Higher Engineering Mathematics

222 GRAPHS

Alternatively, average value

=

sum of mid-ordinates

number of mid-ordinates

Problem 8. Fig. 20.11 shows a sinusoidal out-

put voltage of a full-wave rectifier. Determine,

using the mid-ordinate rule with 6 intervals, the

mean output voltage.

Figure 20.11

One cycle of the output voltage is completed inπ
radians or 180◦. The base is divided into 6 intervals,
each of width 30◦. The mid-ordinate of each interval
will lie at 15◦,45◦,75◦, etc.
At 15◦ the height of the mid-ordinate is
10 sin 15◦= 2 .588 V.
At 45◦ the height of the mid-ordinate is
10 sin 45◦= 7 .071 V, and so on.
The results are tabulated below:

Mid-ordinate Height of mid-ordinate

15 ◦ 10 sin 15◦= 2 .588 V

45 ◦ 10 sin 45◦= 7 .071 V

75 ◦ 10 sin 75◦= 9 .659 V

105 ◦ 10 sin 105◦= 9 .659 V

135 ◦ 10 sin 135◦= 7 .071 V

165 ◦ 10 sin 165◦= 2 .588 V

sum of mid-ordinates=38.636 V

Mean or average value of output voltage

=

sum of mid-ordinates

number of mid-ordinates

=

38. 636

6

=6.439 V

(With a larger number of intervals a more accurate
answer may be obtained.) For a sine wave the actual

mean value is 0.637×maximum value, which in this

problem gives 6.37 V.

Problem 9. An indicator diagram for a steam

engine is shown in Fig. 20.12. The base line has

been divided into 6 equally spaced intervals and

the lengths of the 7 ordinates measured with the

results shown in centimetres. Determine (a) the

area of the indicator diagram using Simpson’s

rule, and (b) the mean pressure in the cylinder

given that 1 cm represents 100 kPa.

Figure 20.12

(a) The width of each interval is

12. 0

6

cm. Using

Simpson’s rule,

area=^13 (2.0)[(3. 6 + 1 .6)+4(4. 0

+ 2. 9 + 1 .7)+2(3. 5 + 2 .2)]

=^23 [5. 2 + 34. 4 + 11 .4]

=34 cm^2

(b) Mean height of ordinates

=

area of diagram

length of base

=

34

12

= 2 .83 cm

Since 1 cm represents 100 kPa, the mean pressure

in the cylinder

= 2 .83 cm×100 kPa/cm=283 kPa.

Now try the following exercise.

Exercise 92 Further problems on mean or

average values of waveforms

1. Determine the mean value of the periodic
   waveforms shown in Fig. 20.13 over a half
   cycle. [(a) 2 A (b) 50 V (c) 2.5 A]