This document provides a report on the design of the main shaft and tower for a Vestas V52-850 kW wind turbine. It begins with an introduction and background review on wind turbine history, components, and design considerations. The objectives are then stated as a detailed design of the main shaft and selection of bearings and fixtures, and a detailed design of the tower. The methodology section outlines the design process flowchart. Free body diagrams are presented for the full wind turbine, tower, and shaft. Finally, stress analyses are shown in the appendices for both the tower and shaft, including calculations for buckling, bending, fatigue, safety factors, and dependent results.

1. REPORT
as a partial requirement
for
the course on
DESIGN OF MACHINE ELEMENTS
ME-3320, C’16
DESIGN PROJECT TITLED:
DESIGN OF WIND TURBINE MAIN SHAFT AND TOWER
Submitted by:
——————————————
Tino Christelis – cschristelis@wpi.edu
——————————————
Jiacheng Liu – jliu4@wpi.edu
——————————————
Daniel A. Ruiz-Cadalso – daruizcadalso@wpi.edu
Submitted to:
Prof. Cosme Furlong
DEPARTMENT OF MECHANICAL ENGINEERING
WORCESTER POLYTECHNIC INSTITUTE
WORCESTER, MA 01609-2280
03/04/2016
Project Score: TOTAL: _______ out of 100%

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ABSTRACT
This is a report that will go into an in-depth analysis and overview of the design of the
main shaft and tower loosely based on the Vestas V52 - 850kW wind turbine. In this report, we
overview some of the basics of wind turbine history and overall design. We then take a close up
look at the design considerations and decisions that are made for our components of
interest. Finally, we apply our knowledge of material covered in class to analyze all of the
stresses for the tower and shaft, which covers shear and bending analysis, buckling analysis,
fatigue failure analysis, and more. Once we consider all of the stresses and calculate safety
factors and endurance limits, we can make a statement about the lifetime and success/failure of
the part. If either of these results do not meet our expectations, we will redesign the part until
we are satisfied with the results. Through the use of material learned in this class and the
computer program MathCAD, we are able to design the shaft and tower such that they meet any
criteria we may have. These MathCAD calculations can be seen in full at the appendix of the
report.

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Table of Contents
ABSTRACT.....................................................................................................................................ii
LIST OF TABLES...........................................................................................................................v
LIST OF FIGURES ........................................................................................................................vi
OBJECTIVES ................................................................................................................................. 1
INTRODUCTION .......................................................................................................................... 2
BACKGROUND REVIEW............................................................................................................ 3
History......................................................................................................................................... 3
Components and Design ............................................................................................................. 3
Rotor........................................................................................................................................ 4
Shaft ........................................................................................................................................ 5
Gears and Generator................................................................................................................ 8
Tower ...................................................................................................................................... 9
METHODOLOGY........................................................................................................................ 12
Flowchart .................................................................................................................................. 12
Design Approach....................................................................................................................... 12
FREE BODY DIAGRAMS .......................................................................................................... 15
Wind Turbine FBD ................................................................................................................... 15
Roll, Yaw, and Pitch............................................................................................................. 17
Drag Force............................................................................................................................. 21
Power .................................................................................................................................... 25
Tower FBD ............................................................................................................................... 27
Shaft FBD ................................................................................................................................. 30
ANALYSIS AND RESULTS ....................................................................................................... 32
Tower ........................................................................................................................................ 32

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Shaft .......................................................................................................................................... 37
CONCLUSION............................................................................................................................. 40
REFERENCES ............................................................................................................................. 41
APPENDICES .............................................................................................................................. 43
Appendix A. Tower Stress Analysis......................................................................................... 43
Appendix A-1. List of Independent Parameters ................................................................... 43
Appendix A-2. Buckling Analysis........................................................................................ 45
Appendix A-3. Normal, Shear, and Moment Diagrams ....................................................... 46
Appendix A-4. Deflection and Slope Diagram..................................................................... 48
Appendix A-5. Principal Stress Calculations........................................................................ 49
Appendix A-6. Static Safety Factor Calculations ................................................................. 51
Appendix A-7. Fatigue Load Analysis ................................................................................. 52
Appendix A-8. S-N Diagram ................................................................................................ 53
Appendix A-9. List of Dependent Values and Results ......................................................... 54
Appendix B. Shaft Stress Analysis ........................................................................................... 57
Appendix B-1. Lis of Independent Parameters..................................................................... 57
Appendix B-2. Shear and Moment Analysis and Diagrams ................................................. 59
Appendix B-3. Principal Stress Calculations........................................................................ 60
Appendix B-4. Static Safety Factor ...................................................................................... 63
Appendix B-5. Stress-Concentration Factor Computation ................................................... 64
Appendix B-5. Fatigue Load Analysis.................................................................................. 65
Appendix B-6. S-N Diagram ................................................................................................ 66
Appendix B-7. Fatigue Safety Factors.................................................................................. 67
Appendix B-8. Vibration Calculations ................................................................................. 68
Appendix B-9. List of Dependent Values and Results ......................................................... 69

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LIST OF TABLES
Table 1: Tentative Tower Properties ............................................................................................ 32
Table 2: Some Carbon Steels with Elongation 15-18%................................................................ 33
Table 3: Initial Dimensions........................................................................................................... 37
Table 4: Independent Parameter Values....................................................................................... 43
Table 5: Dependent Values and Results........................................................................................ 54
Table 6: Table of Independent Parameters................................................................................... 57
Table 7: Dependent Variable Values and Results......................................................................... 69

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LIST OF FIGURES
Figure 1: Vestas V52-850 kW.......................................................................................................... 4
Figure 2: Cross-section of the wind turbine blade at angles: (a) 00, (b) 11o, (c) ........................... 5
Figure 3: Main shaft with main bearing ......................................................................................... 6
Figure 4: Main shaft after being forged.......................................................................................... 7
Figure 5: Planetary spur or epicyclic gearing system .................................................................... 8
Figure 6: Different buckling situations ......................................................................................... 10
Figure 7: Design Process Flowchart ............................................................................................ 13
Figure 8: Side view of wind turbine CAD model showing eccentric buckling. ............................. 15
Figure 9: NASA experimental wind turbines drawn to same scale............................................... 17
Figure 10: Isometric view of wind turbine showing roll (x), yaw (y), and pitch (z) moments ...... 18
Figure 11: Front view of wind turbine CAD model ...................................................................... 19
Figure 12: Top view of wind turbine CAD model ......................................................................... 20
Figure 13: Simplification of the wind turbine blade from the CAD.............................................. 22
Figure 14: “Shape Effects on Drag” ............................................................................................ 23
Figure 15: Tip-speed ratio vs. power coefficient graph................................................................ 26
Figure 16: Tower FBD from X-Y plane view. ............................................................................... 27
Figure 17: Isometric view of wind turbine drawing...................................................................... 28
Figure 18: Drawing of Tower: (a) Top view, (b) Front view, (c) Section view from front view... 29
Figure 19: Shaft Designs, (a) Analysis Shaft, (b) Ideal Shaft. ...................................................... 30
Figure 20: FBD of Shaft................................................................................................................ 30
Figure 21: CAD of shaft, along with the two bearings and the hub.............................................. 31
Figure 22: Section view of the shaft in 2-D drawing. ................................................................... 31
Figure 23: Critical Section and Points.......................................................................................... 34
Figure 24: Analysis Shaft. (a) Parameter Values, (b) Critical Section and Points ...................... 38
Figure 25: Critical Section and Points.......................................................................................... 39
Figure 26: Ideal Shaft Design ....................................................................................................... 40

7. OBJECTIVES
The two objectives for this project are:
1. Detailed design of the main shaft and selection of corresponding bearings and fixtures of
a Vestas’ V52-850 kW wind turbine.
2. Detailed design of the tower for a Vestas’ V52-850 kW wind turbine.

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INTRODUCTION
Wind turbines are one of the most accepted renewable energy technologies from today’s
society. Electricity can be produced by many ways, including solar power, micro hydro, biomass,
etc. but the use of the wind turbine has highly increased lately because, from all the other
technologies used to extract energy from unlimited natural resources (renewable), it is the most
convenient one, generating as much as 5 MW (Mega Watts) while covering much less area than
hydropower dams or geothermal power stations.
Wind turbine design projects are very delicate and complicated, thus, requiring in-depth
analysis on each component. Any relevant mistake can lead to catastrophic disasters such as a
collapse of the whole machine due to failure of the tower. The components of interest for this
project are going to be the main shaft and the tower, specifically for the Vestas V52-850 kW.
This wind turbine is designed to generate 850 kilo Joules of energy per second, and with many
other parameters already being used for the machine, these will be taken for initial analysis.
However, some will be adjusted, if necessary, in order to maximize the design benefits.
Since all of the forces and stresses on these components are due to occurrences happening
outside of the wind turbine, the general approach for this project will consist of analyzing the
external influences, and then using these to further evaluate how they affect the internal
mechanisms.

