Page 64 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

56 Fundamentals of Ocean Renewable Energy

Eq. (3.11) describes a vector, which is orientated towards the centre of the
2
earth, with a magnitude of R E Ω 2 = u /R E , which is called the centripetal
acceleration.
In order to apply the equations of motion in a rotating frame attached to the
Earth, we need to find the relationship between rotating and nonrotating frames.
In general, an object can rotate around three axes in space; therefore, to deal with
a rotational coordinate system in a more general way, we need to formulate a
relationship between inertial and rotational frames of reference in 3D. Consider
a frame of reference at the centre of the Earth (inertial), and a rotational frame of
reference at a point in the ocean (rotates around the Earth’s axis). The rotating
frame may be described by three unit vectors, ˆ i r , ˆ j r , and k r . These unit vectors
ˆ
are rotating with angular velocity Ω around the inertial frame of reference. If
we represent the position of an object in 3D using the position vector R,the
rotational speed will be RΩ, or angular velocity will be Ω × R. Based on the
theory of relative motion in dynamics, for any vector variable A, we can write

dA dA

= + Ω × A (3.12)
dt dt
iner rot
Consequently, the relationship between velocities in the inertial and rotating
frame can be written as

dR dR
= + Ω × R ⇒ u iner = u rot + Ω × R (3.13)
dt iner dt rot
Similarly, if we apply Eq. (3.12) for acceleration (as a vector), it leads to

du iner du iner

= + Ω × u iner (3.14)
dt dt
iner rot

Because, u iner = u rot + Ω × R, therefore,

du iner d(u rot + Ω × R)
= + Ω × (u rot + Ω × R) (3.15)
dt iner dt rot

du rot D Ω

= + 2 Ω × u rot + Ω × ( Ω × R) + × R
dt dt
rot
(3.16)
As can be seen, the acceleration, which is observed in a rotational frame (i.e.
du rot ), is different from the acceleration, which is observed in an inertial frame
dt
(i.e. du iner ). The difference between these accelerations includes three terms:
dt
Coriolis acceleration (−2Ω ×u rot ), the centripetal acceleration −Ω ×(Ω × r),

and, Euler acceleration, the acceleration due to change in the angular velocity