Degarmo's Materials And Processes In Manufacturing
13th Edition
ISBN: 9781119492825
Author: Black, J. Temple, Kohser, Ronald A., Author.
Publisher: Wiley,
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Chapter 5, Problem 83RQ
What are some current goals of the heat treatment industry?
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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 5 Solutions
Degarmo's Materials And Processes In Manufacturing
Ch. 5 - What is heat treatment?Ch. 5 - What types of properties can be altered through...Ch. 5 - Why should people performing hot forming or...Ch. 5 - What is the broad goal of the processing heat...Ch. 5 - Why might equilibrium phase diagrams be useful...Ch. 5 - What are the A1, A3, and Acm lines?Ch. 5 - What are some possible objectives of annealing...Ch. 5 - Why might it be important to include a preceding...Ch. 5 - Describe the cooling conditions of a full anneal.Ch. 5 - Why are the hypereutectoid steels not...
Ch. 5 - Although full anneals often produce the softest...Ch. 5 - What is the major process difference between full...Ch. 5 - Although normalizing is less expensive than a full...Ch. 5 - Prob. 14RQCh. 5 - What types of steel would be candidates for a...Ch. 5 - How might steel composition influence the...Ch. 5 - Other than increasing strength, for what three...Ch. 5 - What are the six major mechanisms that can be used...Ch. 5 - Prob. 19RQCh. 5 - What is required for a metal to be a candidate for...Ch. 5 - What are the three steps in an age�hardening...Ch. 5 - What is the difference between a coherent...Ch. 5 - What is overaging? Why does strength decrease?Ch. 5 - Describe the various aging responses (maximum...Ch. 5 - What is the difference between natural and...Ch. 5 - Why might naturally aging aluminum rivets be...Ch. 5 - Why is it important not to expose precipitation...Ch. 5 - Why is it more difficult to understand the...Ch. 5 - What types of heating and cooling conditions are...Ch. 5 - What are the stable equilibrium phases for steels...Ch. 5 - What are some nonequilibrium structures that...Ch. 5 - Prob. 32RQCh. 5 - What is the major factor that influences the...Ch. 5 - For a given steel, describe the relative strengths...Ch. 5 - Most structure changes proceed to completion over...Ch. 5 - What is retained austenite, and why is it an...Ch. 5 - What types of steels are more prone to retained...Ch. 5 - Why are martensitic structures usually tempered...Ch. 5 - Why does tempering offer a spectrum of possible...Ch. 5 - In what ways is the quench�and�temper heat...Ch. 5 - What is a C�C�T diagram? Why is it more useful...Ch. 5 - What is the critical cooling rate, and how is it...Ch. 5 - What two features combine to determine the...Ch. 5 - What conditions are used to standardize the quench...Ch. 5 - How do the various locations of a Jominy test...Ch. 5 - How do the data collected from a Jominy test...Ch. 5 - What is the assumption that allows the data from a...Ch. 5 - What is hardenability? How is it different from...Ch. 5 - What capabilities are provided by...Ch. 5 - When selecting a steel for an application, what...Ch. 5 - What are the three stages of liquid quenching?Ch. 5 - What are some of the major advantages and...Ch. 5 - Why does brine provide faster cooling than water?Ch. 5 - Why is an oil quench less likely to produce quench...Ch. 5 - What are some of the attractive qualities of a...Ch. 5 - Prob. 56RQCh. 5 - Prob. 57RQCh. 5 - Prob. 58RQCh. 5 - Prob. 59RQCh. 5 - How might the thermally induced residual stresses...Ch. 5 - Prob. 61RQCh. 5 - Prob. 62RQCh. 5 - Prob. 63RQCh. 5 - What is thermomechanical processing?Ch. 5 - Prob. 65RQCh. 5 - Prob. 66RQCh. 5 - Prob. 67RQCh. 5 - Prob. 68RQCh. 5 - Prob. 69RQCh. 5 - Prob. 70RQCh. 5 - Prob. 71RQCh. 5 - Prob. 72RQCh. 5 - Prob. 73RQCh. 5 - Why does a carburized part have to be further...Ch. 5 - Prob. 75RQCh. 5 - Prob. 76RQCh. 5 - Prob. 77RQCh. 5 - Describe the distinguishing features of a box...Ch. 5 - What are some possible functions of artificial...Ch. 5 - Prob. 80RQCh. 5 - Prob. 81RQCh. 5 - Prob. 82RQCh. 5 - What are some current goals of the heat treatment...Ch. 5 - Prob. 1PCh. 5 - Prob. 2PCh. 5 - Prob. 3PCh. 5 - Prob. 4PCh. 5 - Prob. 5PCh. 5 - Prob. 6PCh. 5 - What problems might be expected if the material on...Ch. 5 - Describe some heat treatment processes or...Ch. 5 - Prob. 1.3CSCh. 5 - Prob. 1.4CSCh. 5 - Prob. 1.5CSCh. 5 - Prob. 1.6CSCh. 5 - Prob. 1.7CSCh. 5 - Prob. 1.8CSCh. 5 - Prob. 2.1CSCh. 5 - Prob. 2.2CSCh. 5 - How would you alter the procedures or policies of...
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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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