mathematical modelling and calculation of vertical axis...

AASCIT Journal of Energy 2015 2(3) 9-15

Published online April 30 2015 (httpwwwaascitorgjournalenergy)

Keywords Wind Turbine

Aerodynamic

Force

Torque

Pitch Control

Received March 29 2015

Revised April 14 2015

Accepted April 15 2015

Mathematical Modelling and Calculation of Vertical Axis Wind Turbine Pitch System Using Matlab Tools

Toms Komass

Doctoral student Jelgava Latvia University of Agriculture Faculty of Engineering Institute of

Agricultural Energetic

Email address tf11198llulv

Citation Toms Komass Mathematical Modelling and Calculation of Vertical Axis Wind Turbine Pitch

System Using Matlab Tools AASCIT Journal of Energy Vol 2 No 3 2015 pp 9-15

Abstract In the course of this research the vertical axis wind turbine (VAWT) pitch system

simulation model has been designed and verified using the MATLAB SIMULINK tool

for blade aerodynamic force simulation The research is based on the study of the

Darrieus type VAWT blade force vector calculations which provided the basis for

building the mathematical model It includes the mechanical model of the turbine blade

force and torque simulation with and without the pitch control system The main goal of

the research achieved and the designed model was successfully verified by analytical

results The results show that the simulation is using correct pitch system calculations It

allows analysis of specific VAWT force vectors and torques The calculations can also be

used in transient process simulation for the real turbine with many blades over one rotor

1 Introduction

Use of the renewable power resources has been the subject of active discussions and

research for many years Exorbitant amounts have been invested to research this area

aiming to find the possibility to gain the high quality power from renewable energy

resources (biomass sun wind) with high degree of efficiency [1] Optimisation of the

performance factor in generation and use of energy remains a compelling and as yet

unresolved issue in modern power production systems the same being true of the

renewable power industry [2]

A very important renewable system object of today is the vertical axes wind turbine

with the power range 20 ndash 100kW The reason why these wind turbine type is pushed in

the market is that for the given power range the system is cheaper compared to the

horizontal axes wind turbine however the final solution has not been developed yet for

the VAWT system to achieve the best power performance and to ensure the cheapest

production prices [3] In general VAWT is less productive compared to the HAWT

system yet there is just one final solution how to improve VAWT performance This

solution is the pitch control system for the wind turbine that would control the angle of

attack of the blades [4]

VAWT system is less complex However for future development of the pitch control

system proper analysis of the angle of attack should be performed [5] For the analysis of

the pitch system it is important to perform accurate calculations of the wind turbine rotor

torque real time simulation This allows the system to be used in real time simulations

and helps to find the best solution in developing pitch system control strategy [6]

Pitch system analysis for the VAWT is a difficult task and requires very complex

10 Toms Komass Mathematical Modelling and Calculation of Vertical Axis Wind Turbine Pitch System Using Matlab Tools

aerodynamic calculations The main goal is to describe the

pitch process in a much simpler way and try to calculate the

results with the help of Matlab Software Prior to performing

analysis of the turbine VAWT type the parameters should be

chosen The information about the classic Darrieus turbine

makes it more understandable and easy to analyse [7]

The aim of the paper is to make real time simulation in

MATLAB SIMULINK for VAWT blade angle of attack pitch

angle calculations with all geometrical data set points taken

from geometrical data of a turbine

2 Materials and Methods

In Mart 2015 the experimental research of Darrieus type

vertical axis wind turbine was performed in Jelgava

According to the specification the turbine capacity range is

0-50kW The turbine dimensions are 10m width and 10m

diameter VAWT blade type for simulated wind turbine is

NACA 0018 [8]

Even Though actual measurements were not performed in

the course of this research and results will be compared to

published data During the period of mathematical modelling

of the turbine process information available from other

researchers will be used to analyse the results of

mathematical model before performing the actual pitch

system tests [9]

Aerodynamic efficiency of the wind turbine depends on

the relation between the tangential speed of the tip of a blade

and the actual speed of the wind expressed as the tip speed

ratio (λ) For an experimental wind turbine the value of λ is 3

Any deviation from the most efficient speed relation results

in decrease of the efficiency ratio [10]

R=

V

ωλ sdot (1)

where ω is wind turbine angular speed rads-1

The aerodynamic process is determined by many factors

but the primary ones are the wind turbine solidity and wind

quality This information is described by the Reynolds

number Re Reynolds number is calculated as follows [11]

