sensors provide high resolution measurements because they are immune to electrical and electromagnetic noise. They are also easily configurable for sensing myriads of structural perturbations, and capable of serial multiplexing in applications requiring multiple sensors [2–4]. Guan et al [5] developed a pendulum-based fiber Bragg grating tilt sensor, which consisted of a vertical pendulum hanging inside a steel frame with a free rotational joint at the top and four rigid arms of equal lengths protruded from the pendulum at right angles from each other. The position of the pendulum was kept in equilibrium by the four optical fibers, which were bonded to the four arms at one end and to the steel frame at the other end. Measurement of tilt was accomplished by monitoring the variation of the strains and strain differentials in the FBGs due to movement of the pendulum under applied tilts. By using four FBGs it was possible to compensate for temperature and to determine the magnitude as well as the direction of the inclination. The authors reported accuracy of 0.1° in the measurement of tilts with this sensor.

Zhao et al [6] designed a tilt sensor based on a system consisting of two FBGs, a cantilever and a gravity pendulum. The cross section of the cantilever beam was an isosceles triangle and therefore, when subjected to bending created uniform flexural strains along the entire length of the cantilever due to the tilt related movement of the gravity pendulum. For temperature compensation, FBGs were adhered to opposing faces of the cantilever generating equal wavelength shifts in opposite directions. Tilt measurements were achieved by relating the change in the optical signal power at the output end of the FBGs to the induced tilts. A measurement resolution of 0.002° in the linear measurement range of 0°–6° was achieved for this sensor. With a different design approach, Dong et al [7] developed a tilt sensor with three FBGs that achieved a resolution of 0.02°. The sensor was also able to detect the magnitude as well as the direction of the inclination by measuring the reflected optical powers and bandwidth of the FBGs. The experimental results indicated that the sensor was also insensitive to thermal fluctuations.

Chen et al [8] developed a tilt sensor by using a single pre-strained FBG. Design of the tilt sensor assembly involved use of materials that possessed different coefficients of thermal expansions. In their approach, temperature compensation was achieved by the counter-expansion effects of the sensor materials. The sensor exhibited a tilt angle resolution of 0.0067° and showed temperature stability of 0.33° over the temperature range of 27 °C–75 °C. He et al [9] developed a two-dimensional (2D) tilt sensor based on three FBGs. They were able to use the three-FBG assembly for temperature compensation and achieved a tilt measurement resolution of 0.005°. Bao et al [10, 11] developed two different types of inclinometers by using two FBGs in a mass pendulum assembly. Both inclinometers were capable of compensating for temperature. In one design, the reflected optical powers of the two FBGs were correlated with the tilt angles, and in the other the wavelength separations of the two FBGs were employed for the same purpose.

The objective for the work presented in this study was to investigate development of an FBG-based tiltmeter for field applications, and where real-time monitoring of tilts in structures under low-frequency dynamic displacements becomes important, i.e. in bridges. The proposed sensor design is based on a cantilever beam and lumped mass pendulum structure. In contrast to existing FBG-based tiltmeters, the sensor described herein was designed with capability for damping the high-frequency oscillations of the dynamic system. This attribute precludes post processing of tiltmeter data due to dynamic effects, and allows for real-time monitoring of structural tilts. The investigation involved laboratory experiments under various thermal regimes, and dynamic analysis of the tiltmeter for the evaluation of its damping characteristics. The phase response of the sensor was experimentally determined under low-frequency vibration tests and compared with the theoretical analysis. The experimental and theoretical dynamic analysis of the sensor was instrumental in determining the phase response of the tiltmeter. The sensor design is discussed next.

