on javelin Lift aaad Drag

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1 INTERNATIONAL JOURNAL OF SPORT BIOMECHANICS, 1989, 5, Effect of Vibrations on javelin Lift aaad Drag Mont Hubbard and Christy D. Bergman The theory of crossflow aerodynamics is used to estimate the effect of throwerinduced vibrations on javelin mean lift and drag. Vibrations of all modes increase both lift and drag from the vibration-free condition. Percentage increases in lift and drag are largest at small mean angles of attack, large vibrational amplitudes, and large relative wind speeds. Thus the consequences of vibration effects on aerodynamics may be most significant for elite throwers. In recent years much progress has been made in the understanding of the javelin throw. Simulations of the equations of motion, first by Soong (1975) and later by Red and Zogaib (19771, Hubbard and Rust (1984), and Hubbard and Alaways (19871, have illuminated the effects of release conditions on the flight of the javelin and its range. In addition it has been possible to search in the space of initid conditions for those optimal release variables that maximize the range (Hubbard, 1984). Strides have dso been made in the practical application of this theoretical howledge to the training and coaching of throwers. Bubbard and Alaways (in press) report on the development of a training system based on high-speed 200-Hz video, with which it is possible to measure the release conditions in an actual throw rapidly enough (within several minutes) to be useful in the development of throwing technique. In this work it was found that javelin transverse vibrations, first noted by Ganslen (1967), are a severe detriment to the determination of release conditions, especially velocities of any sort, using position measurements from film or video. Indeed, the significance of vibrations in data analysis had first been realized by Gregor and Pinkc (1985) in their analysis of high-speed video data from a world record throw. Although not explicitly noted in their discussion, vibratory components of acceleration are evident in the data of Miller and Munro (1983). Vibrations have also been observed during javelin wind tunnel tests and in actual throws, by Terauds (1974, 19781, and are mentioned in the review by Bartlett and Best. A general description of the origin of transverse vibrations is given in Bubbasd (1984). During the 150-ms duration of the major acceleration phase of The authors are with the Department of Mechanical Engineering, University of California, Davis, CA

2 JAVELIN LIFT AND DRAG 41 the throw, extremely large forces are applied to the javelinby the thrower. The resultant mean acceleration of the javelin is roughly 40 g's. If these large forces are not applied directly along the javelin's long axis, then transverse vibrations are induced (see Figure 1) that persist until damped by aerodynamic and internal

3 HUBBARD AND BERGMAN Ganslen's (1967) original reference speculates that transverse vibrations cause increased drag and thus sdecrease in range. Terauds (1985, p. 123) cate* gorically states that "greater oscillations [vibrations] increase dragnand decrease lift." In fact, however, there has as yet been no scientific study of the effects of vibrations on javelin aerodynamic characteristics, The purpose of this paper is to calculate the contributions of transverse vibrations to the javelin lift and drag forces. Because die development is somewhat complicated, we will proceed in stages. In the following section the basic quantitative theory & vi~tions is briefly summarized and typical javelin modeshapes and frequencies are calculated. Then the theory of crossflow aerodynamics is described, assuming that the javelin is a rigid body and neglecting the vibration phenomenon. In the section following that, the crossflow theory is modified to account for contributions to relative wind from the vibratory velocities. Finally we present the results of numerical calculations using the model that show the dependence of lift and drag on vibration amplitude, modeshape number, mean angle of attack, and mean wind speed.

4 JAVELIN LIFT AND DRAG 43 nonhomogeneous javelin, however, I(x) and m(x) vary with x and the solution must be computed numerically. To obtain representative numbers, a Held Custom 111 javelin was cut in half (end to end) and the outer diameter 6(x) and wall thickness t(x) were measured. The two functions I(x) and m(x) are then determined from where m,(x) is the mass of the nonstructural weight added to balance the javelin and es is the density of the javelin structural material. The functions I(x) and m(x) were used as data in a nonlinear finite element computer program that calculates the vibratory frequencies y, and the corresponding modeshapes pk(x). The first four of these are shown in Figure 2. (In all practical applications the solution is approximated by a sum of only a small number of modeshapes.) The first two frequencies differ only slightly from the lowest frequencies (26.4 and 72.7 Hz) of a homogeneous free-free beam with the bending stiffness EI and mass density of the javelin at the grip. The unsymmetric modeshapes exhibit features that plausibly result from the weighted and stiffer front end and the lighter, less stiff tail. Table 1 contains numerical values for the first four normalized modeshapes as a function of position along the javelin, with x=o at the front tip. FREQUENCY Figure 2 - First four free-free vibratory modeshapes of a typical javelin and associated frequencies. Modeshapes and frequencies are calculated numerically by considering javelin as an inhomogeneous unsupported beam with varying mass and stiffness per unit length. Although a general vibration contains all modes, the higher frequency modes rapidly diminish in amplitude and importance in real throwerinduced vibrations.

