Let's break down the problem step by step: --- ## **Given Data** - **Shaft material:** AISI 102 cold drawn steel - **Shaft diameter (\(d\))**: 1.5 in - **Torque fluctuates:** - Maximum torque (\(T_{max}\)): 100 lbf·in - Minimum torque (\(T_{min}\)): 250 lbf·in - **Key type:** Square key - **Design safety factor (n):** 2. --- ## **Step 1: Find Allowable Shear Stress for AISI 102 Steel** AISI 102 cold drawn steel typical ultimate tensile strength (\(S_{ut}\)) ≈ 64,000 psi Typical yield strength (\(S_{y}\)) ≈ 54,000 psi For keys (shear is the primary concern), allow shear strength as: \[ \tau_{allowable} = \frac{S_y}{2} = \frac{54,000}{2} = 27,000 \text{ psi} \] With a safety factor (\(n\)) of 2: \[ \tau_{design} = \frac{\tau_{allowable}}{n} = \frac{27,000}{2} = 13,500 \text{ psi} \] --- ## **Step 2: Calculate Maximum Torque the Key Must Transmit** Use the maximum torque for design: \[ T_{max} = 100 \text{ lbf·in} \] --- ## **Step 3: Find Shear Area for Square Key** For a square key (width = height = \(b\)), Shear area = \(b \times L\), where \(L\) is the length of the key. The torque transmitted by the key: \[ T = F \cdot r \] Where \(F\) is the tangential force on the key, \(r\) is the shaft radius. \[ F = \tau_{design} \cdot \text{Area} = \tau_{design} \cdot b \cdot L \] \[ T = F \cdot r = \tau_{design} \cdot b \cdot L \cdot \frac{d}{2} \] \[ T = \tau_{design} \cdot b \cdot L \cdot \frac{d}{2} \] --- ## **Step 4: Select Key Size from Standard Table** For a 1.5" shaft, standard square key sizes (from machinery handbooks) are: - **Width (\(b\)) = Height = 3/8" = .375"** - **Length (\(L\)):** Typically the length of the hub, but let's solve for the required length. --- ## **Step 5: Solve for Required Key Length** \[ T_{max} = \tau_{design} \cdot b \cdot L \cdot \frac{d}{2} \] \[ 100 = 13,500 \cdot .375 \cdot L \cdot \frac{1.5}{2} \] \[ 100 = 13,500 \cdot .375 \cdot L \cdot .75 \] \[ 100 = 13,500 \cdot .28125 \cdot L \] \[ 100 = 3,796.88 \cdot L \] \[ L = \frac{100}{3,796.88} = .263 \text{ in} \] --- ## **Step 6: Check Practical Length** The calculated length is very short. In practice, key lengths are much longer (typically the length of the hub, often 2–5 times the shaft diameter). **Minimum recommended length for 1.5" shaft key:** Standard key length: 3" or 4" --- ## **Step 7: Confirm Key Size is Adequate** Using **standard square key for 1.5" shaft**: - **Width (\(b\)) = Height = 3/8" (.375")** - **Length (\(L\)) = at least 2"** (standard key length for 1.5" shaft is usually 3" or 4", so this is safe) --- ## **Final Answer** ### **Appropriate Key Size:** - **Width = Height = 3/8" (.375")** - **Length = 3" (standard for 1.5" shaft)** --- ### **Summary Table** | Parameter | Value | |------------------|-------------| | Shaft Diameter | 1.5" | | Key Size (Square)| 3/8" x 3/8" | | Key Length | 3" | | Design Factor | 2 | --- **So, use a standard square key: 3/8" x 3/8" x 3" for a 1.5" shaft and these loading conditions.** Let me know if you'd like to see the same calculation for bearing or crushing stress on the key!