Zoom In Zoom Out Reset image size

The proposed tiltmeter consists of a very thin cantilever beam with a mass block at the free end. Schematic diagram of the sensor in global Cartesian coordinates (Y–Z) and local coordinates (Y'–Z') is shown in figure 1. Cantilever end strain due to the force exerted by the lumped mass is correlated against the sensor rotation. The mass block is simplified as a lumped mass at the beam end, and the beam's mass is considered negligible. For the small tilt angles of −2° to 2° considered in the present study, it can be assumed that the displacement of the mass and the beam only occur in the y-direction. The basic flexural relationship for small deformation bending of the beam in global coordinates is given by

$$\begin{equation} k \approx y'' = \frac{{M( z )}}{{EI}}, \end{equation} \tag{ 1 }$$

where y is displacement in the horizontal direction, k is curvature of the beam along the beam (along the z direction), which is approximately equal to the second order derivative of deflection y and M(z) is bending moment in the beam, and that

$$\begin{equation} M( z ) = mg( {y( z ) - y( 0 )} ). \end{equation} \tag{ 2 }$$

With the boundary condition at the fixed end, y(L) = 0 and y'(L) = θ, rotation of any point on the beam is obtained by integrating equation (1):

$$\begin{equation} y'( z ) = \sqrt {\frac{{mg}}{{EI}}( {y( z )^2 - 2y( z )y( 0 )} ) + \theta ^2 } . \end{equation} \tag{ 3 }$$

In a similar manner, displacement of any section of the beam is given by integrating equation (3):

$$\begin{eqnarray} &&\ln\left| {y( z ) - y( 0 ) + \sqrt {( {y( z )^2 - 2y( z )y( 0 )} ) + \frac{{EI}}{{mg}}\theta ^2 } } \right|\nonumber\\ &&- \ln\left| {\sqrt {\frac{{EI}}{{mg}}} \theta - y( 0 )} \right| + \sqrt {\frac{{mg}}{{EI}}} ( {L - z} ) = 0. \end{eqnarray} \tag{ 4 }$$

Equation (4) is implicit for displacement at any point on the beam. When z = 0, displacement of the lumped mass is given by

$$\begin{equation} y( 0 ) = \frac{{1 - {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}{{1 + {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}\sqrt {\frac{{EI}}{{mg}}} \theta . \end{equation} \tag{ 5 }$$

It can be seen that for analysis based on small deformation theory, y(0) is proportional to θ, which means bending moment at the fixed end is proportional to the applied tilt, in which the fixed end bending moment is

$$\begin{equation} M( L ) = - mgy( 0 ). \end{equation} \tag{ 6 }$$

Surface strain, ε, of a beam subjected to bending is a function of the bending moment, M, flexural stiffness, EI, and distance, r, from centroid of cross section to the beam's surface, which is given by

$$\begin{equation} \varepsilon = \frac{M}{{EI}}r. \end{equation} \tag{ 7 }$$

With y(0) given by equation (5) and M(L) given by equation (6), fixed end surface flexural strain is given by the following relationship:

$$\begin{equation} \varepsilon ( L ) = \frac{M}{{EI}}\frac{h}{2} = - \frac{{mg}}{{EI}}y( 0 )\frac{h}{2} = - \frac{h}{2}\frac{{1 - {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}{{1 + {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}\sqrt {\frac{{mg}}{{EI}}} \theta , \end{equation} \tag{ 8 }$$

where h is the thickness of the cantilever beam. Due to symmetry of beam cross section, fixed end surface on the opposing face of the beam is subjected to the same strain with reversed sense, −ε. The flexural strain difference between the two sides of the beam is 2ε. Therefore, the fixed end sensitivity is defined as the ratio of 2ε to θ, which is given by

$$\begin{equation} \frac{{2\varepsilon }}{\theta } = - \frac{{1 - {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}{{1 + {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}\sqrt {\frac{{mg}}{{EI}}} h = - \gamma h, \end{equation} \tag{ 9 }$$

where

$$\begin{equation} \gamma = \frac{{1 - {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}{{1 + {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}\sqrt {\frac{{mg}}{{EI}}} . \end{equation} \tag{ 10 }$$