5 HUBBARD AND BERGMAN Table 1 First Four Javelin Vibratory Frequencies wk and Modeshapes pk(x) Crossflow Aerodynamics With No Vibrations Crossflow principles are a heuristic concept first proposed by Allen (1949) for predicting static aerodynamic forces and moments for bodies of revolution and high fineness ratios (large ratio of length to diameter) at angles of attack much larger than those for which potential-flow aerodynamic theories apply. In the method, a crossflow lift due to viscous crossflow separation is added to the crossflow lift from potential theory. The method has been used mostly to compute the static aerodynamic forces on slender missiles and aircraft, and is comprehensively summarized by Jorgensen (1977).

6 JAVELIN LIFT AND DRAG 45 As the next step in the development of our complete model, we derive the crossflow aerodynamic equations of a javelin for the case of no vibrations. Since javelins are symmetric and the main motion occurs in a vertical plane, we will restrict ourselves to a two-dimensional analysis. Recalling conventional aerodynamics, the Reynolds number, Re, is given by where q denotes air density, v velocity, 6 a characteristic length (here the javelin diameter), and p the coefficient of dynamic viscosity for air. Let vr be the velocity vector of the relative wind inclined to the javelin long axis by angle of attack a, anghavi~ normg and-axial components v, and v,, respectively. That is,tr = v,i + v$ where i and j are unit vectors in the x and y directions, respectively (see Figure 3). Following the basic crossflow principle, we separate the effects of v, and v,. The axial component v, contributes only a skin friction force, typically small when compared with the pressure forces caused by the normal component Vn, especially at large angles of attack. Thus, in what follows we neglect the skin friction drag due to v, and approximate the total aerodynamic force as that produced by v,. Another justification for considering only v, is that the vibrating motions to be considered in the next section are entirely in the normal direction. Although the potential flow term can contribute significantly for blunt based Figure 3 - Schematic of vibration-free javelin inclined to flow at angle of attack a. Crossflow aerodynamics describes production of a differential crossflow force df, through the effect of normal component of velocity vn on the nearly circularly shaped differential element of diameter 6(x) and length dx. The integral of dfn is total normal force.

7 46 HUBBARD AND BERGMAM missile geometries, a body with a sharp aft end like a javelin is adequately described by the crossflow term alone. Thus we make the further simplification of neglecting the potential flow term compared with the crossflow viscous separation term. As shown by Jorgensen (1974, p. 831, the ratio of these terms is typically only a few percent. Because the rate of change of diameter 6(x) with respect to x is so small, each differential element of javelin of length dx may be approximated as cylindrical, and the normal velocity component v, produces a normal force which depends on the Reynolds number. The cross-sectional diameter 6(x) varies with distance along the javelin, and thus the local Reynolds number is given by Thus the normal force on an element of length dx is given by where Cd(Re(x)) is the drag coefficient for a right circular cylinder given by experiment (Daugherty & Franzini, 1977). Then the total normal force, F,, over the length,.!, of the javelin is e Fn = 1 df,(x) (81 and the total lift and drag on the javelin are simply the components of F, perpendicular and parallel to the relative wind, respectively FL = Fn cos a; IFD = Fn sin a. (90) We emphasize that the calculation of the lift and drag (Equations 6 through 9) must be done numerically since the Reynolds number, and hence the drag coefficient, may vary substantially along the length. Crossflow Aerodynamics lncikadiing Vibrations Next we consider the effect of free-free vibrations in the crossflow aerodynamic model. Although several modes of vibration can be present in a throw, for simplicity here we calculate the aerodynamic effect of each mode separately. Each mode results in transverse vibrational velocities of the javelin relative to its undisturbed position of (see Figure 4) Further, the javelin centerline undergoes angular deflection of magnitude B(x,t) = tan-'(dyldx) = tan-' For each differential length dx, the total flow velocity is the vector sum of the mean normal flow, v, (see Figure 4) and the vibratory component of

