This document describes the dynamic analysis of wind and marine turbines. It presents the key steps of the analysis, which are:
1) Idealizing the complex turbine system and making assumptions to derive the governing equations.
2) Deriving the partial differential equation of motion and the associated boundary conditions.
3) Obtaining the natural frequencies by solving the eigenvalue problem of the derived equations.
4) Providing examples of parameter values and non-dimensional forms of the equations.
The goal is to understand the fundamental physics governing the dynamic behavior and design of wind and marine turbine structures through a simplified analytical approach.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
single degree of freedom systems forced vibrations KESHAV
SDOF, Forced vibration
includes following content
Forced vibrations of longitudinal and torsional systems,
Frequency Response to harmonic excitation,
excitation due to rotating and reciprocating unbalance,
base excitation, magnification factor,
Force and Motion transmissibility,
Quality Factor.
Half power bandwidth method,
Critical speed of shaft having single rotor of undamped systems.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
single degree of freedom systems forced vibrations KESHAV
SDOF, Forced vibration
includes following content
Forced vibrations of longitudinal and torsional systems,
Frequency Response to harmonic excitation,
excitation due to rotating and reciprocating unbalance,
base excitation, magnification factor,
Force and Motion transmissibility,
Quality Factor.
Half power bandwidth method,
Critical speed of shaft having single rotor of undamped systems.
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bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
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This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
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loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
This study deals with the active control of the dynamic response of a string with fixed ends and mass
loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
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UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
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1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
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Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
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Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
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In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
JMeter webinar - integration with InfluxDB and Grafana
Dynamics of wind & marine turbines
1. AENG M3102: Dynamic analysis of wind and
marine turbines
S. Adhikari∗
Swansea University, Swansea, U. K.
Contents
Nomenclature 2
1 Introduction 2
2 Equation of Motion and Boundary Conditions 3
2.1 Governing Partial Differential Equation . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Equation of the Natural Frequencies 9
3.1 General Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Numerical Example 13
5 Approximate Natural Frequency Based on SDOF Assumption 21
6 Summary & Conclusions 25
Suggested Reading 27
∗
Professor of Aerospace Engineering, School of Engineering, Swansea University, Singleton Park,
Swansea SA2 8PP, UK, AIAA Senior Member; Web: http://engweb.swan.ac.uk/∼adhikaris, Email:
S.Adhikari@swansea.ac.uk.
1
2. Dynamics of wind and marine turbines (AENG M3102) 2
1 Introduction
Wind and marine turbine structures are long slender columns with a ro-
tor and blade assembly placed on the top. These slender structures vibrate
due to dynamic environmental forces and its own dynamics. Analysis of the
dynamic behavior of wind and marine turbines is fundamental to the stabil-
ity, performance, operation and safety of these systems. For the design and
analysis of real-life systems, detailed finite-element models are often used. A
multi-physics model of a modern wind or marine turbine with (non-linear)
structural dynamics, fluid-structure interaction, soil-structure interaction and
rotor dynamics can easily lead to computational model consisting several mil-
lion degrees-of-freedom. While such a detailed analysis can give incredible
resolution of the dynamic behavior of the system, the understanding of basic
physical principles which govern the overall design may be somewhat difficult
to deduce from such a complex analysis. For this reason, in this note we
will focus on a simplified analysis with the aim of understanding fundamen-
tal physics which underpins the overall dynamic behavior of the system. Our
approach involves the following key steps:
Idealisation of the complex system and related assumptions
Derivation of the equation of motion
Derivation of the boundary conditions
Analytical solution of the eigenvalue problem to obtain natural frequen-
cies of the system
S.Adhikari@swansea.ac.uk
3. Dynamics of wind and marine turbines (AENG M3102) 3
2 Equation of Motion and Boundary Conditions
2.1 Governing Partial Differential Equation
M,J
w(x,t)
x
EI(x),m(x)
F(w)exp [i w t]
P
K
Kl
r
Figure 1: Idealisation of a wind turbine (the first picture is taken from http://www.segen.co.uk)
using Euler Bernoulli beam with a top mass. Flexible springs are assumed to model the soil-structure
interaction. The weight of the rotor-hub and blades are assumed to be P. The forcing due to the
rotation of the blades is assumed to be harmonic in nature with frequency ϖ.
