Higher Engineering Mathematics

PROPERTIES OF LAPLACE TRANSFORMS 637

K

Problem 9. Verify the final value theorem for

the function (2+3e−^2 tsin 4t) cm, which repre-

sents the displacement of a particle. State its final

steady value.

Let f(t)= 2 +3e−^2 tsin 4t

L{f(t)}=L{ 2 +3e−^2 tsin 4t}

=

2

s

+ 3

(

4

(s−(−2))^2 + 42

)

=

2

s

+

12

(s+2)^2 + 16

from (ii) of Table 64.1, page 628 and (ii) of Table 65.1
on page 632.

By the final value theorem,

limit

t→∞

[f(t)]=limit

s→ 0

[sL{f(t)}]

i.e. limit
t→∞

[2+3e−^2 tsin 4t]

=limit

s→ 0

[

s

(

2

s

+

12

(s+2)^2 + 16

)]

=limit

s→ 0

[

2 +

12 s

(s+2)^2 + 16

]

i.e. 2+ 0 = 2 + 0

i.e. 2 = 2 , which verifies the theorem in this case.

The final value of the displacement is thus 2 cm.

The initial and final value theorems are used in pulse

circuit applications where the response of the circuit

for small periods of time, or the behaviour immedi-

ately after the switch is closed, are of interest. The

final value theorem is particularly useful in investi-

gating the stability of systems (such as in automatic

aircraft-landing systems) and is concerned with the

steady state response for large values of timet, i.e.

after all transient effects have died away.

Now try the following exercise.

Exercise 234 Further problems on initial

and final value theorems

1. State the initial value theorem. Verify the
   theorem for the functions (a) 3−4 sint
   (b) (t−4)^2 and state their initial values.
   [(a) 3 (b) 16]


2. Verify the initial value theorem for the voltage
   functions: (a) 4+2 cost(b)t−cos 3tand
   state their initial values. [(a) 6 (b)−1]


3. State the final value theorem and state a
   practical application where it is of use. Verify
   the theorem for the function
   4 +e−^2 t( sint+cost) representing a
   displacement and state its final value. [4]


4. Verify the final value theorem for the function
   3 t^2 e−^4 tand determine its steady state value.
   [0]