Application of Multi-Body Dynamics Software to the Simulation of the Dynamics of Wind

Turbines using the Open Source Software Multi-Body Dynamics and Blender to Visualize the

Results

<Student Name>

<Name and Section # of course>

<Instructor Name>

<Date>

DECLARATION

I hereby declare that I am the sole author of this thesis.

I authorize the University of Waterloo to lend this thesis to other institutions or individuals for

the purpose of scholarly research.

<Student name>

TABLE OF CONTENTS

Abstract 3

List of Figures 6

Scope/aim of the project 8

1. Introduction 9

1.1 Multi-body System 10

1.2 Connection Elements 10

1.3 Kinematic Constraints 10

1.4 Forces 11

1.5 Cartesian Coordinates 11

1.6 Equation of Motion of Multi-Body System 11

1.7 Degree of Freedom 12

1.8 Constraint Conditions 12

2. Equations of Motion 14

2.1 Kinematics 14

2.2 Lagrange Equation of Type One 16

2.3 Lagrange Equation of Type Two 18

2.4 Slider Crank Example 19

2.5 Wheel Suspension Example 21

3. Literature Review 23

3.1 Introduction to Wind Energy and Wind Turbines 23

3.2 Types of Wind Turbines 24

3.3 Components of Wind Turbine 25

3.4 Dynamics of Wind Turbine 26

3.5 Present Wind Turbine Design Codes 29

4. Researches 32

4.1 Nonlinear Dynamics of Wind Turbine Wings 32

4.2 Tutorial on the Dynamics and Control of Wind Turbines and Wind Farms 34

4.3 Advanced Control Structure of Wind Turbines 43

4.4 Dry Friction Models 44

Bibliography 47

LIST OF FIGURES

Figure 1: Slider Crank 19

Figure 2: Five link Wheel Suspension 21

Figure 3: Installed Wind Capacity Worldwide. (2008-2010 are projections) 23

Figure 4: Progression of larger Turbines being used over the last three decades. 24

Figure 5: Vertical and Horizontal-axis configurations of Wind Turbines 24

Figure 6: Components of a Horizontal-axis Wind Turbine 26

Figure 7: Power Curves 27

Figure 8: Wind Turbine Control Block Diagram 34

Figure 9: Sonic and Propeller Anemometers 38

Figure 10: Inside of a 3-bladed CART3 Nacelle 40

Figure 11: Three Pitch Motors of CART3 41

ACKNOWLEDGEMENTS

I would like to thank my supervisor, <Instructor name> for his support, enthusiasm, guidance,

assistance and wealth of ideas. He gave me good advice as a friend and supervisor in many

moments of my life.

SCOPE/AIM OF THE PROJECT

The scope and aim of the project under study is to study the simulation of multi-body

dynamics software in the working of wind turbines and analyze and evaluate the results thus

achieved by the use of blender software. For this, we will have a deeper look to the working and

understanding of multi-body dynamics and its application and use in wind turbines. Finally, the

results we obtain will be tested and verified using blender software to model effective and

efficient wind turbines.

1. INTRODUCTION

With advancements in technology, the present day man has been able to model and

simulate real world applications before actually applying or building them in actual world. The

science of physics has also been revolutionized with this key aspect like other spheres of study

and mechanics associated with it can now be visualized before actually been applied or made in

the real world. Engineers are now able to map real machines and buildings before actually

building them. This process on one hand gives ease of access for designing purposes aiding in

further optimizing the proposed design and on the other hand evaluates the structure of the

building or design of the model verifying its proximity if applied in the real world thus reducing

costs. Moreover, simulations also helps designers and engineers in creating several prototypes

for one project proposing multiple options and choosing the one which seems most efficient.

With large and advanced mechanical and structural systems in place today, the

requirement of designing such complex systems and later on simulating them in real world

environment becomes more important than before. With advancements in technology, powerful

computational resources are available which help in the study of such complex mechanical

systems which involve large motional movements of rigid and flexible bodies such as space

satellites, vehicles, wind turbines etc. One such procedure is the use of multi-body dynamics

system which is the most powerful and efficient computational tool available today that deals

with large transnational or rotational movements. In the preceding chapters, we will highlight the

significance and importance of multi-body dynamics system with a view to analyze their

application in the working of wind turbines.