S.Adhikari@swansea.ac.uk
4. Dynamics of wind and marine turbines (AENG M3102) 4
The Equation
The equation of motion of the beam is given by (see the book by G´eradin
and Rixen [6] for the derivation of this equation):
∂2
∂x2
(
EI(x)
∂2
w(x, t)
∂x2
)
+
∂
∂x
(
P(x)
∂w(x, t)
∂x
)
−
∂
∂x
(
mr2
(x)
∂ ¨w(x, t)
∂x
)
+ m ¨w(x, t) = f(x, t). (1)
Here w(x, t) is the transverse deflection of the beam, t is time, ˙(•) denotes
derivative with respect to time and f(x, t) is the applied time depended load
on the beam. The height of the structure is considered to be L.
The Forcing
The forcing due to the rotation of the blades is assumed to be harmonic
in nature with frequency ϖ. This implies that the system is subjected to a
forcing
F(ϖ) exp[iϖt] (2)
at x = L. We can assume that there is a slight imbalance between the rotor
and the column. Such an imbalance in unavoidable due to construction defects
of such complex systems. Assuming the amount of imbalance is ϵ, we have
F(ϖ) = (Mϖ2
ϵ) due to the centrifugal force. Therefore, the dynamic loading
on the system at x = L can be idealized as
f(x, t) = (Mϖ2
ϵ) exp[iϖt]δ(x − L) (3)
where δ(•) is the Dirac delta function. This equation shows that the magnitude
of the force acting on the system is proportional to the square of the blade
passing frequency.
S.Adhikari@swansea.ac.uk
5. Dynamics of wind and marine turbines (AENG M3102) 5
The Assumptions
The inertial and the elastic properties of the structure are constant along
the height of the structure.
The effect of soil stiffness is elastic and linear and can be captured by a
translational and a rotational spring at the point of contact. In effect,
the soil-structure interaction can be completely captured by the boundary
condition at the bottom of the structure.
The end-mass is rigidly attached to the structure.
The axial force in the structure is constant and remains axial during
vibration.
Deflections due to shear force are negligible and a plain section in the
structure remains plane during the bending vibration (standard assump-
tions in the Euler-Bernoulli beam theory).
The flexible dynamics of the rotor and the blades above the top point is
assumed to be uncoupled with the dynamics of the column.
Forcing to the system is harmonic (e.g., due to any minor imbalance in
the rotor-beam system).
None of the properties are changing with time. In other words, the system
is time invariant.
S.Adhikari@swansea.ac.uk
6. Dynamics of wind and marine turbines (AENG M3102) 6
2.2 Boundary Conditions
Noting that the properties are not changing with x, equation (1) can be
simplified as
EI
∂4
w(x, t)
∂x4
+P
∂2
w(x, t)
∂x2
−mr2 ∂2
¨w(x, t)
∂x2
+m ¨w(x, t) = (Mϖ2
ϵ) exp[iϖt]δ(x−L).
(4)
The four boundary conditions associated with this equation are
Bending moment at x = 0:
EI
∂2
w(x, t)
∂x2
− kr
∂w(x, t)
∂x
= 0
x=0
or EIw′′
(0, t)−krw′
(0, t) = 0. (5)
Shear force at x = 0:
EI
∂3
w(x, t)
∂x3
+ P
∂w(x, t)
∂x
+ ktw(x, t) − mr2 ∂ ¨w(x, t)
∂x
= 0
x=0
or EIw′′′
(0, t) + Pw′
(0, t) + ktw(0, t) − mr2 ∂ ¨w(0, t)
∂x
= 0.
(6)
Bending moment at x = L:
EI
∂2
w(x, t)
∂x2
+ J
∂ ¨w(x, t)
∂x
= 0
x=L
or EIw′′
(L, t) + J
∂ ¨w(L, t)
∂x
= 0.
(7)
S.Adhikari@swansea.ac.uk
7. Dynamics of wind and marine turbines (AENG M3102) 7
Shear force at x = L:
EI
∂3
w(x, t)
∂x3
+ P
∂w(x, t)
∂x
− M ¨w(x, t) − mr2 ∂ ¨w(x, t)
∂x
= 0
x=L
or EIw′′′
(L, t) + Pw′
(L, t) − M ¨w(L, t) − mr2 ∂ ¨w(L, t)
∂x
= 0.
