2012 King Saud University Department of Mechanical Engineering Yousef Rikli
428114302 428114 302
[FINITE ELEMENT ANALYSIS PROJECT] PROJECT] Finite Element Analysis using ANSYS ANSYS of a solid works CAD model of a flat bow of certain dimensions and geometry.
Table of Contents Table of Contents Contents..................................... ........................................................ ...................................... ...................................... ....................................... ........................................ ............................... ...........1 List of Figures .................................... ........................................................ ....................................... ...................................... ...................................... ....................................... ...................................... ..................1 List of Tables.......................................... ............................................................. ...................................... ....................................... ....................................... ...................................... .................................. ...............1 Introduction .................................... ....................................................... ....................................... ....................................... ...................................... ...................................... ...................................... ...................... ...2 1.
Background information information ...................................... ......................................................... ....................................... ....................................... ...................................... ............................ .........2 Identifying the bow: ..................................... ......................................................... ....................................... ...................................... ...................................... ....................................... ........................2 Terminology:...................................... ......................................................... ...................................... ...................................... ....................................... ....................................... .................................. ...............2 Identifying performance criteria and ranges ....................... ........................................... ....................................... ...................................... .................................. ...............3 Calculations:....................................... .......................................................... ...................................... ....................................... ....................................... ...................................... .................................. ...............3
2.
Objective.................................... ........................................................ ....................................... ...................................... ...................................... ....................................... ...................................... ..................4
3.
Assumptions Assumptions..................................... ........................................................ ...................................... ...................................... ....................................... ........................................ ............................... ...........4
4.
Analysis ...................................... .......................................................... ....................................... ...................................... ...................................... ....................................... ...................................... ..................4 Log ...................................... ......................................................... ...................................... ...................................... ....................................... ....................................... ...................................... ............................... ............4 Results and discussion...................................... ......................................................... ...................................... ...................................... ....................................... ...................................... ..................9
5.
Conclusion..................................... ........................................................ ...................................... ...................................... ....................................... ....................................... ................................10 .............10
6.
Addendum..................................... ........................................................ ...................................... ...................................... ....................................... ....................................... ................................10 .............10 Possible improvements improvements .................................... ....................................................... ...................................... ...................................... ....................................... ....................................1 ................100 Extra resources..................................... ........................................................ ...................................... ...................................... ....................................... ........................................ .............................11 .........11
7.
References ..................................... ........................................................ ...................................... ...................................... ....................................... ....................................... ................................11 .............11
List of Figures Figure 1: downloaded bow CAD file .......................................... ............................................................. ....................................... ........................................ ............................... ...........3 Figure 2: screenshot 1 (loading and BC)...................................... ......................................................... ...................................... ...................................... ............................... ............5 Figure 3: screenshot 2 (mesh).................................... ....................................................... ....................................... ....................................... ...................................... ............................... ............6 Figure 4: screenshot 3 (mass, thickness and material) ..................................... ........................................................ ....................................... ............................ ........8 Figure 5: screenshot 4 (safety factor) ................................................ .................................................................... ....................................... ....................................... ......................... .....8 Figure 6: screenshot 5 (strain energy).................................... ....................................................... ...................................... ....................................... ...................................... ..................9
List of Tables Table 1: collected collected bow properties. ...................................... ......................................................... ...................................... ...................................... ....................................... ........................3 Table 2: Tabular Tabular (time) force force values...................................... ........................................................ ...................................... ....................................... ...................................... ..................7 Table 3: comparing comparing parameters parameters of design models...................................... ......................................................... ...................................... .................................. ...............9
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Introduction At the time which I was was surfing the the internet interested interested in bows and their their types and prices, prices, I came across a CAD model by a student for a composite bow on the website grabcad.com, the design was novel and from his imagination, mechanical design is but the critique of designs produced by the unscientific imagination imagination using mathematical calculations. FEA is the practical manifestation of said analytical exact equations in a time-saving, accurate-enough accurate-enough machine compatible method. In this project, I will use FEA based software ANSYS, to validate the novel bow design i posted by Dragos Vasile Mitrofan Mitrofanii on grabcad.com, in terms of structural integrity and its standing as a commercial product.