ReInertialForces V l V l

=ViscouseForces

ρmicro υsdot sdot sdot= = (2)

where l is wind turbine blade chord length in meters (m) υ

=1511 m2s

-1 the kinematic viscosity of the air m

2s

-1 and micro

the dynamic viscosity of the air kgsm-2

The Reynolds number is a dimensionless value that

measures the ratio of inertial forces to viscous forces and

describes the degree of laminar or turbulent flow Systems

that operate with the same Reynolds number will have the

same flow characteristics even if the fluid speed and

characteristic lengths vary [12]

Solidity provides us direct information of swept area is

occupied by the turbine blades Solidity of the wind turbine is

calculated as follows

2

n l

Rσ sdot=

sdot (3)

where n is wind turbine blade count and l the wind turbine

blade length m

The next focus is on forces in the wind turbine blades As

mentioned before the main target is not to makes use of

some unknown efficiencies of the turbine but to operate

directly with the forces in the blades When the wind acts

upon the turbine blade the kinetic energy from the air is

transferred onto the blade which depends on the help of the

area swept by blades [13]

If the turbine operates without a pitch control the angle of

attack α is calculated from the angle of azimuth position of

the rotor The angle α is calculated using the relation

1 sintan [ ]

cos

θαλ θ

minus=+

(4)

where θ ndash azimuth position of the wind turbine rotor deg

Analysis based on the forces in the wind turbine blades is a

very important method since the actual turbine efficiency is

unknown and it is impossible to apply the calculations found

in the classic methods referred to in the literature [14]

Aerodynamic blade shape parameters are the only available

information on the possible efficiency of the blade but not of

the whole turbine not to mention the whole of the vertical

axes wind station [15]

The force acting on the wind turbine blade is calculated in

accordance with the angle of the azimuth and the turbine

rotation speed and wind speed [16]

2 2 2

cos

sinR

R

W R

W V

W W W

τ

τ

θ ωθ

= + sdot= sdot

= + (5)

where Wτ is wind force on tangent axes and WR the wind

force on radial axes and W the total wind force on the blade

The second option is to calculate the angle of attack from

the wind vectors which eventually bring us to the same

equation based on the azimuth position of the rotor[17]

arctan[ ]RW

Wτα= (6)

where α is angle of attack with fixed blade angle

At the time of pitching the additional angle of pitch is in

the system The pitch improve the aerodynamic coefficients

of lift (CL) and drag (CD) for the certain position of the

blade The pitch system angles are calculated from the

relation[18]

β α φ= minus (7)

where φ is turbine blade pitch angle in degrees and β the total

blade angle (in degrees) for CL and CD analysis

AASCIT Journal of Energy 2015 2(3) 9-15 11

The blade lift force (L) is the potentially positive force

which sets the turbine into rotation For calculation of the lift

force projection on the speed vector the following expression

is used

2sin2

c lL CL W

ρα sdot sdot= sdot sdot sdot (8)

where c is wind turbine blade chord length in meters (m) and

CL ndash lift coefficient for NACA0018 [19]

In these circumstances the CL and CD coefficients are

reliable data taken from the global information network site

because the blade profiles The values of these coefficients

have been tested in the laboratories for many years and

thanks to the global web search engines the data is freely

accessible This is the reason why the NACA0018 was

chosen for tests as all information regarding the profile

aerodynamics is easily accessible The CL and CD

coefficients are shown in the angle of attack in the position

from -20degltα gt20deg

Unused force of the wind turbine is the drag force (D) in

terms of energy production its projection on the speed vector

is calculated as follows

2sin2

c lD CD W


where CD is drag force coefficient for NACA0018[19]

The total force F that acts upon the blade in the positive

direction to push the turbine forward is calculated as follows

F L D

T R F

= minus= sdot

(10)

where F is total force in the rotation direction of the blade in

Newtonrsquos (N) and the T is the blade total torque

The turbine total torque T from the force produced by the

wind energy is calculated on the basis of the wind force at the

radius length The final torque shows the actual working

torque of the turbine which can be reduced to the electrical

generator and transformed into the electrical energy The

system should be simulated and checked in the Matlab

Simulink system for curve analysis

3 Results and Discussions

The simulation model is divided into 2 parts Math

calculations referring to the turbine blades were done in

Matlab Simulink for future later use in the VAWT model

simulation in Simulink while the data plot and analysis were

done it the Matlab editor VAWT blade torque simulation

subsystem receives all inputs from the set point written by

the Matlab function running in Matlab Editor The system

calculated the data in accordance with the previously

described equations (Fig1)

