Operation principles of the curvature gauge

Transcription

1 Operation principles of the curvature gauge Y.Z. HE AND A. DJODJEVICH Departent of Manufacturing Engineering and Engineering Manageent City University of Hong Kong Tat Chee Avenue, Kowloon HONG KONG (CHINA) Abstract: - We show that the power loss in curved optical fibres is far greater on its convex boundary than on the opposite, concave boundary. We also report on the insight gained into the physical echaniss contributing to losses inside bent optical fibres. This insight helped us design and produce a loss-odulated optical fibre sensor for easureent of deforation-curvature of structures under echanical loading. As an alternative to strain easureent, such curvature gauge offers a new and radically different concept of onitoring structures and loads. One of its advantages is that it eliinates the need for extree sensitivity strain easureent in bending of thin structures. Moreover, since curvature is a global variable of a structure s cross-section, its easureent eliinates the abiguities associated with strain easureent that are caused by the delaination and icrostructural effects which alter the load transfer onto the straineasuring eleent. Additionally, gauge ay be placed anywhere in the section, including in its iddle where strain is zero and cannot be easured (offering a design flexibility). Key-Words: - Curvature gauge, Distributed easureent, Deforation ode-shapes 1 Introduction A new fibre optic sensor has been proposed with the fibre transissivity sensitive to curvature of loaded structures [1-4]. Naed curvature gauge, it easures inute deflection-curvatures whose radii are in the kiloetre range. The gauge has no echanical coponents attached to it, aking it suitable for integration (ebedent) into structural eleents. Directionality and polarity of easureents are achieved. With two gauges with non-coincidental orientation (not necessarily orthogonal), both, the direction and agnitude of deflection can be deterined. Maxiu Sensitivity Virtual Insensitivity Cantilever Optical fibre Figure 1: Polarity of Measureents 1.1 Advantages of Measuring Curvature The depth at which a curvature sensor is ebedded in a structure's cross-section is of no relevance since curvatures change negligibly within the section (because the structural thickness is negligible in coparison to the deforation-curvature radii that are in the kiloetre range). This is in sharp contrast to strain, which varies within the section fro its axiu negative to its axiu positive value. Hence, ebedding curvature sensors into structures and interpreting the signals obtained are siplified [1,]. Curvature easureents can be ade anywhere in the cross section, including along the neutral axis where there is no strain in bending. Siilarly, because strain agnitudes are inute in thin structures, the need for extree sensitivity is eliinated if curvatures are easured instead. For every strain and every curvature sensor, there exists a breakeven thickness separating the application doains for the two concepts. Moreover, it is an iperative for strain easureents that sensors be unobtrusive. Despite the sall diaeter of optical fibres (of the order of 15-5 icrons), their diaeters are large in coparison with those of typical structural reinforcing fibres (diaeter of graphite fibres is approxiately 5-1 icrons and a single layer of advanced coposite aterial is about 1-14 icrons thick). Although fibre optic sensors ay be

