Dodatek 1:
d2 x dx
2
+ 2� +�0 x = 0
Rozwiązanie równania:
dt2 dt
x = Ae-�t cos(�t + � )
dx
= - A� e-�t cos(�t + � ) - A� e-�t sin(�t + � )
dt
d2 x
2
= A(� -�2 ) e-�t cos(�t + � ) + 2A�� e-�t sin(�t + � )
dt2
2
A(� -�2) e-�t cos(�t + � )+ 2A�� e-�t sin(�t + � )+
2
+ 2�[- A� e-�t cos(�t + � ) - A� e-�t sin(�t + � )]+�0 Ae-�t cos(�t + � ) = 0
2 2 2
A(� -�2 - 2� + �0 ) e-�t cos(�t + � ) = 0
2 2 2
2 2
� = �0 - �
� = �0 - �
Dodatek 2:
d2 x dx
2
+ 2� +�0 x = ą0 sin�t
Rozwiązanie równania:
dt2 dt
dx
d2 x
2
x(t) = Asin(�t + �) = A� cos(�t +�)
= -A� sin(�t +�)
dt
d2 t
2
(�0 -�2)Asin(�t +�) + 2�� Acos(�t +�) = ą0 sin �t
sin(�t +�) = sin�t cos� + cos�t sin�
cos(�t +�) = cos�t cos� - sin�t sin�
2 2
[(�0 -�2)cos� - 2�� sin�]Asin�t +[(�0 -�2)sin� + 2�� cos�]Acos�t = ą0 sin �t
tg� 1
sin� = cos� =
2 2
1+ tg � 1+ tg �
sin� 2�� ą0
= tg� = - A =
2 2 2
cos� �0 -�2 [(�0 -�2)2 + 4� �2]1/ 2
1