As shown in equations (9) and (10), fixed end sensitivity is a function of modulus of elasticity, E, dimensions of the beam such as length, width and thickness of the beam, and the lumped mass. The influence of these parameters on the sensitivity of the tilt measurements is shown in figures 2–6. In general, the sensitivity varies inversely with the width and thickness of the beam and increases with the beam length and lumped mass. As shown in figures 2–6, the correlation between the sensitivity and the aforementioned parameters is nonlinear.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Sensor design can be optimized by appropriate selection of materials and proportioning of the beam and the lumped mass in order to achieve high sensitivity, i.e. large fixed end strains for small rotations. From a practical point of view, however, the size of the tiltmeter is limited by space limitation and ease of installation requirements at the bridge elements. Considering the size limitations and installation requirements, a tiltmeter was fabricated that embodied a lumped mass of 0.5 kg and a cantilever aluminum beam for which the modulus of elasticity, E, was 70 GPa. The length, width and thickness dimensions of the beam were 127.5, 10 and 0.5 mm, respectively. Two FBG sensors with center wavelengths of 1580 nm were pre-strained and adhered to opposing surfaces of the beam at a distance of 101.5 mm from the free end of the beam. The purpose of using two FBG sensors was to automatically compensate for the effect of temperature. This temperature compensation technique was also employed by many others in various sensor configurations.

Static calibration experiments were first performed at ambient temperature of 21.4 °C on a tilt stage. Figure 7 corresponds to the experimental setup. The experimental setup consisted of a platform on a calibrated micro-tilt controller with tilt steps (resolution) of 0.0006°. The fiber optic tiltmeter was attached to the platform, and the FBG sensors were monitored by the fiber optic interrogator. The dynamic range of the interrogator is 25 dB at a sampling rate of up to 500 Hz. Data from the FBG interrogator were acquired and processed in real time. Damping fluid was introduced to eliminate oscillations and expedite data acquisition. The calibration process involved step by step increase in the tilt angle and acquisition of the difference in the wavelength shifts of the two FBG strain sensors. Sensitivity corresponded to the slope of the linear fit to the tilt-wavelength shift response of the tiltmeter.

Zoom In Zoom Out Reset image size

For the ambient temperature condition, the calibration results yielded a sensitivity of 0.1985 nm deg⁻¹, which is larger than the estimated value of 0.18 nm deg⁻¹ from equation (9). The difference is attributed to the influence on the strain transfer of the thickness of the adhesive used for mounting the FBG sensors to the very thin aluminum beam. Thermal strains are compensated by computing the difference in the wavelength shifts in the two FBG sensors:

$$\begin{equation} 2W_\theta = ( {W_\theta + W_T } ) - ( { - W_\theta + W_T } ), \end{equation} \tag{ 11 }$$

where W_θ is the FBG wavelength shift due to the tilt and W_T is the wavelength shift due to temperature. To investigate the temperature compensation capabilities of the sensor, calibration experiments were repeated for the tiltmeter under different temperature regimes. In these experiments, the tiltmeter experiments were performed by wrapping the tiltmeter with electrical heat tape. The nominal temperature regime for the experiments ranged from 21.4 to 38.2 °C. The experimental results are shown in figure 8 indicating linear results at all temperature levels. The results vary within a narrow band of 0.04 nm indicating temperature-related fluctuations within a narrow margin. The calibration curve has an intercept due to the fact that the base wavelengths for the FBG sensors (λ_B) differ from zero.

Zoom In Zoom Out Reset image size

Structural use of tiltmeters requires performance under dynamic motions, such as in situations where they are subjected to vibration in bridges under traffic and truck load impacts. Dynamic response of the tiltmeter under these loading conditions plays a significant role in proper portrayal of displacements and rotations of structural elements. Excessive oscillations in undamped or improperly damped tiltmeters either result in out of phase erroneous interpretation of measurements or require computation time for post processing and signal analysis of the tiltmeter data. Hence, in the design of the tiltmeter proposed in this study damping was introduced by immersing the tiltmeter in a damping fluid. Therefore, this section of the paper pertains to the dynamic response analysis of the damped sensor. Based on the assumption of small deformations, the motion of lumped mass could be regarded as only in the horizontal direction. Global coordinates (Y–Z) are static with origin at the mass point when no tilt happens, z-axis in vertical direction and y-axis is in horizontal direction. Local coordinates (Y'–Z') also have the y-axis and z-axis in the same directions as global coordinates, but the origin is located at the static equilibrium point of the lumped mass when a tilt takes place.

