Dynamics cheat sheet

8.3.3 critically damping forced vibration ξ = c
c r
= 1
The solution is
Where u h = (A + Bt) e −ωt and u p = F k
arctan definition). Hence
u (t) = u h + u p
1
2r
sin (ϖt − θ) where tan θ = (making sure to use correct
√(1−r 2 ) 2 2 1−r
+(2r) 2
u (t) = (A + Bt) e −ωt + F k
1
√(1 − r 2 ) 2 + (2r) 2 sin (ϖt − θ)
where A, B are found from initial conditions
A = u (0) + F k
B = u ′ (0) + u (0) ω + F k
1
√(1 − r 2 ) 2 + (2r) 2 sin θ
1
√(1 − r 2 ) 2 + (2r) 2 (ω sin θ − ϖ cos θ)
8.3.4 over-damped forced vibration ξ = c
c r
> 1
The solution is
where
and
u p (t) = F k
u (t) = u h + u p
u h (t) = Ae p 1t + Be p 2t
1
√(1 − r 2 ) 2 + (2ξr) 2 sin (ϖt − θ)
hence
u = Ae p 1t + Be p 2t + F k
1
√(1 − r 2 ) 2 + (2ξr) 2 sin (ϖt − θ)
where tan θ = 2ξr
1−r 2
Hence the solution is
and
p 1 = − c
√ ( c
2m + 2m
p 2 = − c
2m − √ ( c
2m
) 2 k −
m = −ωξ + ω √
n ξ 2 − 1
) 2 k −
m = −ωξ − ω √
n ξ 2 − 1
(
−ξ+ √ (
ξ
u (t) = Ae
2 −1
)ωt −ξ− √ )
ξ + Be 2 −1 ωt F + β sin (ϖt − θ)
k
u ′ (0) + u (0) ωξ + u (0) ω √ ξ 2 − 1 + F k
((ξ β + √ )
ξ 2 − 1
A =
2ω √ ξ 2 − 1
u ′ (0) + u (0) ωξ − u (0) ω √ ξ 2 − 1 + F k
((ξ β − √ )
ξ 2 − 1
B = −
2ω √ ξ 2 − 1
)
ω sin θ − ϖ cos θ
)
ω sin θ − ϖ cos θ
40