(8)
Assuming harmonic solution (the separation of variable) we have
w(x, t) = W(ξ)exp {iωt} , ξ = x/L. (9)
Substituting this in the equation of motion and the boundary conditions, Eqs.
(4) – (8), results
EI
L4
∂4
W(ξ)
∂ξ4
+
P
L2
∂2
W(ξ)
∂ξ2
(10)
− mω2
W(ξ) +
mr2
ω2
L2
∂2
W(ξ)
∂ξ2
= (Mϖ2
ϵ) exp[iϖt]δ(ξL − L)
EI
L2
W′′
(0) −
kr
L
W′
(0) = 0 (11)
EI
L3
W′′′
(0) +
P
L
W′
(0) + ktW(0) +
mr2
ω2
L
W′
(0) = 0 (12)
EI
L2
W′′
(1) −
ω2
J
L
W′
(1) = 0 (13)
EI
L3
W′′′
(1) +
P
L
W′
(1) + ω2
M W(1) +
mr2
ω2
L
W′
(1) = 0. (14)
S.Adhikari@swansea.ac.uk
8. Dynamics of wind and marine turbines (AENG M3102) 8
It is convenient to express these equations in terms of non-dimensional param-
eters by elementary rearrangements as
∂4
W(ξ)
∂ξ4
+ ν
∂2
W(ξ)
∂ξ2
− Ω2
W(ξ) = pL exp[iϖt]δ(ξL − L) (15)
W′′
(0) − ηrW′
(0) = 0 (16)
W′′′
(0) + νW′
(0) + ηtW(0) = 0 (17)
W′′
(1) − βΩ2
W′
(1) = 0 (18)
W′′′
(1) + νW′
(1) + αΩ2
W(1) = 0 (19)
where
ν = ν + µ2
Ω2
(20)
ν =
PL2
EI
(nondimensional axial force ≈ 0.01 - 0.06) (21)
ηr =
krL
EI
(nondimensional rotational end stiffness ≈ 30 - 200) (22)
ηt =
ktL3
EI
(nondimensional translational end stiffness ≈ 5,000 - 20,000)
(23)
Ω2
= ω2 mL4
EI
(nondimensional frequency parameter) (24)
α =
M
mL
(mass ratio ≈ 0.75 - 2.0) (25)
β =
J
mL3
(nondimensional rotary inertia) (26)
µ =
r
L
(nondimensional radius of gyration 0.1) (27)
pL = ϵ
ML4
EI
ϖ2
(normalized forcing amplitude at the top end) (28)
f0 =
√
EI
mL4
(natural frequency scaling parameter ≈ 1rad/sec - 4rad/sec).
(29)
S.Adhikari@swansea.ac.uk
9. Dynamics of wind and marine turbines (AENG M3102) 9
The natural frequencies of the system can be obtained as
ωj = Ωjf0; j = 1, 2, 3, · · · (30)
3 Equation of the Natural Frequencies
3.1 General Derivation
Natural frequencies of the system can be obtained from the ‘free vibration
problem’ by considering no force on the system. Therefore, we consider pL = 0
in the subsequent analysis. Assuming a solution of the form
W(ξ) = exp {λξ} (31)
and substituting in the equation of motion (15) results
λ4
+ νλ2
− Ω2
= 0. (32)
This equation is often know as the dispersion relationship. This is the equation
governing the natural frequencies of the beam. Solving this equation for λ2
we
have
λ2
= −
ν
2
±
√(
ν
2
)2
+ Ω2
= −
√(
ν
2
)2
+ Ω2 +
ν
2
,
√(
ν
2
)2
+ Ω2 −
ν
2
.