1. Background information Identifying the bow:iii If a bow is of rectangular cross section, it is called a flatbow, if the cross section is a circular or Dshaped cross section, it would become a longbow, both those types are usually as long as the person firing them. If the bow is made of several materials laminated together, it would be a composite/laminated composite/laminated bow. If it is made of a single continuous piece of material (wood) it would become a self-bow. And if the bow curves in the direction away from the archer at its edges it is called a recurved bow. The current design design to be analyzed is and and will stay a flat flat bow. Meaning the 3-part assembly assembly will not not be changed or joined, the material will be single (for the limbs), the cross section rectangular, and the bow limbs profile (side view shape) will not be changed.
Terminologyiv : -
Draw weight (measure) - The number of pounds of force required to draw a bow twentyeight (28) inches. Flatbow (equipment) - A non-recurved bow with a rectangular cross section. Limb (equipment) - The upper and lower arms of a bow (also called limbs). Recurve bow (equipment) - A form of bow in which the unstrung tips curve away from the archer. Grain (unit), a unit of mass equal to 64.79891 milligrams, 1⁄7,000 of an avoirdupois pound. v IBO – International International Bow hunting Organization IBO Speed Rating -The IBO speed rating for a bow is reached by shooting a 350 grain arrow at 70lbs of draw weight and 30 inches of draw length. vi Draw Length - is the distance from the nock point to the throat of the grip plus 1 3/4". vii
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Identifying performance criteria and ranges Property Mass weight
Range 3-4 Kg
Source Online users viii
Draw weight Arrow weight weight IBO speed
60-70 lb. 5-10 grains/lb. grains/lb. draw 260-355 fps 80-108 m/s
web source ix web sourcex 2007 Compound Bow Specification Databasexi Arbitrary (for (for analysis convenience) 2007 Compound Compound Bow Specification Databasexii 2007 Compound Bow Specification Databasexiii
material Axle-to-axle Axle-to-axle length Max Kinetic energy
Ti-alloy, Al-alloy, stainless steel. 25-45 inch inch 55-120 ft-lbs
Notes 1.2-2.3 Kg according accordin g to 2007 database For large men Usually 5 (350 grains) Couldn’t find for
recurve bows. From ANSYS material database Couldn’t find for
recurve bows.
Table 1: collected bow properties. properties.
Figure 1: downloaded bow CAD file
Calculations: -
arrow speed: assuming the whole potential energy of the bow is transferred to the arrow as
√ string tension force on each limb: for a bow of draw weight of
kinetic energy: -
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2. Objective Assessing the the structural integrity and and market readiness readiness for the the bow model. model.
Structural integrity: Safety factor should be 1.2 or higher. Structural error should be less than 0.3. Mesh size, aspect ratio and growth rate is acceptable. Market readiness: should be less than than 4Kg Weight should more than 80m/s. 80m/s. Arrow speed should be more
3. Assumptions a) Since the IBO speed rating over defines the physical unknowns for solving this model; draw length will be dropped from consideration. b) Bow limbs are symmetric and are under the same loading conditions. conditions. c) Flat bows and compound bows are close enough in construction and concept to have their data and specifications interchangeable (shared). d) The bow handle handle is structurally structurally safe and and will not undergo any sort of analysis (except weight). weight). e) Considering the structural error self-imposed limit, any singularities will be ignored.
4. Analysis Log
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Thickness 3mm 3mm and width width 25 mm of bow limb couldn’t parameterize. Set the geometry 2d and imported the bow limb only. Opened geometry and set units to mm Generated, error: attach failed Context: attach feature import1 Reset geometry (3d) and repeated, generated; error; undefined base object same context. Reset geometry, set to 2d and imported whole assembly, set to mm, and generated; error: attach failed. Deleted project, new one (3d), imported and generated, noticed selection for base plane, yet ignored it, error: undefined base object. Retried limb in 2d with base plane, failed. Retried limb in 3d with choosing base plane, success in generation. Selected under surfaces, chose thin feature and thickness=0, generated.
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Tried dimension and modify; modify; couldn’t.
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Opened model, units=mm, went to limb, details, thickness =3 and parameterized it, mass=.268kg and parameterized it. Exited back to engineering data and added; Aluminum alloy, Stainless steel steel and Titanium alloy. Updated model and opened it and changed material to al alloy, mass=9.47e-2kg. Right clicking while geometry allows adding thickness as feature, interesting.
