Page 466 - Mechanical Engineers' Handbook (Volume 2)

3 Block Diagrams 457

Figure 8 Two blocks in cascade. Figure 9 Summing point.

Multiplication. Multiplication is denoted by a symbol as shown in Fig. 10. Here the input
X and the output Y are related by the expression

Y GX

Takeoff Point. If a signal becomes an input to more than one element, then a takeoff point
as shown in Fig. 11 is employed.
A typical block diagram using these elements is shown in Fig. 12.
In the block diagram of Fig. 12 it is assumed that G will have no back reaction (or
2
loading) on G ; G and G usually represent two physical devices. If there are any loading
1
1
2
effects between the devices, it is necessary to combine these components into a single block.
Such a situation is given in Section 3.2.
3.1 Block Diagram Reduction
Any number of cascaded blocks representing nonloading components can be replaced by a
single block, the transfer function of which is simply the product of the individual transfer
functions. For example, the two elements in cascade shown in Fig. 12 can be replaced by a
single element G G G . Some fundamental block diagram reduction rules are shown in
2
0
1
Fig. 13.
In simplifying a block diagram a general rule is to first move takeoff points and summing
points to form internal feedback loops as shown in Fig. 13d. Then remove the internal
feedback loops and proceed in the same manner.
Example 8 Reduce the block diagram shown in Fig. 14.
First, to eliminate the loop G G H , we move H behind block G and therefore obtain
4
1
4
2
3
Fig. 15a. Eliminating the loop G G H , we obtain Fig. 15b. Then eliminating the inner loop
1
4
3
containing H /G , we obtain Fig. 15c. Finally, by reducing the loop containing H , we obtain
2
3
4
the closed-loop system transfer function as shown in Fig. 15d.

Figure 10 Multiplication Y GX. Figure 11 Takeoff point.