Simulation system modelled in the SIMULINK consists of

the mathematical equations and the lookup tables for the

coefficient interpolation The SIMULINK system receives

the simulation starting data from the MATLAB editor what is

calling up the simulation model for one cycle simulation

SIMULINK gives back the main information about azimuth

angle θ angle of attack α pitch angle φ wind torque on

blade T (Fig1)

Fig 1 Internal Structure of lsquoVAWT Blade Torque Calculationrsquo subsystem

The model simulation works in the 2 main loops After the

main parameters of the simulation have been set the first

loop starts to run covering the internal turbine rotation across

the azimuth angle θ from 0 to the maximum of 360deg The

simulation of the θ works with a 1deg step and on each θ step

the next loop starts to run where the pitch angle is changed

for torque calculation The pitch angle φ value ranges from -

20deg to +20deg with the step 01deg After simulation of one

azimuth angle step across the whole pitch angle range the

system does the search for the maximum torque T and saves

the simulation values After the information for each θ

maximum torque is collected the final data plotting is done

(Fig2)


Fig 2 Matlab Editor and SIMULINK simulation system flow chart

In the process of simulation the 2 major information fields

are analysed The first field to compare the simulation code

to the other methods is simulation of the angle of attack α

The second important information is the wind turbine rotor

torque produced by one blade on the rotor with no pitch

control and with the stall angle of attack α

Angle of attack in the period of rotation from 0deg ndash 180deg is

symmetrical to the curve in the period 181deg-360deg (Fig3)

which shows that the calculations are right In the period 0deg-

180deg the first half of the trend is symmetrical at the angle 90deg

which shows that the mathematical calculations performed

by the system are correct

The simulation system is run in 3 ways Description of

each of the three ways is given below

1 The first simulation is run with the pitch angle 0 This is

done to check the simulation system

2 The second simulation is run with the aim to find the

maximum torque The main task of this simulation is to

find the optimal pitch angle to reach the maximal torque

from the wind turbine blade

3 The third simulation is run to search for the maximal

CLCD This simulation is done to find the optimal pitch

angle φ for the maximal CLCD

System simulation with no pitch control system for pitch

angle optimisation shows that the mathematical model is

correct where the torque T and the angle of attack α is in the

correct shape (Fig5) The angle of attack α and CLCD is

symmetrical from 0deg - 180deg to 180deg - 360deg The maximal

torque T received from the wind flow kinetic energy is

mirror-symmetric to the ranges from 0deg - 180deg to 180deg - 360deg

(Fig3)

The results of simulation with no pitch control show that if

the angle of attack is fixed the maximum torque is fixed to

the position of the blade The main solution in the industry is

to develop the pitch control The main target for the pitch

control system is to create the optimal angle of the blade to

reach the maximum performance of the turbine For the best

turbine performance one option is to simulate the system in

order to achieve the two main targets first target being the

optimal torque and second one - the optimal CLCD

Pitch system simulation shows that the pitch angle is

variable in the range -10deg to +10deg which gives the angle of

attack β in the range -16deg to +16deg In total the pitch angle

differs depending on the rotor position azimuth angle θ

(Fig4)

Fig 3 VAWT simulation with no pitch angle φ optimisation


Fig 4 VAWT simulation with pitch angle φ optimisation for maximum torque T

The simulation that is done to find the maximum torque

shows that the pitching angle depends on the azimuth angle θ

For each azimuth angle there is own a most effective pitch

angle The most optimal pitch angle gives the maximum T

value The total torque from pitching is the cumulative value

for the lift and drag force vectors on the positive direction

axe τ The pitching angle is symmetrical in the azimuth angle

range of 0deg-180deg to range 180deg-360deg This is very correct as

the shape of the blade is symmetrical and the force has a

negative value but the vector on the τ gives the positive

force and torque

System simulation that is run to find the maximum CLCD

coefficient shows that the pitch angle varies from -10deg to

+10deg but in a different curve which gives the angle of attack

β -10deg to +10deg Within the azimuth angle range 0-360deg the

following values for the CLCD relation are obtained CLCD

= 80 or CLCD=-80 (Fig5)