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Refreshed geometry yet no thickness, noticing a button in top of (thickness), (thickness), strange since I already added it in properties. Added the thickness thickness in geometry= 3 and refreshed still don’t see thickness for model, continuing anyway. Added fixed reaction for base rectangular of limb. Adding force by component, assumed draw weight is 70lbf, assume each limb carry half=35 lb. =16kg, assume angle at maximum draw is perpendicular to recurve tip surface, force applied at edge=16*9.8=156N edge=16*9.8=156N in +ive x-direction, rest zero. Maybe parameterize later on.
Figure 2: screenshot 1 (loading and BC)
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Insert solutions for max equivalent safety factor, equivalent stress, total deformation, structural error and strain energy. And solving. deformation is large compared compared to the model model bounding bounding box. Verify Getting a warning “ The deformation boundary conditions conditions or consider turning large deflection deflection on. Refer to Troubleshooting Troubleshooting in the Help System for more details.” Noticing that max deformation is 700 mm but animation is very subtle, turning the result scale to 1.0 and seeing that it bends completely, and even extends in length horribly like some piece of gum. Realize that I have to put the force direction changing with deflection to remain perpendicular on surface and represent the bow string tension a bit more accurately. Notice max strain energy is 1.4J, seems very low ; don’t know if this maximum happens on a node, element or what? Notice its place is same place as max error. Try to add total strain energy in solution, add equivalent total strain. Resolve find answer .002mm/mm. Notice structural error is highest (232mJ) in area right after the fixed boundary condition rectangular, need to refine mesh there.
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Min factor of safety is 0.2, very low. Parameterize. Add sizing to mesh , select fixed boundary rectangular surface, set element size 2mm, re solve whole model. Notice solving whole model is fast, so generalize mesh size to all model .
Figure 3: screenshot 2 (mesh)
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tolerance was larger larger than one computed based on size Warning “ The default defeaturing tolerance controls. The mesher has modified the global global defeaturing tolerance tolerance accordingly.”
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Parameterize min factor of safety and max strain energy, go back to parameters and enter thicknesses 4, 5, 6 and 10 and run for all design points.
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Mass increases to .6kg, strangely enough strain energy and max strain doesn’t change one bit,
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need to go back and parameterize max deformation as well. Exit the parameterization study and not save it, lost whole project, thank god for writing down those steps, start over again.
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Change force from constant to tabular (function of time), set intervals at 0.5sec increments, assume first angle at t=o is -90 and force is y=-30N (string tension), 2 nd angle is -67, in which only half half the draw weight is is applied, resulting resulting in y=-30-(156 y=-30-(156/2)cos22.5 /2)cos22.5 and x=(156/2)sin22.5. Finally at angle -45 (full draw) at t=1, y=-30-156cos45 and x=156cos45. Exporting table: Steps
Time [s]
X [N]
Y [N]
Z [N]
1
1
0
0
-30
0
2
1
0.5
30
-102
0
3
1
1
110
-140
0
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Table 2: Tabular (time) force force values.
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Min SF is .54, increasing thickness to 5mm and solving. Min SF =1.5 and weight=0.26Kg. Exporting the strain energy data to an excel sheet and summing the strain energy for all elements the result is a 10150J. We take it 10kJ to compensate for the singularity at the edge of the fixed region (discontinuity in loading).
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Using the equation
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√ and taking an arrow of 350 grains= 0.02268Kg, the velocity
of the arrow would be 939 km/h= 260.9 m/s, around 2.5 times the max arrow velocity for commercial bows!!! Switching to aluminum and solving. I notice that this is only one limb of the bow, so going back, halving my forces and fixing the equation of energy to give for both limbs. Also reducing thickness to 4mm since the load is much lower now. Min SF was 3, so returned r eturned thickness to 3mm and switched material to Al-alloy and solved. Min SF=0.3, switching to Stainless steel and adding thickness=4mm.solving. Min SF is 0.44, increasing thickness thickness 6mm solving. Min SF=1, increasing to 7mm and solving. Min SF is 1.4, total strain energy=460J, mass=0.6kg (for one limb only), max structural error = 0.09.
