2.3 LTI Continuous-Time Systems 135
Analog mechanical systems
Making the analogy shown in Table 2.1 between the different variables and elements in a circuit and
in a mechanical system the differential equations representing mechanical systems are found to be
like those for RLC circuits.
Consider the translational mechanical system shown in Figure 2.7, composed of a massMto which
an external forcef(t)is being applied, and is moving at a velocityw(t). It is assumed that between the
mass and the floor there is a damping with a damping coefficientD. Just as with Kirchhoff’s voltage
law, the applied force equals the sum of the forces generated by the mass and the damper. Thus,
f(t)=M
dw(t)
dt
+Dw(t)
which is analogous to the differential equation of an RL series circuit with a voltage sourcev(t):
v(t)=L
di(t)
dt
+Ri(t)
Exactly the same as with the RL circuit, if the initial velocity and the external force are zero fort<0,
the above differential equation represents a LTI mechanical system.
2.3.4 Application of Superposition and Time Invariance
The computation of the output of an LTI system is simplified when the input can be represented as
the combination of signals for which we know their response. This is done by applying superposition
and time invariance. This property of LTI systems will be of great importance in their analysis as you
will soon learn.
Table 2.1Equivalences in
Mechanical and Electrical
Systems
Mechanical System Electrical System
force f(t) voltage v(t)
velocity w(t) current i(t)
mass M inductance L
damping D resistance R
compliance K capacitance C
FIGURE 2.7
Analog mechanical and electrical systems. Using
the equivalencesR=D,L=M,v(t)=f(t), and
i(t)=w(t), the two systems are represented by
identical differential equations.
f(t)
M
D
w(t)
i(t) R
L
v(t)
+
−