(33)
Because ν2
and Ω2
are always positive quantities, both roots are real with one
negative and one positive root. Therefore, the four roots can be expressed as
λ = ±iλ1, ±λ2 (34)
S.Adhikari@swansea.ac.uk
10. Dynamics of wind and marine turbines (AENG M3102) 10
where
λ1 =
√(
ν
2
)2
+ Ω2 +
ν
2
1/2
(35)
and λ2 =
√(
ν
2
)2
+ Ω2 −
ν
2
1/2
. (36)
From Eqs. (35) and (36) also note that
λ2
1 − λ2
2 = ν. (37)
In view of the roots in equation (34) the solution W(ξ) can be expressed as
W(ξ) = a1 sin λ1ξ + a2 cos λ1ξ + a3 sinh λ2ξ + a4 cosh λ2ξ
or W(ξ) = sT
(ξ)a
(38)
where the vectors
s(ξ) = {sin λ1ξ, cos λ1ξ, sinh λ2ξ, cosh λ2ξ}T
(39)
and a = {a1, a2, a3, a4}T
. (40)
Applying the boundary conditions in Eqs. (16) – (19) on the expression
of W(ξ) in (38) we have
Ra = 0 (41)
S.Adhikari@swansea.ac.uk
11. Dynamics of wind and marine turbines (AENG M3102) 11
where the matrix
R =
s′′
1(0) − ηrs′
1(0) s′′
2(0) − ηrs′
2(0)
s′′′
1 (0) + νs′
1(0) + ηts1(0) s′′′
2 (0) + νs′
2(0) + ηts2(0)
s′′
1(1) − βΩ2
s′
1(1) s′′
2(1) − βΩ2
s′
2(1)
s′′′
1 (1) + νs′
1(1) + αΩ2
s1(1) s′′′
2 (1) + νs′
2(1) + αΩ2
s2(1)
s′′
3(0) − ηrs′
3(0) s′′
4(0) − ηrs′
4(0)
s′′′
3 (0) + νs′
3(0) + ηts3(0) s′′′
4 (0) + νs′
4(0) + ηts4(0)
s′′
3(1) − βΩ2
s′
3(1) s′′
4(1) − βΩ2
s′
4(1)
s′′′
3 (1) + νs′
3(1) + αΩ2
s3(1) s′′′
3 (1) + νs′
3(1) + αΩ2
s3(1)
.
(42)
Substituting functions sj(ξ), j = 1, · · · , 4 from equation (39) and simplifying
we obtain
R =
−λ1ηr −λ2
1
λ3
1 + νλ1 ηt
− sin (λ1) λ1
2
− Ω2β cos (λ1) λ1 − cos (λ1) λ1
2
+ Ω2β sin (λ1) λ1
− cos (λ1) λ1
3
+ ν cos (λ1) λ1 + Ω2α sin (λ1) sin (λ1) λ1
3
− ν sin (λ1) λ1 + Ω2α cos (λ1)
−λ2ηr λ2
2
λ3
2 + νλ2 ηt
sinh (λ2) λ2
2
− Ω2β cosh (λ2) λ2 cosh (λ2) λ2
2
− Ω2β sinh (λ2) λ2
cosh (λ2) λ2
3
+ ν cosh (λ2) λ2 + Ω2α sinh (λ2) sinh (λ2) λ2
3
+ ν sinh (λ2) λ2 + Ω2α cosh (λ2)
.
(43)
The constant vector in equation (41) cannot be zero. Therefore, the equation
governing the natural frequencies is given by
|R| = 0. (44)
This, upon simplification (a Maple code developed for this purpose) reduces
S.Adhikari@swansea.ac.uk
13. Dynamics of wind and marine turbines (AENG M3102) 13
cally.
4 Numerical Example
Turbine Structure Properties Numerical values
Length (L) 81 m
Average diameter (D) 3.5m
Thickness (th) 0.075 mm
Mass density (ρ) 7800 kg/m3
Young’s modulus (E) 2.1 × 1011
Pa
Mass density (ρl) 7800 kg/m3
Rotational speed (ϖ) 22 r.p.m = 0.37 Hz
Top mass (M) 130,000 kg
Rated power 3 MW
Table 1: Material and geometric properties of the turbine structure [7]
S.Adhikari@swansea.ac.uk
14. Dynamics of wind and marine turbines (AENG M3102) 14
The non-dimensional mass ratio can be obtained as
α =
M
mL
=
P
gmL
=
PL2
EI
(
EI
gmL3
)
= ν
(
EI
mL4
)
L/g = νf2
0 L/g (47)
We consider the rotary inertia of the blade assembly J = 0. This is not
a very bad assumption because the ‘point of contact’ is very close to the
center of gravity of the rotor-blade assembly.
