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Optical fibers possess many advantages such as small size, light weight and immunity to electro-magnetic interference that meet the sensing requirements to a large extent. In this investigation, a Mach-Zehnder interferometric optical fiber sensor is used to measure the dynamic strain of a vibrating cantilever beam. A 3 × 3 coupler is employed to demodulate the phase shift of the Mach-Zehnder interferometer. The dynamic strain of a cantilever beam subjected to base excitation is determined by the optical fiber sensor. The experimental results are validated with the strain gauge.

Optical fiber sensors have attracted considerable attention in recent years as powerful measurement devices. They have been used in a variety of engineering applications such as residual strain measurement in composites [

Vibrations have a significant effect on the fatigue life of structures and may even have disastrous consequences. To understand the vibrating behavior of structures, instrumentation for accurate vibration measurement is essential. A number of sensors are available for the measurements of a vibrating structure including strain gauge, piezoelectric transducer, laser vibrometer, accelerometer and optical fiber sensor. Among these measurement devices, optical fiber sensors have received much attention for structural health monitoring applications. They are unique in a number of aspects including small physical size, ease of embedment in structures, immunity to electromagnetic interference and excellent multiplexing capabilities [

In this work, Mach-Zehnder optical fiber interferometric sensor is employed to measure the dynamic strain of a vibrating cantilever beam. The method developed by Brown

The schematic diagram of a Mach-Zehnder interferometer is shown in _{0} is the refractive index of the optical fiber; _{f}_{11} and _{12} are the Pockel’s constants; _{f}_{f}

The integral of the strain in

Thus, once the phase shift Δ

To demodulate phase shift Δ

The three outputs of the 3 × 3 coupler are nominally 120° out of phase with either of its neighbors and can be expressed as:

The DC offset “C” of the output can be obtained by adding the three inputs as follows:

Three new parameters, _{1}, _{2} and _{3} are introduced as follows:

The next step in the processing is to take the difference between each of the three possible pairings of the derivatives (_{2}, _{3}) and multiply this by the third signal (not differentiated):

Summation of

Taking the squares of

Dividing

We can integrate

Thus, the strain in the host structure is readily to be determined by substituting phase shift Δ

In this study, the phase shift demodulation is performed using the commercial software Matlab.

To demonstrate the capability of the proposed methodology in demodulating the phase shift, a numerical example is presented. In the numerical example, the phase shift is assumed to be a sinusoidal function with dual angular frequencies of 34π and 50π as follows:

Substituting

Substituting the numerical data of the three outputs from

A cantilever beam subjected to base excitation is considered in the experimental test. The beam of length L = 285 mm, width b = 20 mm, thickness h = 1 mm is made of copper with elastic modulus E = 120 GPa, density ^{3}. An optical fiber is surface bonded to the middle of the cantilever beam as the sensing fiber of the Mach-Zehnder interferometer. The percentage of the strain in the test specimen actually transferred to the optical fiber is dependent on the bonding length [_{f}_{f}_{f}_{0} = 1.45, pockel’s constants _{11} = 0.12, _{12} = 0.27, radius _{f}_{i}

The first five natural frequencies for the testing cantilever beam are 7.37 Hz, 46.21 Hz, 129.39 Hz, 253.63 Hz and 419.23 Hz, respectively. In the experimental test, the cantilever beam is excited by a shaker with different frequencies.

The cantilever beam is excited by a shaker with the frequency of 7 Hz which is approximate to the first natural frequency of 7.37 Hz.

Substituting the three output signals of the 3 × 3 coupler as shown in

The cantilever beam is excited by the shaker with dual frequencies of 7 Hz and 40 Hz, respectively. The three output signals from the 3 × 3 coupler are plotted in

Substituting these three output signals of the 3 × 3 coupler as shown in

Optical fiber sensors have been demonstrated for their capability to measure the dynamic responses of structures. They permit continuous monitoring of the integrity of the host structures. An optical fiber system has been developed for dynamic sensing in real time. This was done using a Mach-Zehnder interferometer incorporated with a 3 × 3 coupler for strain sensing under dynamic loading. In this work, the phase shift demodulation of the Mach-Zehnder interferometer is carried out using the commercial software Matlab. In the experimental test, the dynamic response measured by the optical fiber sensor for a cantilever beam subjected to base excitation is validated with the result of strain gauge. There is no particular restriction on the frequency of the vibrating structures in the proposed model. However, to measure the high frequency responses, it requires a data acquisition system with high sampling rate. The proposed optical fiber system is simple, inexpensive and easy to implement; moreover, it is high sensitive and accurate. These superior characteristics make it very useful and attractive for dynamic sensing.

The authors gratefully acknowledge the financial support provided by National Science Council of ROC under Grant No. NSC 99-2221-E-155-012 for this work.

Mach-Zehnder interferometer.

Schematic diagram of the Mach-Zehnder interferometric optical fiber sensor.

Block diagram of the phase shift demodulation.

Three outputs of the 3 × 3 coupler.

Comparison of the demodulated phase shift and exact phase shift.

Cantilever beam mounted on a shaker.

Three outputs of the 3 × 3 coupler with excitation frequency of 7 Hz.

Demodulated phase shift with excitation frequency of 7 Hz.

Dynamic strain of a cantilever beam subjected to base excitation frequency 7 Hz.

Three outputs of the 3 × 3 coupler with dual excitation frequencies of 7 Hz and 40 Hz.

Demodulated phase shift with dual excitation frequencies of 7 Hz and 40 Hz.

Dynamic strain of a cantilever beam subjected to dual excitation frequencies of 7 Hz and 40 Hz.