Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 22: Problem 2

In a manner similar to the treatment of Section \(22.2,\) perform a stiffness- to-mass performance analysis on a solid cylindrical shaft that is subjected to a torsional stress. Use the same engineering materials that are listed in Table 22.1 . In addition, conduct a material cost analysis. Rank these materials both on the basis of mass of material required and material cost. For glass and carbon fiber-reinforced composites, assume that the shear module are 8.6 and \(9.2 \mathrm{GPa},\) respectively.

Short Answer

Expert verified
Based on the problem of performing a stiffness-to-mass performance analysis on a solid cylindrical shaft subjected to torsional stress using the engineering materials listed in Table 22.1 and conducting a material cost analysis, provide a short answer discussing the significance of stiffness-to-mass ratio and material cost in selecting materials for engineering applications: The stiffness-to-mass ratio and material cost are crucial factors in selecting materials for engineering applications as they help identify optimal materials for specific applications. A higher stiffness-to-mass ratio indicates that a material provides better structural performance with lower mass, which is desirable in applications where weight reduction is important. Additionally, material cost plays a crucial role in determining the economic feasibility of implementing materials in real-world applications. By considering both factors and ranking materials accordingly, engineers can make informed decisions about material selection, ensuring that they choose the best-suited materials for their specific applications while also considering cost-effectiveness.

Step by step solution

01

Obtain relevant information from Table 22.1

Retrieve material properties such as the shear modulus (G) and the material's cost from Table 22.1.
02

Define the variables related to a solid cylindrical shaft

We are analyzing a solid cylindrical shaft. Let L be the length of the shaft, d be its diameter, G be the shear modulus, and τ be the torsional stress.
03

Calculate the torsional stiffness

The torsional stiffness of a solid cylinder can be calculated using the formula: \(K = \frac{G \cdot J}{L}\), where J is the polar moment of inertia, given by \(J = \frac{\pi d^4}{32}\).
04

Calculate the mass of the shaft

To evaluate the mass, we need to compute the volume and multiply it by the material's density (ρ). Volume can be calculated using the formula: \(V = \pi \frac{d^2}{4} L\). Thus, the mass can be estimated as \(m = ρ \cdot V = ρ \cdot \pi \frac{d^2}{4} L\).
05

Compute the stiffness-to-mass ratio

Divide the torsional stiffness by the mass to obtain the stiffness-to-mass ratio: \(S = \frac{K}{m} = \frac{\frac{G \cdot J}{L}}{ρ \cdot \pi \frac{d^2}{4} L}\). Simplifying this expression, we get \(S = \frac{4 G J}{ρ \pi d^2 L^2}\).
06

Perform the stiffness-to-mass analysis

Using the shear modulus values given for each material, and the density values from Table 22.1, calculate the stiffness-to-mass ratio for each material using the formula derived in Step 5.
07

Conduct a material cost analysis

For each material, compute the cost of the shaft using the given data in Table 22.1. To do this, use the volume of the shaft (\(V = \pi \frac{d^2}{4} L\)) and multiply it by the unit cost of the material.
08

Rank the materials

Using the results from Steps 6 and 7, rank the materials based on the stiffness-to-mass ratio and the material cost separately. Following these steps, you will complete the stiffness-to-mass performance analysis and material cost analysis for the materials under consideration.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A helical spring is to be constructed from a 4340 steel. The design calls for 5 coils, a coil-to-coil diameter of \(12 \mathrm{mm},\) and a wire diameter of \(3 \mathrm{mm}\). Furthermore, a maximum total deflection of \(5.0 \mathrm{mm}\) is possible without any plastic deformation. Specify a heat treatment for this 4340 steel wire in order for the spring to meet the above criteria. Assume a shear modulus of \(80 \mathrm{GPa}\) for this steel alloy, and that \(\tau_{y}=0.6_{y} .\) Note: heat treatment of the 4340 steel is discussed in Section 10.8.