Fig 5 VAWT simulation with pitch angle φ optimisation for maximum CLCD

In aircraft design it is very important to make the

aerodynamic system of the plane to work with the highest

CLCD coefficient ratio This coefficient ratio ensures that

the plane achieves the highest efficiency during the flight In

order to check the wind turbine operation process against the

classical aerodynamics theories it becomes important to

perform simulation by making the system operate at the

maximum CLCD coefficient For the NACA0018 the max

CLCD is 80 or -80 in the opposite direction

Figure 6 presents results of all the 3 simulation process

maximum torques T per one rotor rotation I is clear that the

most effective pitching is at the maximum T while the

CLCD coefficient is not the maximal value For this specific

blade design it is possible to achieve the maximum torque of

up to 9800 Nm while without the pitch control system the

maximal torque is merely 8900 Nm What is most interesting

is that while pitching operates to reach the maximum CLCD

coefficient the maximum torque is just 8000 Nm which is

even less than with no pitching

In the process of wind turbine simulation it is also

important to get the right figures - examine total torque

integration over one rotation of the rotor As we see from the

Figure 6 the most effective control is pitching at the

maximum torque which will bring the highest power per one

rotation One interesting fact is that control of the pitch with

the maximum CLCD gives the lowest integrated torque per

one rotation

The Figure 6 shows the information about the torque

integration over one rotation of the rotor It is important not

just to look at the curves but to see the final values after total


rotation Pitch control for the maximum torque brings the

total value per one rotation 21249106 In case of system

simulation with the pitch form the most effective CLCD in

total integrated torque T is 17345106 When the system is

simulated with no pitch control to optimise the angle of

attack the total integrated torque is 19166106

Fig 6 VAWT simulated system maximum torque T data and integrated torque data

Results of the experiment show that the effect of the pitch

control is in the region of the rotation position 60deg - 120 deg

and 240deg - 290deg In all other regions the pitch control system

is not required But the effect of pitching in those regions in

general brings a 10 higher result comparing to no pitching

system or 184 comparing to pitch system for optimal

coefficient CLCD Since this is just theoretical information

for the pitch system effective results calculation of the

electrical energy consumption during use of the pitch control

was not done

4 Conclusions

1 Pitch control simulation compared to the simulation

without pitch control shows a 10 better results with a

higher torque which can be converted into electric

energy

2 Pitch control system with the maximum torque search

gives better results in comparison to the pitch system

with the maximum CLCD coefficient Pitching at the

maximal torque gives the results that are by 184

better compared to the pitching with the maximal

CLCD

3 Pitch control system demonstrates the highest efficiency

at the azimuth angle being within the range 60deg - 120deg

and 240deg - 290deg and 95 of the achieved 14

improvement fall exactly within these ranges of the

rotation angle

4 The simulation system algorithms can be adjusted to any

turbine dimensions and turbine blade profile The

simulation program is flexible and can be used for future

VAWT development simulation systems

References

[1] Bati A F Brennan F The Modelling Simulation and Control of a 50 kW Vertical Axis Wind Turbine Asian Transactions on Engineering vol2 No4 2012 pp14-24

[2] Taher GAbu-El-Yazied Ahmad MAli Mahdi SAl-Ajmi Islam MHassan Effect of Number of Blades and Blade Chord Length on the Performance of Darrieus Wind Turbine Journal lsquoAmerican Journal of Mechanical Engineering and Automationrsquo Vol2 No1 2015 pp 16-25

[3] Ben Guerts Cario Simao Ferreira Gerarg van Bussel Aerodynamic Analyse of a Vertical Axis Wind Turbine in a Diffuser 3rd EWEA Conference - Torque 2010 The Science of making Torque from Wind Heraklion Crete Greece 28-30 June 2010

[4] Hameed MS Shahid F (2012) Evaluation of Aerodynamic Forces over a Vertical Axis Wind Turbine Blade through CFD analysis J Appl Mech Eng 1116 doi1041722168-98731000116

[5] J J Miau S Y Liang R M Yu C C Hu T S Leu J C Cheng and S J Chen Design and Test of a Vertical-Axis Wind Turbine with Pitch Control Applied Mechanics and Materials Vol 225 (2012) pp 338-343