The moment of inertia of the circular cross section can be obtained as
I =
π
64
D4
−
π
64
(D − th)4
≈
1
16
πD3
th = 0.6314m4
(48)
The mass density per unit length of the system can be obtained as
m = ρA ≈ ρπDth/2 = 3.1817 × 103
kg/m (49)
Using these, the mass ratio α = 0.2495 and the nondimensional axial
force ν = 0.0652. We also obtain the natural frequency scaling parameter
can be obtained as
f0 =
EI
mL4
= 0.9682 s−1
. (50)
The radius of gyration of the wind turbine is given by
r =
√
I
A
=
1
4
√
D2 + (D − th)2 ≈
D
2
√
2
= 1.2374m (51)
Therefore, the nondimensional radius of gyration µ = r/L = 0.0151.
From equation (20) we therefore have
ν = ν + 2.2844 × 10−4
Ω2
≈ ν (52)
We use ηr and ηt as variable parameters and try to understand how they affect
the overall behavior of the system.
S.Adhikari@swansea.ac.uk
15. Dynamics of wind and marine turbines (AENG M3102) 15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL
2
/EI
ω
1
=Ω
1
f
0
(Hz)
real data
η
r
=1.00
η
r
=2.00
η
r
=5.00
η
r
=10.00
η
r
=100.00
η
r
=500.00
(a) ηt = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL
2
/EIω
1
=Ω
1
f
0
(Hz)
real data
η
r
=1.00
η
r
=2.00
η
r
=5.00
η
r
=10.00
η
r
=100.00
η
r
=500.00
(b) ηt = 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
real data
ηr
=1.00
ηr
=2.00
η
r
=5.00
ηr
=10.00
η
r
=100.00
η
r
=500.00
(c) ηt = 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
real data
ηr
=1.00
ηr
=2.00
η
r
=5.00
ηr
=10.00
η
r
=100.00
η
r
=500.00
(d) ηt = 100
Figure 2: The variation of the first natural frequency of the wind turbine with respect to the
nondimensional axial load ν for different values of nondimensional rotational soil stiffness ηr. Four
fixed values of the nondimensional translational stiffness ηt are considered in the four subplots. The
data from the example (ϖ = 0.37Hz and ν = 0.0315) is shown by a ’*’ in the diagram.
S.Adhikari@swansea.ac.uk
16. Dynamics of wind and marine turbines (AENG M3102) 16
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL
2
/EI
ω
1
=Ω
1
f
0
(Hz)
real data
η
t
=1.00
ηt
=2.00
η
t
=5.00
η
t
=10.00
η
t
=100.00
η
t
=500.00
(a) ηr = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL
2
/EIω
1
=Ω
1
f
0
(Hz)
real data
η
t
=1.00
ηt
=2.00
η
t
=5.00
η
t
=10.00
η
t
=100.00
η
t
=500.00
(b) ηr = 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
real data
η
t
=1.00
ηt
=2.00
η
t
=5.00
η
t
=10.00
η
t
=100.00
η
t
=500.00
(c) ηr = 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
real data
η
t
=1.00
ηt
=2.00
η
t
=5.00
η
t
=10.00
η
t
=100.00
η
t
=500.00
(d) ηr = 100
Figure 3: The variation of the first natural frequency of the wind turbine with respect to the
nondimensional axial load ν for different values of nondimensional translational soil stiffness ηt.
Four fixed values of the nondimensional rotational stiffness ηr are considered in the four subplots.
The data from the example (ϖ = 0.37Hz and ν = 0.0315) is shown by a ’*’ in the diagram.
S.Adhikari@swansea.ac.uk
17. Dynamics of wind and marine turbines (AENG M3102) 17
0 20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional rotational stiffness: η
r
= k
r
L/EI
ω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(a) ηt = 1
0 20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional rotational stiffness: η
r
= k
r
L/EIω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(b) ηt = 5
0 20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional rotational stiffness: ηr
= kr
L/EI
ω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(c) ηt = 10
0 20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional rotational stiffness: ηr
= kr
L/EI
ω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(d) ηt = 100
Figure 4: The variation of the first natural frequency of the wind turbine with respect to the
nondimensional rotational soil stiffness ηr for different values of nondimensional axial load ν. Four
fixed values of the nondimensional translational stiffness ηt are considered in the four subplots. The
bade passing frequency ϖ = 0.37Hz is shown by a dashed line in the diagram.
