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EXAMPLE 3-7
Obtain the transfer functions X(s)/U(s) and X2$)/U(s) of the mechanical system shown in
unT;"1lnr"tions
of motion for the system shown in Figure 3-19 are
mrir: -ktxr - krl*, * x2) - b\*y - i2) + u
mzlz: -k*z - kr(*, - xt) - U(*, - *t)
Simplifying, we obtain
mri, + bit + (kt * k)x1 : b*z * k2x2 I u
m2i2 * b*2 + (k2 * k1)x2 = bit + k2xy
Taking the Laplace transforms of these two equations, assuming zero initial conditions, we obtain
lmrs2 + bs + (k, + kr)]xl(s) : (rs + 6)xr1s1+ u(s)
fm2s2 + bs + (k, + h))x21): (rs + k2)xr1)
Solving Equation (3-45) for X2(s) and substituting
into Equation Qa$ and simplifying, we get
from which we obtain
l(mrsz + bs * k,+ k2)(m2s2 .r bs * kz+ h) - (rs + 4)'?)x,1s1
: (m2s2* bs * k2 + h)U (s)
(344)
(3-45)
xr(s) mrsz + bs + kr+ k,
u(s)
(mrs2 + bs + kr+ k2)(m2s2 * bs * k2+ h)-(us + *r)2
(346)
From Equations (3-45) and (3-46) we have
XzG)
u(s)
bs+k2
(m,sz + bs * k1 + kr)(m2s2 + bs + k2 +
t