[6] Chris Ferone Michael Robinson Ying Shi and Vinod Vishwakarma A Control Policy for Maximizing the Power Output of Variable Pitch VAWTs and Reducing Fluctuations in Power with an Battery 2012 College of Engineering Research Symposium Pennsylvania State University University Park PA April 5 2012

[7] Ion NILA Radu BOGATEANU Marcel STERE Small power wind turbine (Type DARRIEUS) Journal INCAS BULLETIN Volume 4 Issue 1 2012 pp 135 ndash 142

[8] Muhammad Mahmood Aslam Bhuttalowast Nasir Hayat Ahmed Uzair Farooq Zain Ali Sh Rehan Jamil Zahid Hussain Vertical axis wind turbine ndash A review of various configurations and design techniques Journal Elsevier - Renewable and Sustainable Energy Reviews 16 (2012) 1926ndash 1939

[9] Andrzej J Fiedler Stephen Tullis Blade Offset and Pitch Effects on a High Solidity Vertical Axis Wind Turbine Journal WIND ENGINEERING VOLUME 33 NO 3 2009 PP 237ndash246

[10] Karl O Merz A Method for Analysis of VAWT Aerodynamic Loads under Turbulent Wind and Platform Motion DeepWind 19-20 January 2012 Trondheim Norway pp44-51


[11] Mazharul Islam_ David S-K Ting Amir Fartaj Aerodynamic models for Darrieus-type straight-bladed vertical axis wind turbines Journal Elsevier - Renewable and Sustainable Energy Reviews 12 (2008) 1087ndash1109

[12] MH Mohamed Performance investigation of H-rotor Darrieus turbine with new airfoil shapes Journal Elsevier - Energy 47 (2012) pp522-530

[13] T Maicirctre E Amet C Pellone Modeling of the flow in a Darrieus water turbine Wall grid refinement analysis and comparison with experiments Journal Elsevier - Renewable Energy 51 (2013) pp497-512

[14] Syed Shah Khalid Zhang Liang Sheng Qi-hu and Zhang Xue-Wei Difference between Fixed and Variable Pitch Vertical Axis Tidal Turbine-Using CFD Analysis in CFX Research Journal of Applied Sciences Engineering and Technology 5(1) 319-325 January 01 2013

[15] Peter J Schubel Richard J Crossley Wind Turbine Blade Design Energies 2012 5 3425-3449 doi103390en5093425 September 06 2012

[16] Gerald M Angle Franz A Pertl Mary Ann Clarke James E Smith Lift Augmentation for Vertical Axis Wind Turbines International Journal of Engineering (IJE) Volume (4) Issue (5) pp 430 ndash 442 December 2010

[17] Sachin Khajuria Jaspreet Kaur Implementation of Pitch Control Of wind Turbine Using Simulink (Matlab) International Journal of Advanced Research in Computer Engineering amp Technology Volume 1 Issue 4 pp196-200 June 2012

[18] Prasad Chougule Sųren Nielsen Overview and Design of self-acting pitch control mechanism for vertical axis wind turbine using multi body simulation approach Journal of Physics Conference Series 524 (2014) 012055 - The Science of Making Torque from Wind 2014 (TORQUE 2014) doi1010881742-65965241012055 June 2014

[19] httpairfoiltoolscomairfoildetailsairfoil=naca0018-il ndash opened 08042015 information about blade profile NACA0018


aerodynamic calculations The main goal is to describe the

pitch process in a much simpler way and try to calculate the

results with the help of Matlab Software Prior to performing

analysis of the turbine VAWT type the parameters should be

chosen The information about the classic Darrieus turbine

makes it more understandable and easy to analyse [7]

The aim of the paper is to make real time simulation in

MATLAB SIMULINK for VAWT blade angle of attack pitch

angle calculations with all geometrical data set points taken

from geometrical data of a turbine

2 Materials and Methods

In Mart 2015 the experimental research of Darrieus type

vertical axis wind turbine was performed in Jelgava

According to the specification the turbine capacity range is

0-50kW The turbine dimensions are 10m width and 10m

diameter VAWT blade type for simulated wind turbine is

NACA 0018 [8]

Even Though actual measurements were not performed in

the course of this research and results will be compared to

published data During the period of mathematical modelling

of the turbine process information available from other

researchers will be used to analyse the results of

mathematical model before performing the actual pitch

system tests [9]