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BACKGROUND REVIEW
History
Wind energy has become the most relied upon form of renewable energy in the past few
decades. Projections suggest that it will be used even more in the future. Civilization has adopted
the use of wind as an energy sources for hundreds, even thousands of years. Sailors used the
wind to help move their ships across the vast oceans, and in around 500-900 A.D., the Persians
began using vertical axis windmills to help with farming. The development of this technology
continued slowly over the flowing years until, in the 14th century horizontal-axis windmills were
created. Following the industrial revolution, the excessive need for water in industrial processes
lead to the development of small machines that were used to pump water in the 19th century.
This marked the beginning of humanity realization that wind could indeed be used to generate
electricity [1]. The first electricity-generating wind turbine was a battery charging machine
installed in July 1887 by Scottish academic James Blyth to light his holiday home in Marykirk,
Scotland Some months later American inventor Charles F. Brush was able to build the first
automatically operated wind turbine in Cleveland, Ohio [2].
Components and Design
Figure 1 shows the technical specifications of the Vestas V52-850 kW along with the
components, or assemblies that are generally used.

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Figure 1: Vestas V52-850 kW
(Vestas Wind Systems, 2007)
Rotor
The three main systems/assemblies in a wind turbine are the nacelle, the rotor, and tower.
The wind turbine is designed to convert the kinetic energy from the wind to electrical power,
through a series of working components, starting with the rotor assembly. The rotor is composed
of the blades, the blade bearings, the blade hub, and the rotor lock system. The blades are
designed in a way such that as wind passes it creates a force (called the lift force in
aerodynamics) that is a function of the wind’s velocity, the wind’s density, the projected area of
the blade from a stand point of view of the wind, the blades depth, and the lift coefficient, which
is proportional angle between the blade and the wind’s direction (see Figure 2).

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The larger angle may contribute to a larger force on the blade, however, there would be
loss of energy (called minor loss in fluid mechanics) due to wind swirls that begin to form
behind the blade, as shown in Figure 2c. Aside from this method of calculating the torque/energy
generated by the blades, an easier, and most likely more straightforward, technique is to compare
the energies of the wind throughout a streamline as it passes through the blades. The blades are
usually made of composite material to avoid huge and unnecessary weight.
(a) (b) (c)
Figure 2: Cross-section of the wind turbine blade at angles: (a) 00
, (b) 11o
, (c)
Shaft
After rotation in blades is achieved, the main shaft (low-speed), which is one of the
components of interest for this project, is responsible for transferring the energy from the blades
to the gearbox. This is the first component of the nacelle, which will be followed by the gearbox,
the high-speed shaft, and the generator. Along with the main shaft’s bearings and fixtures, in-
depth inspection, calculation, and experimentation are critical for this shaft due to the amount of
forces and moments acting on it. Since the shaft is clamped at the end that is connected to the
rotor, and needs to be able to rotate freely, a support with a bearing mechanism is placed, as
shown in Figure 3.

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Figure 3: Main shaft with main bearing
(Tembra)
Analysis for the main shaft include: shear stress, torsion, bending, fatigue, thermal
effects, and other parameters that could potentially lead to component failure. One of the most
common materials used, not only for the shaft but for many other wind turbine components, is
steel. However, there are many types of steel, depending on the heat treatment that is used to
modify the mechanical properties. For the shaft, the material steel is hot forged, and therefore
undergoes tempering and hardening [5]. Tempering makes the metal less brittle and increases
ductility, while hardening creates a hard outer layer that protects the material from receiving
dents, or any other small deformation on the surface. After the shaft is forged, finishing
operations are done, such as surface finishing for higher fatigue strength. Figure 4 shows the
shaft right after being forged [6].

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Figure 4: Main shaft after being forged
(Junda Heavy Industry Forging)
Bearings are subjected to many different loads and moments. Fatigue, however, is one of
the main concerns for this component. A technical paper written by The TIMKEN Company [1]
indicates the importance of the bearing fatigue duty cycle, and explains some methods of
experimentation. “The bearing fatigue duty cycle received from the customer can have a
significant influence on the size and geometry of the mainshaft bearing designs… Duty cycles
usually are generated using design programs to model the wind turbine system, typically with an
output at 20-Hz.” [1]. Fatigue analysis on the bearing was done with high frequency of data that
provided a number of snapshots used to develop a duty cycle. A method to generate duty cycles
from the data for load conditions is called the independent reduction. In this process, each load is
first separated for a specific RPM bin, then load histograms are generated. From the load
histograms, an equivalent load for each of them is calculated. “Finally, a duty cycle can be
constructed with the corresponding combinations of independent equivalent loads.
A list with the recommended order of importance of the data for proper bearing analysis
was generated, as follows:
1. RPM (due to effects on the development of the lubrication film thickness)
2. Pitch moment (MY),
3. Yaw moment (MZ),

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4. Radial load (FZ)
5. Axial load (FX)
6. Radial load (FY).
However, it will be proven on further sections that pitch and yaw moments will not be of
concern, since they are balanced and do minimal damage to the bearings. In addition to load and
speed, other major influencers include: load zone, thermal effects, lubrication effects,
misalignment stresses, fatigue propagation rate, and bearing geometry factors. Main shaft and
bearing require critical and delicate analysis in order to avoid component failure/fracture.
Gears and Generator
Afterwards, energy is transferred from the shaft to the gearbox in order to achieve a high
rotational speed on the high-speed shaft. The main shaft is coupled to the gearbox with the use of
a clamping unit, which uses friction to transfer the torque from the main shaft. The gearbox’s
configuration for the wind turbines is usually planetary/epicyclic (shown in Figure 5), hence, the
gearbox for the Vestas V52-850 kW consists of a 1 planet step 2-step, parallel axle gears [3]. The
high-speed shaft then transfers the energy from the gearbox to the generator in order to convert it
from mechanical to electric energy.
Figure 5: Planetary spur or epicyclic gearing system
(Ragheb A. & Ragheb M.)

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Tower
In order to keep the nacelle and rotor at a certain height, where the wind is usually
stronger, a tower, which is also one of the components of interest in this project, is placed.
Similar to the main shaft and bearing, the tower requires demanding analysis due to occurrences
such as: possible storm winds developing bending stresses, or heavy weights from the nacelle
and rotor creating compression stresses throughout the tower leading to failure due to either
fracture or buckling.
Design requirements or the tower include cross-sectional dimension, local buckling, and
tower top deflection and rotation limits. The height is chosen via a general rule that most
companies follow, which says that it is better to go for as high a tower as makes economic sense.
According to John Corbett Nicholson [8] the cross-section of the outer diameter is chosen to be
not more than 4.5m due to transportation limitations. Additionally, since the material used for the
tower is steel, using standard equipment, blanks can be fairly difficult to roll to sheets with
thickness larger than 40mm, therefore, for most wind turbines the maximum tower wall
thickness is 40mm. The steel used for the tower could be either, but not limited to, AISI 1030
normalized @ 1650 oF, or AISI 1050 hot rolled, and both of them share the same tensile yield
strength (345 MPa used as yield strength by Nicholson [8]).
Load
Wind is distributed non-uniformly throughout the height, consequently, if at a certain
height the velocity is known then a function can be generated for the velocity distribution profile.
This will provide data that is useful for determining forces on the tower produced by the wind.
The force due to this velocity can be determined as a function of the wind velocity, its density,
the frontal area of tower, and the drag coefficient. The frontal area of the tower is the area
projected from the perspective of the wind, while the drag coefficient is a dimensionless number,
as well as the lift coefficient mentioned earlier from the wind turbine blades, that was calculated
through specific analysis (called dimensional analysis in fluid mechanics), and it depends on the
shape of the object that the fluid (wind) is impacting. This force will provide useful calculations
for analysis on subsequent sections.

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Buckling
Buckling, along with local buckling, analysis is critical, and if not done correctly, it can
result in a catastrophic disaster of the whole turbine. Nicholson [8] shared a local buckling stress
method in his 2011 thesis, which involved, “calculating the elastic critical buckling stress of a
cylindrical steel tube, which has modulus of elasticity 𝐸𝑠, wall thickness 𝑡𝑤, a mean radius 𝑟𝑚,
in axial compression… calculating critical stress reduction coefficients for bending and axial
loading… plugging these values into equation 2.1 along with the material’s yield strength 𝑓𝑦 to
obtain the allowable local buckling stress. The maximum principal stress in the structure should
not exceed this allowable local buckling stress value in order to avoid local buckling.” [8]. Local
buckling, however, will not be analyzed in this project; general buckling will be inspected.
Figure 6 [9] shows different situations in which a beam could fail due to buckling.
Figure 6: Different buckling situations
(Robert L. Norton)
Buckling due to external forces is analyzed by using the weight, W, of the nacelle plus
the rotor. Euler’s critical load formula is provided to prevent this type of buckling. The formula
is as it follows:
𝐹 =
𝜋2
𝐸𝐼
( 𝐾𝐿)2

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where F is the maximum or critical force, E is the modulus of elasticity, I is the area moment of
inertia of the cross section of the rod, L is the unsupported length of column, and K is the column
effective length factor (for wind turbines, K= 2.0).
Buckling, along with local buckling, analysis are critical for ensuring the safety of the
wind turbine and the area around it. Furthermore, since the weight of the nacelle plus the rotor is
not concentrated on the axis of the tower, the most likely situation that is going to be seen for the
tower is called “eccentricity”. This will be explained in depth throughout buckling analysis.