√ = 284km/h =79.1m/s
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Plugging into equation for both limbs;
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Switching to Al-alloy and solving; Min SF is 1.8, max structural error is 0.07, mass is 0.2kg (one limb only), total strain energy is 1252J.
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Plugging into equation for both limbs;
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Switching to Titanium and reducing thickness to 3mm and solving SF is 1.09, increasing thickness thickness to 4mm and solving.
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√ = 470km/h=130m/s.
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Figure 4: screenshot 3 (mass, thickness and material)
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SF is 1.96, total strain energy=4952J, energy=4952J, max structural error =0.31, mass is 0.2kg.
Figure 5: screenshot 4 (safety factor)
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Figure 6: screenshot 5 (strain energy)
√ = 935m/h=260m/s.
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Plugging into equation for both limbs;
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Finding mass for whole bow, assigning handle material as stainless steel.
Results and discussion
Factor of safety
mass of one limb (kg)
Total mass (kg)
Max Structural error
Strain energy(J)
Arrow speed(m/s)
7
1.4
0.6
1.1
0.09
460
79.1
7
1.8
0.2
0.79
0.07
1252
130
4
1.96
0.2
0.91
0.31
4952
260
x > 1.2
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x< 4Kg
x < 0.3
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x > 80
Thickness of limb (mm) Stainless steel Aluminum alloy Titanium alloy Set criteria criteri a
Open
Table 3: comparing parameters of design models
We can see in in Table3 that that for thicknesses thicknesses of 7, 7, 7 and 4 for stainless steel, Al-alloy and Tialloy respectively, the CAD model fits (sometimes barely) the structural and market criteria we set in the beginning. beginning. For the mentioned materials and thicknesses we can say that the static safety factor is well over 1.3 and hence safe, we can increase the thickness for stainless steel model if we want a
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bigger margin. All three designs are around a quarter of the designated maximum weight. weight. The structural error error is less than than a third of what what is allowed allowed for stainless stainless steel and and Al-alloy, but a snippet above what’s allowed for Ti -alloy, considering the generous factor of safety that wouldn’t be a problem. The strain energy stored stored by the stainless steel steel model is the least among the three and it’s arrow speed is almost r eaching the set minimum limit, an increase
in thickness would increase the stored energy and hence the arrow speed. The Al-alloy is more than one and a half the required arrow speed, and the Ti-alloy model gives an impressive 260m/s 260m/s arrow, more than three times what is required and by far faster than any arrow shot from a conventional conventional commercial bow. The results for for the arrow speeds are almost almost too good good to be true, true, but that may be due to the assumptions they operate within and not an error in the numerical calculation.
5. Conclusion The project can be summarized summarized in 3 main steps; 1- Researching all there is about bows, their mechanics, performance, and selling points; from users, manufacturers and standard-setting organizations. 2- Setting the geometry and model as simple and accurate as possible, focusing mainly on a good mesh profile. 3- Choosing the important values to parameterize and trying out combinations to generate design models, and putting them against the desired criteria and within the set constraints and assumptions. In the end, if the assumptions for this project are sound, the three design models are structurally safe and are competitive in matters of commercial specifications.
6. Addendum Possible improvements a. Convergence check (for mesh independency). b. Consider changing the cross section into a circle or a D-shape, like a long bow. c. Consider a laminated/composite bow. A simple setup would be to fix two strips of different materials together, the outer good for tension and the inner good for compression. d. Consider an economic analysis, including material and machining cost. e. Could use symmetry for limb, but solving speed was acceptable and symmetry would cause more more calculations load-wise, load-wise, so was neglected neglected here. f. The model could could be solved solved as a 2D model given that we take take the base 2D surface to be the side profile of the limb, such that the variable direction-force would always always be in the 2D plane. g. Refine the area of application of the cord ’s tension force.