8 JAVELIN LIFT AND DRAG 47 Figure 4 - Schematic of javelin with vibration inclined to flow at mean angle of attack a. The differential element dx has vibratory velocity relative to its undisturbed position (the x axis) of magnitude jr(x). The total relative wind is the vector difference of v, and f. Thus the normal component of the relative wind and the local angle of attack are no longer constants but instead vary along the length. velocity, jr) due to oscillations. This total velocity has components transverse and parallel to the local centerline of vt =! v, sin p - y cos 6; vp = v, cos + y sin 6 (12a,b) where the angle P between the local normal and the relative wind direction is given by p = 6(x,t) + a. (13) A consequence of the fact that vt and v are both functions of x and t is that the local angle of attack also varies with &stance along the javelin and with time. This local angle of attack q(x,t) is given by Again applying the crossflow principle, -tion becomes a! (x,t) = tan-l(vt(x,t)/vp(x,t)). (14) 6 for the local Reynolds number and Equation 7 becomes where dfp(x,t) is now the aerodynamic force per unit length acting perpendicular to the center line of a differential element of the vibrating javelin (see Figure 4). The contributions of dfp(x,t) to lift and drag are its components perpen-

9 dicular and parallel to the mean relative wind where Tk = 2~1%. autioned that, although the potation has not yet denoted the, in fact L and D in Equation 19 are fiinctiond of four pasidered; ak, amplitude of the modal vibration; a, mean bgle of attack; and v,, mean relative wind spe may be kept in mind by writing them in the fo L = L (k, ak, a, v,); D = D In the results to be presented below, the role of each of the four variables above will be probed, For the numerical integrations represented by Equation 18 above, the javelin length was divided into 100 segments, each 2.6 cm in length. At every x, javelin diameter &(x) and modeshape pk(x) were computed. Given values for relative wind speed v,, and vibration amplitude ak, Equations 10, 11, $12, 13, and 17 were then evaluated. Finally, the drag coefficient Cd based on the local Reynolds number was computed from a look-up table and the remainind faetors in the ihtegrand of Equation 18 were evaluated. The time averages expressed by Equation 19 were accomplished by integrating Equation 18 at many (typically 20) times evenly spaced in time over one period of vibration of the 'kth mode and then averaging. Results in the evaluation of the instantaneous, Figure 5 portrays some of the crosstimes, t=o and t=n/2wl= 114 or first mode (k= 1): Por <ease in 5 assumed no fnean'ahgle of attdck of the grip a, =0."1 m. hipre 5a

