PRECISION MACHINING TECHNOLOGY PACKAGE
3rd Edition
ISBN: 9780357262788
Author: Hoffman
Publisher: CENGAGE L
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Chapter 7.1, Problem 2RQ
What is the major use of the surface grinder?
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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.
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.
Chapter 7 Solutions
PRECISION MACHINING TECHNOLOGY PACKAGE
Ch. 7.1 - What are three benefits of precision grinding?Ch. 7.1 - What is the major use of the surface grinder?Ch. 7.1 - What are the two spindle types used for surface...Ch. 7.1 - What are the two types of table movements used for...Ch. 7.1 - Label the parts of the surface grinder.Ch. 7.1 - What are three types of cylindrical grinders?Ch. 7.1 - Tool and cutter grinders can be used to _______...Ch. 7.2 - What wheel shapes are commonly used for surface...Ch. 7.2 - What abrasive is commonly used to grind steels?Ch. 7.2 - What abrasive is used to grind nonferrous metals,...
Ch. 7.2 - What does CBN stand for?Ch. 7.2 - What superabrasive is a good choice for grinding...Ch. 7.2 - The grade of a wheel describes its __________ on a...Ch. 7.2 - Prob. 7RQCh. 7.2 - Prob. 8RQCh. 7.2 - Generally, harder wheels should be used to grind...Ch. 7.2 - Wheel A: 32A60H8V Wheel B: 39C100L6V What is the...Ch. 7.2 - Wheel A: 32A60H8V Wheel B: 39C100L6V Which wheel...Ch. 7.2 - Wheel A: 32A60H8V Wheel B: 39C100L6V Which wheel...Ch. 7.2 - Wheel A: 32A60H8V Wheel B: 39C100L6V Which wheel...Ch. 7.2 - Wheel A: 32A60H8V Wheel B: 39C100L6V What type of...Ch. 7.3 - List five surface grinder safety guidelines.Ch. 7.3 - What should be done to every grinding wheel before...Ch. 7.3 - When using a magnetic workholding device, the work...Ch. 7.3 - When using a magnetic chuck to hold a workpiece...Ch. 7.3 - Describe the difference between wheel truing and...Ch. 7.3 - What tool is used to true and dress an aluminum...Ch. 7.3 - How is a CBN wheel trued and dressed?Ch. 7.3 - Depth-of-cut range for surface grinding is: a....Ch. 7.3 - What should be done to a magnetic chuck when it...Ch. 7.3 - What workholding device can extend the magnetic...Ch. 7.3 - What workholding device can be used when grinding...Ch. 7.3 - Briefly describe the method for grinding two...Ch. 7.3 - List three types of workpiece features that can be...Ch. 7.3 - List three possible solutions to eliminate bum...Ch. 7.3 - What can cause scratches on the surface of a...
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- 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).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_forwardI need part all parts please in detail (including f)arrow_forward
- Problem 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_forwardB5 Please help on the attached question.arrow_forward
- B8 Please help on the attached question.arrow_forwardB1 Please help on the attached question.arrow_forwardQu 4 A cylindrical metal specimen 15.0 mm in diameter and 150 mm long is to be subjected to a tensile stress of 50 MPa; at this stress level, the resulting deformation will be totally elastic. If the elongation must be less than 0.072 mm, which of the metals in Table 1 are suitable candidates? If, in addition, the maximum permissible diameter decrease is 2.3 × 10-3 mm when the tensile stress of 50 MPa is applied, which of the metals that satisfy the criterion in part (a) are suitable candidates? see on the tables given part a and b i need to show all work problems formula step by step please make sure is correctly material sciencearrow_forward
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