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

20.2 Analogue simulation 605
y i
y i y o
k
y o
d
a —
2 y
dt 2
1—
a
d 2 y
—
dt 2
dy
– —
1 dt 1
y
Figure20.3
Apotentiometer.
Figure20.4
Firststage in synthesizing Equation(20.4).
toconnectthatpointtothecorrectlysynthesizedvalue.Itismorestraightforwardifthe
assumed point corresponds to the highest derivative. Rearranging Equation (20.4) we
find
a d2 y
dt =f(t)−bdy −cy (20.5)
2 dt
Assuminga d2 y
dt isalreadyavailable,apotentiometercanbeusedtoobtain d2 y
2 dt 2.Integratorscanthenbeusedtoobtainthevariables
dy
dt andy.ThisisshowninFigure20.4.The
nextstageistoobtaintheexpressioncorrespondingtother.h.s.ofEquation(20.5).This
requiresasummertoaddtogethertheindividualtermsandpotentiometerstoallowindividualcoefficientstobeobtained.Aninvertorisalsorequiredtoobtainthecorrectsign
forthevariable dy
dt .Thissimplyconsistsofasummerwithoneinputandagainof1.The
final circuit is shown in Figure 20.5. The three potentiometers allow different values of
a,bandctobeobtained.Ifgainsgreaterthan1wererequiredthenextrasummerswould
be needed, or alternatively the input gains of the existing summers could be adjusted.
The input to the circuit, f (t), can be simulated by applying a signal at the appropriate
point in the circuit. The output of the circuit, y(t), corresponds to the solution of the
differential equation. It may be necessary to include initial conditions corresponding to
y(0)and dy
dt (0).
Although analogue computers remain an interesting historical development in the
simulation of control systems it is now much more common to use digital computers.
Thesystemtobesimulatedcanberepresentedeitherdirectlyintheformofmathematical
equations or by block diagrams similar to that shown in Figure 20.5. The response to
various inputs can be readily obtained.
f(t)
1
–f(t)
1
1
1
a
b dy — dt
d 2 y
—
dt 2
1
1—
a
–b dy — dt
d 2 y
—
dt 2
– dy —
1 dt 1 y
b
cy
c
Figure20.5
Thecomplete circuit to synthesize Equation(20.4).