Page 95 - Modern Control Systems

Section 2.5 The Transfer Function of Linear Systems 69

- • wmmmmmmmm
J
^3
t
LZ Z] Friction £>,
Velocity
M 2
Friction b t LrpJ

M, Vclocitv
FIGURE 2.16
Two-mass Force rtfl [ *',(/)
mechanical system. '

!',(/)
-WV-"
FIGURE 2.17 Current ,
« , < C,"
Two-node electric '(') v
circuit analog
C, = Mi, C 2 = /W 2,
L = 1/fcfti = 1/6,,
fl 2 = 1/fe 2 .

analogous to the node voltages i>j(/) and i> 2(0 of the electrical circuit. The simultane-
ous equations, assuming that the initial conditions are zero, are

M^s) + (b x + Wis) - b yV 2(s) = R(s), (2.47)
and
Ws)
M 2sV 2(s) + fc,(l£(s) - Vy{s)) + k-^- = 0. (2.48)

These equations are obtained using the force equations for the mechanical system
of Figure 2.16. Rearranging Equations (2.47) and (2.48), we obtain

(-bJVfa) + \M 2s + b } +-)V 2(s) = Q,

or, in matrix form,
'M xs + by + b 2 rmi -R(s)~\ (2.49)
k =
M 2s + b x+ - V 2(s) 0
s