Manufacturing Engineering & Technology
7th Edition
ISBN: 9780133128741
Author: Serope Kalpakjian, Steven Schmid
Publisher: Prentice Hall
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Textbook Question
Chapter 1, Problem 27QLP
Two parts have been made of the same material, but one was formed by cold working and the other by hot working. Explain the differences you might observe between the two.
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Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.
An aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.
Chapter 1 Solutions
Manufacturing Engineering & Technology
Ch. 1 - What is the difference between an atom and a...Ch. 1 - Prob. 2RQCh. 1 - Prob. 3RQCh. 1 - Prob. 4RQCh. 1 - Define anisotropy. What is its significance?Ch. 1 - What effects does recrystallization have on the...Ch. 1 - What is strain hardening, and what effects does it...Ch. 1 - Explain what is meant by structure-sensitive and...Ch. 1 - Prob. 9RQCh. 1 - What influence does grain size have on the...
Ch. 1 - What is the relationship between the nucleation...Ch. 1 - What is a slip system, and what is its...Ch. 1 - Explain the difference between recovery and...Ch. 1 - What is hot shortness, and what is its...Ch. 1 - Explain the advantages and limitations of cold,...Ch. 1 - Describe what the orange peel effect is. Explain...Ch. 1 - Some metals, such as lead, do not become stronger...Ch. 1 - Describe the difference between preferred...Ch. 1 - Differentiate between stress relaxation and stress...Ch. 1 - What is twinning? How does it differ from slip?Ch. 1 - Prob. 21QLPCh. 1 - What is the significance of the fact that some...Ch. 1 - Is it possible for two pieces of the same metal to...Ch. 1 - Prob. 24QLPCh. 1 - A cold-worked piece of metal has been...Ch. 1 - What materials and structures can you think of...Ch. 1 - Two parts have been made of the same material, but...Ch. 1 - Do you think it might be important to know whether...Ch. 1 - Explain why the strength of a polycrystalline...Ch. 1 - Describe the technique you would use to reduce the...Ch. 1 - What is the significance of the fact that such...Ch. 1 - Prob. 32QLPCh. 1 - It has been noted that the more a metal has been...Ch. 1 - Is it possible to cold work a metal at...Ch. 1 - Comment on your observations regarding Fig. 1.14.Ch. 1 - Is it possible for a metal to be completely...Ch. 1 - Prob. 37QTPCh. 1 - Prob. 38QTPCh. 1 - Plot the data given in Table 1.1 in terms of...Ch. 1 - A strip of metal is reduced from 30 mm in...Ch. 1 - Prob. 41QTPCh. 1 - How many grains are there on the surface of the...Ch. 1 - Prob. 43QTPCh. 1 - Prob. 44QTPCh. 1 - Prob. 45QTPCh. 1 - A technician determines that the grain size of a...Ch. 1 - If the diameter of the aluminum atom is 0.28 nm,...Ch. 1 - The following data are obtained in tension tests...Ch. 1 - Prob. 50QTPCh. 1 - Prob. 51QTPCh. 1 - Prob. 52QTPCh. 1 - Same as Prob. 1.39, but ASTM no. versus...Ch. 1 - By stretching a thin strip of polished metal, as...Ch. 1 - Draw some analogies to mechanical fiberingfor...Ch. 1 - Draw some analogies to the phenomenon of hot...Ch. 1 - Take a deck of playing cards, place a rubber band...Ch. 1 - Give examples in which anisotropy is scale...Ch. 1 - The movement of an edge dislocation was described...
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- Problem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forwardProblem 1 (35 pts). An elastic string of constant line tension1 T is pinned at x = 0 andx = L. A constant distributed vertical force per unit length p (with units N/m) is appliedto the string. Under this force, the string deflects by an amount v(x) from its undeformed(horizontal) state, as shown in the figure below.Force equilibrium in the string requires thatdfdx − p = 0 , (1)where f(x) is the internal vertical force in the string, which is given byf = Tdvdx . (2)(a) [10pts] Write down the BVP (strong form) that the string deflection v(x) must satisfy.(b) [2pts] What order is the governing PDE in the BVP of (a)?(c) [3pts] Identify the type (essential/natural) of each boundary condition in (a).(d) [20pts] Find the analytical solution of the BVP in (a).arrow_forwardProblem 2 (25 pts, (suggested time 15 mins). An elastic string of line tension T andmass per unit length µ is pinned at x = 0 and x = L. The string is free to vibrate, and itsfirst vibration mode is shown below.In order to find the frequency of the first mode (or fundamental frequency), the string isdiscretized into a certain number of linear elements. The stiffness and mass matrices of thei-th element are, respectivelyESMi =TLi1 −1−1 1 EMMi =Liµ62 11 2 . (2)(a) [5pts] What is the minimum number of linear elements necessary to compute the fundamental frequency of the vibrating string?(b) [20pts] Assemble the global eigenvalue problem and find the fundamental frequency ofvibration of the stringarrow_forward
- I need part all parts please in detail (including f)arrow_forwardProblem 3 (10 pts, suggested time 5 mins). In class we considered the mutiphysics problem of thermal stresses in a rod. When using linear shape functions, we found that the stress in the rod is affected by unphysical oscillations like in the following plot E*(ux-a*T) 35000 30000 25000 20000 15000 10000 5000 -5000 -10000 0 Line Graph: E*(ux-a*T) MULT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Arc length (a) [10pts] What is the origin of this issue and how can we fix it?arrow_forwardanswer the questions and explain all of it in words. Ignore where it says screencast and in class explanationarrow_forward
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