2 ebedded relatively easily in lainated aterials, these inclusions have been shown by any authors to cause local stress/strain concentrations. For exaple, an optical fibre included in a fibre/resin coposite causes foration of the resin rich areas representing local discontinuities in aterial properties [5]. These local strain perturbation-effects have received considerable attention in the literature as they adversely affect the accuracy of strain easureent. Furtherore, selection of the optical fibre coating also plays an iportant role as it cushions strain transission onto the fibre itself. Hence, optical fibres ust be regarded as ultiphase entities [5] and this further coplicates interpretation of the signals obtained. This is particularly noticeable if optical fibres are glued to a etal surface or ebedded into concrete where it is desirable to retain the fibre's protective coating. Measureents with the curvature sensor, on the other hand, are NOT based on strain easureents and for that reason are insensitive to local strain perturbations. It is the structure's change in curvature with bending which is the easured quantity; this is a variable NOT local to the sensitised zone of the fibre. The optical fibre by itself renders little resistance to bending and readily confors to the shape of the structure. 1. Sensitivity Curvature with the radius approaching 4 k has been easured. (Please note that curvature is defined as the reciprocal of its radius and very sall curvatures have very large radii.) Such sall curvatures correspond to those occurring when the free-end of a 5 c long cantilevered bea is deflected by.7. This nuerical figure pertains to the gauge s axiu sensitivity direction and varies with the plane of bending (Figure 1) the gauge exhibits polarity. For exaple, if the bea is supported in the chuck of a lathe and its free-end is deflected a constant aount in the horizontal plane, sinusoidal response is obtained while turning the chuck. This is equivalent to deflecting the stationary bea in different planes. 1.3 Torsional and Axial Loads Curvature sensor ay also be used as an axial force and torque sensor [3], Figure. The change in fibre curvature during either of these loadings is functionally related to an equivalent structural deforation. Figure : Application of Curvature Gauges for axial and torsion loads.. Matheatical derivations.1 Motivation and concept outline While the early work with the curvature gauge was experiental, further progress was constrained by the lack of deep understanding of how the gauge operated. Hence, theoretical analysis was carried out, results of which are suarized in this paper. Since the curvature-gauge in operation is essentially a curved wave-guide, the well established wave equation is expressed in the curved-cylindrical coordinate syste of the curved fiber and is copared with the standard Maxwell s equation. The distribution of an effective refractive index across the fiber is then calculated as a function of the fiber curvature radius. This allowed us to forulate the ray equation and to copute the incident ray angles at the boundary. Our calculations show atheatically that the fiber curvature alters incident angles in the way that is equivalent, in ters of the loss variation, to having a straight fiber with a total internal reflection angle that is higher on its convex side and lower on the opposite side. Most of the power loss therefore occurs on the convex boundary. Our results are supported by experiental findings.. Wave equation in local fiber coordinates Figure 3 illustrates a global cylindrical syste of reference (, ψ, z) as well as a curved-cylindrical syste (r, φ, ς) local to the fiber. The origin O of the global coordinate syste is in the center of the fiber curvature; is the radius of this curvature; the origin O 1 of the local fiber syste is soe fixed reference point on the fiber axis; point O is the intersection between the fiber axis and the noral

3 section of the fiber through soe point of interest, P, (the plane of this noral section contains point O as well); ζ is the curved coordinate along the fiber axis fro O 1 to O and z is perpendicular to the plane OO 1 O of the fiber curvature. O Figure 3: Curved-cylindrical syste of the fiber The functional relationships between the local curved-cylindrical coordinate syste (r, φ, ς) and the global syste (, ψ, z) are: = r cos φ z = r sin φ (1) ς ψ = In global cylindrical coordinates (, ψ, z), the wave equation is obtained in the following for [6]: r 1 (cos z 1 r φ ς φ r n k 1 r r 1 sin φ r E = ) φ () where n=n for the fiber core, n=n d for the cladding, k is the wave nuber and E is electric field. We are the first to report the wave equation () in the local coordinates of the curved fiber (r, φ, ς)..3 efractive index distribution in a curved optical fiber We have shown [6] that for sall fiber curvature (deflection-curvature radius is very large), Equation (3) reduces to: P r O O 1 P p x 1 1 E r r φ r r ς (3) rcosφ n (1 ) k E = Equation (3) differs fro the standard wave equation for the straight fibre [7] only in the very last ter. They becoe identical if the following effective refractive-index distribution is introduced for the curved fibre: r n eff = 1 cos φ n (4) In other words, a curved fiber can be thought of as a straightened fiber with an effective index distribution (4). This has also been verified experientally [6]. If x is to substitute rcosφ (x=rcosφ, Figure 3), Equation (4) reduces to: x n eff = ( 1 ) n (5) Expression (5) reveals a stratified distribution of the effective index in fibers bent to a radius : this index varies only while oving along the global axis (x= ), Figure 3..4 Incidence angles in curved optical fibers Based on Equation (5), the variation of n eff fro the concave to convex fiber side of the straightened fiber (along the x axis) ay be viewed as if the ediu traversed has been stratified into layers of constant refractive index. Such non-hoogeneity affects light propagation through the fibre, resulting in the curved ray trajectory. It has been shown [6] that the incidence angles of rays vary according to the following Equation: π ( B π δ dq 1 dh tan( ) = ± (6) sin θ dξ sinθ dξ where ξ cos( πξ Q = 3 [1 B ( π sin ( πξ ] H = ξ cos θ x U = [1 ( B B ( π U cos( πξ π sin ( πξ ] 3 x =B cos(πξ/l) is the th (=1,,3 ) deflection ode shape for a structure of length L, at its coordinate ξ, x o =x(ξ=), and the upper/lower sign pertains to the convex/ concave fibre boundary.

