MATERIALS AND SCIENCE IN SPORTS. Edited by: EH. (Sam) Froes and S.J. Haake. Dynamics


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1 MATERIALS AND SCIENCE IN SPORTS Edited by: EH. (Sam) Froes and S.J. Haake Dynamics Analysis of the Characteristics of Fishing Rods Based on the LargeDeformation Theory Atsumi Ohtsuki, Prof, Ph.D. Pgs TIMS 184 Thorn Hill Road Warrendale, PA (724)
2 Analysis of the Characteristics of Fishing Rods Based on the LargeDeformation Theory Atsumi OHTSUKI, Prof., Ph.D. Department of Mechanical Engineering, Meijo University, Shiogamaguchi, Tempakuku, Nagoya, Aichi , Japan Abstract There are very few scientific studies about fishing rods in the field of the socalled sportsleisure. This study deals with large deformation of fishing rods that would be useful in the development of the characteristic design of fishing rods. In this report, based on the nonlinear large deformation theory, new fundamental equations are introduced for thin, straight tapered fishing rods with a circular crosssection under concentrated loads at the free end. As a result, it is found that the large deformations of fishing rods can be described by nondimensional load parameters, ratios of rod diameters and supporting angles. Furthermore, the experimental verification of this analysis is carried out using a flexible rod model. The theoretically predicted results are in fairly good agreement with the experimental data. Consequently, the new deformation theory is proved to be of practical use. Key words : Fishing, Rod deformation, Large deformation, Tapered beam, Mechanics, Elastica. Materials and Science in Sports Edited by F.H. (Sam) Froes IMS (The Minerals, Metals & Materials Society),
3 Introduction In recent years, leisure activities, such as tennis, golf, skiing, fishing, etc. are very popular. The performance of various instruments used in such leisure activities is also greatly improved. Scientific analysis on the function of such instruments is necessary in order to develop practical instruments in the field of sportsleisure. Various experiential studies that clarify the characteristics of fishing rods have been published. However, there are very few scientific studies. The behavior of fishing rods can be characterized by large deformation. The rod where the lower end is supported by hand is bent circularly due to the resistance force of the fish on the upper end. The fishing rod is very slender and flexible. The fishing rod is thus assumed to undergo large deformation. Ultimately, the analysis of fishing rod behavior is attributed to the large deformation problem of a cantilever beam with a taper. Some similar examples on large deformation are given in a polevaulting pole [13], archery [4,5] and Japanese bow [6,7]. The purpose of this study is to theoretically clarify the large deformation characteristics of the fishing rod. The structure of a fishing rod is classified into two types. A typical type is a combination rod with joints, made up of several parts with changes in crosssection, and the other is a tapered rod, which does not have any joints. In this research, the fishing rod with a taper, which is a prototype of a fishing rod, is analyzed theoretically. Though there is an analytic solution on the straight beam with a constant crosssection [8,9], the beam with a step or taper like a fishing rod has not yet been analyzed. Here, a fundamental equation that determines the deformed shape of tapered beam model is derived from an equilibrium state under a concentrated load at the free end. Moreover, in order to confirm the applicability of the presented theory, a large deformation experiment is carried out and the experimental data are compared with the theoretical results. Largedeformation theory In order to predict large deformation occurring in fishing rods, a simplified feature is given in Fig.l. The mathematical model is a tapered cantilever beam with a circular crosssection, subjected to tensile load at the tip. The lower end is supported at an arbitrary angle fi. A >7 loaded shape 0 y \i Figure 1: Large deformation of a fishing rod under load applied at the tip. 163