Aerodynamic efficiency of the wind turbine depends on

the relation between the tangential speed of the tip of a blade

and the actual speed of the wind expressed as the tip speed

ratio (λ) For an experimental wind turbine the value of λ is 3

Any deviation from the most efficient speed relation results

in decrease of the efficiency ratio [10]

R=

V

ωλ sdot (1)

where ω is wind turbine angular speed rads-1

The aerodynamic process is determined by many factors

but the primary ones are the wind turbine solidity and wind

quality This information is described by the Reynolds

number Re Reynolds number is calculated as follows [11]

ReInertialForces V l V l

=ViscouseForces

ρmicro υsdot sdot sdot= = (2)

where l is wind turbine blade chord length in meters (m) υ

=1511 m2s

-1 the kinematic viscosity of the air m

2s

-1 and micro

the dynamic viscosity of the air kgsm-2

The Reynolds number is a dimensionless value that

measures the ratio of inertial forces to viscous forces and

describes the degree of laminar or turbulent flow Systems

that operate with the same Reynolds number will have the

same flow characteristics even if the fluid speed and

characteristic lengths vary [12]

Solidity provides us direct information of swept area is

occupied by the turbine blades Solidity of the wind turbine is

calculated as follows

2

n l

Rσ sdot=

sdot (3)

where n is wind turbine blade count and l the wind turbine

blade length m

The next focus is on forces in the wind turbine blades As

mentioned before the main target is not to makes use of

some unknown efficiencies of the turbine but to operate

directly with the forces in the blades When the wind acts

upon the turbine blade the kinetic energy from the air is

transferred onto the blade which depends on the help of the

area swept by blades [13]

If the turbine operates without a pitch control the angle of

attack α is calculated from the angle of azimuth position of

the rotor The angle α is calculated using the relation

1 sintan [ ]

cos

θαλ θ

minus=+

(4)

where θ ndash azimuth position of the wind turbine rotor deg

Analysis based on the forces in the wind turbine blades is a

very important method since the actual turbine efficiency is

unknown and it is impossible to apply the calculations found

in the classic methods referred to in the literature [14]

Aerodynamic blade shape parameters are the only available

information on the possible efficiency of the blade but not of

the whole turbine not to mention the whole of the vertical

axes wind station [15]

The force acting on the wind turbine blade is calculated in

accordance with the angle of the azimuth and the turbine

rotation speed and wind speed [16]

2 2 2

cos

sinR

R

W R

W V

W W W

τ

τ

θ ωθ

= + sdot= sdot

= + (5)

where Wτ is wind force on tangent axes and WR the wind

force on radial axes and W the total wind force on the blade

The second option is to calculate the angle of attack from

the wind vectors which eventually bring us to the same

equation based on the azimuth position of the rotor[17]

arctan[ ]RW

Wτα= (6)

where α is angle of attack with fixed blade angle

At the time of pitching the additional angle of pitch is in

the system The pitch improve the aerodynamic coefficients

of lift (CL) and drag (CD) for the certain position of the

blade The pitch system angles are calculated from the

relation[18]

β α φ= minus (7)

where φ is turbine blade pitch angle in degrees and β the total

blade angle (in degrees) for CL and CD analysis





is used

2sin2

c lL CL W

















2sin2

c lD CD W





F L D

T R F

= minus= sdot

(10)



























blade T (Fig1)














(Fig2)

































(Fig3)














(Fig4)














force and torque






= 80 or CLCD=-80 (Fig5)




























one rotation





















was not done

4 Conclusions




energy






CLCD





rotation angle





References

























is used

2sin2

c lL CL W

















2sin2

c lD CD W





F L D

T R F

= minus= sdot

(10)



























blade T (Fig1)














(Fig2)

































(Fig3)














(Fig4)














force and torque






= 80 or CLCD=-80 (Fig5)




























one rotation





















was not done

4 Conclusions




energy






CLCD





rotation angle





References





















































(Fig3)














(Fig4)














force and torque






= 80 or CLCD=-80 (Fig5)




























one rotation





















was not done

4 Conclusions




energy






CLCD





rotation angle





References

































force and torque






= 80 or CLCD=-80 (Fig5)




























one rotation





















was not done

4 Conclusions




energy






CLCD





rotation angle





References






































was not done

4 Conclusions




energy






CLCD





rotation angle





References





















mathematical modelling and calculation of vertical axis...

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