2009020076 박문규
2011018753 박준연
2011020404 박현수
CONTENTS
- Bungee Jump Equation
- Matlab Code
- Improved Euler’s methods
- Runge-Kutta methods
- Numerical Result
- Improved Euler’s methods
- Runge-Kutta method
MATHEMATICAL MODELING
- More...
MATHEMATICAL MODELING
BUNGEE JUMP EQUATION
●
dx
=v
dt
●
dv
cd 2
=g− v
dt
m
MATHEMATICAL MODELING
● v 0 = 0 ,x 0 = 0
where g = 9.81 , cd = 0.25
, m = 68.1
MATHEMATICAL MODELING
ANALYTICAL APPROACH
Set 𝑘 =
𝑑𝑣
𝑔 − 𝑘𝑣 2
−1
𝑔𝑘
𝑐𝑑
𝑚
. Then
=
𝑑𝑡
𝑘
+ 𝐶)] = (− 𝑔𝑘)𝑡 + 𝐶
𝑔
[ tanh−1 (
1
𝑣 𝑘
= tanh{(− 𝑔𝑘)𝑡 + 𝐶}
𝑔
v(t) = − 𝑘 ln | cosh{(− 𝑔𝑘)𝑡}|
1
x(t) = 𝑘 ln | cosh{(−
𝑔𝑘)𝑡}|
MATHEMATICAL
𝑣=
𝑔
tanh{(−
𝑘
𝑥=
𝑣 𝑑𝑡 =
MODELING
𝑔𝑘)𝑡 + 𝐶}
𝑔
𝑘
sinh{(− 𝑔𝑘)𝑡}
cosh{(− 𝑔𝑘)𝑡}
dt =
𝑔
𝑘
1
𝑑𝑡
𝑢
=
1
𝑘
ln | cosh{(− 𝑔𝑘)𝑡}| + 𝐶
MATHEMATICAL MODELING
NUMERICAL APPROACH
𝒉
𝟐
𝒙 𝟐 = 𝒙 𝟎 + [𝒇𝟏 𝒕𝟎 , 𝒙𝟎 + 𝒇𝟏 (𝒕𝟎 + 𝒉, 𝒙𝟎 + 𝒉𝒇𝟏 (𝒙𝟎 , 𝒕𝟎 ))]
=0+
𝒉
𝟐
𝒗 𝟎 +𝒗 𝟎+𝒉
𝒉
𝟐
= 𝒗 𝟎 + 𝒉 =v(2)
𝒙 𝟒 =𝒙 𝟐 +
𝒉
𝒗 𝟐 +𝒗 𝟐+𝒉
𝟐
= 𝟐𝒗 𝟐 + 𝒗 𝟒 ≈ 𝟔𝟖. 𝟒𝟐𝟓𝟎
𝒉
𝒗 𝟐 = 𝒗 𝟎 + 𝒇𝟐 𝒕𝟎 , 𝒗𝟎 + 𝒇𝟐 𝒕𝟎 + 𝒉, 𝒗𝟎 + 𝒉𝒇𝟐 𝒕𝟎 , 𝒗𝟎
𝟐
= 𝟎+
=
𝒉
𝟐
𝒉
𝟐
𝒈+𝒈−
𝟐𝒈 −
𝒄𝒅
𝒎
𝒄𝒅 𝟐 𝟐
𝒉 𝒈
𝒎
𝒗 𝟎 + 𝒉𝒈
≈ 𝟏𝟖. 𝟐𝟎𝟔𝟖
𝟐
MATHEMATICAL MODELING
NUMERICAL APPROACH
𝒉
𝒉
𝒌𝟏
𝒙 𝟐 = 𝒙 𝟎 + [𝒇𝟏 𝒕𝟎 , 𝒙𝟎 + 𝟐𝒇𝟏 𝒕𝟎 + 𝒉, 𝒙𝟎 +
+ ⋯]
𝟔
𝟐
𝟐
𝟏
𝟑
= 𝟎 + [𝒗 𝟎 + 𝟐𝒗 𝟏 + 𝟐𝒗 𝟏 + 𝒗 𝟐 ]
=
𝟏
𝟑
𝟒𝒗 𝟏 + 𝒗 𝟐
≈ 𝟏𝟗. 𝟏𝟔𝟓𝟔
𝒉
𝒄𝒅 𝒈 𝟐
𝒄𝒅
𝒄𝒅 𝒈 𝟐 𝟐
𝒗 𝟐 = 𝒗 𝟎 + 𝒈 + 𝟐(𝒈 − ( ) + 𝟐 𝒈 − (𝒈 − ( ) )
𝟔
𝒎 𝟒
𝒎
𝒎 𝟒M A T H E M A T I C A L
+(𝒈 − 𝒈 −
𝒄𝒅
𝒄𝒅 𝒈 𝟐 𝟐 𝟐
(𝒈 − ( ) ) )]
𝒎
𝒎 𝟒
≈ 𝟏𝟖. 𝟕𝟐𝟓𝟔
MODELING
MATLAB CODE
Improved Euler’s methods
MATHEMATICAL MODELING
MATHEMATICAL MODELING
MATLAB CODE
Runge - Kutta methods
MATHEMATICAL MODELING
MATHEMATICAL MODELING
RESULT
Time(s)
2
4
6
8
10
IE
18.2068
32.0113
40.7275
45.7404
48.5024
RK
18.7256
33.0995
42.0547
46.9345
49.4027
ODE45
18.7292
33.1118
42.0763
46.9575
49.4214
MATHEMATICAL MODELING
MATHEMATICAL MODELING
RESULT
Time(s)
2
4
6
8
10
IE
18.2068
68.4250
141.1637
227.6316
321.8744
RK
19.1669
71.9277
147.9317
237.4792
334.1312
ODE45
19.1663
71.9303
147.9461
237.5104
334.1782
MATHEMATICAL MODELING
MATHEMATICAL MODELING
MORE..
•
•
We want to model the vertical dynamics of a jumper connected to a
stationary platform with a bungee cord. F=ma
Forces:
– mg
(gravity, g = acceleration due to gravity)
– cd v2 (drag force, cd = drag coefficient, v = velocity)
(need to always retard v, so if falling (v>0) need force neg,
if rising (v<0) need force pos to reduce dv/dt)
(use sign(v) for drag force)
– k (x-L) (spring force, x = distance measured down from
platform, L = rest length of cord)
MATHEMATICAL MODELING
– γv
(damping force, γ is damping coefficient of cord)
MATHEMATICAL MODELING
MORE...
k =  = 0 if
xL
If weight rises, then the force which defy free fall will decrease.
MATHEMATICAL
MODELING
Thus, both x(t) and v(t) increase faster.
MATHEMATICAL MODELING
THANK YOU
MATHEMATICAL MODELING
*MATLAB CODE
MATHEMATICAL MODELING
MATHEMATICAL MODELING