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METHODOLOGY
Flowchart
Figure 7 a flowchart introducing the key topics and order of the design methods used for
this project.
Design Approach
Before specifically analyzing the components of interest, the wind turbine will be looked
at as a single body, and used in a free body diagram to evaluate the external forces acting on
such. The reason for this is to identify the way these forces influence the components of interest,
as a means to have all the needed information to be able to continue with further analysis. There
are two main parts for the design and analysis of a component: static and dynamic stress
analysis. Before this, however, assumptions have to be made for the material and geometry of the
part that is not fixed. After this, the static part of the analysis may be started.
The component will have to be verified against buckling, in case this is more demanding
than fracture due to compressive stresses. If the component does not survive against buckling,
the assumptions made for the material and geometry will be manipulated until it does. After, this,
normal, shear and bending diagrams will be drawn as a means to locate the section where the
maximum stresses are found, and later specify the points in that section where the maximum
uniaxial and shear stresses reside. These points will be chosen based on the weak areas of the
tentative material. Sharp corners, or any discontinuity on the part geometry plays a big role in the
design process, since the stress flow in the component gets disrupted around these
discontinuities. Stress concentrations are created around these areas, and these increase the value
of the nominal stresses. Therefore, the component will be verified against stress concentrations,
and the nominal stresses will be corrected to the actual ones. With this, principle and Von Mises
stresses are able to be evaluated, and this will provide enough information to conclude the static
stress analysis by computing static safety factors.

19. Figure 7: Design Process Flowchart

20. The component will then be checked regarding fatigue considerations, which is all the
dynamic stress analysis is about. Fatigue will happen to a component when time-varying forces
or stresses are present. The behavior of these with respect to time is usually idealized as
sinusoidal waves, which have both a mean (if any) and an amplitude value. The type of stresses
will be determined, whether they are fluctuating, fully-reversed, or repeated. For full-reversed
stresses there will not be any mean value. This will be used in further calculations during fatigue
load analysis. First, the static stress-concentration factors that were calculated before will need to
be converted to fatigue. This is because stress concentrations have a slightly different effect on
time-varying forces and stresses. These stress-concentration factors, however, will only apply to
amplitude stresses. If the component is undergoing fluctuating stresses, the mean stress will be
needing its own stress-concentration factor in order to be corrected. Forthwith, fatigue loads will
be analyzed, and mean and amplitude Von Mises equivalent stresses are computed. An S-N
curve will be drawn next and compared to the amplitude equivalent stress. This will inform on
the life of the part concerning fatigue. In order to construct such diagram, the material needs to
be characterized as either ferrous or non-ferrous. Ferrous metals are those that contain what is
called a “knee” in the S-N curve, which distinguish from the non-ferrous by having a range of
amplitude-stress values to which the material will never fail from fatigue. Ferrous materials will
contain an endurance limit, while non-ferrous will have a specific fatigue strength at N = 5*108
cycles. For simpler communication, fatigue limit will be the name used for describing both
values in this section. The fatigue limit may be easily related to and extracted from the material’s
physical and mechanical properties, however, this would be an uncorrected value. To correct
this, correction factors will need to be determined. Most correction factors are due to the
geometry of the part, the load it is subjected to, and the setting around it. After the fatigue limit is
corrected, it will be plotted in comparison to the amplitude Von Mises equivalent stress. Finally,
fatigue safety factors will be computed, and the design process will be concluded if the
component met all expectations.

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FREE BODY DIAGRAMS
Wind Turbine FBD
In order to analyze forces and moments acting on the components of interest, we would
have to determine first all of the external forces acting on the wind turbine as a whole. As seen in
Figure 8, the only external influence is the air, however, forces such as drag and lift leading to
various types of stresses are generated.
Figure 8: Side view of wind turbine CAD model showing eccentric buckling.
As mentioned, the air’s velocity as a function of the height was generated, and since its
forces are a function of the velocity, this will be taken into account when calculating the
variances throughout both the tower and the blades (where large heights are covered). Assuming
nominal speed (16 m/s [3]), although it will actually be varying between 4 and 25 m/s with time,
at the hub height, we can determine a function for the velocity as follows:
𝑢 𝑟 = 16
𝑚
𝑠
, 𝑦𝑟 = 49 𝑚 , 𝛼 = 0.1 (Neutral stability conditions are assumed)
𝑢( 𝑦) =
16
490.1
∗ 𝑦0.1
≈ 10.84 ∗ 𝑦0.1

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Adjustments on the angle of the rotor with respect to the ground are made depending on
the information gathered from the wind velocity distribution. If the velocity at the tip of a blade
varies drastically while the blade rotates, then adjustments are made to counteract the effects
done by this phenomenon. The velocity distribution is parabolic, therefore the change in velocity
will be clearer the lower the height. With this in mind, there is a specific section below some
specific height where such change in velocity is clear, thus, if the rotor covers too much of this
area, said adjustments are made. Along with said section, there is a range for allowable wind
speeds. These are usually called “cut-in” and “cut-out” wind speed, which are the speeds at
which the wind is either travelling too slow or too fast, respectively, in order to generate power.
It is clear to see how slow wind speed would be unproductive towards power generation, but not
as so for fast wind speeds. There is a rotational speed called “optimum rotational speed”, in
which the rotor is spinning at a rotational speed that is profitable for generating power, however,
beyond that point the blades will be spinning at speeds fast enough to be depicted as a solid “flat
disk” to the wind that will create drastic disturbances as it penetrates [reference]. This is related
to the wind speed via a dimensionless ratio called the “tip-speed ratio”, which will be discussed
later in this section.
A series of wind turbine models were developed by NASA (shown in Figure 9, along
with the U.S. Department of Energy, taking into account the different types of environments
[10]. Model Mod-5A, since the tip of the blade reaches a very low height, the angle of the rotor
is adjusted due to the variances of wind velocities. To determine if this modification has to be
made on the Vestas V52-850kW, the wind velocities at the extreme ends would have to be
inspected. The height at the bottom of the rotor diameter:
ℎℎ𝑢𝑏 −
𝐷 𝑟𝑜𝑡𝑜𝑟
2
= 23 𝑚 → 𝑢(23) = 10.84 ∗ (23)0.1
= 14.8
𝑚
𝑠
At the top:
ℎℎ𝑢𝑏 +
𝐷 𝑟𝑜𝑡𝑜𝑟
2
= 75 𝑚 → 𝑢(75) = 10.84 ∗ 750.1
= 16.7
𝑚
𝑠

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Figure 9: NASA experimental wind turbines drawn to same scale
(Wikipedia)
As seen, the difference is approx. 2 m/s, and since the cut-in and cut-out wind speed are 4
and 25 m/s, respectively, such adjustments will not have to be made. Recall that this situation
involves nominal wind speed at hub height, and beyond the cut-out wind speed the brake is
going to be applied in order to stop the rotor from rotating, however, designs for the components
of interest (main shaft and tower) should consider wind speeds as high as those generated by
storms (maximum of approx. 69 m/s for hurricane of category 4 [11]), even though they are
occasional. Therefore, drag forces that are calculated for analysis on the wind turbine will be
done so allowing for a time-varying maximum speed of 69 m/s at hub height.
Roll, Yaw, and Pitch
Before calculating forces, a balance (if any) in moments about every axis needs to be
determined and proved. This is a familiar concept with airplanes, where any rotation due to
disequilibrium of moments about the ‘x’ axis is called “roll”, about the ‘y’ axis is called “yaw”,
and about the ‘z’ axis is called “pitch” (as shown in Figure 10). On a wind turbine, equilibrium in
moments about any axis is preferable because this spares unnecessary stresses throughout the
body.

