Several algebraic solutions in BlockSim were used in the prior examples in this chapter of the on-line reference. BlockSim constructs and displays these equations in different ways, depending on the options chosen. As an example, consider the complex system shown in Figure 4.29.
Figure 4.29: Complex RBD.
When BlockSim constructs the equation internally, it does so in what we will call a "symbolic mode." In this mode, portions of the system are segmented. For this example, the symbolic (internal) solution is shown in Figure 4.30 and composed of the terms shown in Eqns. (11, 12, 13, 14).
Figure 4.30: BlockSim's symbolic solution for the RBD shown in Figure 4.29.
Or:
(11)
(12)
(13)
(14)
These terms use tokens to represent portions of the equation. In the symbolic equation setting, one reads the solution from the bottom up, replacing any occurrences of a particular token with its definition. (Note: It is possible, on rare occasions, for tokens to duplicate since they are arbitrarily chosen when displayed. If a token repeats, then the new token definition should be used for items above the recast token.) In this case then, and to obtain a system solution, one begins with Eqn. (14). Eqn. (14) contains the token I11, which is defined in Eqn. (13). This is then substituted into Eqn. (14) yielding:
(15)
Now, Eqn. (15) contains the token D1, which is defined in Eqn. (12). The next step is to substitute Eqn. (12) into Eqn. (15). Doing so yields Eqn. (16), shown next.
(16)
Eqn. (16) contains the token D1,which is defined in Eqn. (12). The last step is then to substitute Eqn. (12) into Eqn. (16):
(17)
When the complete equation is chosen, BlockSim's equation viewer performs these token substitutions automatically. It should be pointed out that the complete equation can get very large. While BlockSim internally can deal with millions of terms in an equation, the equation viewer will only format and display equations up to 64,000 characters. BlockSim uses a 64K memory buffer for displaying equations.
Figure 4.31: BlockSim's complete solution for the RBD shown in Figure 4.29.
When computing the system equation with the "Use IBS" option selected, BlockSim looks for identical blocks (blocks with the same failure characteristics) and attempts to simplify the equation, if possible. The symbolic solution for the system in the prior case, with the "Use IBS" option selected and setting equal reliability block properties, is:
Figure 4.32: BlockSim's symbolic solution, with the IBS option selected, for the RBD shown in Figure 4.29.
When using IBS, the resulting equation is invalidated if any of the block properties (e.g. failure distributions) have changed since the equation was simplified based on those properties.
See Also:
RBDs and Analytical System Reliability
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