Displacement of the mass point in global coordinates is considered as y_t, displacement in local coordinates is y and displacement of the local coordinates to global coordinates is y_g. The relationship between y_t, y and y_g is given by the following equation:

$$\begin{equation} y_t = y + y_g . \end{equation} \tag{ 12 }$$

Referring to the lumped mass cantilever beam system in figure 1, the tiltmeter is simplified as a single degree of freedom system when mass of the cantilever beam is neglected, and the dynamic equation is given by

$$\begin{equation} m\ddot y_t + c\dot y + ky = 0 \end{equation} \tag{ 13 }$$

which can be rewritten as

$$\begin{equation} m\ddot y + c\dot y + ky = - m\ddot y_g , \end{equation} \tag{ 14 }$$

where m is mass of the system, c is the viscous damping coefficient and k the stiffness.

$\ddot y_g$ is the acceleration of the local coordinate and, according to equation (5), it is

$$\begin{equation} \ddot y_g = \frac{{1 - {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}{{1 + {\rm e}^{2\sqrt {\frac{{mg}}{{EI}}} L} }}\sqrt {\frac{{EI}}{{mg}}} \ddot \theta . \end{equation} \tag{ 15 }$$

Free vibration test of the tiltmeter was first performed in air, which can be considered as a damping-free system, and the measured natural frequency was 1.656 Hz. After damping fluid was introduced to eliminate oscillation in the sensing system, the measured natural frequency was reduced to 1.302 Hz, and the damping ratio was computed to be 0.248 by using the log-decrement approach. The relationship between natural frequency of the damped system and the damping-free system is given by the following equation:

$$\begin{equation} \omega _D = \omega \sqrt {1 - \xi ^2 } , \end{equation} \tag{ 16 }$$

where ω_D is the natural frequency of the damped system, ω is the natural frequency of the damping-free system and ξ is the damping ratio. Besides adding damping to the sensor system, damping fluid also added extra mass to the lumped mass. According to equation (16), the natural frequency of the damping-free system with added mass was computed to be 1.345 Hz, which is lower than 1.656 Hz.

Considering applied tilt with oscillations of sinusoidal form

$$\begin{equation} \theta = \theta _0 + \theta _1 \sin ( {\bar \omega t} ), \end{equation} \tag{ 17 }$$

where θ₀ is the nominal tilt, θ₁ is the amplitude of oscillation, and $\bar \omega$ is the frequency of oscillation. From equation (14), the steady-state response for the dynamics equation in local coordinates is

$$\begin{eqnarray} &&y( t ) = \frac{{\gamma m\theta _1 \bar \omega ^2 }}{k}\left[ {\frac{1}{{( {1 - \beta ^2 } )^2 + ( {2\xi \beta } )^2 }}} \right]\nonumber\\ &&\times [ {( {1 - \beta ^2 } )\sin ( {\bar \omega t} ) - 2\xi \beta \cos ( {\bar \omega t} )} ], \end{eqnarray} \tag{ 18 }$$

where frequency ratio $= {{\bar \omega }} /\omega$ and, according to equation (5), motion of the local coordinates is given by

$$\begin{equation} y_g = \gamma ( {\theta _0 + \theta _1 \sin ( {\bar \omega t} )} ). \end{equation} \tag{ 19 }$$

Relative motion of lumped mass to static equilibrium point with tilt θ₀ is given by

$$\begin{equation} y_r ( t ) = y_g ( t ) + y( t ) - \gamma \theta _0 . \end{equation} \tag{ 20 }$$

Equations (19) and (20) can however be combined as follows:

$$\begin{eqnarray} &&y_r ( t ) = \frac{{\gamma \theta _1 \bar \omega ^2 }}{{\omega ^2 }}\bigg[ {\frac{1}{{( {1 - \beta ^2 } )^2 + ( {2\xi \beta } )^2 }}} \bigg] \nonumber\\ &&\times\bigg[ \bigg[ ( {1 - \beta ^2 } ) + \frac{{\omega ^2 }}{{\bar \omega ^2 }}{\rm }[ {( {1 - \beta ^2 } )^2 + ( {2\xi \beta } )^2 } ] \bigg] \sin( {\bar \omega t} ) \nonumber\\ &&- 2\xi \beta \cos ( {\bar \omega t} ) \bigg]. \end{eqnarray} \tag{ 21 }$$

