Principles of Engineering - Simple Machines: Screw Calculations

System Overview and Specifications

• System Description:

  • A $7/16\,\text{in}$ nut driver with a $1\,1/2\,\text{in}$ ($1.5\,\text{in}$) diameter handle is used to turn and install a $1/4\text{''}-20\,\text{UNC}$ bolt into a robotic arm assembly.

• Key Hardware Components and Dimensions:

  • Tool: Nut driver ($7/16\,\text{in}$ size).

  • Handle Diameter ($d$): $1.5\,\text{in}$ ($1\,1/2\,\text{in}$).

  • Handle Radius ($r$): $r = \frac{d}{2} = \frac{1.5\,\text{in}}{2} = 0.75\,\text{in}$.

  • Fastener Specification: $1/4\text{''}-20\,\text{UNC}$.

  • Major Diameter: $1/4\,\text{in}$ ($0.25\,\text{in}$).

  • Threads Per Inch (TPI): $20\,\text{threads/in}$.

  • Thread Series: Unified National Coarse (UNC).

Circumference of Effort Application

• Concept:

  • The effort distance for a simple machine screw system driven by a handle is equal to the circular distance traveled by the outer effort boundary (handle radius) during one complete revolution ($360^{\circ}$).

• Formula:

  • $C = 2\pi r$

  • Where:

  • $C$ = Circumference of effort circle

  • $r$ = Radius of handle ($0.75\,\text{in}$)

• Calculation and Substitution:

  • $C = 2 \times \pi \times 0.75\,\text{in}$

  • $C = 4.71239\,\text{in}$

• Final Answer:

  • $C = 4.71\,\text{in}$

Pitch Determination

• Concept:

  • Pitch ($P$) is defined as the distance measured parallel to the axis of a screw between corresponding points on adjacent thread forms. It represents the linear distance advanced by the screw in a single full turn.

• Formula:

  • $P = \frac{1}{\text{TPI}}$

  • Where:

  • $P$ = Thread pitch

  • $\text{TPI}$ = Threads per inch ($20\,\text{threads/in}$)

• Calculation and Substitution:

  • $P = \frac{1}{20\,\text{threads/in}}$

  • $P = 0.05\,\text{in}$

• Final Answer:

  • $P = 0.05\,\text{in}$

Ideal Mechanical Advantage (IMA)

• Concept:

  • The Ideal Mechanical Advantage (IMA) of a screw system is the theoretical force multiplication factor calculated as the ratio of effort distance (circumference per turn, $C$) to resistance distance (pitch per turn, $P$).

• Formula:

  • $\text{IMA} = \frac{C}{P}$

  • Where:

  • $C$ = Effort distance per revolution ($4.71\,\text{in}$)

  • $P$ = Resistance distance per revolution ($0.05\,\text{in}$)

• Calculation and Substitution:

  • $\text{IMA} = \frac{4.71\,\text{in}}{0.05\,\text{in}}$

  • $\text{IMA} = 94.2$

• Final Answer:

  • $\text{IMA} = 94.2$

Theoretical Resistance Force Output

• Concept:

  • In an ideal system operating without frictional losses, $\text{IMA} = \frac{F_R}{F_E}$. The theoretical load or resistance force ($F_R$) that can be overcome is equal to the applied effort force ($F_E$) multiplied by the ideal mechanical advantage.

• Formula:

  • $\text{IMA} = \frac{F_R}{F_E} \implies F_R = F_E \times \text{IMA}$

  • Where:

  • $F_E$ = Applied effort force ($5\,\text{lb}$)

  • $F_R$ = Output resistance force overcome

  • $\text{IMA}$ = Ideal mechanical advantage ($94.2$)

• Calculation and Substitution:

  • $F_R = 5\,\text{lb} \times 94.2$

  • $F_R = 471\,\text{lb}$

• Final Answer:

  • $F_R = 471\,\text{lb}$