Mathematical modelling in the field of kites

• Calculation of kites strength, shape and aerodynamic coefficients Material Science, Solid

mechanics, Aerodynamics, Aeroelasticity

• Finding kites trajectory, speed and attitude Multi-body dynamics

• Laddermill control Stability, optimization and

optimal control

• Problems of aerodynamics Airflow around the kite Finding the aerodynamic

coefficients• Problems of aeroelasticity

Finding the shape of the kite under air load

Finding the aerodynamic coefficients

• Problems of solid mechanics Prediction of fatigue and failure

of kite materials and structure Prediction of life cycle length

Mathematical modelling in the field of kites

0

0

2

1

0

,,

,

s

klijklij

ijjiij

ijij

M

w

C

uu

f

Mathematical modelling in the field of kites

Mathematical modelling in the field of kites

• What’s next?

Courtesy of Rensselaer Polytechnic Institute

Courtesy of University of California in Irwing

Courtesy of Ecole Polytechnique de Montréal

Step 32

Step 155

Step 806

The pictures of the airflow around 6*3 pt plate in a 30*30 pt region

Mathematical modelling in the field of kites

)()()(2

112213 ijijijijiiLiiiij vdvdVcSrL

ijiiDiiiij vVcSrD )()(2

13

001

011

iii

ijiiiiiiij RtR

rtRRAET

)(1

31 ijjiijijiji

ij rgTTLDmdt

dv

)( ijjijij rwv

dt

dr

0tir 0tiVInitial conditions: and .

Here:

Equations of motion:

Mathematical modelling in the field of kites

Stability of energy production

PEEPPwwP

EEww

here w is wind speed and E is energy production

,min)(

2tu

.PEEP

PwwP

)(

1

0

001 maxtu

t

t

dttt VT

xuxux,

,,f

dt

td

u10 ttt

.

,

max1

PEEP

PwwP w

tu

Flight optimization

T

a

Δ

Δ

a

1. Creating the table of tests

2. Determining Pareto-set

3. Choosing constraints on objectives.

Choice of test pointsEquidistant grid LPτ: all coordinates of all points are different

0 0.4 0 .8

0

0.4

0 .8

0 0.2 0 .4 0 .6 0 .8 1

0

0.2

0 .4

0 .6

0 .8

1

N

j

M

kk

kjj

sq

1 1 2iM 2log1

M

klkljl

kj

irs

322

Mathematical modelling in the field of kites

• Calculation of kites strength, shape and aerodynamic coefficients Material Science, Solid

mechanics, Aerodynamics, Aeroelasticity

• Finding kites trajectory, speed and attitude Multi-body dynamics

• Laddermill control Stability, optimization and

optimal control

• Problems of aerodynamics Airflow around the kite Finding the aerodynamic

coefficients• Problems of aeroelasticity

Finding the shape of the kite under air load

Finding the aerodynamic coefficients

• Problems of solid mechanics Prediction of fatigue and failure

of kite materials and structure Prediction of life cycle length