Thinking Like an Engineer: An Active Learning Approach (4th Edition)
4th Edition
ISBN: 9780134639673
Author: Elizabeth A. Stephan, David R. Bowman, William J. Park, Benjamin L. Sill, Matthew W. Ohland
Publisher: PEARSON
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Chapter 6, Problem 2ICA
Decide on the type of problem for each question.
- a. Given human reaction times as a function of age and information regarding the braking distances of cars as a function of speed, what are appropriate following distances for driving?
- b. Parachute manufacturers provide the fall speed of an open chute as a function of person weight. If we open our chute at 3000 ft, how long will it take to land?
- c. Which will cool more quickly, one cup of coffee or two cups half full?
- d. Estimate the force required to accelerate a mass of 61.4 kilograms at a rate of 2.9 m/s2.
- e. At a party, we blow up a balloon. After three quick breaths, we pause and then wonder how much greater the surface area will be if we add another three breaths.
- f. Tom drives from his home to a nearby town that is 20 miles away at 30 miles per hour. He wishes to drive back home at a speed that will allow him to average 60 miles per hour for the round trip. How fast should he drive on the return trip?
- g. To construct a concrete patio, the soil is leveled and gravel is laid on the area. Concrete is poured on top of this. The thickness of the gravel layer needs to be determined based on the gravel size. For small pebble size (average size 4 mm), the layer must be the same thickness as the concrete. For medium size (average 16 mm), it must be two times thicker, and for large size (64 mm), it must be three times thicker. We need to prepare a design chart that will allow us to select the proper thickness of concrete immediately as the gravel is delivered. (We donʼt know what the gravel size will be until the shipment arrives.) Once on site, a quick test will provide the gravel size; we do know, however, it will be somewhere between 4 millimeters and 50 millimeters.
- h. A firefighter aims a stream of water just over the top of a 50-foot flagpole. If the firefighter stands 60 feet away from the pole, about how far does the water stream travel before hitting the ground?
- i. The drag generated by a parachute (which keeps us from falling faster) depends on the falling speed squared. If a 100-pound person falls at 10 feet per second, how fast will a 200-pound person fall?
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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 6 Solutions
Thinking Like an Engineer: An Active Learning Approach (4th Edition)
Ch. 6.2 - We use SOLVEM to complete this problem in the...Ch. 6.2 - State the objective and any relevant observations...Ch. 6.2 - Create a list of variables and constants for the...Ch. 6.2 - Create a list of equations for the following...Ch. 6.3 - Manipulate and solve for the following problem,...Ch. 6 - Decide on the type of problem for each question....Ch. 6 - Decide on the type of problem for each question....Ch. 6 - Final Assignment of this ICA: You have done...Ch. 6 - A hungry bookworm bores through a complete set of...Ch. 6 - Two cargo trains each leave their respective...
Ch. 6 - Water drips from a faucet at the rate of 3 drops...Ch. 6 - During rush hour, cars back up when the traffic...Ch. 6 - Suppose that the earth were a smooth sphere and...Ch. 6 - Chapter 6 Review Questions Analyze the following...Ch. 6 - 2. A circus performer jumps from a platform onto...Ch. 6 - 3. Your college quadrangle is 85 meters long and...Ch. 6 - 4. I am standing on the upper deck of the football...Ch. 6 - 5. A 1-kilogram mass has just been dropped from...Ch. 6 - 6. Neglect the weight of the drum in the following...
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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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