10 JAVELIN LIFT AND DRAG 49 plots the transverse velocity component vt on which the calculation of the crossflow force is based. At t=o the javelin is straight and points in the direction of the relative wind, but all points except the two stationary nodes have maximum vibratory velocity. The largest velocity, 38 mls (even larger than the mean relative wind speed!), is that of the tail (at x=2.6 m), but even the grip has a substantial transverse velocity of - 15 d s. After a quarter of a period, however, the javelin has fully deformed into the first modeshape and is motionless (the cosine factor in Equation 10 is zero). At this time the transverse velocity is simply the transverse component of the relative wind, and varies with length because the slope of the centerline does also. In the front of the javelin (x=o) the y displacement is negative (the tip lies below the x axis) and the relative wind approaches the tip from "above," resulting in a negative value for the transverse velocity of about 10 mls and a negative local angle of attack. A point near the grip is roughly 0.1 m above the x axis but has zero slope so the local angle of attack and transverse velocity are zero there. The tail has a larger negative deflection than the tip, but a positive slope (also larger than that at the tip) and a corresponding transverse velocity of about 15 d s. Figure 5b portrays the differential perpendicular crossflow force per unit length dfp at the same two times. The shape of the force results from its dependence on the product of the square of the crossflow velocity and the diameter, both of which vary with position along the axis of symmetry. Because the grip adds roughly 20% to the maximum javelin diameter, its effect on the crossflow force is clearly visible at x= 1.2 m when t=o, a time when the grip has nonzero velocity. Even though the tail diameter is only about 20% of that near the grip, the crossflow force is largest at the tail at t=o because the square of its large velocity more than compensates for the diameter deficit. At t= 114 period, the force profile is much smaller and less complicated because the velocities everywhere are smaller (Figure Sa). Figure 6 shows the crossflow force per unit length for the same mean relative wind speed as in Figure 5, but for a nonzero mean angle of attack, a = 30". The force for the no-vibration case is compared to that at three times (t=o, 114 period, 112 period) in a vibration of the fundamental mode with amplitude 0.1 m. When there is no vibration (dashed curve), the transverse velocity is constant along the length and the crossflow force is nearly exactly proportional to the javelin shape (diameter is fat in front with an abrupt sharp point, but tapers gradually in the rear), the difference being the small dependence of Cd on the local Reynolds number which varies because of the variations in diameter. Again the shape of the grip is clearly evident. The javelin is straight at both t=o and t=112 period but has maximum velocity of opposite signs at these two times. Near the grip the vibratory transverse velocity nearly completely cancels the tranverse component of the mean relative wind when t=o but adds to it at t= 112 period, resulting in enormous variations in the crossflow force magnitude and shape over the course of one cycle. Interestingly, at t = 114 period when the javelin is completely motionless, the relatively narrow tail has a larger crossflow force per unit length than the thicker fore section because the deformed shape adds to the mean angle of attack behind the grip but subtracts from it in the front,

11 HUBBARD AND BERGMAN DFP VS. X AT TIME-O AND 114 PERIOD AWW=@DEC).aiMP=O. l(m),wialkss(kl/5),file=bjfp.plc Figure 5 - Crossflow variables as a function of length along javelin at two times (&=O and 114 period) during a vibration of the fundamental mode (lr=l) with zero mean angle of attack, viloaation amplitude of 0.1 m, and mean rdative wind speed of 30 m/s. (5a) Transverse velocity; velocity at t =0 is largest at tail and is entirely due to vibratory velocity; after 114 period the javelin is deformed but motionless and the transverse velocity is simply the transverse component of the wind speed. (5b) Differential crossflow force per unit length. When 8=0, note the clear effect of the larger grip at x=1.2 m and that the force is largest at the tail.

12 JAVELIN LIFT AND DRAG 51 DFP VS. X WITH AND WITHOUT VlBRATlONS ~-~o@fd)a~p=o QII o.i(u~.mdrsoiy/s~,nu=auip.mc Figure 6 - Differential crossflow force per unit length as a function of position along javelin for no vibration and at three times (t =0, 114, and 112 period) during a 0.1-m amplitude vibration of the fundamental mode, all with mean angle of attack of 30" and mean relative wind speed of 30 mls. At t =O and 112 period, the javelin is straight but has substantial vibratory velocity. At the grip this velocity cancels that of the wind when t =O but adds to it at t = 112 period. With no vibrations, the crossflow force roughly mirrors the cross-sectional shape of the javelin, fat with a sharp point in front but gradually tapering in the rear. Again note the effect of the grip. Figures 5 and 6 certainly support the expectation that vibrations will change the lift and drag from those experienced in rigid flight. In order to see the total effect, however, the differential crossflow force given by Equation 16 and shown graphically in Figures 5b and 6 must be integrated with respect to length and averaged over the period of vibration. Shown in Figure 7 are the results of these calculations; mean lift and mean drag are plotted versus mean angle of attack for several values of vibration amplitude of the first mode. For mean angles of attack up to 0.85 rad (45"), a range that includes all angles of attack likely to be encountered in a typical flight, vibrations increase both lift and drag. Although the effect on drag is in conformity with intuition, the effect on lift is surprising and contrary to the speculations of previous authors (Terauds, 1985). When the results in Figures 7a and 7b are viewed over the entire range of angles of attack, vibrational effects seem not so large, being only a 10 to 15 % increase at larger angles of attack. Their importance can be assessed more accurately, however, by realizing that the effect of aerodynamics (and especially drag) on range is largest near the beginning of the flight and that flights almost always begin at small angles of attack. Lift and drag increase and decrease the range by augmenting and decrementing the vertical and horizontal velocities, respec-

