MISG12_FishingPoleDesign

FISHING POLE DESIGN

Mulalo Nengome, Marijke Rademeyer, Akinlotan MorenikejiDeborah, Gideon fareo

January 07, 2012

Mulalo Nengome, Marijke Rademeyer, Akinlotan Morenikeji Deborah, Gideon fareoMathematics in Industry Study Group (MISG) 2012

Outline

Introduction


Outline

Introduction

Objectives


Outline

Introduction

Objectives

A sketch of The Beam’s Structure


Outline

Introduction

Objectives


Scaling


Outline

Introduction

Objectives


Scaling

Dimensionless ODE Formulation


Outline

Introduction

Objectives


Scaling


Progress


Outline

Introduction

Objectives


Scaling


Progress

Simplification of Coefficients


Outline

Introduction

Objectives


Scaling


Progress


Graphical Analysis


Outline

Introduction

Objectives


Scaling


Progress


Graphical Analysis

Comparism of Different Rod Materials


Outline

Introduction

Objectives


Scaling


Progress


Graphical Analysis


Length comparisons


Outline

Introduction

Objectives


Scaling


Progress


Graphical Analysis


Length comparisons

Comparison of Radii


Outline

Introduction

Objectives


Scaling


Progress


Graphical Analysis


Length comparisons

Comparison of Radii

Conclusions


Introduction

We consider a cylindrical solid pole modelled as a beam based onthe Euler-Bernoulli beam theory.

The partial differential equation describing the displacement,u(x , t), of the beam as a function of position and time is given by

∂2u

∂t2+

Ek2

ρ

∂4u

∂x4= 0, 0 < x < L, t > 0. (1)

subject to spatial boundary condition

u(0, t) = a sin(ωt), ux (0, t) = 0, uxx(L, t) = 0, uxxx(L, t) = 0

and initial condition

u(x , 0) = 0, ut(x , 0) = 0


Objectives

Our aims are to formulate a model that can:

determine the fishing rod (or pole) that can achieve an optimalcasting distance in order to optimize the fishing process.


Objectives

Our aims are to formulate a model that can:

determine the fishing rod (or pole) that can achieve an optimalcasting distance in order to optimize the fishing process.

determine the major factors that determines the dynamics of afishing pole, such as the length factor and materials.




Scaling

It is always better to simplify a mathematical model by scaling itwhere possible.

The advantages of scaling are:

Parameter reduction.


Scaling




Highlight of essential dynamical features.


Scaling





Exposure of important dimensionless groups.


Scaling






Simplified and dimensionless resulting equations.


Scaling







Clear understanding of the problem and easier interpretation.


Scaling







Clear understanding of the problem and easier interpretation.

Thus, we scale the variables as below:

v(ξ, τ) =u

a, ξ =

x

L, τ = ωt.


Scaling

With this scale, (1) becomes:

∂2v

∂τ2+ J

∂4v

∂ξ4= 0, 0 < ξ < 1, τ > 0. (2)

v(0, τ) = sin(τ), vξ(0, τ) = 0, vξξ(1, τ) = 0, vξξξ(1, τ) = 0

with initial conditions: v(ξ, 0) = 0, vτ (ξ, 0) = 0

The dimensionless parameter J = Ek2

ρω2L4 is the most important

response here and it describes the dynamic behaviour of the fishingpole, where E is the Young’s modulus, ρ is the density, K is theradius of gyration and ω is the angular frequency of oscillation.



After scaling the parameters of the PDE, we look for a solution ofthe form:

u(x , t) = X (ξ) sin(τ).