S.Adhikari@swansea.ac.uk
18. Dynamics of wind and marine turbines (AENG M3102) 18
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: ηt
= kt
L
3
/EI
ω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(a) ηr = 1
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: ηt
= kt
L
3
/EIω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(b) ηr = 5
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: η
t
= k
t
L3
/EI
ω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(c) ηr = 10
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: η
t
= k
t
L3
/EI
ω
1
=Ω
1
f
0
(Hz)
blade passing frequency: 0.37Hz
ν=0.001
ν=0.03
ν=0.10
ν=0.25
ν=0.50
ν=1.00
(d) ηr = 100
Figure 5: The variation of the first natural frequency of the wind turbine with respect to the
nondimensional translational stiffness ηt for different values of nondimensional axial load ν. Four
fixed values of the nondimensional rotational soil stiffness ηr are considered in the four subplots. The
bade passing frequency ϖ = 0.37Hz is shown by a dashed line in the diagram.
S.Adhikari@swansea.ac.uk
19. Dynamics of wind and marine turbines (AENG M3102) 19
0 0.2 0.4 0.6 0.8 10 50 100 150 200
0
0.1
0.2
0.3
0.4
0.5
0.6
ηr
= kr
L/EI
ν = PL
2
/EI
ω
1
=Ω
1
f
0
(Hz)
(a) ηt = 1
0 0.2 0.4 0.6 0.8 10 50 100 150 200
0
0.1
0.2
0.3
0.4
0.5
0.6
ηr
= kr
L/EI
ν = PL
2
/EIω
1
=Ω
1
f
0
(Hz)
(b) ηt = 5
0 0.2 0.4 0.6 0.8 10 50 100 150 200
0
0.1
0.2
0.3
0.4
0.5
0.6
ηr
= kr
L/EI
ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
(c) ηt = 10
0 0.2 0.4 0.6 0.8 10 50 100 150 200
0
0.1
0.2
0.3
0.4
0.5
0.6
ηr
= kr
L/EI
ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
(d) ηt = 100
Figure 6: The variation of the first natural frequency of the wind turbine with respect to the
nondimensional axial load ν and nondimensional rotational soil stiffness ηr. Four fixed values of the
nondimensional translational stiffness ηt are considered in the four subplots.
S.Adhikari@swansea.ac.uk
20. Dynamics of wind and marine turbines (AENG M3102) 20
0
50
100
150
200
0
50
100
150
200
0
0.1
0.2
0.3
0.4
0.5
0.6
η
t
= k
t
L3
/EI
η
r
= k
r
L/EI
ω
1
=Ω
1
f
0
(Hz)
(a) ν = 0.001
0
50
100
150
200
0
50
100
150
200
0
0.1
0.2
0.3
0.4
0.5
0.6
η
t
= k
t
L3
/EI
η
r
= k
r
L/EI
ω
1
=Ω
1
f
0
(Hz)
(b) ν = 0.1
0
50
100
150
200
0
50
100
150
200
0
0.1
0.2
0.3
0.4
0.5
0.6
η
t
= k
t
L3
/EI
η
r
= k
r
L/EI
ω
1
=Ω
1
f
0
(Hz)
(c) ν = 0.03
0
50
100
150
200
0
50
100
150
200
0
0.1
0.2
0.3
0.4
0.5
0.6
η
t
= k
t
L3
/EI
η
r
= k
r
L/EI
ω
1
=Ω
1
f
0
(Hz)
(d) ν = 0.5
Figure 7: The variation of the first natural frequency of the wind turbine with respect to the
nondimensional translational stiffness ηt and nondimensional rotational soil stiffness ηr. Four fixed
values of the nondimensional axial load ν are considered in the four subplots.
