Page 62 - Biosystems Engineering

Biosystems Analysis and Optimization 43

where f (t) is the force exerted by the actuator on mass m . By elimi-
a 2
nating the actuator force f (t), we obtain a differential equation for the
a
relation between the positions of both masses:
2
2
dx t () dx t () dx t ()
−m 2 = m 1 + c 1 + kx t () (2.29)
2 2 1 2 1
dt dt dt
Because both masses are coupled by the actuator, the position x (t) of
2
the second mass m (e.g., the implement) with respect to the solid
2
body (e.g., the ground) is equal to the sum of the position of the first
mass x (t) and the length of the actuator x(t):
1
xt () = x t () + x t () (2.30)
2 1
In this mechanical subsystem, the length of the actuator is x(t).
Because we want to know the position x (t) of the second mass (e.g.,
2
implement) with respect to the solid body (e.g., ground) and are not
so much interested in the position x (t) of the first mass m , we can
1 1
eliminate the latter by combination of Eqs. (2.29) and (2.30):
2
dx () dx () t dx () t dx t ()
2
t
(m + m ) 2 + c 2 + kx () t = m + c + kx t () (2.31)
1 2 2 2 1 2
dt dt d dt dt
In the time domain, it is only possible to rewrite this equality of two
differential equations as an equation giving the output signal x (t) as a
2
function of the input signal x(t) by solving the differential equation
[Eq. (2.31)], which is often a tedious task. Moreover, combining the
mechanical subsystem model [Eq. (2.31)] with the hydraulic actuator
subsystem [Eq. (2.26)] to eliminate the actuator length x(t) further com-
plicates the input–output relation. In the Laplace domain, isolation and
combination of both subsystem transfer functions becomes incredibly
easy. Let us transform the hydraulic actuator subsystem model
[Eq. (2.26)] to the Laplace domain by applying the integration theorem
[Eq. (2.15)] and rewrite it as a transfer function G (s):
1
Xs() K
Gs() = = (2.32)
1
Qs() s
where X(s) and Q(s) are, respectively, the Laplace transform of the
actuator length signal x(t) and the oil-flow signal q(t).
Similarly, the mechanical subsystem model [Eq. (2.31)] can be
transformed to the Laplace domain by applying the linearity and dif-
ferentiation theorems [Eqs. (2.12) to (2.14)]. When we assume the initial
conditions to be zero, we obtain the transfer function G (s) for the
2
mechanical subsystem as follows:
+
2
Xs() ms + cs k
Gs() = 2 = 1 (2.33)
+
2 Xs() ( m + m s ) 2 + cs k
1 2