EBK MANUFACTURING ENGINEERING & TECHNOL
7th Edition
ISBN: 8220100793431
Author: KALPAKJIAN
Publisher: PEARSON
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Chapter 11, Problem 58SDP
It sometimes is desirable to cool metals more slowly than they would be if the molds were maintained at room temperature. List and explain the methods you would use to slow down the cooling process.
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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 11 Solutions
EBK MANUFACTURING ENGINEERING & TECHNOL
Ch. 11 - Describe the differences between expendable and...Ch. 11 - Name the important factors in selecting sand for...Ch. 11 - What are the major types of sand molds? What are...Ch. 11 - List important considerations when selecting...Ch. 11 - What is the function of a core?Ch. 11 - What is the difference between sand-mold and...Ch. 11 - What are composite molds? Why are they used?Ch. 11 - Describe the features of plaster-mold casting.Ch. 11 - Name the type of materials typically used for...Ch. 11 - What are the advantages of pressure casting over...
Ch. 11 - List the advantages and limitations of die...Ch. 11 - What is the purpose of a riser? What is a blind...Ch. 11 - Explain the purpose of a vent and a runner in a...Ch. 11 - How are shell molds produced?Ch. 11 - What keeps the mold together in vacuum casting?Ch. 11 - What is squeeze casting? What are its advantages?Ch. 11 - What are the advantages of the lost-foam casting...Ch. 11 - Prob. 18RQCh. 11 - What are the reasons for the large variety of...Ch. 11 - Why are risers not as useful in die casting as...Ch. 11 - Describe the drawbacks to having a riser that is...Ch. 11 - Why can blind risers be smaller than open-top...Ch. 11 - Why does die casting produce the smallest cast...Ch. 11 - Why is the investment-casting process capable of...Ch. 11 - What differences, if any, would you expect in the...Ch. 11 - Recently, cores for sand casting have been...Ch. 11 - Would you recommend preheating the molds used in...Ch. 11 - Give reasons for, and examples of, using die...Ch. 11 - Referring to Fig. 11.3, do you think it is...Ch. 11 - Explain why squeeze casting produces parts with...Ch. 11 - How are the individual wax patterns attached on a...Ch. 11 - Describe the measures that you would take to...Ch. 11 - You have seen that, even though die casting...Ch. 11 - How are hollow parts with various cavities made by...Ch. 11 - It was stated that the strength-to-weight ratio of...Ch. 11 - How are risers and sprues placed in sand molds?...Ch. 11 - In shell-mold casting, the curing process is...Ch. 11 - Why does the die-casting machine shown in Fig....Ch. 11 - Prob. 39QLPCh. 11 - What are the benefits to heating the mold in...Ch. 11 - The slushy state of alloys refers to that state...Ch. 11 - Can a chaplet also act as a chill? Explain.Ch. 11 - Rank the casting processes described in this...Ch. 11 - Estimate the clamping force for a die-casting...Ch. 11 - In sand casting, it is important that the...Ch. 11 - If an acceleration of 100 g is necessary to...Ch. 11 - A jeweler wishes to produce 24 gold rings in one...Ch. 11 - Describe the procedures that would be involved in...Ch. 11 - The optimum shape of a riser is spherical to...Ch. 11 - Sketch and describe a casting line consisting of...Ch. 11 - Outline the casting processes that would be most...Ch. 11 - Make a list of the mold and die materials used in...Ch. 11 - Write a brief paper on the permeability of molds...Ch. 11 - Light metals commonly are cast in vulcanized...Ch. 11 - It sometimes is desirable to cool metals more...Ch. 11 - The part shown in Fig. P11.59 is a hemispherical...Ch. 11 - Porosity that has developed in the boss of a...Ch. 11 - In Fig. II.1b the gemstones have been cast in...
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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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