7.13: Quick Check

Quick Check

1. Classify each as analysis or design and explain why: (a) Compute current through 470 Ω at 12 V. (b) Choose a resistor so a 9 V circuit stays below 0.25 W. (c) Compute cable tension at 45° for 2000 kg. (d) Select cable angle and cross-section for 3000 N load with SF ≥ 2.5.
2. Write a strong design objective for an LED circuit at 9 V with 0.06 W maximum LED power. Include the evaluation metric.
3. A rod supports 8000 N. Allowable stress = 160 MPa. Required SF = 2.5. Find minimum area in mm².
4. Using \(P = V^2/R\), find the minimum resistance so power stays below 0.25 W for a 12 V source.
5. Given two alternatives: A has 20 mm² (SF = 3.2, simple to manufacture) and B has 15 mm² (SF = 2.4, harder to manufacture). Weight: mass minimization 50%, manufacturing simplicity 25%, safety margin 25%. Build a decision matrix and determine which alternative wins.
6. Explain why iteration is a feature of design rather than a sign of failure. Give one example of a productive iteration cycle.
7. A bracket analysis gives actual stress of 8 MPa against a working limit of 40 MPa. What does this suggest, and what action would you take?
8. [Bridge Challenge prep] List the hard constraints for the Bridge Challenge. For each, explain what design choices it rules out immediately.

Answers - click to expand

1. Classify each as analysis or design.
   1. Analysis: Computing current through a known \(470 \ \Omega\) resistor at \(12 \text{ V}\) uses known inputs to find an output. The component has already been selected.
   2. Design: Choosing a resistor so a \(9 \text{ V}\) circuit stays below \(0.25 \text{ W}\) requires selecting a component that satisfies a constraint.
   3. Analysis: Computing cable tension at \(45°\) for a \(2000 \text{ kg}\) load uses a known load and geometry to calculate the resulting tension.
   4. Design: Selecting cable angle and cross-section for a \(3000 \text{ N}\) load with \(SF \ge 2.5\) requires choosing design variables that satisfy strength and safety constraints.
2. Strong design objective: Design a resistor-protected LED circuit powered by a \(9 \text{ V}\) source so that the LED power remains at or below \(0.06 \text{ W}\) during normal operation while still producing acceptable brightness.

   Evaluation metric: A suitable metric is LED power margin, defined by how far the calculated LED power is below the \(0.06 \text{ W}\) limit. A second useful metric could be circuit current or brightness if LED specifications are provided.

3. Minimum rod area:

   Since \(1 \text{ MPa} = 1 \text{ N/mm}^2\), the minimum area is \(A = \dfrac{F}{\sigma_\text{design}} = \dfrac{8000 \text{ N}}{64 \text{ N/mm}^2} = 125 \text{ mm}^2\).

   Answer The rod needs a minimum cross-sectional area of \(125 \text{ mm}^2\).

4. Minimum resistance for \(P \le 0.25 \text{ W}\):

   For \(V = 12 \text{ V}\) and \(P_\text{max} = 0.25 \text{ W}\), \(R_\text{min} = \dfrac{(12 \text{ V})^2}{0.25 \text{ W}} = \dfrac{144}{0.25} = 576 \ \Omega\).

   Answer The resistance must be at least \(576 \ \Omega\). In practice, choose the next higher standard resistor value to provide margin.

5. Decision matrix: One reasonable scoring method is to score each category from 1 to 5, where 5 is best. Since lower mass is better, the smaller area receives the higher mass score. Since higher safety factor is better, the larger safety factor receives the higher safety score.

   Alternative	Mass score (50%)	Manufacturing score (25%)	Safety score (25%)	Weighted total
   A: \(20 \text{ mm}^2\), \(SF = 3.2\), simple	3	5	5	\(3(0.50)+5(0.25)+5(0.25)=4.00\)
   B: \(15 \text{ mm}^2\), \(SF = 2.4\), harder	5	2	2	\(5(0.50)+2(0.25)+2(0.25)=3.50\)

   Answer Alternative A wins with a weighted score of \(4.00\) compared with \(3.50\) for Alternative B. More importantly, B has \(SF = 2.4\), which may fail if the design requires \(SF \ge 2.5\). If \(SF \ge 2.5\) is a hard constraint, B should be rejected before scoring.

6. Iteration is a feature of design because design problems have constraints, trade-offs, and incomplete information. A first design rarely satisfies every requirement perfectly. Iteration allows engineers to test an idea, identify weaknesses, revise the design, and improve performance.

   Example: A bridge prototype may be strong enough but too heavy. The team can revise the truss layout, remove unnecessary members, retest the load capacity, and compare the new strength-to-weight ratio. That loop is not failure; it is how the design improves.

7. Bracket stress interpretation: If the actual stress is \(8 \text{ MPa}\) and the working limit is \(40 \text{ MPa}\), then the bracket is using only \(8/40 = 0.20\), or \(20\%\), of the allowed stress.

   This suggests the bracket may be overdesigned for strength. That is not automatically bad, but it may mean the design is heavier, more expensive, or larger than necessary.

   Action: I would check whether other constraints control the design, such as deflection, buckling, fatigue, impact loading, manufacturability, or required safety factor. If strength is the only concern, I would iterate by reducing material or changing geometry, then rerun the analysis.

8. Bridge Challenge prep: Answers will depend on the exact rules provided for the course version of the Bridge Challenge. A strong answer should identify hard constraints and explain what each one rules out.

   Common hard constraints might include:

   • Span requirement: The bridge must cross a specified gap, so designs that are too short are immediately ruled out.
   • Load requirement: The bridge must support a required load, so designs that cannot carry the test load are ruled out even if they are lightweight.
   • Material limit: Only approved materials may be used, so metal, glue types, fasteners, or reinforcement methods outside the rules are not allowed.
   • Mass or material quantity limit: The bridge cannot exceed a maximum weight or material amount, so simply adding more material everywhere is not an acceptable strategy.
   • Dimensional envelope: The bridge must fit within height, width, or clearance limits, so overly tall, wide, or obstructive designs are ruled out.
   • Connection rules: If joint types or adhesive locations are restricted, designs that depend on prohibited connections must be revised.
   • Testing interface: The bridge must accept the loading fixture, so designs that leave no place to apply the load are ruled out.

   Answer Hard constraints define the boundary of acceptable designs. Any design that violates a hard constraint should be rejected before comparing performance metrics such as strength-to-weight ratio, cost, or ease of construction.