Dynamics of Machines - Part II - IFS.pdf

Permanent Solution and Steady-State Analysis – The permanent solution takes into
account the right side of the differential equation (see eq.(60)), or in other words, the effect of
the force on the system. In case of harmonic excitation, one can write:
A˙zp(t) + Bzp(t) = fe jωt
(60)
zp(t) = Ae jωt , (assumption!) (61)
˙zp(t) = jωAe jωt
The assumption adopted in eq.(61) and its derivative are introduced into the differential equation
(60), leading to:
[jωA + B]Ae jωt = fe jωt ⇒ A = [jωA + B] −1 f (62)
The permanent solution of the equation of motion is given by:
zp(t) = Ae jωt ⇒ zp(t) = [jωA + B] −1 fe jωt
General Solution – The general solution of a linear differential equation is achieved by adding
the homogenous and the permanent solutions, and by sequentially defining the initial conditions
of the movement. This solution will provide information about the transient and steady-state
response of the mechanical model. Considering that the order of the mechanical model is correct
(in this case, the two degree-of-freedom system), the solution of the linear differential equation
will be useful for predicting the dynamical behavior of the physical system, if the coefficients
of the differential equation (M, D and K, or A and B) are properly chosen, either by using
theoretical or experimental information or both. The general solution of the differential equation
of motion is given by:
z(t) = C1u1e λ1t + C2u2e λ2t + C3u3e λ3t + C4u4e λ4t + Ae iωt
where z(t) gives information about the displacement and velocity of the coordinates y1 and y2.
Introducing the initial conditions of displacement and velocity
zini = { v1ini
v2ini y1ini y2ini }T
into eq.(64), when t = 0, one obtains
z(0) = zini = C1u1e λ10 + C2u2e λ20 + C3u3e λ30 + C4u4e λ40 + Ae iω0
or
⎧
⎪⎨
zini = C1u1 + C2u2 + C3u3 + C4u4 + A = [ u1 u2 u3 u4 ]
⎪⎩
C1
C2
C3
C4
(63)
(64)
(65)
(66)
⎫
⎪⎬
+ A (67)
⎪⎭
Solving the linear system by inverting the modal matrix U = [ u1 u2 u3 u4 ] one achieves the
vector c = { C1 C2 C3 C4 } T :
34