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Figure 10: Isometric view of wind turbine showing roll (x), yaw (y), and pitch (z) moments
In order to evaluate moments about the ‘x’ axis, we would have to consider the moment
about such axis produced by all forces. The weight of the nacelle, the hub, and the tower will not
be considered for calculations because, since they lie on the x-y plane, they create zero moments
about the ‘x’ axis. Any drag force by the wind will be disregarded also because they only
produce moments about the ‘y’ and ‘z’ axis.
Supposing Blade 1 from Figure 11 is at an angle of θ degrees from the horizontal axis (‘z’
axis), the remaining angles for the other two blades can be calculated. The angle between the
blades is 120 degrees, therefore, the angle of Blade 2 will be 60 – θ degrees, and while for Blade
3 the angle will be 60 + θ (all angles are measured with respect to the ‘z’ axis, as seen in Figure
11. Now, moments with respect to the ‘x’ axis were calculated as shown in the following
calculations (keep in mind, the weight is assumed to be the same for all of the blades):
∑ 𝑀 𝑥 = 𝑊𝐵𝑙𝑎𝑑𝑒 ( 𝑟1 ∗ 𝑐𝑜𝑠𝜃) − 𝑊𝐵𝑙𝑎𝑑𝑒 [𝑟2 ∗ cos(60− 𝜃)] − 𝑊𝐵𝑙𝑎𝑑𝑒 [𝑟3 ∗ cos(60 + 𝜃)]

25. 19 | P a g e
Figure 11: Front view of wind turbine CAD model
Since the blades are all made to share the same shape:
Therefore, for any angle θ, any weight of blade, and any distance ‘r’ from origin to center
of mass (as long as consistent in every blade), the sum of all moments about the ‘x’ axis (roll) is
0. Keep in mind, however, that this only applies for the wind turbine in a fully equilibrium state
(i.e. the blades are not moving). In the case where the wind turbine blades are rotating, there is
torque applied by the wind and, ergo, not 0 anymore.

26. 20 | P a g e
For evaluating moments about the ‘y’ axis (yaw), forces on the x-z plane have to be
determined, although it is not necessary to calculate it. Shown in Figure 12, the top view of the
wind turbine would give a best sense on visualizing the moments about the ‘y’ axis. Drag forces
on the blades will be the only influences on ‘yaw’ moments, and as shown in Figure 12, they are
located at some arbitrary distances x1, x2, and x3 from the ‘x = 0’ axis. Looking back to Figure
11, and assuming that for this situation Blade 1 is also at an angle 𝜃, said distances can be written
in terms of 𝑟𝐷 (distance from origin to the point on the blade where the drag force is
concentrated) and 𝜃:
Figure 12: Top view of wind turbine CAD model
𝑥1 = 𝑟𝐷 ∗ cos( 𝜃), 𝑥2 = 𝑟 𝐷 ∗ cos(60 + 𝜃), 𝑥3 = 𝑟𝐷 ∗ cos(60 − 𝜃)
The value 𝑟𝐷 is not known specifically, because it is dependent on the frontal area of the
blades from a perspective point of view of the wind. However, it should be consistent on every
blade and, thus, the same for all three blades. Moments about the ‘y’ axis were calculated as
follows:
∑ 𝑀𝑦 = 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒
( 𝑟𝐷 𝑐𝑜𝑠𝜃) − 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 ∗ 𝑟𝐷 cos(60 − 𝜃) − 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 ∗ 𝑟𝐷 cos(60 + 𝜃)
= 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 ∗ 𝑟𝐷 ∗ ( 𝑐𝑜𝑠𝜃 − cos(60 − 𝜃) − cos(60 + 𝜃))
= 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 ∗ 𝑟𝐷 (𝑐𝑜𝑠𝜃 − cos(60) 𝑐𝑜𝑠𝜃 − sin(60) 𝑠𝑖𝑛𝜃 − cos(60) 𝑐𝑜𝑠𝜃 + sin(60) 𝑠𝑖𝑛𝜃)

27. 21 | P a g e
= 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 ∗ 𝑟𝐷 ∗ ( 𝑐𝑜𝑠𝜃 − 2 ∗ (
1
2
) ∗ 𝑐𝑜𝑠𝜃) = 0
It can be seen that, similar to “roll” moments, moments about the ‘y’ axis are balanced
for any angle 𝜃, any drag force 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 applied on the blades, and any distance 𝑟𝐷 (as long as the
blades are of the same shape.
As seen in Figure 8, the center of mass of the assembly consisting of the nacelle and
the rotor is at some distance ‘e’ (term called “eccentricity in buckling analysis) from the ‘y-z’
plane, and therefore, causes moments about the ‘z’ axis, in addition to the moment created by the
drag force from the wind. However, the bolts that clamp the tower to the ground play a huge role
in balancing the pitch moments because it induces bending stresses throughout the tower. These
bending stresses will have to be accounted for stress analysis on the tower. On the other hand, it
would be possible for a balance of moments, without creating bending stresses throughout the
tower, in this direction if the wind would come to travel fast enough that it would create a
sufficiently-big drag force that will counteract the moment created by the weight. But this would
be an occasional occurrence, and thus, will not be included in further analysis.
Drag Force
Drag forces on the wind turbine usually are not of concern, unless wind velocities are as
high as they are during a storm (hurricane, tornado, tropical storm, etc.). Therefore, as an
engineer, designing with considerations of the relatively-worst conditions is desirable. This
information will be important for further calculations on the shaft, because since the rotor is
completely connected to the shaft, the forces that the blades endure will be transferred to the
shaft. As mentioned, the wind velocity that will be used on further calculations in this section is
70 m/s. Recall the drag force equation:
𝐹𝐷 =
1
2
𝐶 𝐷 𝜌𝑣2
𝐴
As explained, the parameter ‘A’ in the equation represents the frontal area of the object
through which the fluid (air) if impacting. The object in this case is the blade, and as seen in
Figure 13, a rectangular shaped area can be used to replace the frontal area of the blade, which is
too complex to calculate.

28. 22 | P a g e
Figure 13: Simplification of the wind turbine blade from the CAD
The drag coefficient depends solely on the object’s geometry and is evidently
proportional to the drag force. This is why drag forces on wind turbine blades are usually
irrelevant, since the drag coefficient is very small. The “Danish Wind Industry Association”
[reference] explained the effects of drag forces on objects and commented on drag coefficients
for airfoil shaped objects. “An airfoil shape used on aircraft wings or rotor blades, typically has
an extremely small CD about 0.04.” [12]. Shown in Figure 14 from NASA at Glenn Research
Center a figure called “Shape Effects on Drag” [13] that shows various drag coefficients
corresponding to object shapes was developed, and determined the drag coefficient of a typical
airfoil to be 0.045. Additionally, the hub, even though small, endures drag forces. Its drag
coefficient is the same as for the bullet shape shown in Figure 14.

29. 23 | P a g e
Figure 14: “Shape Effects on Drag”
(NASA)
With the following information, the drag force on the blades can be determined, as shown
(Note: Some of the values obtained for dimensions were determined by inspection, and are but
approximations):
𝐶 𝐷,𝐵𝑙𝑎𝑑𝑒 = 0.045 , 𝐶 𝐷,𝐻𝑢𝑏 = 0.295 , 𝜌 = 1.225
𝑘𝑔
𝑚3 , 𝑣 = 70
𝑚
𝑠
, 𝐷 𝐻𝑢𝑏 =
26
13.8
= 1.88 𝑚 ,
𝐷 𝐵𝑙𝑎𝑑𝑒 ≈
3
4
∗ 𝐷 𝐻𝑢𝑏 = 1.41 𝑚 , 𝐿 𝐵𝑙𝑎𝑑𝑒 = 𝑅 𝑅𝑜𝑡𝑜𝑟 −
𝐷 𝐻𝑢𝑏
2
= 25.05 𝑚
Then, the forces on the blades and hub were calculated:
𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 =
1
2
(0.045)(1.225)(702)(
𝐷 𝐵𝑙𝑎𝑑𝑒
2
)(
𝐷 𝑅𝑜𝑡𝑜𝑟
2
−
𝐷 𝐻𝑢𝑏
2
) = 2386.08 𝑁
𝐹𝐷,𝐻𝑢𝑏 =
1
2
(0.295)(1.225)(702)(
𝜋
4
)( 𝐷 𝐻𝑢𝑏
2 ) = 2457.7 𝑁
Therefore,
𝐹𝐷 ,𝑅𝑜𝑡𝑜𝑟 = 3 ∗ 𝐹𝐷,𝐵𝑙𝑎𝑑𝑒 + 𝐹𝐷,𝐻𝑢𝑏 = 9615.9 𝑁
Since these calculations assume the blades in static equilibrium, a different method will
now be used to calculate the drag force, and will then be compared to the first method. This
method will consist of determining the change of energy of the wind as it penetrates through the

