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Chapter 14: Problem 29

A helical extension spring, loaded in fatigue, has been designed for infinite life with the following data. \(C=9, d=8 \mathrm{~mm}\), working deflection \(=50 \mathrm{~mm}\), unpeened chrome-silicon wire, \(F_{\max }=935 \mathrm{~N}, F_{\min }=665 \mathrm{~N}, F_{\text {init }}=235 \mathrm{~N}, 13.75\) active coils. Find the safety factors for failure in the standard hooks. State all assumptions and sources of empirical data used.

Short Answer

Expert verified
To provide the short answer, figure out the numerical outputs from each step above and Input the information into the safety factor formula in the final step to find out the two significant safety factors for failure in the standard hooks.

Step by step solution

01

Direct Shear Stress Calculation

Calculate the direct shear stress using the equation, \( \tau_{direct} = \frac{P}{A} = \frac{P}{\pi * \frac{d^2}{4}} \), where P is the maximum force \( F_{\max} \) and d is the diameter of the wire. For \( F_{\max} = 935 \mathrm{~N} \) and \( d = 8 \mathrm{~mm}\), the direct shear stress \( \tau_{direct}\) can be computed.
02

Torsional Shear Stress Calculation

Compute the torsional shear stress using the formula \( \tau_{torsion} = \frac{8 * F_{\max} * D * C}{\pi * d^3} \), where D is the mean coil diameter (compute this from \( D = C * d \)), \( F_{\max} \) is the maximum force, C is the spring index, and d is the wire diameter. Calculate \( \tau_{torsion} \) with \( D = 9 * 8 \mathrm{~mm}\), \( F_{\max} = 935 \mathrm{~N}\), \( C = 9 \), and \( d = 8 \mathrm{~mm}\).
03

Equivalent Shear Stress Calculation

Calculate the equivalent shear stress using the von Mises Stress Criterion, also known as the Maximum Distortion Energy Theory. The equation is \( \tau_{eq} = \sqrt{\tau_{direct}^2 + 3 * \tau_{torsion}^2} \). Substitute the values obtained from Steps 1 and 2 to compute \( \tau_{eq} \).
04

Safety Factor Calculation

Compute the safety factor using the formula \( n = \frac{\tau_{allowed}}{\tau_{eq}} \) where \( \tau_{allowed} \) is the allowed shear stress for unpeened chrome-silicon wire (use empirical values, such as \( \tau_{allowed} = 1200 \mathrm{MPa} \)), and \( \tau_{eq} \) is the equivalent shear stress (obtained from Step 3). Solve for n to find the safety factor for failure in the standard hooks.

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Most popular questions from this chapter

A linear spring is to give \(200 \mathrm{~N}\) at its maximum deflection of \(150 \mathrm{~mm}\) and \(40 \mathrm{~N}\) at its minimum deflection of \(50 \mathrm{~mm}\). What is its spring rate?

Design a helical compression spring for a static load of \(400 \mathrm{~N}\) at a deflection of 45 \(\mathrm{mm}\) with a safety factor of \(2.5\). Use \(C=8\). Specify all parameters necessary to manufacture the spring. State all assumptions.

A spring with ends squared and ground, wire diameter \(d=4 \mathrm{~mm}\), outside diameter \(D_{o}=40 \mathrm{~mm}, 18\) total coils, and free length \(L_{f}=140 \mathrm{~mm}\) has been chosen for an application where the initial deflection is \(15 \mathrm{~mm}\) and the working deflection is 50 \(\mathrm{mm}\). Determine minimum working length, shut height, clash allowance, spring index, and spring rate for this spring.

Given the following data for a helical compression spring loaded in fatigue, design the spring for infinite life. State all assumptions and sources of empirical data used. \(C=8.5, d=8 \mathrm{~mm}, 625 \mathrm{rpm}\), working deflection \(=20 \mathrm{~mm}, 15 \%\) clash allowance, unpeened music wire, squared ends, preset, \(F_{\max }=450 \mathrm{~N}, F_{\min }=225 \mathrm{~N}\).

Design a helical extension spring with standard hooks to handle a dynamic load that varies from \(300 \mathrm{lb}\) to \(500 \mathrm{lb}\) over a 2 -in working deflection. Use chrome-vanadium wire. The forcing frequency is \(1000 \mathrm{rpm}\). Infinite life is desired. Minimize the package size. Choose appropriate safety factors against fatigue, yielding, and surging.
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