Dynamics of Machines - Part II - IFS.pdf

The analytical solution can be divided into three steps: (I) homogeneous solution (transient analysis);
(II) permanent solution (steady-state analysis) and (III) general solution (homogeneous
+ permanent), as mentioned in section 1.7.3. Introducing the initial conditions of displacement
and velocity
zini = { v1ini
one gets
v2ini v3ini y1ini y2ini y3ini }T
z(t) = C1u1e λ1t + C2u2e λ2t + C3u3e λ3t + C4u4e λ4t + C5u5e λ5t + C6u6e λ6t + Ae iωt
⎧
⎪⎨
⎪⎩
C1
C2
C3
C4
C5
C6
where
λ1 = −ξ1ωn1 − ωn1 1 − ξ2 1 · i
and u1
λ2 = −ξ1ωn1 + ωn1 1 − ξ2 1 · i
and u2
λ3 = −ξ2ωn2 − ωn2 1 − ξ2 2 · i
and u3
λ4 = −ξ2ωn2 + ωn2 1 − ξ2 2 · i
and u4
λ5 = −ξ3ωn3 − ωn3 1 − ξ2 3 · i
and u5
λ6 = −ξ3ωn3 + ωn3 1 − ξ2 3 · i and u6
⎫
A = [jωA + B] −1 f
⎪⎬
= [ u1 u2 u3 u4 u5 u6 ] −1 { zini − A}
⎪⎭
1.8.3 Programming in Matlab – Theoretical Parameter Studies and Experimental
Validation
52
(83)
(82)