(a) Using the expression developed for stiffness performance index in Problem 22.D3(a) and data contained in Appen\(\operatorname{dix} \mathrm{B},\) determine stiffness performance indices for the following polymeric materials: high- density polyethylene, polypropylene, poly(vinyl chloride), polystyrene, polycarbonate, poly(methyl methacrylate),poly(ethylene terephthalate), polytetrafluoroethylene, and nylon \(6,6 .\) How do these values compare with those of the metallic materials? (Note: In Appendix B, where ranges of values are given, use average values.)(b) Now, using the cost database (Appen\(\operatorname{dix} C),\) conduct a cost analysis in the same manner as Section \(22.2 .\) Use cost data for the raw forms of these polymers. (c) Using the expression developed for strength performance index in Problem \(22 . \mathrm{D} 3(\mathrm{a})\) and data contained in Appendix B, determine strength performance indices for these same polymeric materials.

Consider the plate shown below that is supported at its ends and subjected to a force \(F\) that is uniformly distributed over the upper face as indicated. The deflection \(\delta\) at the \(L / 2\) position is given by the expression $$\delta=\frac{5 F L^{3}}{32 E w t^{3}}$$ Furthermore, the tensile stress at the underside and also at the \(L / 2\) location is equal to $$\sigma=\frac{3 F L}{4 w t^{2}}$$ (a) Derive stiffness and strength performance index expressions analogous to Equations 22.9 and 22.11 for this plate (Hint. solve for \(t\) in these two equations, and then substitute the resulting expressions into the mass equation, as expressed in terms of density and plate dimensions.) (b) From the properties database in Appendix \(\mathrm{B},\) select those metal alloys with stiffness performance indices greater than 1.40 (for \(E\) and \(\rho\) in units of \(\mathrm{GPa}\) and \(\mathrm{g} / \mathrm{cm}^{3}\) respectively). (c) Also using the cost database (Appendix C), conduct a cost analysis in the same manner as Section \(22.2 .\) Relative to this analysis and that in part b, which alloy would you select on a stiffness- per-mass basis? (d) Now select those metal alloys having strength performance indices greater than 5.0 (for \(\sigma_{y}\) and \(\rho\) in units of \(\mathrm{MPa}\) and \(\mathrm{g} / \mathrm{cm}^{3}\) respectively \(),\) and rank them from highest to lowest \(P\).(e) And, using the cost database, rank the materials in part d from least to most costly. Relative to this analysis and that in part d, which alloy would you select on a strength-per-mass basis? (f) Which material would you select if both stiffness and strength are to be considered relative to this application? Justify your choice.

(a) Using the procedure as outlined in Section 22.2 ascertain which of the metal alloys listed in Appendix B have torsional strength performance indices greater than \(10.0\left(\text { for } \tau_{f} \text { and } \rho \text { in units of } \mathrm{MPa} \text { and } \mathrm{g} / \mathrm{cm}^{3}\right.\) respectively \(),\) and, in addition, shear strengths greater than \(350 \mathrm{MPa}\). (b) Also using the cost database (Appendix C), conduct a cost analysis in the same manner as Section \(22.2 .\) For those materials that satisfy the criteria noted in part a, and, on the basis of this cost analysis, which material would you select for a solid cylindrical shaft? Why?

One of the critical components of our modern video cassette recorders (VCRs) is the magnetic recording/playback head. Write an essay in which you address the following issues: (1) the mechanism by which the head records and plays back video/audio signals; (2) the requisite properties for the material from which the head is manufactured; then (3) present at least three likely candidate materials, and the property values for each that make it a viable candidate.
See all solutions

Recommended explanations on Physics Textbooks

Fields in Physics

Read Explanation

Quantum Physics

Read Explanation

Fluids

Read Explanation

Space Physics

Read Explanation

Radiation

Read Explanation

Waves Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free