082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

604 Chapter 20 Ordinary differential equations II
y ic
y 1
10
y 1
1
y 2
1
y o
y 2
1
y o
y 3
1
y 3
10
Figure20.1
An integrator.
Figure20.2
A summer.
technique is known as analogue simulation. Prior to the mass availability of powerful
digital computers this was a common approach in engineering design because it providedtheengineerwithanautomatedmethodtosolvearangeofmathematicalmodels.
Special-purposecomputers,knownasanaloguecomputers,werehistoricallyusedwhich
had the electronic circuits already incorporated, thus making iteasier tosimulate a particular
differential equation.
Three basic types of circuit are required to enable ordinary differential equations
with constant coefficients to be simulated. The first type is an integrator which has already
been discussed in Section 13.2. The usual symbol for such a circuit is shown in
Figure 20.1.
If an analogue computer is used it is common for there to be a gain of only 1 or 10
ontheinputvoltage.Againabove10isbetterachievedbylinkingtogethertwocircuits
in series. In addition there is usually a facility to allow initial conditions to be set. The
equation forthe circuitof Figure 20.1 is
v o
=−
∫ t
0
(10v 1
+ v 2
+ v 3
)dt − v ic
(20.1)
where v ic
indicatestheinitialoutputvoltage.Notethatthegainisusuallywrittenonthe
input line.
The second type of circuit is the summer. It has the symbol shown in Figure 20.2.
The equation forthe circuitof Figure 20.2 is
v o
= −(v 1
+ v 2
+10v 3
) (20.2)
Finally, a circuit is required to allow gains to be varied. This is simply a potentiometer
andisillustratedtogetherwithitssymbolinFigure20.3.Theequationforthecircuitof
Figure 20.3 issimply
v o
=kv i
0k1 (20.3)
The potentiometer only allows the gain to be varied between 0 and 1. If a variable gain
greater than 1 is required, the potentiometer must be placed in series with a circuit that
increases gain, for example a one-input summer withagain of 10.
Thesecircuitscanbecombinedtoobtainthesolutionsofdifferentialequations.This
isbestillustratedbymeansofanexample.Considerthegeneralformofasecond-order
differential equation with constant coefficients:
a d2 y
dt +bdy +cy=f(t) (20.4)
2 dt
The general approach to obtaining the circuit for a particular differential equation is to
assume a certain point in the circuit corresponds to a particular term and then arrange