13 52 HUBBARD AND BW

14 JAVELIN LIFT AND DRAG 53 LlFl VS. ALPHA. PARAMETERIZED BY AMPUTUDE(M) n z 4- ~E=l.WINI)r23[M~).FLE=UYKl V29VALPlC - w=om 1' --- w=om /' AMP= AMP AMP W=O.lO.*./- (c) ALPHA (RADIANS) and drag. For small angles of attack consistent with typical release conditions (d0.2 rad), substantial vibration of 8-cm amplitude can more than double the lift (7c) and more than triple the drag (7d).

15 54 HUBBARD AND BERGMAN tivelyi In each givenramonnt,af-rli&bor,drag acting for a fixed At adds a given amount,to.the~velocity, When the.lift,oridrag acts early in the flight, the added (or decreased) velocity endures for a longer time and thus affects the range more. In this sense, the effect of aerodynamics is largest near the beginning of the flight. Figures 7c and 7d show the same data as 7a and 7b on an expanded horizontal scale, for mean angles of attack up to o d(11.5"). Atanangle of attack of 0.1 rad (5,7"), a vibration amplitude o quintuples the lift and increases the drag by roughly a factor of lo! Although a vibrational amplitude of 0.1 m is large (probably a reasonable upper bound), it must be agreed that the effects of vibrations on javelin aerodynamics at small angles of attack are enormous. The-increased drag would cause decreases in range but the increased lift would be beneficial. Whether the lift will more than outweigh the effects of drag can only be answered by simulating the flight while accounting for the increased lift and drag due to vibrations. It may even turn out that the decades-old speculation that wibrations are uniformly harmful to the flight will be borne out, but that is beyond the scope of this paper. Figures 8a and 8b show the lift and drag, respectively, versus amplitude of vibration md parzmeterized by angle of attack. At all memrangles of attack, increasing the amplitude of vibratibn increases bo4h tlge lift and the drbg. In both cases, the percent increase is largest at sinall angles of attack and large vibration amplitudes. Presented in Figures 9a and 9b are the lift and drag for the second vibrationd mode, plw versus mean ring% df attaclc and param1!terizeti by drnfilitude of vibration. In thismodeprecall-that thewfrequency is 2.4 times the fundamental frequency. Also, because there are three stationary nodes rather than two (see Table 1 and Figure 2), a given vibrational amplitude results in larger slopes of the fully vibrationally deformed javelin away from the straight configuration. For these two reasons, the percent increases in lift and drag due to a given amplitude vibration are larger in the second mode than in the fundamental (compare to Figures 7a and 7b). This trend should be expected td continue for even higher modes. This is not to say, however, that the higher harinonics become ever more important in real throws because, as mentioned previously, there is less of the higher-modes present. Although we have not measured the relative amplitudes of the first two modes in actual throws, we expect the ratio to decrease faster than the increases in the lift and drag of the modes so that the fundamental will always dominate in its aerodynamic effects. Shown in Figures 10a and lob are the lift and drag for the fundamental vibrational tiiode, plotted versus mean angle of attack and parameterized by the mean relative wind velocity. Although the effect~~are somewhat obscured by the fact that vibration-free lift and drag are themselves quadratic Enctions of relative wind speed, a detailed inspection of Figure 10 reveals that the vibrational increases in both lift and drag are larger at larger velocities. This means that elite athletes with the largest release velocities may pay-the highest penalties for, or reap the highest benefits-from, thrower-induced vibrations (depending on whether vibrations help or hurt Overall). It is therefore more importa

16 JAVELIN LIFT AND DRAG 55 DRAG VS. AMPUTUDEB PARAMETERIZED BY ALPHA(DEGREES) WESHAPE=l.VWIND=~~(M /S).FILE=DAHK~VZW&PIC Figure 8 - Mean lift (8a) and drag (8b) versus amplitude of vibration of the first mode parameterized by mean angle of attack. The percent increases in lift and drag become less as angle of attack increases because the zero vibration lift and drag are monotonic functions of angle of attack.