S.Adhikari@swansea.ac.uk
21. Dynamics of wind and marine turbines (AENG M3102) 21
5 Approximate Natural Frequency Based on SDOF
Assumption
In the first mode, we can replace the distributed system by a single-degree-
of-freedom (SDOF) system with equivalent stiffness ke and equivalent mass Me:
The first natural frequency is given by
k
M
e
e
Figure 8: Equivalent single-degree-of-freedom system for the first bending mode of the
turbine structure.
ω2
1 =
ke
Me
(53)
Following Blevins [Table 8-8, case 1, page 158, 5] for a perfectly cantilever
column one has
Me = M + 0.24Mb = (α + 0.24)mL (54)
For our case the column is standing on elastic springs and also have an axial
force. Therefore the coefficient 0.24 needs to be modified to take these effects
into account. We suppose that the equivalent mass can be represented by
Me = (α + γm)mL (55)
where γm is the mass correction factor.
It is useful to express ke normalized by the stiffness term, kCL = EI/L3
.
S.Adhikari@swansea.ac.uk
22. Dynamics of wind and marine turbines (AENG M3102) 22
Therefore, the first natural frequency can be expressed as
ω2
1 ≈
ke
Me
=
ke
kCL
EI/L3
(α + γm)mL
=
EI
mL4
γk
(α + γm)
(56)
or ω1 ≈f0
√
γk
α + γm
(57)
where the stiffness correction factor γk is defined as
γk =
ke
kCL
. (58)
We only need to obtain γk and γm in order to apply the expression of the first
natural frequency in equation (56).
These correction factors can be obtained (using the procedure developed
in reference [4]) as:
γk =
λ3
ηt (ηr cos (λ) − λ sin (λ))
ηrηt (sin (λ) − λ cos (λ)) + λ2 (ηt sin (λ) + ηr cos (λ) − λ2 sin (λ))
and γm =
3
140
11 ηr
2
ηt
2
+ 77 ηt
2
ηr + 105 ηr
2
ηt + 140 ηt
2
+ 420 ηrηt + 420 ηr
2
9 ηr
2 + 6 ηr
2ηt + 18 ηrηt + ηr
2ηt
2 + 6 ηt
2ηr + 9 ηt
2
(59)
where
λ =
√
ν (60)
Substituting these expressions in the approximate formula (57) gives the com-
plete parametric variation of the first-natural frequency in terms of ν, α, ηr
and ηt.
S.Adhikari@swansea.ac.uk
23. Dynamics of wind and marine turbines (AENG M3102) 23
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL
2
/EI
ω
1
=Ω
1
f
0
(Hz)
exact: η
t
=1.00
approx: ηt
=1.00
exact: η
t
=10.00
approx: η
t
=10.00
exact η
t
=500.00
approx: η
t
=500.00
(a) ηr = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL
2
/EIω
1
=Ω
1
f
0
(Hz)
exact: η
t
=1.00
approx: ηt
=1.00
exact: η
t
=10.00
approx: η
t
=10.00
exact η
t
=500.00
approx: η
t
=500.00
(b) ηr = 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
exact: η
t
=1.00
approx: ηt
=1.00
exact: η
t
=10.00
approx: η
t
=10.00
exact η
t
=500.00
approx: η
t
=500.00
(c) ηr = 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional axial force: ν = PL2
/EI
ω
1
=Ω
1
f
0
(Hz)
exact: η
t
=1.00
approx: ηt
=1.00
exact: η
t
=10.00
approx: η
t
=10.00
exact η
t
=500.00
approx: η
t
=500.00
(d) ηr = 100
Figure 9: Approximation of the first natural frequency of the wind turbine with respect to the
nondimensional axial load ν for different values of nondimensional translational soil stiffness ηt. Four
fixed values of the nondimensional rotational stiffness ηr are considered in the four subplots.