On Substituing this into the PDE, we obtained a dimensionlessODE:

Jd4X

dx4− X = 0, J =

EK 2

ρω2L4(3)

with boundary conditions:

X (0) = 1

X ′(0) = 0

X ′′(1) = 0

X ′′′(1) = 0

(4)

For the resulting ODE, we also look for a solution of the form:

X (ξ) = Aeλξ

Where E is the Young’s modulus, K is the radius of gyration of theMulalo Nengome, Marijke Rademeyer, Akinlotan Morenikeji Deborah, Gideon fareoMathematics in Industry Study Group (MISG) 2012

Progress

From this, we obtain the general solution:

X (ξ) = C1eJ−1/4

+ C2e−J−1/4ξ + C3 cos(J−1/4ξ) + C4 sin(J−1/4ξ)

Using hyperbolic trigonometric functions, we have:

New Solution Form:

χ(ξ) = C ∗1 cosh(λξ) + C ∗

2 sinh(λξ) + C ∗3 sin(λξ) + C ∗

4 cos(λξ)

whereJ−1/4 = λ


Simplification of coefficients

C ∗1 =

1

2

1 + coshλ cosλ+ sinλ sinhλ

1 + coshλ cosλ

C ∗2 = −

1

2

sinλ+ cos λ sinhλ

1 + coshλ cos λ

C3 =1

2

cosλ sinhλ+ sinλ coshλ

1 + coshλ cosλ

C4 =1

2

1 + coshλ cosλ− sinλ sinhλ

1 + coshλ cosλ

(5)

For the equations above to blow up, the denominator1 + coshλ cosλ must be zero.

Let the denominator of the solution be

g(λ) = 1 + cos(λ) cosh(λ)

=⇒ g(J) = 1 + cos(J−14 ) cosh(J−

14 )


Graphical Analysis

Behaviour of the denominator g(λ) of solution:Graph of g(λ) against λ

20 30 40 50

-1´1015

-5´1014

5´1014

From the graph, we can see that there are infinitely manysingularities for some values of λ > 30. This implies that there areinfinitely many values of the response J for which the solutionU(x , t) is undefined.


Graphical Analysis

An emphasised Graph of g(λ) against λ ∈ (0, 5)

1.8751 4.6941

1 2 3 4 5

-10

10

20

The smallest singularity of the solution is at λ = 1.8751, followedby λ = 4.6941, as shown above. Since λ and J are inverselyproportional, these values corresponds to their equivalent values ofJ for which the rod will attain its resonant frequency which resultsin a maximum amplitude (deflection) that can cause a break in therod as the base of the pole is been moved in a sinusoidal motion offrequency ω and amplitude a.


Graphical Analysis

Graph of Amplitude X (x) of the oscillation at the rod’s tip(i.e. at x = L = 1m) against λ

2 3 4 5 6 7 8Λ

-5

5

XH1LHx,tL


Graphical Analysis

Graph of Amplitude X (x) of the oscillation at the rod’s tip(i.e. at x = L = 1m) against J

0.02 0.04 0.06 0.08 0.10J

-20

-10

10

20

30

XH1LHx,tL


Graphical Analysis

Graph of Curvature of the rod against λ

2 4 6 8 10x

50

100

150

200

uHx,tL

The curvature (bending moment) of the rod at its base (i.e atx = 0) goes to infinity at the values of λ that results in the rod’sresonant frequency.


Graphical Analysis

Graph of Curvature of the rod against J

0.02 0.04 0.06 0.08 0.10J

-100

-50

50

100

X''H0LHx,tL

The curvature (bending moment) of the rod at its base (i.e atx = 0) goes to infinity at the values of J that results in the rod’sresonant frequency.


Graphical Analysis

Blue line:Graph of Deflection U(x , t = pi4) against x

Red line: Graph of Curvature X ′′(x) against x

uJx,Π

4)

X''

0.2 0.4 0.6 0.8 1.0x

5

10

15

Using a value of λ = 1.8 (quite close to the singularity of thesolution), the graphs above shows that the curvature of the rod ishigh around its base and it tends to zero at the tip of the rod (1.e.at x = L = 1m).


