Linear generator in this study is used as a power take off mechanism. using Ocean waves as source of energy, linear generators are used as direct electrical drive system that can extract power from wave directly into electrical form without a mechanical intervention between them. This will save a lot of maintenance and repatriation for its lack of many components like gearbox and others. Figure 3.1 shows a simple diagram of this three phase permanent magnet linear generator:

Figure 3.1: Modeling of three phase permanent magnet

            The wave exciting force and the hydro mechanical force acting upon the floating structure (i.e. translator) will lead the translator to move inside the stator. This movement of magnet will lead to a change of the magnetic flux within the stator and by faradays law this will induce a current in the winding coils by a phenomena called electromagnetic induction.    This generator is consisted of highly conducting cables, structure steel, light weight floating structure, and permanent magnet embedded with iron and coal. The magnet material used will be described in section 4.3. However, how the stator core is fixed? Many studies show the stator should be moored to the seabed and this need strong ropes that can last for a long time and bear this stator under all weather and environmental changes. Linear generator will have a pretty output of voltage and current as long as the magnetic flux is changing within the stator. The relative motion between the stator and the translator becomes very crucial in a three phase generator. To achieve this, a number of springs should be added in addition to the hinges between the part of the floating structure and the new design is becoming as shown in the following figure 3.2:

Figure 3.2: Detailed model of linear generator

            These will not prevent from using ropes to fix the system in one position. In the above design when the translator moves up the stator will moves down by the help of the sticks attached to the hinge.  For more clarity, the floating structure- attached to the stator- is stacked to an arm where rotation is allowed. This is the same between the arm and the floating structure attached to the translator.

3.2 Mechanical Modeling of the arm

As mentioned above, the arm is attached to the stator and the translator. This arm is formed of two revolute joints for rotation from its ends and one prismatic joint in the middle connecting the two links of the arm for linear displacement between them. Therefore the arm is made up of 3 joints each with a single degree of freedom of motion and up of 4 links. The arm is shown below figure 3.3:

Figure 3.3: Modeling of the arm and joints

“A” represents the floating structure attached to the stator and “B” to the translator. In this section, the relative motion of “B” to “A” and the position and the orientation is determined in terms of the joint variables. This is so called kinematic study where the motion is considered without studying the causes of motion i.e. forces and torques.   

Let’s first use the Denavit-Hartenberg (DH) convention that uses four basic transformations [4]:

=

=

=

:        is the angle of rotation

:        is the distance of displacement

= distance along  from the intersection of the   and  axes to .

= distance along  from  to the intersection of the  and  axes. If joint i is prismatic then  is variable and denoted by .

= the angle from  to  measured about .

= the angle from  to  measured about . If joint i is revolute,  is variable and denoted by .

Keeping in mind the actuating of the joint and therefore of the links is covered by the forces of ocean water. And for some considerations of rules of DH, The system is considered as following

Figure 3.4: Stator-link-translator relative motion

The DH-table is given by the following:

Link

1

 This is considered for the moment identity matrix

2

             0

          +90

0

3

             0

-90

0

4

0

0

Table 3.1: DH-table

=I

=     ;     =     

    =    

Then the transformation is obtained by:

=    

In fact  is not an identity matrix, but we are considering a special situation where the floating structure attached to the stator is fixed. However this is not the case.

Figure 3.5: Representing the system with respect to a fixed reference axis

In order to solve the problem, the transformation matrix obtained should be multiplied by the following:

=

The new homogenous transformation matrix relating the translator to the base frame is given by:

= =

Through this transformation, we found the translator motion in terms of joint variables ( ,  ,  and ) in the base frame.

Now remembering the new motion of the translator relative to the world frame can be obtained from adding the new force of arm on the translator:

Figure 3.6: Modeling of translator system

Knowing that  is the force exerted by arm on the floating structure fastening to the translator.  is considered a constant determined due the length of the arm.

The new dynamical modeling of the translation generator motion is given by:

Where  is a function of joint variables so the body motion is determined by the Jacobian and therefore:

 where, q is the joint variable.

The  column of the Jacobian matrix corresponds to the  joint whether its prismatic or revolute so for example the last revolute joint nears the translator is given by:

And

It’s enough to stop there where further advanced studies can be done, and in the coming sections those form a basis under some estimations for calculations. 

3.3 Dynamical Model

            In this section there will be a studying of the relation between the applied forces and the resulting motion of the cross-section part of the system where it can be then generalized to the whole system. However, it’s important to show some drawbacks that arises from the study like low rigidity ( elasticity in the structure and the joints), potential unknown parameters(mass, inertia, dimensions…..), dynamic coupling among links…In addition there is friction, dead zones and other in actuation system that should be taken into consideration.

By following Euler-Lagrange equations, one can achieve the desired model even this is applicable to some constraints. Lagrangian of the system is the difference between the kinetic energy and the potential energy. And this is given by:

                         L = K – P

the Euler- Lagrangian equation is derived as:

Where q is the mechanical parameters and is shown on the part of link is shown in the following fig.3.7:

Figure 3.7: Dynamic Modeling

The kinetic energy of the first floating structure, assume it to be translator:

The kinetic energy of the first link arm:

 The kinetic energy of the Second link arm:

The kinetic energy of second floating structure, and then could stator:

Where  can be derived as:

The Potential Energy of translator:

The Potential Energy of first link arm attached to translator:

The Potential Energy of second link arm attached to stator:

The Potential Energy of Stator:

The Kinetic Energy for the whole system is given by:

K=

The potential energy for the whole system is given by:

P=

Applying the Lagrangain Euler equation: