Mechanics of a Simple Bow

Transcription

1 Mechanics of a Simple Bow Mar French, Assistant Professor Brendan Curtis, Undergraduate Student Vinh Pham, Undergraduate Student Department of Mechanical Engineering Technology Purdue University West Lafayette, IN 4797 rmfrench@purdue.edu Abstract The simple bow is one of the first machines made by man. It is still a fascinating device since its simplicity results in complex behavior. The mechanics involved are highly non-linear and sensitive to even small changes in the geometry of the limbs. To experimentally investigate the behavior of simple bows, we made a shooting machine designed to shoot with little variation in results. In addition to experimental results, we show a discrete analytical model that captures the static behavior of the bow. Introduction A bow is simply a device that turns strain energy into inetic energy. The strain energy stored in the limbs (the flexible elements) is converted to both inetic energy in the limbs and inetic energy in the arrow. A simple bow is one that doesn t use pulleys or levers to produce a mechanical advantage. The geometry of a symmetric simple bow is shown in Figure. Figure - Geometry of a Bow Being Drawn

2 Analytical Model We chose to model the bow limbs simply as an assembly of rigid elements connected by linear torsion springs [,]. The geometry of the math model is shown in Figure. Figure - Geometry of Limb Discretization The x and y locations of the th grid point in the limb is given by x x = ; x = Lesin( θj)for n () i = j= y ; y = Le; y = Lecos( θj) + Lefor n () i = where q = and n is the number of grid points in the limb discretization. The notation in Equations () and () becomes very cumbersome when derivatives are introduced. To eep things clear, two intermediate variables are defined j= S = sin θ i (3) C = cos θi (4) The derivatives of S and C with respect to q i are

3 S j C j S = j > ; = cos θi j (5) j C = j > ; =sin θi j (6) j Equations () and () can be more simply written as x = x = ; x= Le Si for n (7) y = ; y = Le ; y = Le Ci + Lefor n (8) For simplicity, I have assumed that the element lengths, L e, are constant. The deformed shape of the bow is found by minimizing the strain energy in the bow limbs using the shape variables θ i as the design variables [3]. The potential energy of the limb is n ( ) PE..= Ki θi θ i (9) where the superscript indicates the angles between the individual elements that are present before the bow is strung. For a straight longbow, θ o is zero for all elements. The deformed shape of the bow limb is that which minimizes the strain energy. Thus, finding the deformed shape can readily be cast as a minimization problem. To avoid the need for specialized optimization software, a Lagrange multiplier approach can be used to minimize the potential energy function [4]. This allows the use of a commonly-available non-linear equation solver to address the minimization problem developed here. The method appends constraints to the objective function to form a Lagrange function. Minimization of this function results in a solution of the constrained minimization problem. For the single constraint problem examined here, the Lagrange function is of the form L = F + λ g () where λ is a constant called a Lagrange multiplier and g is a constraint. The function L is then minimized by setting the gradient of L taen with respect to the design variables and λ equal to zero and solving the resulting system for the design variables and λ. L M L = {} () L n λ where and L F λ g = + () i i i

4 L λ The derivative of the potential energy expression with respect to θ i is = g (3) PE = K i( θi θ i ) θi (4) i The bow can exist statically in three states: unstrung, strung and drawn. Only in the drawn state are any external forces acting on it. The shape of the unstrung bow is nown a priori and defined by the vector {q o }. The shape of the bow when strung is determined by minimizing the total strain energy subject to the constraint that the tip of the limb can be no more than the length of the string, L s, from the x-axis. The constraint is and the derivative with respect to the i th design variable is n g = yn Ls = Le Ci + Le Ls = (5) g n =Le Si (6) There are two possible ways to string a bow. The first is to bend the limbs forward and the second is to bend the limbs bac. Obviously, the limbs bac solution is the desired one (although for an idealized straight longbow, it doesn't really matter). To ensure the correct solution, one can use a limbs-bac shape for the initial design vector. The system of equations to be solved for the stringing problem is f n F g f = + λ = K( θ θ) θ λles = n F g = + λ M = Kn ( θn θn) θn λle Si = n n n f = L C + L L = n e i e s (7) The only change for the drawing problem is the form of the constraint. Once the bow has been strung, a different constraint must be applied, namely that the distance from the tip of the limb to the point at which the string is being drawn must equal the length of the string. This constraint is n n g = yn + ( xs x ) n Ls = Le Ci + Le xs Le Si Ls i + i = (8) = = where x s is the point to which the string is drawn. The derivative with respect to the i th design variable is