Graphical Analysis

Blue line:Graph of Deflection U(x , t = pi4) against x

Red line: Graph of Curvature X ′′(x) against x

u(x,Π

4)

X''(x)

0.2 0.4 0.6 0.8 1.0x

50 000

100 000

150 000

200 000

250 000

300 000

Using the singularity value λ = 1.8751, the graphs above shows thecurvature of the rod which is highest around the base of the rodresults in an extremely high deflection (U(x , pi

4) of the rod which

can cause a break in it as the base of the pole is being moved in asinusoidal motion.Mulalo Nengome, Marijke Rademeyer, Akinlotan Morenikeji Deborah, Gideon fareoMathematics in Industry Study Group (MISG) 2012

Graph of U(x , t) against x for J = 0.1

t=0

t =Π

2

t =3 Π

2

J=0.1

0.2 0.4 0.6 0.8 1.0x

-6

-4

-2

2

4

6

uHx,tL


Graph of U(x , t) against x for J = 0.01

t=0

t =Π

2

t =3 Π

2

J=0.01

0.2 0.4 0.6 0.8 1.0x

-1.0

-0.5

0.5

1.0

uHx,tL


Comparism of Different Rod Materials(Fixed Geometricaldimensions)

We consider the various parameters in the expression for J for 3different materials and their properties. Their corresponding J valueand the ratio E

ρ determines which material is best suited.

MATERIAL ρ (kg/m3) E (kgm−1s−2) Eρ (m

2s−2) J value

Bamboo 0.3 × 103 0.39 × 1011 1.3 × 108 6.5859

Fibre Glass 0.8 × 103 0.95 × 1011 1.1 × 108 6.0286

Carbon Fibre 1.8 × 103 1.7 × 1011 0.94 × 108 4.7621

Graphite Fibre 2.53 × 103 2.07 × 1011 0.82 × 108 4.1547


Length Comparison for Bamboo

Bamboo, (Graph of U(x , t) against x at timet = π

2,K = 0.014, ω = 20π, E

ρ = 1.3 × 108)

L=5

L=1

L=1.2L=1.5

0.2 0.4 0.6 0.8 1.0x

0.7

0.8

0.9

1.0

1.1

uHx,tL


Length Comparison for Bamboo

Bamboo, (J = Eρ

K2

ω2L4 ),Eρ = 1.3 × 108)

L=1

L=1.2

L=1.5

L=2

0.2 0.4 0.6 0.8 1.0x

1.05

1.10

1.15

1.20

uHx,tL


Length Comparison for Graphite Fibre

Graphite fibre, (J = Eρ

K2

ω2L4 ),Eρ = 0.82 × 108

L=2

L=1

L=1.2

L=1.5

0.2 0.4 0.6 0.8 1.0x

1.05

1.10

1.15

1.20

1.25

1.30

1.35

uHx,tL


Comparison of different radii

Graph of U(x , t) against x for varying radius

r , t = π2, L = 1, ω = 20π, E

ρ = 1.3 × 108,K (r) = r√

2

2

r=0.02

r=0.04

r=0.06

r=0.08

0.2 0.4 0.6 0.8 1.0x

1.002

1.004

1.006

1.008

1.010

uHx,tL


Comparison of All The Different Rod Materials

Graph of U(x , t) against x witht = π

2, L = 1, ω = 20π,K = 0.014 and varying E

ρ

Bamboo

graphite fibre

fibreglass

carbonfibre

0.2 0.4 0.6 0.8 1.0x

1.005

1.010

1.015

1.020

1.025

1.030

uHx,tL


Conclusion

In conclusion, a meticulous comparison of all the materials showsgraphite fibre to have the lowest J value, which implies the longestrod length (which greatly favours on shore angling), andconsequently produces the largest deflection at the tip of the rodfor a given displacement at the base. From all our observations, wesuggest that for a one to achieve an optimal casting distance inorder to optimize the fishing process, a long graphite fibre pole maybe best.


The End

THANK YOU!


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