Dynamics of Machines - Part II - IFS.pdf

⎧ ⎫
⎪⎨
˙y1(t)
⎪⎬
˙y(t) ˙y2(t)
z(t) = =
y(t) ⎪⎩
y1(t) ⎪⎭
y2(t)
→ velocity
→ velocity
→ displacement
→ displacement
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).
Homogeneous Solution and Transient Analysis – The homogeneous differential equation
is achieved when the right side of the equation is set zero (see eq.(55)), or in other words, when
no force is acting on the system.
A˙zh(t) + Bzh(t) = 0 (55)
(54)
zh(t) = ue λt , (assumption) (56)
˙zh(t) = λue λt
The assumption (56) and its derivative are introduced into the differential equation (55), leading
to an eigenvalue-eigenvector problem:
[λA + B]ue λt = 0 ⇒ [λA + B]u = 0 (57)
• Eigenvalues λi can be calculated by using eq.(58):
determinant(λA + B) = 0 ⇒ λ1 , λ2 , λ3 , λ4 (58)
• Eigenvectors ui can be calculated by using eq.(59):
λ1Au = −Bu ⇒ u1
λ2Au = −Bu ⇒ u2
λ3Au = −Bu ⇒ u3
λ4Au = −Bu ⇒ u4
• The homogeneous solution can be written as:
zh(t) = C1u1e λ1t + C2u2e λ2t + C3u3e λ3t + C4u4e λ4t
where C1, C2, C3 and C4 are constants depending on the initial displacement and velocities of
the coordinates y1 and y2, when t = 0.
33
(59)