[/math] units must fail for the system to fail. However, the component with the highest reliability in a parallel configuration has the biggest effect on the system's reliability, since the most reliable component is the one that will most likely fail last. [/math], {{R}_{s}}=\underset{i=1}{\overset{n}{\mathop \prod }}\,P({{X}_{i}})\,\! As the number of components connected in series increases, the system's reliability decreases. Consider a system with three components. From reliability point of view, a series system (Fig. 0000005891 00000 n Chapter 3: RBDs and Analytical System Reliability, More Resources: BlockSim Examples Collection, Download Reference Book: System Analysis (*.pdf), Generate Reference Book: File may be more up-to-date. & +{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}} \\ x�bf]������� Ā B@16�= It should be pointed out that the complete equation can get very large. {{R}_{Computer1}}= & ({{R}_{Power\,Supply}}\cdot {{R}_{Processor}}\cdot {{R}_{HardDrive}} Using the system's reliability equation, the corresponding time-to-failure for a 0.11 unreliability is 389.786 hours. \end{align}\,\! & -{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}} \\ \\ Complex systems are discussed in the next section. 2.2 Parallel System . E�#��k�82���Q�!����H��"Zl�D�\�"�ʨw@I�� #+êy� ��ܧ�|��h¶.�y��7���tK}���y�U��Kf� .��. For a parallel configuration, as the number of components/subsystems increases, the system's reliability increases. Example: Effect of a Component's Reliability in a Series System. Example: Effect of the Number of Components in a Parallel System. In the first figure below, Subdiagram Block A in the top diagram represents the series configuration of the subsystem reflected in the middle diagram, while Subdiagram Block G in the middle diagram represents the series configuration of the subsubsystem in the bottom diagram. X7= & A\overline{BC}-\text{Units 2 and 3 fail}\text{.} parallel components for the system to succeed. [/math], then substituting the first equation above into the second equation above yields: When using BlockSim to compute the equation, the software will return the first equation above for the system and the second equation above for the subdiagram. This page uses frames, but your browser doesn't support them. In the symbolic equation setting, one reads the solution from the bottom up, replacing any occurrences of a particular token with its definition. 0000054580 00000 n \end{align}\,\! [/math], ${{R}_{System}}={{R}_{Computer1}}\cdot {{R}_{Computer2}} \ \,\! The reliability of the component is 95%, thus the reliability of the system is 95%. Consider a system of 6 pumps of which at least 4 must function properly for system success. \\ For this reason, we will first look at different component/subsystem configurations, also known as structural properties (Leemis [17]). 0000036160 00000 n & -{{R}_{9}}\cdot {{R}_{8}}\cdot {{D}_{1}}-{{R}_{5}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{9}}+{{R}_{2}}\cdot {{R}_{10}} \\ However, in the case of independent components, equation above becomes: Or, in terms of individual component reliability: In other words, for a pure series system, the system reliability is equal to the product of the reliabilities of its constituent components. Figure 9.$, ${{R}_{s}}={{R}_{1}}{{R}_{2}}+{{R}_{3}}-{{R}_{1}}{{R}_{2}}{{R}_{3}}\,\! 1a) is such, which fails if any of its elements fails. Example: Effect of the Number of Components in a Series System. Parallel Configuration Systems 5. However, as individual items fail, the failure characteristics of the remaining units change since they now have to carry a higher load to compensate for the failed ones. What is the overall reliability of the system for a 100-hour mission? One block within the container must be operating, otherwise the container will fail, leading to a system failure (since the container block is part of a series configuration in this example).$, $-{{R}_{9}}\cdot {{R}_{5}}\cdot ({{R}_{7}}\cdot (-{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}}-{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}+{{R}_{5}}+{{R}_{6}}+{{R}_{4}}))\,\! In this case, the resistance for the resistor is infinite and the equivalent resistance is: If two resistors fail open (e.g., #1 and #2), the equivalent resistance is: Thus, if [math]{{r}_{1}}\,\! X6= & \overline{A}B\overline{C}-\text{Units 1 and 3 fail}\text{.}$, ${{R}_{2}}=80%\,\!