Dynamics cheat sheet

8.1.3 critically damped free vibration ξ = c
c r
= 1
The solution is
u (t) = (A + Bt) e −( cr
2m
)
t
= (A + Bt) e −ωt
where A, B are found from initial conditions A = u (0),B = u ′ (0) + u (0) ω, hence
8.1.4 over-damped free vibration ξ = c
c r
> 1
The solution is
where A, B are found from initial conditions.
u (t) = ( u (0) + ( u ′ (0) + u (0) ω ) t ) e −ωt
u (t) = Ae λ 1t + Be λ 2t
A = u′ (0) − u (0) λ 2
2ω √ ξ 2 − 1
B = −u′ (0) + u (0) λ 1
2ω √ ξ 2 − 1
where λ 1 and λ 2 are the roots of the characteristic equation
8.2 Constant input F
8.2.1 Undamped case ζ = 0
λ 1 = − c
√ ( c
2m + 2m
λ 2 = − c
2m − √ ( c
2m
) 2 k −
m = −ξω + ω√ ξ 2 − 1
) 2 k −
m = −ξω − ω√ ξ 2 − 1
mu ′′ + ku = F
u ′′ + ω 2 u = F
u (t) = u h + u p
Where u h = A cos ωt + B sin ωt and u p = F k
, the solution is
Applying initial conditions gives
u (t) = A cos ωt + B sin ωt + F k
And complete solution is
A = u (0) − F k
B = u′ (0)
ω
u (t) = F (
k + u (0) − F )
cos ωt + u′ (0)
sin ωt
k
ω
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