5 g n n n Ci Le Ci + Le Le xs Le Si L i i i = + = = = n n Le Ci + Le xs Le Si i + n e S i (9) Missing from the discussion so far is a means for finding the draw weight of the bow for a given draw length. Again, an energy approach seemed simplest. Assuming negligible losses, the strain energy stored in the bow is the integral of the applied force over the distance in which it is applied plus the energy stored during stringing PE = PE + f(x)dx () x Thus, the force required to hold the bow at any point x' is the rate at which potential energy is increasing at that point x d PE x x= x' = f ( x') () Note that for very small models, a Monte-Carlo approach can be used to simply map design space. Figure 3 shows the design space for a two variable problem. The minimum potential energy state, and thus the actual strung shape occurs when θ is approximately deg and θ is approximately 5 degrees. Figure 4 shows the predicted strain energy as a function of draw length. Figure 3 Design Space for -DOF Model

6 Strain Energy (in-lbf) Normalized Strain Energy Strain Energy Vs. Draw Length String:67 in Brace height: in String:7 in Brace height:6.6 in String:64 in Brace height:.5 in Draw Length (in) Figure 4 Calculated Results for Straight Longbow Using 3-DOF Model Experiment Using A Recurve Bow Efficiency is a measure of how much of the energy introduced to a system is transmitted and how much is lost. Not all the inetic energy is in the arrow, a significant amount can be transferred to the limbs as they spring forward. E = KE + KE () strain arrow limbs The expression for bow efficiency is η = E strain m V arrow arrow (3) It's important to note that this is true for any ind of bow. It doesn't matter whether there are pulleys in the system somewhere or not. From an efficiency standpoint, the purpose of the pulleys in a compound bow is to allow the use of shorter, stiffer bending elements. These compact bending elements undergo relatively little motion during, so more of the strain energy is converted into inetic energy in the arrow. Figure 5 shows a measured draw curve for a simple recurve bow. Strain energy the area under the curve. The area under the draw weight curve from 9.5 in (the brace height) to 3" is inch-pounds

7 6 5 Draw Weight (Pounds) Figure 5 Measured Draw Forces For a Recurve Bow Draw Length (Inches) The arrow weighed about 5 grains and the measured arrow velocity at release averaged 66 feet/second. The resulting efficiency was 64.6% Shooting Machine Perhaps the most serious problem in maing measurements on bows is the variation between shots. Even an accomplished archer cannot exactly duplicate the same conditions over a large number of tests. To improve repeatability, we made a simple shooting machine with replaceable limbs as shown in Figure 6. Figure 6 Shooting Machine, Unstrung

8 The string is drawn and held in position with a mechanical release connected to the fixture with a strain gauge lin as shown in Figure 7. This allowed us to measure an accurate draw force curve. Figure 7 Strain Gauge Lin for Shooting Machine The dimensions of the bow were measured and recorded. Then using a strain gage linage, the strain was measured at different draw lengths. The results are summarized in Table. Table Summary of Draw Force Data Strain Energy (in-lbf) Strain Energy (ft-lbf) Position Draw Length (in) strain, ε (in/in) Draw Force (lbf) E E E E E E E E E (draw length measured from string position at no draw) Draw Length from front of bow (in) Draw force was plotted against draw length. This was fit to a linear equation as seen in Figure 8. Strain energy was found by integrating the linear force equation and is shown in Figure 9. Note the strong correlation between the measured strain energy curve and the predicted shapes presented in Figure 4.

9 Force vs Draw 7. y = 3.59x Draw Force (lbf) Draw Length (in) Figure 8 Draw for Curve for Shooting Machine Strain Energy vs Draw Length 7 6 y =.659x x - 7E-6 5 Strain Energy (in-lbf) Draw Length (in) Figure 9 Strain Energy Curve for Shooting Machine

10 The arrow s weight was measured and high speed video was taen of the arrow s path after being fired from full draw. A frame from the video is shown in Figure. From this video, the arrow s velocity, after leaving the string was determined. The results are summarized in Table Table Summary of Shooting Results Weight (grams) Weight (lbm) Velocity (in/sec) Velocity (ft/sec) Velocity (mph) Kinetic Energy (ft-lbf) Kinetic Energy (in-lbf) Arrow This bow s efficiency was determined to be 56.4% which is approximately correct for long bows. Figure Frame From High Speed Video of Arrow Leaving Shooting Machine

11 Summary We have presented an analytical model for static deformation of a simple bow along with experimental data showing qualitative correlation with both a simple recurve bow and a shooting machine. Ongoing wor will replace the low curvature limbs on the shooting wor with ones having a more pronounced recurve and will quantitatively correlate the discrete model with measured results. Acnowledgement The authors gratefully acnowledge the assistance of Tom Kir in this effort, particularly during the construction of the shooting machine. References. Marlow, W.C.; Bow and Arrow Dynamics, American Journal of Physics; Vol. 49, No. 4, April 98.. Klopsteg, P.E.; "Physics of Bows and Arrows"; American Journal of Physics; Vol., No 4, August 943, pp Tauchert, T.R.; "Energy Principles in Structural Mechanics"; Krieger Publishing, Vanderplaats, Garret N.; "Numerical Optimization Techniques For Engineering Design"; McGraw-Hill, New Yor, 984.