17 HUBBARD AND BERGMAN DRAG VS. ALPHA, PARAMETERIZED BY AMPLITUDE(M) MODESHAPE=Z,VWIND=~~(M/S).~ILE=DAMK~V~~VAL.P~C 4 ALPHA (RMIAWS) Figure 9 - Mean lift (9a) and drag (9b) versus mean angle of attack parameterized by amplitude of vibration of the second mode. Second mode vibration again increases both lift and drag, but more than first mode at the same amplitude (compare Figure 7).

18 JAVELIN LIFT AND DRAG 57-5 I I I I I I I I ALPHA GwlANSl DRAG VS. ALPHA, PARAMETERIZED BY VELOCITY(M/S) LK)DESHLIPE=I,AHPUTUDE=O.~~(M).FILE=DVM~ A~-ZYUIPIC Figure 10 - Mean lift (10a) and drag (lob) versus mean angle of attack parameterized by mean wind speed for a constant amplitude of vibration of the fmt mode of 0.05 m.

19 58 HUBBARD AND BERGMAN Nubbad and daways (in press) report on the development of a high-speed video system to measure javelin rigid-body release conditions (release velocity, release angle, angle of attack, etc.) in a throw rapidly enough to be of use in thrower feedback. It was found that javdin transverse vibrations mst be explicitly accounted for in the delemination of release conditions using position measurements from Nm or video. Because the vibrations must be identified in order to be removed, a measure of the vibration severity is provided for free, so to speak. The video system can thus be used as a diagnostic tool in the improvement of the vibrational aspects of the throw as well as the rigid-body aspects. Once the consequences of the increased vibrational lift and drag on the trajectory have been determined through simulation, it will be possible to say what the desired vibrationd state is at release and to use the video system in the correction or optimization of thrower-induced vibrations. Before ending, we would be remiss not to address the extremely important pitching moment of the aerodynamic force about the center of mass. Crossflow principles are an approximation for the flow field and its effects which are able to predict mean lift and drag reasonably accurately. But they apparently are boo crude to allow calculation of higher moments of the pressure distribution. One of us (M.N.) has tried unsuccessffially to apply crossflow techniques in the prediction of pitching moment even in the simpler nonvibratory case. Conclusions A detailed model has been presented of the effects of thrower-induccd vibrations on the aerodynamic forces experienced by a javelin in flight. The model is based on crossflow aerodynamic principles and accounts for the perturbations of the relative wind speed and direction about their mean values due to the vibrational velocities of points along the entire length of the javelin. Vibrations of all nnodcs increase both the lift and drag over vibration-free conditions. The percentage increases in lift and drag are largest at small mean angles of attack, large vibrational ampliludes, and large mean relative wind speeds, implying that ahc consequences of such vibrations may be most significant for elite throwers. Allen, N.4. (1949). Estimation of the forces and morncnts acting on inclined bodics of revolution of high fineness ratios, NACA Technical Report RM A9126. Bartlett, R.M., & Best, R.J. (in press). The biomechanics of javelin throwing: A review. Journal of Sport Sciences. Daugherty, R.L., & Franzini, J.B. (1977). Fluid 1~7ecllarzic.s wit11 et~giireerit~g r~pplitntions. New York: McGraw Hill. Ganslen, R.V. (1967). Javelin aerodynamics. Track TecI~rziqu~, 30, Gregor, R.J., &Pink, M. (1985). Wiomechanical analysis of a world record javelin throw: A case study. Intert7ational Journal of Sport Bi~~rneclir~nics, 1, Hubbard, M. (1984). Optimal javelin trajectories. Jounzul (~Rior~~~c:lr~~/~i~~.s, 17, Hubbard, M., & Alaways, L. (1987). Optimal release conditions for the new rules javelin. International Jourtzal of S'lort Bionzechmzio~, 3, Hubbard, ha., & Alaways, L. (in press). Accurate estimation of release conditions in the javelin throw..lournal of' Biomechanics.

20 JAVELIN LIFT AND DRAG 59 Hubbard, M., & Rust, H.J. (1984). SimuIation of javelin erimental aerodynamic data. Jounurl of Biomechanics, 17, 769- Jorgensen, L.H. (1977). Prediction of static aerodynamic bodies alone and with lifting surfaces to very high angle of attack. (NASA Technical lied Mechanics, 44; 4