Dynamics of Machines - Part II - IFS.pdf

1 yini − A
det
λ1 vini − A
C2 = =
1 1
det
−λ1(yini − A) + (vini − iωA)
λ2 − λ1
λ1 λ2
Summarizing, below is the analytical solution of second order differential equation, which is
responsible for describing the movements of the mass-damping-spring system in time domain,
as a function of the excitation force and initial condition of displacement and velocity:
y(t) = C1e λ1t + C2e λ2t + Ae iωt
where
λ1 = −ξωn − ωn 1 − ξ2 · i
λ2 = −ξωn + ωn 1 − ξ2 · i
A =
f/m
−ω 2 + ω 2 n + i2ξωnω
C1 = λ1(yini − A) − (vini − iωA)
λ2 − λ1
C2 = −λ1(yini − A) + (vini − iωA)
λ2 − λ1
Numerical Solution According to Taylor’s expansion, an equation can be approximated by:
f(t) ⋍ f(t0) + df
dt
t=t0
(t − t0) + d2 f
dt 2
t=t0
(t − t0)... + dn f
dt n
t=t0
(40)
(t − t0) (41)
Assuming a very small time step t −t0 ≪ 1, the higher order terms of eq.(42) can be neglected.
It turns:
f(t) ⋍ f(t0) + df
dt
t=t0
(t − t0) (42)
Knowing the initial conditions of the movement when t = 0,
y(0) = y0 = yini
˙y(0) = ˙y0 = vini
and the equation of motion, which has to be solved,
¨y(t) = − 2ξωn ˙y(t) − ω 2 ny(t) + f
m eiωt
13
(43)