To begin with, we will show some basic concepts that would be discussed in detail later

in the paper. These concepts are important for the understanding of the basic terms and the way

the study will be progressed in the paper.

1.1 Multi-Body System

Large mechanical systems which are composed of rigid or flexible bodies interconnected

by connection elements can be termed as multi-body systems. In other words, a multi-body

system can be defined as a system which is used to model and simulate the dynamic attributes

and behavior of interconnected rigid or flexible bodies undergoing large translational and

rotational displacements.

1.2 Connection Elements

The connection elements as iterated above play an important role as they are responsible

for interaction between the bodies within the system and between the outer environment and the

system. There are primarily two types of connection elements namely material or immaterial.

Example of material connection elements is spherical joints of vehicles. Interactions between the

system and the environment are due to magnetic or gravitational fields which are characterized

as immaterial connection elements.

1.3 Kinematical Constraints

Restricting the relative movement of connected bodies is another important functionality

of the connection elements. This is known as kinematical constraints. In case of interconnected

rigid bodies with no tree structure of the system, formulating kinematical constraints becomes

very complex.

1.4 Forces

Several forces can be exerted on the interconnected bodies by or due to the connection

elements. Prescribed forces can be exerted by the connection elements. Constraint forces which

are essential in upholding kinematical constraints of the connection are another example. Newton

and Euler stated laws which dealt with the forces exerted by the connection elements on each of

the interrelated bodies. According to them, the forces thus exerted by the connection elements

were “to the linear acceleration of the center of mass and to the angular acceleration of that

body”. Moreover, bodies in a multi-body system also undergo huge rotations. Details of these

forces, rotations and their affects will be studied later in the study.

1.5 Cartesian Coordinates

In multi-body systems, every component is positioned using fixed coordinate frames

which also help in specifying the kinematic constraints. In this regard, several formalisms have

been studied and suggested for using coordinate sets such as Natural (Garcia de Jalon & Bayo,

1994), Cartesian (Nikravesh, 1988) and Relative Coordinates (Nikravesh & Gim, 1993). Each of

these three has their own advantages and disadvantages and are equally applied in the field

depending on the specific objective of the researcher. In our case, we will be using Cartesian

coordinates to describe and illustrate multi-body systems.

1.6 Equations of Motion of Multi-Body System

Euler – Lagrange equations coupled with the principle of virtual works as illustrated by

Nikravesh (1988) are used to formulate the equations of motion of multi-body systems. Lagrange

multipliers are used to add the kinematic constraints to the equilibrium equations. These

equations along with acceleration constraint equations are further formulated to produce system

accelerations. Variable time integration algorithm is then used to integrate time in system

variables. Details of the formulation of equations of motion of multi-body systems will be

discussed in the preceding chapters.

1.7 Degree of Freedom

In multi-body systems, the degree of freedom may be characterized as the number of

possible movements that the body may have. Considering a rigid body in spatial motion, there

are six possible degrees of freedom. On the other hand, in case of planar motion, degrees of

freedom are only three. It is pertinent to mention that in spatial motion, the six degrees of

freedom consist of three translational and three rotational degrees of freedom whereas in planar

motion having three degrees of freedom, two are translational and one is rotational.

Examples of above stated degrees of freedom are as follows:

Spatial Motion: An example could be an aircraft having six degrees of freedom;

movements as well as rotation along the 3 axis.

Planar Motion: An example can be that of a computer mouse. Left, right, up and

down are the translational movements whereas rotation about the vertical axis is

its rotational movement.

1.8 Constraint Conditions

In multi-body systems, constraint conditions refer to restraining or limiting the degrees of

freedom of rigid or flexible bodies. There are two types of constraints: Classical constraints and

Non-classical constraints. An algebraic equation identifying translational or rotational motions or

velocities between two bodies is basically classical constraint, for example rolling disc, whereas

non-classical constraints does not necessarily restrict the degrees of movement of the bodies. In

this case, there is a possibility that a new coordinate is introduced such that some point of the

body makes movement next to the surface of another body. An example of non-classical

constraint is a sliding joint.

2. EQUATIONS OF MOTION

In this chapter, we will discuss some of the concepts stated above in more detail and will

try to formulate equations explaining how the dynamics of a multi-body system could be

modeled.