30. 24 | P a g e
entire area swept by the blades when they rotate. As the wind passes through the rotor, energy is
absorbed, and thus, a pressure change can be calculated from the streamline of the wind. The
following equation is Bernoulli’s equation of conservation of energy along a streamline:
𝑃1 +
1
2
𝜌𝑣1
2
+ 𝛾𝑧1 = 𝑃2 +
1
2
𝜌𝑣2
2
+ 𝛾𝑧2
where P is pressure, v is velocity, and z is height. Since the height does not change, it can be
cancelled out. P2 – P1 will represent the pressure absorbed by the rotor as the wind penetrates it.
The change in velocity is determined by a factor called the “axial induction factor” (AIF) in
aerodynamics. From background research on this topic, it was found that on wind turbines the
factor ranges from 0.2 to 0.9 [14]. A factor of 0.6 will be assumed for these calculations. For this
method, the cut-out wind speed (25 m/s) will be used, in order to calculate the maximum force
from the time-varying drag forces. This factor gives a final velocity of 15 m/s after it penetrates
the rotor. From plugging the values in Bernoulli’s equation, a change of pressure of 245 Pa was
computed, which in turn, gives a drag force of 662,480 N. This value is much larger than the one
calculated in the previous method, and on top of that, it assumes the blades are rotating, which is
ideal. This total drag force on the rotor will be useful for calculations on both the main shaft, and
the tower by creating uniaxial stress concentrations and bending stresses, respectively.
Since this will be a time-varying drag force, due to the fact that the velocity varies from 4
to 25 m/s, a function of this of velocity has to be determined. If the AIF is given the symbol “a”,
whose function is as follows:
𝑎 =
𝑣1 − 𝑣2
𝑣1
then the pressure change may be written as:
∆𝑃 =
1
2
𝜌( 𝑣1
2
− 𝑣2
2)
and plugging in the final velocity in terms of AIF and the initial velocity will give:
∆𝑃 =
1
2
𝜌( 𝑎2
𝑣1
2
− 2𝑎𝑣1 )
and finally, the drag force as a function of velocity is as follows:

31. 25 | P a g e
𝐹𝐷 (𝑣) =
1
2
𝜌𝐴( 𝑎2
𝑣2
− 2𝑎𝑣)
Power
There are many ways of calculating the power generated by the wind turbine as it absorbs
the wind’s energy, however, the most direct and accurate one was derived from Betz’s Law,
which is an “aerodynamic cycle efficiency” [15] stating that no wind turbine can convert more
than 59.3% of the kinetic energy carried by the wind. The power content that can be absorbed
from a free flowing stream can be expressed as the change in kinetic energy, and through a series
of manipulation using fluid mechanics theories, an equation was determined that showed the
power of the wind as a function of its velocity. With the aid of Betz’s limit, a coefficient called
the “coefficient of performance” was defined. [15] This a ratio of the mechanical power, which
is extracted by the wind turbine from the wind, to the wind power that was just discussed. Said
coefficient and wind power are defined as follows:
𝐶 𝑃 =
𝑃 𝑀𝑒𝑐ℎ
𝑃 𝑊𝑖𝑛𝑑
, 𝑃 𝑊𝑖𝑛𝑑 =
1
2
𝜌 (
𝜋𝐷 𝑅𝑜𝑡𝑜𝑟
2
4
) 𝑣3
𝑃 𝑀𝑒𝑐ℎ =
1
2
𝐶 𝑃 𝜌(
𝜋𝐷 𝑅𝑜𝑡𝑜𝑟
2
4
) 𝑣3
Therefore, this can be re-written as:
𝑃 𝑀𝑒𝑐ℎ = 𝑇 ∗ (
𝜆𝑣
𝑟𝑅𝑜𝑡𝑜𝑟
) => 𝑇 =
𝑟𝑅𝑜𝑡𝑜𝑟
2𝜆
𝐶 𝑃 𝜌𝑣2
(
𝜋𝐷 𝑅𝑜𝑡𝑜𝑟
2
4
)
With this equation, determining the torque that a wind turbine rotor generates from wind
flowing through a stream at velocity ‘v’ is fairly straightforward. The tip-speed ratio and the
coefficient of performance may not seem forthright to determine, nonetheless, these two ratios
can indeed be related. A recent article developed a series of curves plotted in a tip-speed ratio vs.
power coefficient (same as coefficient of performance), as shown in Figure 16 [16]. The
maximum power than can be calculated from any tip-speed ratio is at a coefficient of
performance of ~0.45, with a tip-speed ratio of 6.0. This will be implemented in analysis on the
components of interest in subsequent sections.

32. 26 | P a g e
Figure 15: Tip-speed ratio vs. power coefficient graph.
(M. Ragheb)

33. 27 | P a g e
Tower FBD
A free body diagram was drawn for the tower on the X-Y plane, and shown in Figure-----
. An isometric view for the wind turbine was drawn (see Figure ----), along with all the forces
that act on it, in order to show the remaining forces and moments that cannot be seen from only
one plane view.
Figure 16: Tower FBD from X-Y plane view.

34. 28 | P a g e
Figure 17: Isometric view of wind turbine drawing

35. (a) (b) (c)
Figure 18: Drawing of Tower: (a) Top view, (b) Front view, (c) Section view from front view.
(Reference dimensions were used)

36. Shaft FBD
Two shaft designs were created: an “analysis shaft” design (Figure 19a) and an “ideal
shaft” design (Figure 19b). Figure 21 shows the free body diagram of the shaft. It is assumed
cantilever because of the fact that the gearbox delivers a counteractive torque. However, on other
situations, it will be nothing more than a bearing with a vertical reaction force.
(a) (b)
Figure 19: Shaft Designs, (a) Analysis Shaft, (b) Ideal Shaft.
Figure 20: FBD of Shaft

37. 31 | P a g e
Figure 21: CAD of shaft, along with the two bearings and the hub.
Figure 22: Section view of the shaft in 2-D drawing.

38. 32 | P a g e
ANALYSIS AND RESULTS
Tower
All calculations described in this section were done in MathCAD. Screenshots of these are found
in Appendix A. For a complete list of results, and other dependent variable values, see Appendix
A-9.
Table 1: Tentative Tower Properties
Properties Variable Value Units
SAE 1060 Q&T @ 800 o
F
𝑆 𝑦
𝑆 𝑢𝑡
669
965
MPa
MPa
Tower Wall Thickness 𝑡 0.04 m
Top Outer Diameter 𝑑 𝑇 2 m
BottomOuter Diameter 𝑑 𝐵 2.5 m
Information that is going to be needed throughout this section is listed in Appendix A-1.
A tentative material, along with dimensions, has to be chosen before starting stress analysis. See
Table 1 for a list of tentative information that was assumed for analysis of the tower. This
material was chosen to have an elongation percentage between 15 and 18%, because the metal
needs to deform a certain amount before it fails. From these few steels gathered up in Table 2,
the tentative material was chosen to be SAE 1060 quenched and tempered @ 1000 F, having a
yield strength of 669*106 and an ultimate tensile strength of 965*106. Since sheet metal requires
rolling of the metal as the previous operation, it is going to be assumed that the quenched and
tempered steel is rolled (either hot or cold). The condition of the metal is extremely important,
since it defines the characteristics of such. This steel is categorized as tempered martensite.
Martensite is made when carbon steel is heated up until it becomes austenite, and then suddenly
quenched with either air or water (preferably water). Martensite is one of the strongest steels
created; however, extremely brittle. On the other hand, with the process of tempering, which is
heating the steel up to just below the eutectoid temperature and leaving it for some period of
time, martensite becomes ductile enough while maintaining its strength. This process is not easy
nor hard to do, and it will not be an economic metal, but not too expensive also.

39. 33 | P a g e
On the static part of the stress analysis, the component needs to be verified against
buckling before anything else. Buckling could happen to any beam-like component subjected to
compressive stresses. Whether or not the part fails due to buckling is determined by a ratio called
the “slenderness ratio”, which depends solely on the geometry of a component. If this ratio is less
than 10, then the component will not buckle, however, if larger than 10, the part may be
subjected to buckling. The slenderness ratio was found to be 150.484, therefore, buckling
became of concern. Buckling analysis is shown in Appendix A-2, and the load to which the
component will buckle (the critical load) was found to be 5.124*106, which is higher than the
load that the tower is subjected to, and thus, it will not buckle.
Table 2: Some Carbon Steels with Elongation 15-18%
(Robert L. Norton)
SAE/AISI
Number
Condition Yield Strength
(MPa)
Ultimate Tensile
Strength (MPa)
Elongation (%)
1020 Cold rolled 393 469 15
1030
Quench & temper
@ 400 o
F
648 848 17
1035 Hot rolled 276 496 18
1040 Hot rolled 290 524 18
1045 Hot rolled 310 565 16
1050 Hot rolled 345 621 15
1060
Normalized @
1650 o
F
421 772 18
1060
Quench and
temper @ 800 o
F
669 965 17
Since the tower is essentially clamped to the ground like a cantilever, reaction forces and
moments on and about every axis is expected, as seen in the free body diagram (see Figure 17).
The variation of drag forces on the whole system is going to be taken into consideration due to
the fact that it subjects the tower to fatigue. Recall that drag forces calculated for this project are
calculated using the change in pressure derived from Bernoulli’s equation as the wind penetrates
the rotor. The reactions at the fixed end were calculated as a function of the wind velocity, and
from this, normal, shear and moment functions were calculated as functions of both tower height