$, \begin{align} Just multiply them. The following graphic demonstrates the RBD for the system. = & 1-\underset{i=1}{\overset{n}{\mathop \prod }}\,(1-{{R}_{i}}) \ Reliability Block Diagram (RBD) Reliability Block Diagram (RBD) is a graphical repppyresentation of how the components of a system are connected from reliability point of view. P(s|A)={{R}_{2}}{{R}_{3}} = & 1-[(1-{{R}_{1}})\cdot (1-{{R}_{2}})\cdot ...\cdot (1-{{R}_{n}})] \\ With this technique, it is possible to generate and analyze extremely complex diagrams representing the behavior of many subsystems in a manageable way. & +{{R}_{5}}+{{R}_{6}}+{{R}_{4}} \, -{{R}_{9}}\cdot {{R}_{8}}\cdot ({{R}_{7}}\cdot (-{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}}-{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}+{{R}_{5}}+{{R}_{6}}+{{R}_{4}}))\,\! This is a 2-out-of-3 configuration. \end{align}\,\! The primary advantage of the analytical solution is that it produces a mathematical expression that describes the reliability of the system. \end{align}\,\! \end{align}\,\! & +{{R}_{A}}\cdot {{R}_{B}}\cdot {{R}_{C}}\cdot {{R}_{2/3}}\cdot {{R}_{E}}\cdot {{R}_{F}} \\ Configuration with inherited subdiagrams. While BlockSim internally can deal with millions of terms in an equation, the System Reliability Equation window will only format and display equations up to 64,000 characters. Doing so yields [math]{{I}_{7}}\,\! & +{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}} \\ The following operational combinations are possible for system success: The probability of success for the system (reliability) can now be expressed as: This equation for the reliability of the system can be reduced to: If all three hard drives had the same reliability, [math]R\,\! Recommended for you 8.02x - Lect 16 - Electromagnetic Induction, Faraday's Law, Lenz Law, SUPER DEMO - Duration: 51:24. In mirrored blocks, the duplicate block behaves in the exact same way that the original block does. The simplest case of components in a k-out-of-n configuration is when the components are independent and identical. Mathematically, the reliability of this series system, R ss, is given by (1.1) where R 1, R 2, , R n are the reliabilities of Unit 1, Unit 2, ..., Unit n, respectively. {{X}_{1}}=1,2\text{ and }{{X}_{2}}=3 & -{{R}_{2}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{D}_{1}}-{{R}_{2}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{8}}\cdot {{D}_{1}} \\ Series systems with correlated random safety margins distributed according to the normal distribution play an important role in recent developments of methods of structural reliability theory. P(s| A)= &{{R}_{B}}{{R}_{F}}\left[ 1-\left( 1-{{R}_{C}} \right)\left( 1-{{R}_{E}} \right) \right] \\, \begin{align} Each pump has an 85% reliability for the mission duration. The reason that BlockSim includes all items regardless of whether they can fail or not is because BlockSim only recomputes the equation when the system structure has changed. If a component in the system fails, the "water" can no longer flow through it. This is primarily due to the fact that component [math]C\,\! The container serves a dual purpose. I. Bazovsky,Reliability theory and practice, Prentice-Hall Inc., Eaglewood Cliffs, New Jersey, U.S.A. (1961). It is widely used in the aerospace industry and generally used in mission critical systems., {{r}_{3}}\,\! The relays are situated so that the signal originating from one station can be picked up by the next two stations down the line. The RBD is analyzed and the system reliability equation is returned. \end{align}\,\! There is a saying that a chain is only as strong as its weakest link. 116 0 obj<>stream Contents 1. 32 2 R 1 1 0.94 2 0.9865 ps, {{R}_{2}}\,\ = 70%\,\! The reliability-wise configuration of components must be determined beforehand. \end{align}\,\! As a result, the reliability of a series system is always less than the reliability of the least reliable component. Series System Reliability Property 2 for Parts in Series The upper series of images relate to head pulleys used on conveyor belt systems. into equation above: When the complete equation is chosen, BlockSim's System Reliability Equation window performs these token substitutions automatically. RBD is used to model the various series-parallel and complex block combinations (paths) that result in system successblock combinations (paths) that result in system success. \end{align}\,\! Series System Failure Rate Equations Consider a system consisting of n components in series. 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