……….

2.2 Lagrange Equations of Type One

Considering the first approach of modeling as stated above, we will be using the

Lagrange multipliers λ and variable p to throw light on the dynamics of multi-body systems. The

Lagrange Equation of Type One or equation for constrained mechanical motion can thus be

derived as following:

𝑀 𝑝 𝑝′′ = 𝑓 𝑝, 𝑝′ , 𝑡 − 𝐺𝑇 𝑝 ƛ

0 = g (p)

where

𝑀 𝑝 ∈ 𝑅𝑛𝑝 𝑥 𝑛𝑝

𝑓 𝑝, 𝑝′ , 𝑡 ∈ 𝑅𝑛𝑝

In the above stated two descriptions, 𝑀 𝑝 ∈ 𝑅𝑛𝑝 𝑥 𝑛𝑝 implies the mass matrix whereas

𝑓 𝑝, 𝑝′ , 𝑡 ∈ 𝑅𝑛𝑝 denotes the applied and internal forces vector.

………

Equation of Motion

2.3 Lagrange Equations of Type Two

Let us now consider the second approach of modeling where we will try to figure out a

minimum set of coordinates y and use it to figure out a system of differential equations. To

achieve this, we will modify the Hamilton principle equation stated above and insert the

expression of coordinate transformation p = p (y (t)) into it, or alternatively, we can use this

expression into the equation of motion of Type One. We observe that due to the property g (p(y))

≡ 0, the Lagrange multipliers and the constraints are cancelled and thus the result is the

Lagrange’s Equations of Type Two:

𝑀 𝑦 𝑦 = 𝑓 (𝑦, 𝑦′ , 𝑡)

Taking into consideration the null space matrix N, we derive the following equations for

acceleration and velocity vectors:

𝑑

𝑑𝑡 𝑝 𝑦 = 𝑁 𝑦 𝑦′

𝑑2

𝑑𝑡2 𝑝 𝑦 = 𝑁 𝑦 𝑦′′ +

𝜕 𝑁 (𝑦)

𝜕𝑦 (𝑦′ , 𝑦′)

Now inserting these equations into Lagrange Equation of Type One as iterated above reveals that

the mass matrix 𝑀 is basically the projection of M which is done on a constraint manifold:

𝑀 𝑦 = 𝑁𝑇 𝑦 𝑀 𝑝 𝑦 𝑁 𝑦 ∈ 𝑅𝑛𝑦 𝑥 𝑛𝑦

Similarly, the force vector f (p, p’, t) Є Rnp

would be represented as following:

𝑓 𝑦, 𝑦′ , 𝑡 = 𝑁𝑇 𝑦 𝑓 𝑝 𝑦 ,𝑁 𝑦 𝑦′ , 𝑡 − 𝑁𝑇 𝑦 𝑀 (𝑝 )) 𝜕 𝑁 𝑦

𝜕𝑦 (𝑦′ , 𝑦′)

Equation of Motion

…………….

2.4 Slider Crank Example

In this example, we will consider the dynamics of a slider crank. Figurative interpretation

of the subject to be discussed in the preceding paragraphs is as follows:

Figure 1: Slider Crank

As evident from the pictorial representation, the slider crank consists of primarily three

bodies which are the crank, connecting rod and the sliding block. There are total four joints

(three revolute joints and one translational joint) which perform the duty of connecting the

bodies and constrain their motion as well.

There are three coordinates that illustrate the motion of the bodies which are stated as

following:

𝑝 = (𝛼1, 𝛼2, 𝑧3)𝑇

……………

2.5 Wheel Suspension Example

This example illustrates the differential algebraic model of a five link wheel suspension.

Pictorial representation of the discussion to be done in the preceding paragraphs is as follows:

Figure 2: Five link Wheel Suspension

We have discussed in the first chapter about spatial multi-body systems. Our current

example of wheel suspension can be taken as one of a kind of spatial multi-body systems. It

comprises of seven bodies which are the wheel, wheel carrier and connection elements between

car body and the carrier, i.e. five rods. It also has number of spherical and universal joints along

with a spring damper which acts as a shock absorber between the car body and rod number five.

Overall, the five link wheel suspension is a rear suspension system of high class automobiles

providing extra comfort to the occupants of the vehicle.