Fundamentals - Preventive Biomechanics - page 16

Fig. 3.1 Trajectory of the
net magnetization vector in
the fixed reference frame

M ? B 0 ¼ð M x e x þ M y e y Þ B 0 e z ¼ B 0 ð M y e x M x e y Þ
M ? ¼ M x e x þ M y e y ;

ð 3 : 18 Þ

M II ¼ M z e z :

Substituting ( 3.17 ) and ( 3.18 )in( 3.16 ) and taking ( 3.9 ) 2 into account, leads
after comparison of coordinates in e x ; e y and e z -direction to the following
coupled scalar-valued linear first-order system of differential equations for the
coordinates M x ; M y and M z of the net magnetization vector M

M x ¼ x L M y 1

T 2 M x

M y ¼ x L M x 1

T 2 M y

ð 3 : 19 Þ

M z ¼ 1
T 1

ð M 0 M z Þ:

The solution of both coupled differential equations ( 3.19 ) 1 and ( 3.19 ) 2 is
obtained, for instance, by eliminating one of the respective moment coordinates
and an ordinary differential equation of the form M k þð 2 = T 2 Þ M k þ
ð x L þ 1 = T 2 Þ M k ¼ 0 with (k = x, y) is obtained. Its solution using M k ð t Þ¼ Ce kt
leads to the characteristic polynomial of the form k 2 þð 2 = T 2 Þ k þ x L þ 1 = T 2 ¼ 0
with the roots k 1 ; 2 ¼ 1 = T 2 ix L (the solution may also be generated using
matrices formulations). Substitution of both integration constants by M x ð 0 Þ and
M y ð 0 Þ; leads to (solution ( 3.20 ) 3 is obtained for instance from ( 3.19 ) 3 by separation
of variables)

M x ð t Þ¼ e T 2 ½ M x ð 0 Þ cos x L t M y ð 0 Þ sin x L t
M y ð t Þ¼ e T 2 ½ M y ð 0 Þ cos x L t þ M x ð 0 Þ sin x L t
M z ð t Þ¼ M z ð 0 Þ e T 1 þ M 0 ð 1 e T 1 Þ:

ð 3 : 20 Þ

Using ( 3.13 ) together with ( 3.17 ), the net magnetization vector M is given by
(cf. also Fig. 3.1 )