Design of Limb in a takedown re-curve bow plays critical role in making the bow acceptable to the archer. Detailed and step-by- step design procedure of limb is explained.

1. Design of Limb of a Takedown Re-curve Bow
T
Manohar M Hegde

2. Contents
1. Principle of Archery – Energy Storage and Release
3. Problem Statement
4. Minimum Requirements and Performance Criteria
2. Typical Specification of a Re-curve Bow
6. Importance of Re-curve Limb Shape in a Bow
7. Product Benchmarking – Physical and Digital
10. Flowchart of Overall Process
5. Anatomy of a Typical Takedown Bow ; Function Analysis
9. Product Validation
8. Design Methodology and Analysis

3. Archer in Action

4. Energy is Stored in Bow When Drawn
1. Principle of Archery – Potential Energy to Kinetic Energy
Stored Energy is Transferred to
Arrow When Released
Drawn Released
L = Draw Length
F = Draw Weight
L
F

5. Type : Re-curve, take down
Draw Weight : 42 lbs
String Height or Bow Length : 64”
Brace Height: 6”-7”
Draw Length : 32”
Bow Weight : 1.5 Kgs
Riser length : 25”
Riser Material : Aluminium / Magnesium /Composite
Limb length : Short,Medium,Long
Limb Material : Composite
ILF (International Limb Fitment)
2. Typical Specification of a Re-curve Bow

6. 3. Problem Statement
3. Smoothness of draw indicated by draw weight
4. Stability indicated by the torsional and lateral stiffness
“To design a Re-curve Bow Limb Fulfilling Performance Goal,
for a Given Draw Length and Within the Given Constraints “
5. Easy to handle indicated by the physical weight
6. Robustness and long life indicated by the stress distribution
1. Draw length is the primary input (decided by the user)
2. Performance goal is to maximise the energy stored during draw
“Lower Values of Draw Weight, Stiffnesses, Physical Weight
and Stresses are Desirable “

7. 4. Minimum Requirements and Performance Criteria
1. Arrow Speed
2. Stability – Accuracy and Consistency of Shooting
3. Smoothness - Draw Quality and Feel
4. Easy Handling - Physical Weight
5. Affordability - Manufacturing Cost.
6. Durability – Minimum no. of Shooting Operations Before Failure
7. Reliability – Performance and Life Under Actual Usage Conditions
8. Flexible to Change and Ease of Operation
9. Aesthetically Pleasing and Ergonomic (Comfortable Grip)

8. Simple Bow And Re-curve Bow
Simple Bow / Long Bow Re-curve Bow
Limb With
Re-curve

9. 5. Main Parts of a Re-curve Bow And Their Functions
Riser
Limb
String
Riser – Acts
as a Frame
Limb – Stores
Energy During
Draw
String – Transfers
Draw Force to
Limbs
Arrow

10. Primary Function of Limb
Before Draw During Draw During Release
T
“Limb Works Like a Spring, Storing and Releasing Potential Energy”

11. Draw Force and Draw Length
Un-braced Braced Fully Drawn
Draw
Force
Draw Length

12. Potential Energy Stored in a Bow
“Area Under The Curve Represents The Energy Stored -
Higher The energy Stored Higher is The Arrow Speed”
Draw Length
DrawForce/DrawWeight
Area Under
The curve
Higher Arrow
Speed

13. 6. Importance of Limb Shape in a Bow
“Re-curve Influences the Bow Behaviour During Draw and Release”
High Re-curve Low Re-curveMedium Re-curve
Un-braced Braced

14. “Design of the Limb Decides the Amount of Energy Stored”
Different Bows and The Energy Storage Curves
Primitive Bow
Long Bow
Re-curve Bow-1
Re-curve Bow-2
Draw Length
DrawWeight

15. Problem Statement re-stated
“ Design a Re-curve Bow Limb To Maximise The Area Under
The Curve for a Given Set of Draw Weight and Draw Length.”
Target Energy Storing
Curve With
Tolerance Band
Draw Length
DrawWeight
“ Rapid Rise and Then Flattening Trend of the Curve
Results in Smooth Drawing Experience.”