4 \ A thin, straight tapered beam with a circular crosssection. y A schematic configuration of a loaded cantilever beam. Figure 2: Mathematical model mathematical model is shown in Fig.2. As shown in Fig.2(a), the length of the beam is / and diameters of large edge (: lowed end, fixed end) and small edge (: upper end, free end, tip) are D l and D 2, respectively. The origin of the coordinate system is at point O, and x represents the horizontal displacement, y the vertical displacement, and 9 the deflection angle (see Fig.2(b)). In denoting the arc length by s, the radius of curvature by R and the bending moment by M, the relationship among R, M, s, x, y and 6 are expressed as: _ R" El" ds dx = ds cos0, dy = ds si (1) where E is Young's modulus and /, the second area moment of the cross section about the neutral axis. The diameter D and the second area moment / at an arbitrary position Q(x,y) are given by: D n 64 J (2) where /j is the second area moment at the large edge. Therefore, the fundamental equation of large deflection is derived from Eqs.(l) and (2). E\l 1 DJI f ds (3) 164
5 In order to facilitate the common understanding of large deformation behavior, the following nondimensional variables are introduced:, / Z) 2 EL ia^ (4) From Eqs.(l),(3) and (4), the nonlinear differential equation (5) is ultimately obtained. /, ZM } 4 d 2 9./, /Mr, /, /M I 3 d0 2  l 1 \TI\ r 4 1 Jl 1 \TI\ = acos0 (5) v; v DJ L \ dn \ DJ\ \ DI) L \ dn DI In general, it is difficult to obtain an analytical solution of the nonlinear equation (5). Therefore, in this research the RungekuttaGill (RKG) method is adopted in order to obtain the necessary solution numerically. The outline of RKG method is as follows: do = z (6) Boundary conditions : n = 0, 9 = p at the fixed end n = 0, ^ = 0 at the tip (9) dn (8) Then, x, y coordinates at an arbitrary position Q(x,y) can be calculated respectively from the following equations (10) and (11).  =fcosodri (10) (11) Especially, the vertical displacement yl at the tip is an important physical quantity. In order to compare with the result from the exact nonlinear largedeformation theory, the calculation formula based on the classical linear beam theory is shown in the following equation. Vi 1 = sinjs   a cos fi (12) i O As a special condition, the buckling load P cr of the beam sustained at the vertical position (i.e. /3=9Q ) is obtained from the following equation. 165
6 In other words, the nondimensional buckling load a cr is expressed as (14) Variable t/> of Eqs. (13) and (14) is determined as the value which satisfies the following transcendental equation. tantp ~ /Z>2 (15) On the other hand, the curvature I//? at an arbitrary position Q(x,y) is given by : R I (16) Moreover, the maximum bending stress a at the outer fiber of the beam is expressed as: (17) This maximum bending stress is importantly related to the breakage of fishing rods. Theoretical calculations and experimental results In order to confirm the applicability of the proposed analytical theory, experiment is performed on a commercially available fishing rod. The fishing rod is composed of five parts connected continuously. For the experiment, the tip section (length / =800 mm, weight W=15.2 g, large edge diameter D l =5.26 mm, small edge diameter D 2 =0.98 mm, the ratio of diameter D 2 /D 1 =0.186 with Young's modulus E =31.85 Gpa) is used. A schematic view of the experimental setup is shown in Fig.3. The rod is fixed at the large edge and placed on the 1mm Grid paper Load pan Figure 3: Experimental setup 166
7 A A = Exact largedeformation theory Experiments a =1.13 6, =13.6 a = , =58.1 a = , =74.2 a = , = xll Figure 4: Effect of the nondimensional loads, a, on large deformation of a rod. Exact largedeformation theory o Experiments 0.6 j max Tip 0.6 a = 5.63 A/ A =0.186 Figure 5: Effect of the supporting angles, /3, on large deformation of a rod. 167
8 Exact largedeformation theory o Experiments A/A = o.i 6, = 88.4 A/A = , =83.5 A/A = 0.3 9,= 75.7, A/A = 0.5 e, Figure 6: Effect of the ratios of diameter, D 2 /D 19 on l ar e deformation of a rod. vertical position. The rod deforms in a vertical plane. Loading is applied by hanging dead weights from a system of cables attached to the small edge (:tip) of the rod. In the experiment, horizontal and vertical displacements Jt, y caused by the application of vertical load P at an arbitrary position Q(x,y) are read directly off a sheet of standard grid paper (1 mm scale) affixed to the vertical plane. Figure 4 shows some deformed shapes of a rod with the ratio D2/D1 =0.186 supported at the angle /3= 60. The effect of the nondimensional load a, ranging from a light load to a heavy load, under large deformation is well known. The deformation increases gradually as the nondimensional load a increases. And, the peak (shown by X) represents the maximum vertical displacement (: ymax), which tends to move towards the fixed edge, as the nondimensional load a increases. It can be seen from Fig.4 that the actual values from the experiment are very close to the theoretical computed values. Figure 5 shows some deformed shapes of a rod with D 2 ID l =0.186 when the constant nondimensional load a = 5.63 is applied to the