Here, M, is combined mass of the IP02 cart, weight and additional 'mass' due to the IP02
cabling, B, is the additional damping observed by the active cart, By is the damping caused
by the linear flexible joint between the two-masses, Ky is the stiffness due to the compression
spring, x, is the position of the IP02 cart in the x-direction, x2 is the position of the LFJ cart
in the x-direction, fe is the external force applied to the IP02 cart at the motor pinion, and (
represents time derivatives of the respective variables.
iv. LFJ Cart Position
The fourth and final equation of motion can be obtained from the sum of the forces on the
LFJ cart as,
Maxz + By(x2 - x1) + Bzx2 + Ky(x2 - x) =0
(6)
where, M2 is combined mass of the LFJ cart and weights, and B2 is the additional damping
observed by the passive cart.
v. Combined Transfer Function
To combine these equations of motion via substitution, Eq. 3 can be rewritten to isolate fe
as,
JmNeg + BmNeg = Kia - fed
(7a)
Jmpeg - Bmy Ogt - Ktla = fc
(7b)
By substituting Eq. 7b into Eq. 5, and realizing that the angle of the motor pinion gear may
be written as 0. = =, Eq. 5 may be rewritten as,
Mix, + By(x1 - 12) + Ky(x1 - x2) + Bix = -Jmrs
-Bg - Bmeg+
NKtia
(8a)
(Mi + / m = ) $1 + (B, + By + Bmz 1 1 + Kyx1 - Byx2 - K,x2 = =Kia
(8b)