40. 34 | P a g e
and wind velocity. Diagrams for these functions at the maximum wind velocity were drawn and
shown in Appendix A-3.
As seen and expected, the maximum shear and bending stresses are located at the fixed
spot, to which it is called a “critical section”. As seen in Figure 24, there are two moments, one
of which is the roll moment generated by the blades, and the other one is the moment about ‘z’
that created the bending stresses calculated earlier, and these will both create normal and
transverse-shear stresses. Both of them were compared in order to distinguish the areas where
said stresses are maximum, and concluded that the demanding moment is the moment about the
‘z’ axis. The assumption that was made to choose the tentative material allowed to specify the
weak areas of the material, and consequently, two critical points on said section were
determined; one where the maximum shear stress is located, and the other where the maximum
normal stress is located.
Figure 23: Critical Section and Points
The said stresses that were calculated are considered nominal, and thus, they needed to be
modified to the actual ones by considering stress concentrations. Static stress concentrations
were assumed to be 1.4 for shear and 1.7 for normal. With this, the nominal stresses were

41. 35 | P a g e
corrected, and this provided enough information to evaluate the principal stresses at each point,
along with the Von Mises effective stresses.
To conclude the static part of the analysis, safety factors for ductile materials were
computed following the distortion energy theory for both uniaxial stress and pure shear, and the
maximum shear and normal stress theory. The lowest safety factor from these was found to be
1.794, which is considered “safe” since it is larger than 1.
Wind velocities, as mentioned, vary throughout a period of time, and thus, will produce
time-varying drag forces. Since the velocity as a function of time cannot be sketched precisely,
the curve was drawn as sinusoidal waves, having wind velocity as a sinusoidal function of time.
The wind was specified to travel at speeds varying from 59 to 69 m/s, having a mean velocity of
64 m/s, and an amplitude velocity of 5 m/s. These values will be used throughout this part.
Recall that the reaction forces and moments were calculated as functions of velocity. This
allowed the easy manipulation of further functions, and made it easy to retrieve mean and
amplitude normal and shear stresses.
Before analyzing fatigue loads, the static stress concentration factors were converted to
fatigue with the aid of the notch sensitivity. This notch sensitivity is a function of Neuber’s
Constant, which depends only on the material, and the notch radius, which was assumed to be 12
inches. This allowed for the computation of the normal and shear fatigue stress concentration
factors that were 1.692 and 1.396, respectively. These factors are only viable for determining the
actual amplitude stress, however, mean fatigue stress-concentration factors were also calculated
to fix the nominal mean stresses. Afterwards, Von Mises equivalent amplitude and mean stresses
were evaluated for subsequent calculations.
To finalize the dynamic part of the stress analysis, an S-N curve, followed by fatigue
safety factors, was drawn and compared to the equivalent amplitude stress to verify if, and for
how many cycles, the component survived against fatigue. For this, the uncorrected endurance
limit of the material was determined and correction factors, some assumed, were used to correct
such endurance limit. Notice the term “endurance limit” is being used, and this is because steel is
a ferrous metal. Data and calculations for this part are found in Appendix B-9. After correcting
the endurance limit using the correction factors, the S-N diagram was drawn (shown in Appendix
B-9), and compared to the Von Mises equivalent amplitude stress. It was found that for the

42. 36 | P a g e
tentative material containing the tentative dimensions listed in Table 1 the component will never
fail due to fatigue.
With the corrected endurance limit, a fatigue safety factor may be computed for
fluctuating stresses. There are many cases, however, for which to assume about the behavior of
the wind velocity under occasional circumstances. Concerning calculations of drag forces, the
hurricane wind speeds are assumed to stay present over the life of the wind turbine. On top of
this, it is also assumed that the mean velocity of the wind stays constant at 64 m/s while the
maximum and minimum velocities may increase and decrease under service conditions. From
the four cases for fluctuating stresses, Case 2, will be assumed, which requires the mean stress to
stay constant, but the amplitude stress may increase. The fatigue safety factor was calculated to
be 1.275, which is considered “safe”.

43. 37 | P a g e
Shaft
All calculations described in this section were done in MathCAD. Screenshots of these are found
in Appendix B. For a complete list of results, and other dependent variable values, see Appendix
B-9.
An analysis shaft design was created in order to simplify calculations while achieving
not-so-distinct results, while the ideal shaft design would be the final design if it were to be
manufactured. With the exception of stress concentrations, notch radii, and any other stress-
influencing parameter that depends solely on the specific geometry of the part, all MathCAD
analysis is done on the “analysis shaft”. See Table 3 for initial dimensioning of the analysis
shaft.
Table 3: Initial Dimensions
Length a 0.500 m
Length b 1.400 m
Length L 1.500 m
Location Bearing 1 0.125 m
Location Bearing 2 1.250 m
Diameter da 0.200 m
Diameter db 0.500 m
Diameter dL 1.000 m
Filet Radius 0.250 m

44. 38 | P a g e
(a) (b)
Figure 24: Analysis Shaft. (a) Parameter Values, (b) Critical Section and Points
The critical section of the shaft was found to be situated at the location of the second
bearing. The specific critical points of interest are points B and C (see Figure 26). At these
critical points, we calculate the true, principle stresses that these points are subjected to (and Von
Mises stresses, and so on). The greater of the two Von Mises equivalent stresses end up being at
point C, with a maximum normal effective stress of 22.93 MPa and a maximum shear stress of
11.47 MPa.
From here, we must move from looking at an “idealized” shaft to a more realistic shaft,
and take into account geometry and the fact that there will be fluctuating / repeated
stresses. Firstly, we must take a closer look at areas of the shaft with unique geometry, which in
this case would be the fillet before the end of the shaft. The filet ends up producing a stress
concentration factor of 1.172 and 1.018 under static considerations for bending and torsion,
respectively, and a factor of 1.239 and 1.011 under dynamic normal and dynamic shear stresses
respectively. None of these values, when multiplied into stresses 1.4 meters into the shaft (At
the filet location) produce a stress larger than those found at the critical section found
previously. Future calculations will continue to use the previous results for the original critical
section. The smallest static safety factor calculated was found to be 8.103, which is considered
safe, and with this the dynamic part of the design process is begun. For specific safety factors
(including the Distortion Energy Theory, Maximum Shear, etc.) see Appendix B-4.

45. 39 | P a g e
Now interested in the lifetime of the shaft, we begin our fatigue analysis, starting by
finding the alternating stress, which is the amount the stresses vary by when subjected to
dynamic loads. These values are then used to calculate the Von Mises alternating stress, which
comes out to be 1.733 MPa. Now, we calculate the coefficients of correction to find the corrected
endurance limit to see if our shaft will live forever or for a non-infinite amount of cycles. The
corrected endurance limit is calculated to be 52.65 MPa, which is well above the alternating
stresses the shaft is subjected to. Our part has an infinite lifetime, and is in no need of dimension
changes.
After verifying and concluding the stress analysis on the shaft, the program ANSYS was
used to corroborate the deformation of the shaft (see Figure 26). Also, vibrations throughout the
shaft were inspected to verify that the critical frequency and the maximum frequency of the shaft
are far apart, in order to evade resonance. “If the forcing frequency happens to coincide with one
of the element’s natural frequencies, then the amplitude of the vibratory response will be much
larger than the amplitude of the driving function.” [17]. The ratio of these two values was found
to be 166.308, which is considered safe for the component. The spring constant was found using
a formula given by Tongue, B. H., in “Principles of Vibration” [18].
Figure 25: Critical Section and Points

46. 40 | P a g e
CONCLUSION
After many dimension revisions and stress analysis implementations, the shaft and tower
are both designed for long life and excellent durability. The shaft has a tapered design such that
there are no “steps”, thus minimizing stress concentrations. A single roller bearing is located at
the clamp end of the shaft, and a thrust roller bearing towards the rotor end. The thrust bearing is
responsible for taking any forces transmitted axially to the shaft and grounds them to the nacelle,
and so is placed as closely to the rotor end as possible to minimize length of shaft that is
subjected to normal stresses. For the immediate change in diameter at the rotor end that is
required to interface the rotor with the shaft, a filet must be used to minimize stress
concentrations. After a complete stress analysis of the part, the design of the shaft has proven to
be sufficient and optimally engineered. See Table 6 for a complete list of dimension values.
Figure 26: Ideal Shaft Design
The tower is designed to withstand extreme-wind conditions, and thus able to survive for
a hypothetical infinite life while subjected to varying loads. It is designed such that its diameter
decreases with height to better support its weight and prevent against buckling. While simple,
this design of the tower proves to be both reliable and sturdy. See Table 4 for a complete list of
dimension values. For complete results, calculations, and analysis via Mathcad and ANSYS, see
the appendix.