rod. According to this figure, it is possible to observe the effect of the supporting angle 13 on the states of large deformation. With an increase of the angle /J, the rod greatly deforms, and the peak (: ymax) tends to shift towards the tip. Here too, the theoretical computations agree well with the experimental measurements. Figure 6 shows some deformed shapes of a rod supported at the angle 13 = 60 when the constant nondimensional load a = 5.0. The effect of the ratio D 2 /D l on the states of large deformation can be investigated. The deformation is reduced gradually as the ratio D 2 /D l increases, which means that a rod with a larger ratio is difficult to bend in comparison with that of a smaller one. This tendency is remarkably observed in a rod in the tip side beyond the peak position. Here, the peak (: y max ), tends to move towards the fixed edge, as the ratio D 2 /D l 168
9 Exact largedeformation theory Linear theory 1.0 p = = 30 c L Figure 7: Relationship between the non dimensional vertical displacement, y max //, at the tip and the nondimensional load, a, when the rod is supported at /3. increases. Figure 7 exemplifies the variation of the nondimensional vertical displacement y max // at the tip for several supporting angles 13 under the particular condition in which the ratio D 2 /D l =0.1. When the rod is fixed at the angle f$ = 90, the rod is only compressed by a load smaller than the critical load, but the rod does not bend at all. When a load exceeds the abovementioned critical load, the rod buckles and begins to bend. This critical load (: P cr ) is called as Euler's buckling load. In this figure, the discontinuity of the line (when p = 90 ) shows the critical load (which corresponds to the nondimensional buckling load a cr =0.804). The computations based on the linear beam theory and largedeformation theory are both shown in Fig.7 to show the limit of application of the linear beam theory. For example, the linear beam theory is applicable until a nondimensional load a takes a value of about 0.3, if the error between the exact theory and the simple linear theory is limited within 2 %. Conclusions For effective use of a fishing rod, it is essential to understand the large deformation 169
10 behavior of a fishing rod under a variety of loads. In this paper, the nonlinear large deformation response of a simplified rod model is analyzed theoretically. The fundamental equations can be obtained under concentrated loads at the tip and calculated strictly by means of numerical solution method. The numerical solutions are obtained for the representative flexural quantities such as the horizontal displacement, vertical displacement. Furthermore, a large deformation experiment is carried out to confirm the applicability of the present large deformation theory. The following conclusions are drawn based on the results of the theoretical and experimental analyses. (1) The large deformation behavior of a rod is affected by deflection parameters such as the nondimensional load a, ratio of diameter D 2 /D l and supporting angle /3. Especially, the deformed shape of a rod depends on D 2 /D 1 considerably. (2) The maximum vertical displacement (: ymax) tends to move towards the fixed edge, as a, D 2 /D^ /3 increases, respectively. (3) The vertical displacement of the tip y t is also a function of a and D 2 ID V. (4) The results predicted theoretically are found to be in very good agreement with the experimental data. Acknowledgements The author wish to acknowledge the contributions of Mr. T. Takeuchi of the Meijo University, Japan, for the data processing. References 1. Hubbard, M., "Dynamics of the Pole Vault", J. Biomechanics. Vol.13, No.ll, 1980, Ohtsuki, A. and Ohshima S., "Analysis of Mechanism of Pole Vaulting", Trans. JSME. Vol.64, No.623,1998, Ohtsuki, A. and Ohshima S., "Fundamental Study on Large Deformations of Poles and PoleVaulting Characteristics", Journal of Japan Society of Sports Industry. Vol.8, No.2, 1998, Hay, J. G., The biomechanics of Sports Techniques ( Prentice Hall, Inc., 1973), Hickman, C. N., " The dynamics of a bow and arrow", Journal of Applied Physics. Vol.8, 1937, Hosoya, S., Miyaji, C. and Kobayashi, K., "Computer simulation of restitution of Japanese Bows, Journal of Japan Society of Sports Industry, Vol.5, No.2, 1995, Ohtsuki, A. and Ohshima S., "Analysis of Large Deformation of Japanese Bows", The engineering of Sports (Blackwell Science Ltd., 2000), Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity (Dover Publications, Inc., 1944), Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability (McGrawHill, Inc., 1959),
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