4 esults based on such calculations are shown in next Section where they are copared to their counterparts for the straight fiber as the reference case for evaluation of the loss variation with fiber curvature. 3 Coputations Based on Expression (6), for a particular fibre used, the graphs of Figures 4 and 5 show that effects of the fiber curvature on rays directed towards the convex fiber side are opposite to those directed towards the concave fiber side. Figure 4 shows the change of the rays incidence angles at the boundary with the fiber deforation agnitude B. The upper (lower) line pertains to the ray propagating towards the concave (convex) boundary. Only for the straight fiber (B =) the incidence angle at the boundary reains equal to its initial value at the launch point. With the increasing fiber curvature, the incidence angle at the concave boundary increases. It decreases at the convex boundary. In either case, it differs fro its initial value at the launch point in proportion to the fiber deflection agnitude. Incidence angle [ ] concave side convex side Fiber deflection agnitude [µ] Figure 4. Incidence angles at boundary vary with fibre curvature As the incidence angles increase on one side and decrease on the opposite side of cross sections of curved fibers, the losses on corresponding boundaries are no longer balanced. They increase on the convex and decrease on the concave fiber side. With the fiber s deflection agnitude set to B=1µ, the graph of Figure 5 shows the incidence angle variation at the boundary with the initial incidence angle of the rays at the launch point (θ ). The larger the initial value, the greater the change at the boundary. It is again evident that this variation is exactly opposite for the rays directed towards the concave and convex fiber sides: while of equal agnitude, such change is positive (negative) for the ray approaching the concave (convex) fiber boundary. Incidence angle [ ] concave side convex side Incidence angle at launch [ ] Figure 5: Incidence angles at fibre boundaries for different launch angles Because the incidence angles decrease for rays propagating towards the convex fiber side, soe of these rays no longer satisfy the condition for their total internal reflection at the boundary. Consequently, such rays escape fro the fiber, giving rise to the radiant loss on the convex side. As the curvature increases, so does this loss, which explains the odulation principle of the curvature gauge [1-4]. On the other hand, rays that in the straight fiber satisfied the condition for total internal reflection would do so at the concave side of the curved fiber with an even greater argin because their incidence angles have increased further. For the sae reason, if there were rays that in the straight fiber were contributing to the radiant losses (for exaple close to the input end of the fiber), they ight no longer do so at the concave boundary of the curved fiber. The concave curvature tends to help in guiding such rays that do not satisfy the condition for the total internal reflection in a straight fiber. This loss reduction would tend to offset the loss contribution by the convex boundary. In order to prevent this decrease in its sensitivity, curvature gauge should be built further along the fiber where there are few such rays and where the equilibriu ode distribution ay be assued. Such findings have been confired experientally [6].

5 4 Conclusions We show atheatically that the incident angles decrease on the convex and increase on the concave boundary of curved optical fibres relative to the situation with the straight fiber as the reference case. In ters of the loss variation, this is equivalent to having a straight fiber with a total internal reflection angle that is higher on its one (convex) side and lower on the opposite side. Most of the power loss therefore occurs on the convex boundary. Our analysis is supported by experiental findings. esults provide an insight into the operation of the optical fiber curvature gauge reported recently. This gauge is the first to easure inute deflection-curvatures of loaded structures. Advantages of the curvature (over strain) easureent concept are presented in the Introduction. Acknowledgeent: This work was supported by the University Grants Council of Hong Kong under the grant nuber CityU 1/99E. Help by Dr. Svetislav Savovic, esearch Assistant, is gratefully acknowledged. eferences: [1] A. Djordjevich and M. Boskovic, Curvature Gauge, Sensors and Actuators (Physical), Vol. 51, 1996, pp [] A. Djordjevich and Y.Z. He, Thin Structure Deflection Measureent, IEEE Transactions on Instruentation and Measureent, Vol. 48 No.3, 1999, pp [3] A. Djordjevich, Curvature Gauge as Torsional and Axial Load Sensor, Sensors and Actuators (Physical) Vol. 64, 1998, pp [4] A. Djordjevich, J.K. Ki and Y.Z He, Curvature Gauges Ebedded in Coposites, Materials Science esearch International, Vol.5 No.3, 1999, pp [5] J. S. Sirkis, Interpretation Of Ebedded Optical Fiber Sensor Signals, in Applications of Fiber Optic Sensors in Engineering Mechanics, F. Ansari, Ed. New York: Aer. Soc. Civil Eng., 1993, pp [6] Y.Z. He, Operation Principles Of Curvature Gauges, PhD Thesis, City University of Hong Kong, 1, in print. [7] D. Marcuse, Influence of Curvature on the Losses of Doubly Clad Fibers, Applied Optics, Vol.1 No.3, 198, pp