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Calculate the reliability of the sonar system to work 100 hours failure-free. The components of an underwater sonar system are given in the block diagram below: The failure rate of one unit of each component is estimated below using available data tables:

Subsystem 	Failure rate λ/106 hours
Transducer 	1,000
Power Supply 	20 Heat Exchanger 	20
Transmitter/Receiver Switch 	6
Power Amplifier 	130
Pre-Amplifier 	32
Front-end Processor 	400
System Controller (parallel) 	10
SDLC Bus (parallel) 	15
Signal Processor 	400
Display Processor (parallel) 	130
Display Monitor (parallel) 	50
Audio Processor

To calculate the reliability of the sonar system, we need to use the failure rates of each subsystem and apply the exponential reliability formula:

\[
R(t) = e^{-\lambda t}
\]
where:
- \( R(t) \) is the reliability over time \( t \),
- \( \lambda \) is the total system failure rate (failures per hour),
- \( t \) is the time the system is required to operate failure-free (100 hours in this case).

### Steps:

1. **Identify the failure rates of each component** from the table. Each component has its failure rate \( \lambda \) provided in units of \( 10^6 \) hours. 
2. **Handle parallel systems**: For components connected in parallel, the overall failure rate \( \lambda_{\text{parallel}} \) is smaller than that of each individual component. For two components in parallel, the equivalent failure rate is given by:
   \[
   \lambda_{\text{parallel}} = \frac{\lambda_1 \cdot \lambda_2}{\lambda_1 + \lambda_2}
   \]
   This formula accounts for the increased reliability provided by parallel systems.

3. **Sum the failure rates of all subsystems** (after adjusting for parallel configurations).

4. **Calculate the total system failure rate**, and then use the reliability formula over 100 hours.

Let's proceed with these steps. I'll calculate the failure rate for each component and apply the parallel system formula where necessary.The total failure rate of the sonar system is approximately **0.00216 failures per hour**. Over 100 hours, the reliability of the system, or the probability that it will operate without failure, is approximately **80.57%**.

Would you like more details or explanations on any part of this process?

Here are 5 related questions that build on this topic:
1. How does a parallel system configuration improve reliability?
2. What would happen to the reliability if the operating time increased from 100 to 200 hours?
3. How could you improve the overall reliability of this sonar system?
4. What is the impact of components with high failure rates on the overall system reliability?
5. How does the failure rate change with different operational environments (e.g., underwater pressure)?

**Tip:** Parallel configurations are used to enhance reliability by providing redundancy—if one component fails, the other can take over, reducing the chance of system failure.

Calculate the reliability of the sonar system to work 100 hours failure-free. The components of an underwater sonar system are given in the block diagram below: The failure rate of one unit of each component is estimated below using available data tables.

Learn how to calculate the reliability of an underwater sonar system operating failure-free for 100 hours. This problem involves key reliability engineering concepts such as failure rate calculations and exponential reliability formulas. Suitable for undergraduate engineering students.

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