Applied Fluid Mechanics (7th Edition)
7th Edition
ISBN: 9780132558921
Author: Robert L. Mott, Joseph A. Untener
Publisher: PEARSON
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Chapter 4, Problem 4.54PP
Use Fig. 4.54. The surface is
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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 4 Solutions
Applied Fluid Mechanics (7th Edition)
Ch. 4 - figure 4.2 shows a vacuum tank with a flat...Ch. 4 - The flat left end of the tank shown in Fig. 4.21...Ch. 4 - An exhaust system for a room creates a partial...Ch. 4 - A piece of 14 -in Schedule 40 pipe is used as a...Ch. 4 - A pressure relief valve is designed so that the...Ch. 4 - A gas-powered cannon shoots projectiles by...Ch. 4 - The egress hatch of a manned spacecraft is...Ch. 4 - A tank containing liquid ammonia at 77F has a flat...Ch. 4 - The bottom of a laboratory vat has a hole in it to...Ch. 4 - A simple shower for remote locations is designed...
Ch. 4 - Calculate the total force on the bottom of the...Ch. 4 - If the length of the tank in Fig. 4.24 is 1.2m,...Ch. 4 - An observation port in a small submarine is...Ch. 4 - A rectangular gate is installed in a vertical wall...Ch. 4 - '4.15 A vat has a sloped side, as shown in Fig....Ch. 4 - The wall shown in Fig. 4.28 is 20 ft long, (a)...Ch. 4 - If the wall in Fig. 4.29 is 4m long, calculate the...Ch. 4 - Refer to Fig. 4.30Ch. 4 - Refer to Fig. 4.31Ch. 4 - Refer to Fig.4.32Ch. 4 - Refer to Fig 4.33Ch. 4 - Refer to Fig. 4.34Ch. 4 - Refer to Fig. 4.35 (?Ch. 4 - Swimming poo!WilierGlasswindow2 ft diameterFigure...Ch. 4 - 4.25 Refer to Fig 4.37Ch. 4 - Refer to Fig.4.38Ch. 4 - Refer to Fig.4.39Ch. 4 - Refer to Fig.4.40Ch. 4 - Refer to Fig 4.41Ch. 4 - figure 4.42i5 shows a gasoline tank filled into...Ch. 4 - If the tank in Fig. 4.42 is filled just to the...Ch. 4 - If the tank in Fig. 4.42 is only half full of...Ch. 4 - For the water tank shown in Fig. 4.43, compute the...Ch. 4 - For the water tank shown in Fig. 4.43, compute the...Ch. 4 - For the water tank shown in Fig. 4.43, compute the...Ch. 4 - For the orange-drink tank shown in Fig. 4.32,...Ch. 4 - For the orange-drink tank shown in Fig. 4.32,...Ch. 4 - For the oil tank shown in Fig. 4.35, compute the...Ch. 4 - For the oil tank shown in Fig. 4.35; compute the...Ch. 4 - figure 4.44 shows a rectangular gate holding water...Ch. 4 - figure 4.45 shows a gate hinged at its bottom and...Ch. 4 - figure 4.46 shows a tank of water with a circular...Ch. 4 - Repeat Problem 4.19(Fig. 4.31), except that the...Ch. 4 - Repeat Problem 4.22 (Fig. 4.32), except that the...Ch. 4 - Repeat Problem 4.26 (Fig. 4.38 ). except that the...Ch. 4 - Repeat Problem 4.28 (Fig. 4.40 ), except that the...Ch. 4 - Use Fig 4.47. The surface is 2.00m long.Ch. 4 - Use Fig.4.48. The surface is 2.50m long.Ch. 4 - Use Fig.4.49. The surface is 5.00 ft longCh. 4 - Use Fig.4.50. The surface is 4.50 ft long.Ch. 4 - Use Fig.4.51.The surface is 4.00 m long.Ch. 4 - Use Fig .4.52. The surface is 1.50m longCh. 4 - Use Fig. 4.53. The surface is 1.50m long.Ch. 4 - Use Fig. 4.54. The surface is 60 in longCh. 4 - Repeat Problem 4.47 using Fig. 4.47, except that...Ch. 4 - Repeat Problem 4.48 using Fig. 4.48, except that...Ch. 4 - The tank in Fig. 4.55 has a view port in the...Ch. 4 - Insulated concrete forms (ICFs) are becoming more...Ch. 4 - Lacks are installed in rivers to allow boats to...Ch. 4 - When a dam is installed in a river that has...Ch. 4 - A wealthy eccentric is interested in having an...Ch. 4 - A pneumatic cylinder like the one shown in Fig....Ch. 4 - Determine the magnitude and the location of the...Ch. 4 - For the hinged gate shown in Fig. 4.61, determine...Ch. 4 - Prob. 4.65PPCh. 4 - Write a program to solve Problem 4.41 with any...Ch. 4 - Write a program to solve Problem 4.42 (Fig. 4.46)...Ch. 4 - Write a program to solve curved surface problems...Ch. 4 - For Program 1, cause the depth h to vary over some...
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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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