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Extra resources
Lab experiment to determine arrow speed experimentally ( http://www.physics.csbsj http://www.physics.csbsju.edu/RPEG/no u.edu/RPEG/no_paper/handouts/ _paper/handouts/arrow.shot.h arrow.shot.html tml ) ) Research paper studying non-linearity in compound bows ( http://physics.mercer.edu/petep http://physics.mercer.edu/petepag/combow.ht ag/combow.html ml ) ) Autodesk Mechanical Mechanical Event Event used to optimize optimize a compound compound bow ( http://www.algor.com/pro http://www.algor.com/products/analysis_repl ducts/analysis_replays/compound ays/compoundbow/default.a bow/default.asp sp ) ) Olympic recurve bow parts and functions ( http://www.wellesley.edu http://www.wellesley.edu/Activities/ho /Activities/homepage/archery/ mepage/archery/Equipment/equip Equipment/equipment.html ment.html ) Types and configurations configurations of recurve bows bows ( http://www.squidoo.com/rec http://www.squidoo.com/recurvebows urvebows ) ) Specs and nomenclature (compound bow) ( http://mathewsinc.com/bows/mathews-101/specs-101/#pw ) http://mathewsinc.com/bows/mathews-101/specs-101/#pw ) thorough consideration of the parameters and their effect on compound bow performance ( http://www.huntersfriend. http://www.huntersfriend.com/bowselect com/bowselection.htm ion.htm ) ) Online calculators for Kinetic Energy, Arrow Weight and FOC (forward of Center Balance) ( http://www ( http://www.stickemarchery.c .stickemarchery.com/stickemcart/arc om/stickemcart/archery-calculators.aspx hery-calculators.aspx , http://www.goldtip.com/ca http://www.g oldtip.com/calculators.aspx lculators.aspx ) ) Critique of the difference between IBO speeds and real arrow speeds ( http://www.bghi.us/index.php?x=bowspeed http://www.bghi.us/index.php?x=bowspeed ) World archery rule book ( http://www.archery.org/content.asp?id=1023&me_id=827 http://www.archery.org/content.asp?id=1023&me_id=827 ) Archery Trade Association Association ( http://www.archerytrade.org/index.php http://www.archerytrade.org/index.php ) AMO (Archery (Archery Manufacturer's Manufacturer's Organization, Organization, currently ATA) ATA) full bow bow manufacturing manufacturing standards, 1968. ( www.outlab.it/doc/amostd.p www.outlab .it/doc/amostd.pdf df ) )
7. References i
http://grabcad.com/library/composite-bow-simp http://grabcad.com/library/composite-bow-simple le http://grabcad.com/dragos.vasile.mitrofan-1 http://grabcad.com/dragos.vasile.mitrofan-1 iii http://en.wikipedia.org/wiki/Bow_shape http://en.wikipedia.org/wiki/Bow_shape iv http://en.wikipedia.org/wiki/List_of_archery_term http://en.wikipedia.org/wiki/List_of_archery_termss v http://en.wikipedia.org/wiki/Grain_(unit) http://en.wikipedia.org/wiki/Grain_(unit) vi http://mathewsinc.com/bows/mathews-101/specs-1 http://mathewsinc.com/bows/mathews-101/specs-101/#pw 01/#pw vii http://www.yeoldearche http://www.yeoldearcheryshoppe.com/drawleng ryshoppe.com/drawlength.php th.php viii http://www.archeryinterchange.com/f12/bo http://www.archeryinterchange.com/f12/bow-mass-weight-vs-holding-weight-27040 w-mass-weight-vs-holding-weight-27040/ / ix http://www.yeoldearcher http://www.yeoldearcheryshoppe.com/drawlengt yshoppe.com/drawlength.php h.php x http://www.huntingnet.com/staticpages/stati http://www.huntingnet.com/staticpages/staticpage_detail.aspx?id=15 cpage_detail.aspx?id=15 xi http://www.huntersfriend.com/2007-Bow-Reviews/compoun http://www.huntersfriend.com/2007-Bow-Reviews/compound-bows-sorted-by-ke-output d-bows-sorted-by-ke-output.htm .htm xii http://www.huntersfriend.com/2007-Bow-Reviews/comp http://www.huntersfriend.com/2007-Bow-Reviews/compound-bows-sorted-by-axleto ound-bows-sorted-by-axletoaxle.htm axle.htm xiii http://www.huntersfrien http://www.huntersfriend.com/2007-Bow-Reviews/compoun d.com/2007-Bow-Reviews/compound-bows-sorted-by-ke-output.h d-bows-sorted-by-ke-output.htm tm ii
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