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

664 Chapter 21 The Laplace transform
Solution The transfer function for each equation is found and combined into one block diagram
(see Figure 21.11). The three blocks are simplified to a single block as shown in Figure
21.12. The overall systemtransfer function is
G(s) = Z(s)
F(s) = 1
(2s −1)(s +3)(s +1) .
F(s)
1
2s – 1
X(s)
1
s + 3
Y(s)
1
s + 1
Z(s)
F(s)
1
(2s – 1)(s + 3)(s + 1)
Z(s)
Figure21.11
Block diagram forthe system given in
Example 21.30.
Figure21.12
Simplified block diagram for
Example 21.30.
Engineeringapplication21.6
Transportlaginasystemtosupplyfueltoafurnace
Transportlagisatermusedtodescribethetimedelaythatmaybepresentincertain
engineering systems. A typical example would be a conveyor belt feeding a furnace
with coal supplied by a hopper (see Figure 21.13). The amount of fuel supplied to
the furnace can be varied by varying the opening at the base of the hopper but there
is a time delay before this changed quantity of fuel reaches the furnace. The time
delaydependsonthespeedandlengthoftheconveyor.Mathematically,thefunction
describing the variation in the quantity of fuel entering the furnace is a time-shifted
version of the function describing the variation in the quantity of fuel placed on the
conveyor (see Figure 21.14).
Let
u(t)q(t) = quantity offuelplacedon the conveyor,
whereu(t) isthe unit stepfunction,
Coal
Conveyor velocity, y
l
Conveyor
Furnace
Figure21.13
Coalisfed intothe furnacevia the conveyor
belt.