S.Adhikari@swansea.ac.uk
24. Dynamics of wind and marine turbines (AENG M3102) 24
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: ηt
= kt
L
3
/EI
ω
1
=Ω
1
f
0
(Hz)
exact: ν=0.00
approx: ν=0.00
exact: ν=0.25
approx: ν=0.25
exact ν=1.00
approx: ν=1.00
(a) ηr = 1
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: ηt
= kt
L
3
/EIω
1
=Ω
1
f
0
(Hz)
exact: ν=0.00
approx: ν=0.00
exact: ν=0.25
approx: ν=0.25
exact ν=1.00
approx: ν=1.00
(b) ηr = 5
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: η
t
= k
t
L3
/EI
ω
1
=Ω
1
f
0
(Hz)
exact: ν=0.00
approx: ν=0.00
exact: ν=0.25
approx: ν=0.25
exact ν=1.00
approx: ν=1.00
(c) ηr = 10
20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Nondimensional translational stiffness: η
t
= k
t
L3
/EI
ω
1
=Ω
1
f
0
(Hz)
exact: ν=0.00
approx: ν=0.00
exact: ν=0.25
approx: ν=0.25
exact ν=1.00
approx: ν=1.00
(d) ηr = 100
Figure 10: Approximation of the first natural frequency of the wind turbine with respect to the
nondimensional translational stiffness ηt for different values of nondimensional axial load ν. Four
fixed values of the nondimensional rotational soil stiffness ηr are considered in the four subplots.
Both these plots show that the results from this simple closed-form expres-
sion is in excellent agreement with the exact results. Therefore, the formula
shown in equation (57) should be used for all practical purposes.
S.Adhikari@swansea.ac.uk
25. Dynamics of wind and marine turbines (AENG M3102) 25
6 Summary & Conclusions
Dynamics of flexible (marine and wind) turbine structures on elastic end
supports has been investigated using the Euler Bernoulli beam theory
with axial load, elastic support stiffness and top mass with rotary inertia.
The physical assumptions behind the simplified model have been ex-
plained in details.
The non-dimensional parameters necessary to understand the dynamic
behavior are: nondimensional axial force (ν), nondimensional rotational
soil stiffness, (ηr), nondimensional translational soil stiffness, (ηt), mass
ratio between the building and the turbine (α), nondimensional radius of
gyration of the turbine (µ).
The characteristic equation governing the natural frequency of the system
is obtained by solving the associated eigenvalue problem.
One of the key conclusion is that the first natural frequency of the turbine
structure will decrease with the decrease in the stiffness properties of the
(soil) support and increase in the axial load in the column. This means
that one needs to check the condition of the underlying soil and weight
of the rotor-blade assembly. The first natural frequency of the system
should be well separated from the the blade passing frequency.
Based on an equivalent single-degree-of-freedom system assumption, a
simple approximate expression of the first natural frequency is given.
S.Adhikari@swansea.ac.uk
26. Dynamics of wind and marine turbines (AENG M3102) 26
Numerical verifications confirm that the results from this simple closed-
form expression is in excellent agreement from the results obtained via
numerical solution of the complex transcendental frequency equation over
a wide range of parameter values.
Using this expression, designers could estimate the first natural frequency
for various parameter values and design the turbine structure such that
the resulting natural frequency does not come close to the blade passing
frequency.
S.Adhikari@swansea.ac.uk
27. Dynamics of wind and marine turbines (AENG M3102) 27
Suggested Reading
[1]S. Adhikari and S. Bhattacharya, Dynamic analysis of wind turbine towers
on flexible foundations, Shock and Vibraion (2011), in press.
[2]S. Adhikari and S. Bhattacharya, Vibrations of wind-turbines considering
soil-structure interaction, Wind and Structures, An International Journal
(2011), in press.
[3]S. Bhattacharya and S. Adhikari, Experimental validation of soil-structure
interaction of offshore wind turbines, Soil Dynamics and Earthquake Engi-
neering (2011), in press.
[4]S. Bhattacharya, S. Adhikari and N. A. Alexander, A simplified method
for unified buckling and dynamic analysis of pile-supported structures in
seismically liquefiable soils, Soil Dynamics and Earthquake Engineering 29
(2009), 1220–1235.
[5]R. D. Blevins, Formulas for Natural Frequency and Mode Shape, Krieger
Publishing Company, Malabar, FL, USA, 1984.
[6]M. G´eradin and D. Rixen, Mechanical Vibrations, John Wiely & Sons, New
York, NY, second edition, 1997, translation of: Th´eorie des Vibrations.
[7]D.-P. Tempel and D.-P. Molenaar, Wind turbine structural dynamics - A
review of the principles for modern power generation, onshore and offshore,
Wind Engineering 26 (2002), 211–220.
S.Adhikari@swansea.ac.uk