47. 41 | P a g e
REFERENCES
[1] Illustrated History of Wind Power Development, n.d., from
http://ww.telosnet.com/wind/early.html
[2] Trevor, P., 2009, “Blyth, James (1839–1906)”, Oxford DNB, from
http://dx.doi.org/10.1093/ref:odnb/100957
[3] Vestas Wind Systems A/S, 2007, “V52-850 kW”, Denmark, from
http://www.vestas.cz/files/V52-850.pdf
[4] TEMBRA GmbH & Co. KG, n.d., from
http://www.tembra.com/index.php/drive-train.html
[5] Stiesdal, H., 1999, “The Wind Turbine Components and Operations”, Bonus Energy A/S.
[6] Junda Heavy Industry Forging, n.d., from
http://en.hi-junda.com/
[7] Ragheb, A., and Ragheb, M., 2010, “Wind turbine gearbox technologies”, 1st International,
pp. 1-8, from
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5462549&tag=1
[8] Nicholson, J. C., 2011, “Design of wind turbine tower and foundation systems: optimization
approach”, MS thesis, University of Iowa, from
http://ir.uiowa.edu/etd/1042/
[9] Norton, R. L., 2014, Machine Design, Worcester Polytechnic Institute, Worcester, MA,
Chap. 4.
[10] Wikipedia, 2015, from
https://en.wikipedia.org/wiki/NASA_wind_turbines
[11] 1998, Hurricanes, World Book Encyclopedia, Chicago, IL, pp. 452-456.
[12] Danish Wind Industry Association, n.d., “Wind Power”, from
http://xn--drmstrre-64ad.dk/wp-content/wind/miller/windpower%20web/en/tour/wtrb/drag.htm
[13] NASA, n.d., from
http://www.grc.nasa.gov/WWW/K-12/airplane/shaped.html
[14] Akon, A. F., 2012, "Measurement of axial induction factor for a model wind turbine." from
http://ecommons.usask.ca/bitstream/handle/10388/ETD-2012-08-585/AKON-
THESIS.pdf?sequence=4
[15] Ragheb, M., 2014, “Wind Energy Conversion Theory, Betz Equation”, from

48. 42 | P a g e
http://mragheb.com/NPRE%20475%20Wind%20Power%20Systems/Wind%20Energy%20Conv
ersion%20Theory%20Betz%20Equation..pdf
[16] Ragheb, M., 2014, “Optimal Rotor Tip Speed Ratio”, from
http://users.wpi.edu/~cfurlong/me3320/DProject/Ragheb_OptTipSpeedRatio2014.pdf
[17] Norton, R. L., 2014, Machine Design, Worcester Polytechnic Institute, Worcester, MA,
Chap. 10.
[18] Tongue, B. H., 2002, Principles of Vibration, Oxford University Press, NY, Chap. 1.

49. 43 | P a g e
APPENDICES
Appendix A. Tower Stress Analysis
Appendix A-1. List of Independent Parameters
Table 4: Independent Parameter Values
Parameters Variables Values Units
Wind Information
Air Density 𝜌 𝑎𝑖𝑟 1.225 Kg/m3
Relative MaximumVelocity 𝑢 𝑟𝑚𝑎𝑥 69 m/s
Relative Amplitude Velocity 𝑢 𝑟𝑎 5 m/s
Relative Mean Velocity 𝑢 𝑟𝑚 64 m/s
Tower Information
Tower Height 𝐻 49 m
Tower Wall Thickness 𝑡 0.04 m
Top Outside Diameter 𝑑 𝑇 2 m
BottomOutside Diameter 𝑑 𝐵 2.5 m
Tower Avg. Outer Diameter 𝑑 𝑜𝑢𝑡 2.25 m
Tower Avg. Inner Diameter 𝑑𝑖𝑛 2.17 m
Tower Weight 𝑤𝑡 480525.812 N
Tower Drag Coefficient 𝐶 𝐷𝑇 0.82 /
Notch Radius 𝑟𝑛𝑜𝑡𝑐ℎ 12 in
Fatigue Load Information
Amplitude Drag on Rotor 𝐹𝑎𝑅𝑜𝑡𝑜𝑟 49.061 N
Mean Drag on Rotor 𝐹𝑚𝑅𝑜𝑡𝑜𝑟 1.395*105
N
MaximumDrag on Rotor 𝐹 𝑚𝑎𝑥𝑅 4.146*105
N

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MinimumDrag on Rotor 𝐹 𝑚𝑖𝑛𝑅 6.634*104
N
Amplitude Drag on Tower 𝐹𝑎𝑇𝑜𝑤𝑒𝑟 1.221*104
N
Mean Drag on Tower 𝐹𝑚𝑇𝑜𝑤𝑒𝑟 2.328*104
N
MaximumDrag on Tower 𝐹 𝑚𝑎𝑥𝑇 6.922*104
N
MinimumDrag on Tower 𝐹 𝑚𝑖𝑛𝑇 1.107*104
N
Other Information
Weight of Nacelle + Rotor 𝑊 323619.45 N
Roll Moment 𝑀 𝑅𝑜𝑙𝑙 4.87*105
N-m

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Appendix A-2. Buckling Analysis

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Appendix A-3. Normal, Shear, and Moment Diagrams

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Appendix A-4. Deflection and Slope Diagram

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Appendix A-5. Principal Stress Calculations

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Appendix A-6. Static Safety Factor Calculations

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Appendix A-7. Fatigue Load Analysis

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Appendix A-8. S-N Diagram

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Appendix A-9. List of Dependent Values and Results
Table 5: Dependent Values and Results
Parameters Variables Values Units
Eccentricity 𝑒 2 m
Effective Length 𝐿 𝑒𝑓𝑓 117.6 m
Radius of Gyration 𝑘 0.516 m
Slenderness Ratio 𝑆 𝑟 227.739 /
Slenderness Ratio at Point D 𝑆 𝑟𝐷 100 /
Eccentricity Ratio 𝐸𝑟 5.625 /
Tower Avg. Moment of Inertia (z) 𝐼𝑧 0.049 m4
Tower Avg. Moment of Inertia (x) 𝐼𝑥 0.049 m4
Tower Avg. Cross-Sectional Area 𝐴 𝑐𝑠 0.183 m2
Tower Frontal Area 𝐴 𝐹 220.5 m2
Blade Frontal Area 𝐴 𝐹𝐵𝑙𝑎𝑑𝑒 17.667 m2
Hub Frontal Area 𝐴 𝐹𝐻𝑢𝑏 2.776 m2
Buckling Critical Load 𝑃𝑐𝑟 5124 kN
Compressive Stress on Bottomof Tower by
Weight of Wind Turbine 𝜎 𝑤 7.00 MPa
MaximumShear Stresson Bottomof Tower 𝑉𝑚𝑎𝑥 12.19 MN
MaximumBending Stress at Bottomof Tower 𝑀 𝑚𝑎𝑥 12.73 MN-m
Stress-Concentration Factor(Normal Stress) 𝐾𝑡 1.7 /
Stress-Concentration Factor(Shear Stress) 𝐾𝑡𝑠 1.4 /
MaximumShear at Point C 𝜏 𝐶𝑚𝑎𝑥 160.1 MPa

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MaximumShear at Point B 𝜏 𝐵𝑚𝑎𝑥 186.5 MPa
MaximumNormal Stress at any Point in the
Critical Section 𝜎 𝑚𝑎𝑥 320 MPa
Von Mises Equivalent Stress 𝜎′ 323.3 MPa
MaximumShear Stressat any Point in the
Critical Section 𝜏 𝑚𝑎𝑥 186.5 MPa
Safety Factor(Distortion Energy Theory –
Uniaxial Stress) 𝑁 𝐷𝑈 2.069 /
Safety Factor(Distortion Energy Theory –
Shear Stress) 𝑁 𝐷𝑆 2.07 /
Safety Factor(MaximumShear StressTheory) 𝑁𝑆 1.794 /
Safety Factor(MaximumNormal Stress
Theory) 𝑁 𝑁 2.091 /
MinimumStatic Safety Factor 𝑁𝑆𝑡𝑎𝑡𝑖𝑐 1.794 /
Notch Sensitivity at Notch Radius 𝑞 𝑆(𝑟𝑛𝑜𝑡𝑐ℎ) 0.989 /
Fatigue Stress-Concentration Factor
(Normal)
𝐾𝑓 1.692 /
Fatigue Stress-Concentration Factor (Shear) 𝐾𝑓𝑠 1.396 /
MaximumFatigue Normal Load Stress 𝜎𝑓𝑚𝑎𝑥 318.4 MPa
MinimumFatigue Load Normal Stress 𝜎𝑓𝑚𝑖𝑛 225.1 MPa
Uncorrected Amplitude Stress 𝜎𝑎′ 15.85 MPa
Uncorrected Mean Stress 𝜎 𝑚′ 159.5 MPa
MaximumFatigue Load Shear Stress 𝜏𝑓𝑚𝑎𝑥 260.3 MPa
Von Mises MaximumFatigue Load Stress 𝜎 𝑚𝑎𝑥
′ 552 MPa
Fatigue Stress-Concentration Factor for
Mean Normal Stress
𝐾𝑓𝑚 1.692 /
Fatigue Stress-Concentration Factor for
Mean ShearStress
𝐾𝑓𝑠𝑚 1.396 /
Amplitude Stress 𝜎𝑎 26.82 MPa
Mean Stress 𝜎 𝑚 269.9 MPa

