Possibly Useful Equations
f¯ = mā
f (t) =
I0 ᾱ =
X
M̄0
an =
bn =
cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b)
a0 =
e±iωt = cos(ωt) ± i sin(ωt)
x(t) = aeiωn t + be−iωn t
δL
δ q̇i
+
Z
ω0
π
Z
ω0
2π
Z
2π
ω0
i = 1, . . . , n
K − ω 2 M X̄ = 0
2π
ω0
Z
p
[C(ω, ζ)]2 + [S(ω, ζ)]2
ζ≈
δh
2ωn
E = mc2
−1
t
f (τ )h(t − τ )dτ
x(t) =
A=
0
Z
f (t)dt
0
2 + 2 = 2 × 2 = 22 = 0b0010 = 0x2
√
f (t) sin(nω0 t)dt
0
x(0)
x(N tp )
x(0)
σ = ln
x(tp )
σ
ζ=√
4π 2 + σ 2
σ
ζ=
2π
p
ωp = ωn 1 − 2ζ 2
M Ẍ + C Ẋ + KX = F
i≡
f (t) cos(nω0 t)dt
0
1
σ = ln
N
ωn2 ȳ
sin(ωt)
ωn2 − ω 2
det K − ω 2 M = 0
2π
ω0
ω0
π
x(t) = c + e−ζωn t [a cos(ωd t) + b sin(ωd t)]
X
δL
δRD
−
= Qi ,
δ q̇i
δqi
bn sin(nω0 t)
1
X̃i = q
Xi
XiT M Xi
−1
1
a b
d −b
=
c d
a
ad − bc −c
x(t) = e−ζωn t [a cos(ωd t) + b sin(ωd t)]
√
−b ± b2 − 4ac
x=
2a
Z
Z
u dv = uv − v du
d
dt
∞
X
n=1
V (ω, ζ) = e−ζωtn
x(t) = a cos ωn t + b sin ωn t
x(t) =
an cos(nω0 t) +
n=0
sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b)
δoc V = ∀
∞
X
t
0
−K
−K −C
B=
−K 0
0
M
f (t − τ )h(τ )dτ
x(t) =
0
1
U T M U Ḧ + U T KU H = U T F cos ωt
Possibly Useful Equations
¯
¯
f
f
(ωn2 − ω) m
(2ζωωn ) m
cos
ωt
+
sin ωt
x(t) = − 2
(ωn − ω 2 )2 + (2ζωωn )2
(ωn2 − ω 2 )2 + (2ζωωn )2
"
#
f¯
1
2ζωωn
−1
p
=
cos (ωt − φ), where φ = tan
m (ωn2 − ω 2 )2 + (2ζωωn )2
ωn2 − ω 2
s
ωn4 + (2ζωωn )2
eiφ , where
(ωn2 − ω 2 )2 + (2ζωωn )2
2ζωωn
−1 2ζω
−1
φ = φ1 − φ2 , where φ1 = tan
and φ2 = tan
ωn
ωn2 − ω 2
g(ω) =
s
1 + (2ζΩ)2
eiφ , where
(1 − Ω2 )2 + (2ζΩ)2
2ζΩ
−1
−1
φ = tan (2ζΩ) − tan
1 − Ω2
g(Ω) =
1
p
e−iφ , where
m (ωn2 − ω 2 )2 + (2ζωωn )2
2ζωωn
−1
φ = tan
ωn2 − ω 2
g(ω) =
x(t) =
eβω 2
cos(ωt − φ),
(ωn2 − ω 2 )2 + (2ζωωn )2
2ζωωn
−1
where φ = tan
ωn2 − ω 2
p
eβΩ2
p
cos(ωt − φ), where
(1 − Ω2 )2 + (2ζΩ)2
2ζΩ
−1
φ = tan
1 − Ω2
x(t) =
2