Dynamics of Machines - Part II - IFS.pdf

one can get the initial acceleration, when t = 0, on the basis of initial conditions:
t0 = 0
˙y0
y0
⎫
⎬
⎭ ⇒ ¨y0 = −2ξωn ˙y0 − ω 2 ny0 + f
m eiωt0
The first predicted values of displacement, velocity and acceleration in time t1 = ∆t , using the
approximation given by eq.(42), are:
t1 = ∆t
˙y1 = ˙y0 + ¨y0∆t
y1 = y0 + ˙y1∆t
¨y1 = −2ξωn ˙y1 − ω 2 ny1 + f
m eiωt1
The second predicted values of displacement, velocity and acceleration in time t2 = t1 + ∆t ,
using the approximation given by eq.(42), are:
t2 = 2∆t
˙y2 = ˙y1 + ¨y1∆t
y2 = y1 + ˙y2∆t
¨y2 = −2ξωn ˙y2 − ω 2 ny2 + f
m eiωt2
The N-th predicted values of displacement, velocity and acceleration in time tN = tN−1 + ∆t ,
using the approximation given by eq.(42), are:
tN = N∆t
˙yN = ˙yN−1 + ¨yN−1∆t
yN = yN−1 + ˙yN∆t
¨yN = −2ξωn ˙yN − ω 2 nyN + f
m eiωtN (44)
Plotting the points [y1, y2, y3, ..., yN] versus [t1, t2, t3, ..., tN], one can observe the numerical
solution of the differential equation, which describes the displacement of the mass-dampingspring
system in time domain. Plotting the points [ ˙y1, ˙y2, ˙y3, ..., ˙yN] versus [t1, t2, t3, ..., tN] or
[¨y1, ¨y2, ¨y3, ..., ¨yN] versus [t1, t2, t3, ..., tN] one can also observe velocity and acceleration of the
mass-damping-spring system in time domain. The analytical and numerical solutions eq.(43) of
the second order differential equation are illustrated using a Matlab code.
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