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Amplitude ShearStress 𝜏 𝑎 8.822 MPa
Mean ShearStress 𝜏 𝑚 158.2 MPa
Von Mises Equivalent Amplitude Stress 𝜎𝑎
′ 30.87 MPa
Von Mises Equivalent Mean Stress 𝜎 𝑚
′ 384.7 MPa
Uncorrected Endurance Limit 𝑆 𝑒
′ 482.5 MPa
Correction Factor for Load 𝐶𝑙𝑜𝑎𝑑 0.7 /
Correction Factor for Size 𝐶𝑠𝑖𝑧𝑒 0.6 /
Correction Factor for Surface 𝐶𝑠𝑢𝑟𝑓 0.46 /
Correction Factor for Temperature 𝐶𝑡𝑒𝑚𝑝 1 /
Correction Factor for Reliability 𝐶𝑟𝑒𝑙𝑖𝑎𝑏 0.702 /
Fatigue Strength at 103
Cycles 𝑆 𝑚 723.7 MPa
Corrected Endurance Limit 𝑆 𝑒 65.44 MPa
Fatigue Safety Factor 𝑁𝑓 1.275 /

63. 57 | P a g e
Appendix B. Shaft Stress Analysis
Appendix B-1. Lis of Independent Parameters
Table 6: Table of Independent Parameters
Parameters Variables Values Units
Wind Information
Air Density 𝜌 1.225 Kg/m3
Relative MaximumVelocity 𝑢 𝑟𝑚𝑎𝑥 25 m/s
Relative Amplitude Velocity 𝑢 𝑟𝑎 10.5 m/s
Relative Mean Velocity 𝑢 𝑟𝑚 14.5 m/s
Shaft Information
Shaft Length 𝐿 1.5 m
Length “a” (see Figure 25) 𝑎 0.5 m
Length “b” (see Figure 25) 𝑏 1.4 m
Diameter at first bearing 𝑑 𝑎 0.2 m
Diameter at second bearing 𝑑 𝑏 0.5 m
Diameter at rotor connection 𝑑 𝐿 1 m
Young’s Modulus 𝐸 206.8*109
Pa
Density of Steel 𝛾 7.8*103
Kg/m3
Notch Radius 𝑟 10 in
Fatigue Load Information
Amplitude Drag on Rotor 𝐹𝑎𝑅𝑜𝑡𝑜𝑟 49.061 N
Mean Drag on Rotor 𝐹𝑚𝑅𝑜𝑡𝑜𝑟 8.038*103
N
MaximumDrag on Rotor 𝐹 𝑚𝑎𝑥𝑅 9.343*103
N
MinimumDrag on Rotor 𝐹 𝑚𝑖𝑛𝑅 7.989*103
N
MaximumTorque 𝑀 𝑅(𝑢 𝑟𝑚𝑎𝑥) 5.716*103
N-m

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Mean Torque 𝑀 𝑅(𝑢 𝑟𝑚) 1.923*103
N-m
Alternating Torque 𝑀 𝑅(𝑢 𝑟𝑎) 1.008*103
N-m
Other Information
Weight of Nacelle + Rotor 𝑊 323619.45 N
Area of Rotor 𝐴 𝑅𝑜𝑡𝑜𝑟 2.124*103
m2
Axial Induction Factor 𝑎 𝐴𝐼𝐹 0.6 /

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Appendix B-2. Shear and Moment Analysis and Diagrams

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Appendix B-3. Principal Stress Calculations

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68. 62 | P a g e

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Appendix B-4. Static Safety Factor

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Appendix B-5. Stress-Concentration Factor Computation

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Appendix B-5. Fatigue Load Analysis

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Appendix B-6. S-N Diagram

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Appendix B-7. Fatigue Safety Factors

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Appendix B-8. Vibration Calculations

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Appendix B-9. List of Dependent Values and Results
Table 7: Dependent Variable Values and Results
Parameters Variables Values Units
MaximumMechanical Power Absorbed 𝑃 𝑀𝑒𝑐ℎ 9.146*106
N
MaximumTorque Generated 𝑀 𝑅 1.189*106
N-m
MaximumShear Stress 𝑉𝑚𝑎𝑥 1.063*105
N
MaximumBending Stress 𝑀 𝑚𝑎𝑥 2.589*104
N-m
MaximumSlope 𝜃 𝑚𝑎𝑥 6.877*10-5
rad
MaximumDeflection 𝛿 𝑚𝑎𝑥 0.01576 mm
Stress-Concentration Factor(Bending) 𝐾𝑡 1.253 /
Stress-Concentration Factor(Torsion) 𝐾𝑡𝑠 1.012 /
MaximumShear at Point C 𝜏 𝐶𝑚𝑎𝑥 49.14 MPa
MaximumShear at Point B 𝜏 𝐵𝑚𝑎𝑥 49.8 MPa
MaximumNormal Stress at any Point in the
Critical Section
𝜎 𝑚𝑎𝑥 52.64 MPa
Von Mises Equivalent Stress 𝜎′ 86.28 MPa
MaximumShear Stressat any Point in the
Critical Section
𝜏 𝑚𝑎𝑥 49.8 MPa
Safety Factor(Distortion Energy Theory –
Uniaxial Stress) 𝑁 𝐷𝑈
9.353 /
Safety Factor(Distortion Energy Theory –
Shear Stress) 𝑁 𝐷𝑆 9.351 /
Safety Factor(MaximumShear StressTheory) 𝑁𝑆 8.103 /
Safety Factor(MaximumNormal Stress
Theory) 𝑁 𝑁 15.329 /
MinimumStatic Safety Factor 𝑁𝑆𝑡𝑎𝑡𝑖𝑐 8.103 /
Notch Sensitivity at Notch Radius 𝑞 𝑆(𝑟𝑛𝑜𝑡𝑐ℎ) 0.987 /

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Fatigue Stress-Concentration Factor
(Normal)
𝐾𝑓 1.25 /
Fatigue Stress-Concentration Factor (Shear) 𝐾𝑓𝑠 1.012 /
MaximumFatigue Normal Stress (Point C) 𝜎𝐶𝑛𝑜𝑚 5.588 MPa
MinimumFatigue Normal Stress (Point C) 𝜎𝐶𝑚𝑖𝑛 2.02 MPa
MaximumFatigue Normal Stress(Point B) 𝜎𝐵𝑛𝑜𝑚 3.478 MPa
MinimumFatigue Normal Stress (Point B) 𝜎𝐵𝑚𝑖𝑛 0.08904 MPa
MaximumFatigue Shear Stress (Point C) 𝜏 𝐶𝑛𝑜𝑚 48.44 MPa
MinimumFatigue ShearStress (Point C) 𝜏 𝐶𝑚𝑖𝑛 1.24 MPa
MaximumFatigue Shear Stress (Point B) 𝜏 𝐵𝑛𝑜𝑚 49.17 MPa
MinimumFatigue Shear Stress (Point B) 𝜏 𝐵𝑚𝑖𝑛 0.5181 MPa
Fatigue Stress-Concentration Factor for
Mean Normal Stress (Point C)
𝐾𝐶𝑓𝑚 1.25 /
Fatigue Stress-Concentration Factor for
Mean ShearStress (Point C)
𝐾𝐶𝑓𝑠𝑚 1.012 /
Fatigue Stress-Concentration Factor for
Mean Normal Stress (Point B)
𝐾 𝐵𝑓𝑚 1.25 /
Fatigue Stress-Concentration Factor for
Mean ShearStress (Point B)
𝐾 𝐵𝑓𝑠𝑚 1.012 /
Von Mises Equivalent Amplitude Stress 𝜎𝑎
′ 42.68 MPa
Von Mises Equivalent Mean Stress 𝜎 𝑚
′ 43.59 MPa
Uncorrected Endurance Limit 𝑆 𝑒
′ 562 MPa
Correction Factor for Load 𝐶𝑙𝑜𝑎𝑑 0.7 /
Correction Factor for Size 𝐶𝑠𝑖𝑧𝑒 0.814 /
Correction Factor for Surface 𝐶𝑠𝑢𝑟𝑓 0.251 /
Correction Factor for Temperature 𝐶𝑡𝑒𝑚𝑝 1 /
Correction Factor for Reliability 𝐶𝑟𝑒𝑙𝑖𝑎𝑏 0.702 /

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Fatigue Strength at 103
Cycles 𝑆 𝑚 843 MPa
Corrected Endurance Limit 𝑆 𝑒 56.31 MPa
Case 1 Fatigue Safety Factor 𝑁𝑓1 17.535 /
Case 2 Fatigue Safety Factor 𝑁𝑓2 1.268 /
Case 3 Fatigue Safety Factor 𝑁𝑓3 1.255 /
Case 4 Fatigue Safety Factor 𝑁𝑓4 1.188 /
Total Spring Constant 𝑘 𝑇 1.003*1010
N/m
Critical Frequency 𝜔 𝑛 1.279*103
rad/sec
MaximumFrequency 𝜔 𝑓 7.692 rad/sec
Critical/MaximumSpeed Ratio 𝐶𝑀 𝑅𝑎𝑡𝑖𝑜 166.308 /