College aantekeningen Bewegingsanalyse 1 (BWB220)

Formuleblad

Fourier analyse
2𝜋
𝑦(𝑡) = 𝑎0 + ∑∞
𝑛=1[𝑎𝑛 cos(𝑛𝜔𝑡) + 𝑏𝑛 sin (𝑛𝜔𝑡)] met 𝑛 ε ℕ
+
𝜔= = 2𝜋𝑓
𝑇
1 𝑇 2 𝑇 2 𝑇
𝑎0 = ∫ 𝑦(𝑡) 𝑑𝑡 𝑎𝑛 = ∫ 𝑦(𝑡)cos (𝑛𝜔𝑡) 𝑑𝑡 𝑏𝑛 = ∫ 𝑦(𝑡)sin (𝑛𝜔𝑡) 𝑑𝑡
𝑇 0 𝑇 0 𝑇 0

𝑦(𝑡) = 𝑐0 + ∑∞
𝑛=1[𝑐𝑛 sin(𝑛𝜔𝑡 + 𝜑𝑛 )] 𝑎0 = 𝑐0 𝑎𝑛 = 𝑐𝑛 sin(𝜑𝑛 ) 𝑏𝑛 = 𝑐𝑛 cos(𝜑𝑛 )
𝑎
𝜑𝑛 = arctan (𝑏𝑛) |𝑐𝑛 | = √𝑎𝑛2 + 𝑏𝑛2
𝑛




Differentiaalvergelijkingen
ÿ + 𝜔02 𝑦 = −𝑥̈
𝜔 2
(
𝜔0
) 𝜔 2
𝐻(𝜔) = 𝜔 2
𝜔02 𝐻(𝜔) = 1 − (𝜔 ) 𝑚𝑒𝑡 𝜔0 = √𝑘/𝑚
|1−( ) | 0
𝜔0

𝑑𝑈𝑜𝑢𝑡 1 1
𝑈𝑖𝑛 = 𝑅𝐶 + 𝑈𝑜𝑢𝑡 𝐻(𝜔) = 2
𝑚𝑒𝑡 𝜔𝑐 = 𝑅𝐶
𝑑𝑡
√1+( 𝜔 )
𝜔𝑐

𝜔
𝑑𝑈𝑜𝑢𝑡 𝑑𝑈𝑖𝑛 𝜔𝑐 1
𝑅𝐶 + 𝑈𝑜𝑢𝑡 = 𝑅𝐶 (𝜔) = 2
𝑚𝑒𝑡 𝜔𝑐 = 𝑅𝐶
𝑑𝑡 𝑑𝑡
√1+( 𝜔 )
𝜔𝑐




Goniometrische relaties
sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽
cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽


tan 𝛼 + tan 𝛽
tan(𝛼 + 𝛽) =
1 − tan 𝛼 tan 𝛽


tan 𝛼 − tan 𝛽
tan(𝛼 − 𝛽) =
1 + tan 𝛼 tan 𝛽


sin(2𝛼) = 2 sin 𝛼 cos 𝛼