Amplitude of response due to oscillation is

$$\begin{equation} \rho = \gamma \theta _1 \left[ {\frac{{1 + ( {2\xi \beta } )^2 }}{{( {1 - \beta ^2 } )^2 + ( {2\xi \beta } )^2 }}} \right]^{1/2} . \end{equation} \tag{ 22 }$$

Phase shift φ is

$$\begin{equation} \varphi = \arctan \left( {\frac{{2\xi \beta ^3 }}{{1 - \beta ^2 + 4\xi ^2 \beta ^2 }}} \right). \end{equation} \tag{ 23 }$$

Amplification factor is defined as the ratio of ρ to γθ₁:

$$\begin{equation} a = \frac{\rho }{{\gamma \theta _1 }} = \left[ {\frac{{1 + ( {2\xi \beta } )^2 }}{{( {1 - \beta ^2 } )^2 + ( {2\xi \beta } )^2 }}} \right]^{1/2} . \end{equation} \tag{ 24 }$$

From equation (23), the relationship between amplification factor and frequency ratio for the damped system is computed and shown in figure 9. When ξ = 0, the relationship between the amplification factor and the frequency ratio for the undamped system is given by

$$\begin{equation} a = \frac{\rho }{{\gamma \theta _1 }} = \left| {\frac{1}{{1 - \beta ^2 }}} \right|.\end{equation} \tag{ 25 }$$

Zoom In Zoom Out Reset image size

The frequency response of the tiltmeter was experimentally determined by attaching the tiltmeter to an I-beam in a closed-loop structural testing frame. The hydraulic actuators were programmed to apply sinusoidal displacements in the range 0.5–2.5 Hz at the mid span of the beam at constant displacement amplitudes for the entire frequency range. Time domain sensor signals were employed in the computation of the experimental amplification factors. Furthermore, the measured amplification factors are compared with the analysis results from equation (24) and shown in figure 9. The experimental and theoretical amplification factors show close agreement at frequencies outside resonance. At resonance, the theoretical values are asymptotic considering that the limits approach infinity. In practice, however, displacements are large but bound to limits. This difference is attributed to the mismatch at resonance. Introduction of the damping is significant for tiltmeters that operate in applications requiring low-frequency tiltmeters, such as in bridges. Even at slower rates (i.e. β < 1), the undamped system displays unstable behavior when the bridges operate near their natural frequencies and the dynamic response of the tiltmeter can be much larger than the actual tilts experienced by the structure. Moreover, as indicated by the phase angle, φ in figure 10, for β > 1 the undamped tiltmeter approaches the phase angle of 180° much faster than the damped tiltmeter. At these frequencies, the displacement response of the undamped tiltmeter will be out of phase with respect to the structural displacements.

Zoom In Zoom Out Reset image size

The research described in this paper pertains to the development and fabrication of an FBG-based tiltmeter for applications in civil structures. The mechanism for transduction of structural tilts is based on monitoring the strains in a cantilever beam–lumped mass design. Temperature compensation is achieved by subtracting the output of the two FBGs that measure the cantilever end strains on opposing surfaces of the beam. Theoretical relationships were formulated for the analysis of the sensor response both under static as well as in dynamic excitations. Design of the tiltmeter involved a sensitivity analysis considering the size as well as the mechanical characteristics of the cantilever-lumped mass system. The experimental program involved static calibrations under ambient as well as a thermal regime ranging from 21.4 to 38.2 °C. The static calibration results exhibited linear response over the designed tilt band of ±2° with a nominal sensitivity of 0.1985 nm deg⁻¹. The tiltmeter attained a resolution of 0.005° with an accuracy of ±0.06°. Damping fluid was used to enhance the dynamic response of the tiltmeter. Dynamic response analysis of the tiltmeter indicated that damping improved the response in suppressing excessive oscillations. The damped system also exhibited phase stability at higher rates of displacements.

This material is based upon work supported by the National Science Foundation under grant no. 0730259.