16. 7. Benchmarking
The Desired Qualities of the Proposed Bow Have to be
Arrived at Based on Benchmarking of Competitive Models.
1. Weight of the Limb
2. Draw Weight vs Draw Length
3. Lateral Force vs Draw Length
The Typical Quality Characteristics are :
4. Location of C.G.
5. Selling Price
6. ……
“Each of the Above Qualities Influences the Desirability of
the Bow by the User in Some Way or the Other”

17. 7. Benchmarking - Physical
Tabulate the data as follows :
Prepare benchmarking plan for selected brands of bows
Brand / Model →
A B CAttribute /
Characteristics ↓
Limb Weight
Limb Length
Limb Thickness
Limb deflection in
X- direction
Limb deflection in
Y-direction
……………….
“ Benchmarking Requires Recording of Physical
Features, Dimensions, and Performance Characteristics,
Using Appropriate Tools and Equipment “

18. 7. Benchmarking - Digital
Tabulate the results as follows :
Use The Data From Physical Benchmarking to Prepare 3D
Model of The Bow ; Predict The Performance By Simulation
Brand / Model → A B C
TargetPerformance
Characteristics ↓
Meas
ured
Predi
cted
Meas
ured
Predi
cted
Measu
red
Predic
ted
Limb Weight
Riser Weight
Limb deflection in
x- direction
Limb deflection in
y-direction
Draw Weight vs
Draw Length
Center of Gravity
“ Benchmarked Data is Used to Set Initial Targets”

19. 8. Design Methodology
Attributes of The Bow :
Attributes of The Riser :
Attributes of The Limb :
Bow length, draw length(Input), draw force(output), overall weight
Riser length, Limb seat angle, Riser weight
Limb length, Limb Shape, Limb widths, Limb thicknesses,
Material density, Modulus of Elasticity.
Design Attributes
“Listing of Design Attributes at the Product Level and
Component Level is the Starting Point of the Design Process”

20. Design and Analysis
1. Shape of the limb
2. Dimensions of the limb
For a Given Set of Draw Length and Draw Weight, Keeping
the Riser Same, The Potential Energy Stored Depends
upon :
3. Material properties of the limb material
Limb Design Variables

21. Limb Size and Bow Size
“Bow Length is Made-up of
Riser Length and Length of
Two Limbs”
Bow
Length
Riser
Limb
Length Limb
String

22. Sizes of Limb and sizes of Riser
Limb
Length
Riser
Length
Bow
Length
“4 sizes of Risers Together
with 4 Sizes of Limbs will
Make a Total of 16 Bow Sizes”
Riser
Length
Short
Limbs
Medium
Limbs
Long
Limbs
21” 62” 64” 66”
23” 64” 66” 68”
25” 66” 68” 70”
27” 68” 70” 72”

23. Variations in Bow Size – Same Limb Different Risers
“One Size of Limb Can go With Many Sizes of Riser”
27”
Medium
25”
Medium
21”
Medium
Bow Length = 64” Bow Length = 68” Bow Length = 70”

24. “One Size of Riser Can go With Many Sizes of Limb”
25”
Medium
Bow Length = 66” Bow Length = 68” Bow Length = 70”
Short
25”
Long
25”
Variations in Bow Size – Same Riser Different Limbs

25. Bow Length and Draw Length
“Required Draw Length is Decided by Archer”
Bow
Length- 62”
Draw Length -22”
Draw Length -24” Bow Length, inches
DrawLengthinInches
48 72686456 6052
14323028262422201816

26. Limb Design – Primary Inputs
“ Bow size and Draw Lengths Are The Primary Design Inputs”
Riser
Length
Short
Limbs
Medium
Limbs
Long
Limbs
21” 62” 64” 66”
23” 64” 66” 68”
25” 66” 68” 70”
27” 68” 70” 72”
Limb
Length
Bow Length, inches
DrawLengthinInches
48 72686456 6052
14323028262422201816

27. Forces in the Bow During Drawing
θ
α
Draw Force, F
String Tension,T
Limb Deflection Force, f
H
Draw Length
“ The Stiffness of the Limb Resists the Pull by the String “

28. Primary Load – In The String Plane
Global Co-ordinate System
Z
Y
X
Z
Y
X
T String Pull
“String Pull T Decides The Potential Energy Stored”
Local Co-ordinate System

