
EBK MANUFACTURING ENGINEERING & TECHNOL
7th Edition
ISBN: 8220100793431
Author: KALPAKJIAN
Publisher: PEARSON
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Chapter 3, Problem 15QLP
To determine
Which of the properties described in this chapter are important for (a) pots and pans, (b) cookie sheets for baking, (c) rulers, (d) paper clips, (e) music wire, and (f) beverage cans? Explain your answers.
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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.
Problem 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).
Problem 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 string
Chapter 3 Solutions
EBK MANUFACTURING ENGINEERING & TECHNOL
Ch. 3 - List several reasons that density is an important...Ch. 3 - Explain why the melting point of a material can be...Ch. 3 - What adverse effects can be caused by thermal...Ch. 3 - Prob. 4RQCh. 3 - What is the piezoelectric effect?Ch. 3 - Prob. 6RQCh. 3 - Prob. 7RQCh. 3 - What is the difference between thermal...Ch. 3 - What is corrosion? How can it be prevented or...Ch. 3 - Explain stress-corrosion cracking. Why is it also...
Ch. 3 - Prob. 11RQCh. 3 - Prob. 12RQCh. 3 - What is the fundamental difference between...Ch. 3 - Describe the significance of structures and...Ch. 3 - Prob. 15QLPCh. 3 - Note in Table 3.1 that the properties of the...Ch. 3 - Rank the following in order of increasing thermal...Ch. 3 - Prob. 18QLPCh. 3 - Explain how thermal conductivity can play a role...Ch. 3 - What material properties are desirable for heat...Ch. 3 - Prob. 21QLPCh. 3 - Prob. 22QLPCh. 3 - Two physical properties that have a major...Ch. 3 - Which of the materials described in this chapter...Ch. 3 - Which properties described in this chapter can be...Ch. 3 - If we assume that all the work done in plastic...Ch. 3 - The natural frequency, f, of a cantilever beam is...Ch. 3 - Plot the following for the materials described in...Ch. 3 - It can be shown that thermal distortion in...Ch. 3 - Add a column to Table 3.1 that lists the...Ch. 3 - Prob. 31SDPCh. 3 - Prob. 32SDPCh. 3 - Prob. 33SDPCh. 3 - Prob. 34SDPCh. 3 - Prob. 36SDPCh. 3 - Prob. 38SDPCh. 3 - Prob. 40SDPCh. 3 - Prob. 41SDPCh. 3 - Prob. 42SDPCh. 3 - Prob. 43SDP
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