29. Free Body Diagram(FBD) of Limb and Analysis
θ
α
Draw
Force, F
T
f
H
Draw Length, L
h(θ)
f
αH
Z
X
Z
X
“Limb Experiences Comparatively High Deflection”

30. Form Design Parameters of Limb
w1 w2
t1
t2
S(l )
l
“Limb’s Long and Slender Body Results in Non-Linear
Deflection Pattern ”
Width, Thickness and Shape

31. Form Design of Limb
“ Every Proposed Shape of Limb Used For Iteration
Needs to be Mathematically Uniquely Identified For
Future References”
Z
XS(l )
S(l ) = f(X,Y )
Limb Shape Options
y = 8E-14x10.118
0
1
2
3
4
5
6
7
0 5 10 15 20 25
Option Example-1 Option Example-2
y = 7E-19x13.837
y = -0.0377x + 0.6521
0
1
2
3
4
5
6
7
0 5 10 15 20 25

32. Typical Specification of a Limb
Sl
No
Attribute Symbol Type of Parameter Remarks
1 Limb Length t Design constraint Fixed parameter
2 Limb Shape s(l ) Design Variable Iterative
3 Limb width1 w1 Design Variable Iterative
4 Limb width2 w2 Design Variable Iterative
5 Limb Thickness1 t1 Design Variable Iterative
6 Limb Thickness2 t2 Design Variable Iterative
7 Stiffness K Resultant Parameter Modulus of Elasticity
(E )
8 Weight W Resultant Parameter Material Density (ρ)
“ Values of Parameters Under Attributes 2 to 6 are
Finalised by Iterations ”

33. Deflection and Stress Analysis of The Limb
Limb Is a Slender Beam With Taper in Both Directions, Hence
Undergoes Large Strain. No Closed Form Solution For Deflection
Exists
“Finite Element Analysis (FE) is Required”
α
T
H
Un-braced
Braced
Fully Drawn

34. Z
Y
X
Z
Y
X
T (String Pull)
Secondary Load – Torsional
Twisting Moment
M
Global Co-ordinate
System
“Torsional Stiffness is a Measure of The Stability of the Bow”
Local Co-ordinate
System

35. Deflection and Stress Analysis of The Limb
Finite Element Analysis
T
String Pull
Twisting Moment
M
Limb Body Made of
Composite Structure
Fixed End
Loaded End
“ Non-Linear FE Analysis is
Required”

36. Analysis Results and Their Interpretation
Sl No Measure Units Method of Qualifying
1 Static Bending
Stress in the limb
body
N/mm2 Limb manufacturer’s norm ,
comparison with results of
physical and digital
benchmarking.
2 Shear Stress at
String
Attachment Point
N/mm2 Limb manufacturer’s norm ,
comparison with results of
physical and digital
benchmarking.
3 Static Torsion
Stress in the
body
N/mm2 Limb manufacturer’s norm ,
comparison with results of
physical and digital
benchmarking.
4 Energy Stored by
The Limb
N-mm Compare the Draw Weight
vs Draw curve with
benchmarked data
“Factor Of Safety(FOS) Can be Arrived at by Comparing
Analysis Results with Benchmarked Data”

37. 9. Design Validation
1. Lab Validation
2. Users’ Validation
a. Static bending
b. Static twisting
c. Reverse bending
d. Reverse twisting
e. Full bow Draw Weight vs Draw Length characteristics
a. By experts under standard conditions
b. By experts under adverse conditions
c. By trainees under standard conditions
d. By trainees under adverse conditions
“Results Are Compared With Benchmarked Data”

38. 10. Typical Design and Development Process
Select target bows and
compile their
specifications
Benchmark competitors’ bow
models and establish
reference data
Benchmark individual
risers and limbs of
competitors’ bow Models
Work out preliminary
engineering calculations,
free-body diagrams(FBD).
Create feasible concepts of
basic form and shape of
proposed limb
Finalise the target
specification of proposed
limb, verification and
testing specifications
Design FMEA ; refinement
of verification and testing
specification
Design verification using CAE /
Simulation
Design iterations
Performance testing, lab
testing and user trials
Prototype manufacturing of
components and assembly
Prepare detailed
manufacturing drawings
Iteration
Iteration
Serial
Production
“Multiple Reviews on Design and Manufacturing
Aspects Can be Held as Required”